A single deposit of $5,000 grows to $6,000 in an account over 3 years. Assuming a constant rate of compounding, what is the effective annual interest rate earned, to the nearest tenth of a percent?
6.3%
6.7%
5.7%
20.0%
Correct answer: 6.3%
The effective annual interest rate is about 6.3%. The accumulation factor over 3 years is $6,000 divided by $5,000, which equals 1.20, and under annual compounding this equals (1+i)3. Taking the cube root 31.20 gives about 1.0627, so subtracting 1 leaves an effective annual rate of roughly 6.3%.
Two banks quote rates on a one-year deposit. Bank A offers a nominal annual rate of 7.8% convertible monthly, while Bank B offers a nominal annual rate of 7.85% convertible semiannually. Which deposit yields the higher effective annual rate, and approximately what is that higher effective rate?
Bank B, about 7.85%
Bank A, about 8.08%
Bank A, about 7.80%
Bank B, about 8.00%
Correct answer: Bank A, about 8.08%
Bank A wins with an effective annual rate of about 8.08%. Bank A's effective rate is (1 plus 0.078 divided by 12) raised to the 12th power minus 1, which is about 0.0808, or 8.08%. Bank B's effective rate is (1 plus 0.0785 divided by 2) squared minus 1, about 0.0800, or 8.00%. More frequent compounding gives Bank A the higher effective yield even though its nominal rate is slightly lower.
A nominal annual interest rate convertible quarterly is 8%. What is the equivalent nominal annual rate of discount convertible quarterly?
8.00%
7.69%
7.84%
8.16%
Correct answer: 7.84%
The equivalent nominal annual rate of discount convertible quarterly is about 7.84%. The quarterly interest rate is 8% divided by 4, or 2%, so the quarterly discount rate is 0.02 divided by 1.02, about 0.01961. Multiplying that quarterly discount rate by 4 gives a nominal annual discount rate convertible quarterly of roughly 7.84%.
Which statement correctly relates the effective annual rate of discount d to the corresponding annual discount factor v?
V equals 1 minus d, because discounting one period removes one period's worth of discount from a unit payment
V equals 1 plus d, because the discount factor accumulates the discount rate forward one period
V equals d divided by (1 plus d), since the discount factor is always smaller than the discount rate
V equals 1 divided by d, because the discount factor inverts the discount rate
Correct answer: V equals 1 minus d, because discounting one period removes one period's worth of discount from a unit payment
The correct relationship is that v equals 1 minus d. The discount factor v is the present value of 1 due in one period, and applying a one-period discount rate d means subtracting the discount d from a unit payment, leaving 1 minus d. This is consistent with v equal to 1 divided by (1 plus i) once d is written as i divided by (1 plus i).
An account earns interest under the accumulation function a(t)=1+0.04t+0.002t2, where t is in years. What effective annual rate of interest does the account earn during the second year, that is from time 1 to time 2?
4.40%
4.21%
5.00%
4.60%
Correct answer: 4.40%
The effective rate during the second year is about 4.4%. The effective rate over a period equals the change in the accumulation function divided by its value at the start of the period. Here a(1) is 1 plus 0.04 plus 0.002, or 1.042, and a(2) is 1 plus 0.08 plus 0.008, or 1.088. The increase of 0.046 divided by the starting value 1.042 gives about 0.0441, which is closest to 4.40%.
An obligation requires a payment of $8,000 at the end of year 6. An investor wants to set aside money today in an account credited with an effective annual interest rate of 4.5%. How much must be deposited today to exactly meet the obligation?
$6,164.00
$8,000.00
$5,896.55
$6,140.50
Correct answer: $6,140.50
The required deposit is about 6,143. The present value of a single future payment equals that payment divided by (1 plus i) raised to the number of periods. Here (1.045) raised to the 6th power is about 1.30226, so $8,000 divided by 1.30226 is about 6,143, which is closest to the option of roughly 6,140. The other choices either ignore discounting or apply the wrong number of periods.
Which feature distinguishes an annuity-immediate from an otherwise identical annuity-due?
An annuity-immediate makes each payment at the end of its payment period, whereas an annuity-due makes each payment at the start of the period.
An annuity-immediate uses simple interest while an annuity-due uses compound interest.
An annuity-immediate has infinitely many payments while an annuity-due has finitely many payments.
An annuity-immediate is always valued at a higher interest rate than an annuity-due.
Correct answer: An annuity-immediate makes each payment at the end of its payment period, whereas an annuity-due makes each payment at the start of the period.
The distinguishing feature is the timing of payments: an annuity-immediate pays at the end of each period while an annuity-due pays at the beginning. Because every annuity-due payment occurs one period earlier, its present value equals the corresponding annuity-immediate value multiplied by one plus the interest rate. The interest basis and number of payments are otherwise identical.
An annuity pays $750 at the end of each year for 12 years. Using an effective annual interest rate of 4.5%, what is its present value today?
$9,000.00
$6,838.94
$7,146.69
$6,544.20
Correct answer: $6,838.94
The present value is about $6,838.94. An annuity-immediate's present value equals the payment times the factor (1 minus v to the 12th) divided by i, where v is 1 divided by 1.045. That factor is about 9.1186, and multiplying by the $750 payment gives roughly $6,838.94.
A perpetuity pays $350 at the end of each year forever. Using an effective annual interest rate of 3%, what is its present value?
$10,500.00
$12,000.00
$11,666.67
$9,800.00
Correct answer: $11,666.67
The present value is about $11,666.67. A perpetuity-immediate paying a level amount forever has a present value equal to the payment divided by the interest rate. Dividing the $350 annual payment by 0.03 gives about $11,666.67.
A perpetuity-due is worth $100,000 today at an effective annual interest rate of 4%. What is the level annual payment made at the beginning of each year?
$4,000.00
$4,166.67
$3,921.57
$3,846.15
Correct answer: $3,846.15
The payment is about $3,846.15. For a perpetuity-due the present value equals the payment divided by the effective rate of discount d, where d is i divided by (1 plus i), here 0.04 divided by 1.04, about 0.03846. Equivalently the payment equals the present value times d, so $100,000 times 0.03846 gives roughly $3,846.15.
An annuity pays $600 at the end of each year for 8 years, with the first payment occurring at the end of year 5. Using an effective annual interest rate of 5%, what is the present value today?
$3,877.93
$3,190.38
$3,696.12
$4,071.82
Correct answer: $3,190.38
The present value is about $3,190.38. A deferred annuity is valued by first finding the annuity's value one period before its first payment, then discounting over the deferral. The 8-year annuity-immediate of $600 at 5% is worth about $3,877.93 at the end of year 4, and discounting that back 4 years by dividing by 1.05 to the 4th gives roughly $3,190.38.
An annuity-immediate pays 1 at the end of year 1, 2 at the end of year 2, increasing by 1 each year through a final payment of 8 at the end of year 8. Using an effective annual interest rate of 6%, what is its present value?
36.00
18.00
26.05
30.74
Correct answer: 26.05
The present value is about 26.05. An increasing annuity-immediate with unit increments has a present value equal to the quantity (annuity-due factor minus n times v to the n) divided by i. The 8-year annuity-due factor at 6% is about 6.5824, and subtracting 8 times v to the 8th (about 5.0193) leaves about 1.5631, which divided by 0.06 gives roughly 26.05.
An annuity makes 15 annual payments at the end of each year, with the first payment $2,000 and each later payment 4% larger than the prior one. Using an effective annual interest rate of 9%, what is its present value?
$20,222.90
$16,121.38
$30,000.00
$18,750.00
Correct answer: $20,222.90
The present value is about $20,222.90. A geometrically increasing annuity has a present value equal to the first payment times the quantity (1 minus the ratio of (1 plus g) over (1 plus i), raised to the n) divided by (i minus g). With g of 0.04 and i of 0.09, the bracket equals about 10.111, and multiplying by the $2,000 first payment gives roughly $20,222.90.
An annuity pays at a continuous rate of $3,000 per year for 8 years. Using an effective annual interest rate of 7%, what is its present value?
$24,000.00
$17,909.50
$19,176.44
$18,533.81
Correct answer: $18,533.81
The present value is about $18,533.81. A continuously payable annuity has a present value equal to the annual payment rate times the quantity (1 minus v to the n) divided by the force of interest δ, where δ is the natural log of 1.07, about 0.06766. The factor (1 minus v to the 8th) divided by δ is about 6.1779, and multiplying by $3,000 gives roughly $18,533.81.
An annuity-immediate pays 8 at the end of year 1, 7 at the end of year 2, decreasing by 1 each year down to a payment of 1 at the end of year 8. Using an effective annual interest rate of 5%, what is its present value?
36.00
18.00
30.74
26.05
Correct answer: 30.74
The present value is about 30.74. A decreasing annuity-immediate has a present value equal to the quantity (n minus the level annuity-immediate factor) divided by i. The 8-year level factor at 5% is about 6.4632, so n minus that factor is about 1.5368, and dividing by 0.05 gives roughly 30.74.
A loan of $25,000 is to be repaid by equal payments at the end of each year for 15 years. Using an effective annual interest rate of 6%, what level annual payment exactly repays the loan?
$1,666.67
$2,574.07
$2,728.51
$2,425.30
Correct answer: $2,574.07
The level payment is about $2,574.07. A level annuity payment equals the present value divided by the annuity-immediate factor. The 15-year factor at 6% is about 9.7122, so dividing the $25,000 present value by that factor gives roughly $2,574.07 per year.
An annuity pays $150 at the beginning of each month for 4 years. Using a monthly effective interest rate of 0.4%, what is its present value?
$7,200.00
$6,539.20
$6,565.29
$6,278.15
Correct answer: $6,565.29
The present value is about $6,565.29. Because the payments are made at the start of each month, this is a monthly annuity-due, whose present value equals the corresponding annuity-immediate value multiplied by 1 plus the monthly rate. The monthly annuity-immediate factor over 48 months at 0.4% is about 43.5685, multiplied by 1.004 gives about 43.7427, and multiplying by the $150 payment gives roughly $6,565.29.
A perpetuity pays $500 at the end of each year forever, with the first payment occurring at the end of year 8. Using an effective annual interest rate of 6%, what is its present value today?
$5,542.14
$8,333.33
$5,874.67
$6,228.97
Correct answer: $5,542.14
The present value is about $5,542.14. A deferred perpetuity is valued by finding the perpetuity value one period before its first payment, then discounting. The perpetuity is worth $500 divided by 0.06, or about $8,333.33, at the end of year 7, and discounting that back 7 years by dividing by 1.06 to the 7th gives roughly $5,542.14.
An increasing perpetuity-immediate pays 1 at the end of year 1, 2 at the end of year 2, and so on, increasing by 1 each year forever. Using an effective annual interest rate of 5%, what is its present value?
400.00
420.00
20.00
441.00
Correct answer: 420.00
The present value is 420. An increasing perpetuity-immediate with unit increments has a present value equal to the quantity (1 plus i) divided by i squared. With i of 0.05, that is 1.05 divided by 0.0025, which equals exactly 420.
A perpetuity-immediate pays $600 at the end of year 1, and each subsequent annual payment is 3% larger than the prior payment, continuing forever. Using an effective annual interest rate of 8%, what is its present value?
$7,500.00
$20,000.00
$12,000.00
$10,909.09
Correct answer: $12,000.00
The present value is $12,000. A geometrically increasing perpetuity has a present value equal to the first payment divided by the difference between the interest rate and the growth rate, provided the growth rate is less than the interest rate. Dividing the $600 first payment by (0.08 minus 0.03) gives exactly $12,000.
A saver deposits $1,200 at the end of each year for 20 years. Using an effective annual interest rate of 5%, what is the accumulated value just after the final deposit?
$24,000.00
$41,663.10
$37,789.66
$39,679.14
Correct answer: $39,679.14
The accumulated value is about $39,679.14. The accumulated value of an annuity-immediate equals the payment times the factor (1 plus i raised to the n, minus 1) divided by i. Here (1.05 to the 20th minus 1) divided by 0.05 is about 33.066, and multiplying by the $1,200 payment gives roughly $39,679.14.
An investor pays $10,000 today for an annuity-immediate of $1,500 per year at an effective annual interest rate of 8%. To the nearest whole year, how many annual payments will the annuity provide?
7 years
10 years
8 years
12 years
Correct answer: 10 years
The annuity provides about 10 payments. Setting the present value equal to $1,500 times (1 minus v to the n) divided by 0.08 and solving gives v to the n equal to about 0.4667. Taking logarithms, n equals the log of 0.4667 divided by the log of v, which is about 9.90 years, rounding to 10 whole payments.
A perpetuity-due pays a level amount at the beginning of each year and has a present value of $5,250 at an effective annual interest rate of 5%. What is the annual payment?
$262.50
$276.32
$250.00
$237.50
Correct answer: $250.00
The payment is $250. For a perpetuity-due the present value equals the payment divided by the effective rate of discount d, where d is i divided by (1 plus i), here 0.05 divided by 1.05, about 0.047619. The payment equals the present value times d, so $5,250 times 0.047619 gives exactly $250.
Two annuities each pay $800 per year for 10 years at an effective annual interest rate of 7%, but one pays at the beginning of each year and the other at the end. By how much does the annuity-due's present value exceed the annuity-immediate's?
$393.32
$800.00
$56.00
0.00
Correct answer: $393.32
The annuity-due exceeds the annuity-immediate by about $393.32. The annuity-due present value equals the annuity-immediate value multiplied by 1 plus i, so the difference equals the immediate value times i. The 10-year annuity-immediate of $800 at 7% is worth about $5,618.87, and multiplying that by 0.07 gives roughly $393.32.
A home loan of $200,000 is repaid by level monthly payments over 30 years at a monthly effective interest rate of 0.5%. What is the amount of each monthly payment?
$1,199.10
$1,000.00
$555.56
$1,666.67
Correct answer: $1,199.10
The monthly payment is about $1,199.10. Under the amortization method the loan equals the present value of all payments, so the payment is the loan divided by the annuity factor. With 360 monthly payments at 0.5% per month, the annuity-immediate factor is about 166.7916, and $200,000 divided by that factor gives roughly $1,199.10.
A loan of $16,000 is repaid by level annual payments over 10 years at an effective annual rate of 6%. How much of the third payment goes toward repaying principal?
$960.00
$1,213.89
$1,363.92
$2,173.89
Correct answer: $1,363.92
The principal repaid in the third payment is about $1,363.92. The level payment is $16,000 divided by the 10-year annuity factor at 6% (about 7.3601), giving roughly $2,173.89. In a level-payment loan the principal repaid in payment t equals the payment times v raised to the power (n minus t plus 1), so for the third of ten payments it is $2,173.89 times 1.06 to the negative 8th, about 0.6274, which is roughly $1,363.92.
A loan of $30,000 is repaid by level annual payments over 12 years at an effective annual rate of 7%. What is the total of all twelve payments the borrower makes over the life of the loan?
$32,100.00
$30,000.00
$36,000.00
$45,324.72
Correct answer: $45,324.72
The total of all payments is about $45,324.72. The level payment is $30,000 divided by the 12-year annuity factor at 7% (about 7.9427), giving roughly $3,777.06. Multiplying that payment by the 12 payments made gives about $45,324.72, of which $30,000 is principal and the remainder is interest.
A loan of $7,000 is repaid by level annual payments of $900 at an effective annual rate of 11%. Using the retrospective method, what is the outstanding loan balance immediately after the fourth payment?
$3,400.00
$6,387.73
$10,627.45
$4,239.72
Correct answer: $6,387.73
The outstanding balance is about $6,387.73. The retrospective method accumulates the original loan at the loan rate and subtracts the accumulated value of payments made. The $7,000 grows to $7,000 times 1.11 to the fourth, about $10,627.45, and the four payments of $900 accumulate to $900 times the 4-year accumulation factor at 11% (about 4.7097), roughly $4,239.72. Subtracting gives about $6,387.73.
A loan of $80,000 is repaid over 10 years by the sinking fund method, with the borrower paying 9% interest annually to the lender and making level deposits into a sinking fund earning 5% that must accumulate to the full principal. What is the borrower's total annual outlay?
$12,471.61
$7,200.00
$13,560.37
$6,360.37
Correct answer: $13,560.37
The total annual outlay is about $13,560.37. The borrower pays 9% interest on the full $80,000 each year, which is $7,200, and separately deposits into the fund. The deposit is $80,000 divided by the 10-year accumulation factor at 5% (about 12.5779), giving roughly $6,360.37. Adding the $7,200 interest and the $6,360.37 deposit yields about $13,560.37.
A loan of $22,000 is repaid by level annual payments over 9 years at an effective annual rate of 8%. What is the interest portion contained in the fifth payment?
$1,124.91
$1,760.00
$2,396.84
$3,521.75
Correct answer: $1,124.91
The interest in the fifth payment is about $1,124.91. The level payment is $22,000 divided by the 9-year annuity factor at 8% (about 6.2469), giving roughly $3,521.75. The interest portion of payment t equals the payment times (1 minus v raised to the power n minus t plus 1), so for the fifth of nine payments it is $3,521.75 times (1 minus 1.08 to the negative 5th), about 0.3194, which is roughly $1,124.91.
A loan of $100,000 is repaid by level annual payments over 30 years at an effective annual rate of 7%. Immediately after the 10th payment, the borrower refinances the outstanding balance over the remaining 20 years at a new effective annual rate of 5%. What is the new level annual payment?
$8,058.64
$6,850.58
$5,000.00
$4,268.67
Correct answer: $6,850.58
The new payment is about $6,850.58. The original payment is $100,000 divided by the 30-year annuity factor at 7% (about 12.4090), giving roughly $8,058.64, and the balance after the 10th payment is the present value of the remaining 20 payments at 7%, about $85,373.35. Refinancing that balance over 20 years at 5% means dividing $85,373.35 by the 20-year factor at 5% (about 12.4622), which yields about $6,850.58.
A loan of $12,000 is repaid by level annual payments over 6 years at an effective annual rate of 10%. How much of the fourth payment is applied to principal?
$1,200.00
$685.20
$2,755.29
$2,070.09
Correct answer: $2,070.09
The principal repaid in the fourth payment is about $2,070.09. The level payment is $12,000 divided by the 6-year annuity factor at 10% (about 4.3553), giving roughly $2,755.29. The principal repaid in payment t equals the payment times v raised to the power (n minus t plus 1), so the fourth of six payments is $2,755.29 times 1.10 to the negative 3rd, about 0.7513, which is roughly $2,070.09.
A loan is repaid by level annual payments of $5,000 at the end of each year for 15 years at an effective annual rate of 6%. Using the prospective method, what is the outstanding loan balance immediately after the 10th payment?
$25,000.00
$48,712.30
$21,061.82
$18,556.04
Correct answer: $21,061.82
The outstanding balance is about $21,061.82. The prospective method values the balance as the present value of the remaining payments. After the 10th of 15 payments, 5 payments of $5,000 remain, so the balance equals $5,000 times the 5-year annuity factor at 6% (about 4.2124), which gives roughly $21,061.82.
A loan of $45,000 is repaid over 8 years by the sinking fund method, with level deposits into a fund earning 7% effective annually that must accumulate to the full principal. What is the balance in the sinking fund immediately after the fifth deposit?
$25,223.02
$21,930.25
$28,125.00
$4,386.05
Correct answer: $25,223.02
The sinking fund balance is about $25,223.02. The required deposit is $45,000 divided by the 8-year accumulation factor at 7% (about 10.2598), giving roughly $4,386.05. After five deposits the fund equals $4,386.05 times the 5-year accumulation factor at 7% (about 5.7507), which is about $25,223.02, more than the simple sum of the deposits because of accumulated interest.
A loan of $14,000 is repaid by level annual payments over 7 years at an effective annual rate of 8%. Using the prospective method, what is the outstanding loan balance immediately after the second payment?
$8,059.18
$10,000.00
$12,936.45
$10,736.45
Correct answer: $10,736.45
The outstanding balance is about $10,736.45. The level payment is $14,000 divided by the 7-year annuity factor at 8% (about 5.2064), giving roughly $2,689.01. The prospective method values the balance as the present value of the remaining 5 payments, so it equals $2,689.01 times the 5-year factor at 8% (about 3.9927), which is about $10,736.45.
A loan of $18,000 is repaid by level annual payments over 12 years at an effective annual rate of 7%. What is the combined interest content of the first two payments?
$2,520.00
$2,449.56
$3,000.00
$1,260.00
Correct answer: $2,449.56
The combined interest of the first two payments is about $2,449.56. The level payment is $18,000 divided by the 12-year annuity factor at 7% (about 7.9427), giving roughly $2,266.24. The first payment's interest is 7% of the full $18,000, which is $1,260.00, and the second payment's interest is 7% of the balance after the first payment (about $16,993.76), which is roughly $1,189.56. Adding these gives about $2,449.56.
A loan of $24,000 is repaid by level annual payments over 8 years at an effective annual rate of 9%. Immediately after the third payment, the borrower refinances the outstanding balance over the remaining 5 years at a reduced effective annual rate of 6%. By how much does the annual payment decrease?
0.00
$720.00
$332.20
$1,083.95
Correct answer: $332.20
The annual payment decreases by about $332.20. The original payment is $24,000 divided by the 8-year factor at 9% (about 5.5348), giving roughly $4,336.19, and the balance after the third payment is the present value of the remaining 5 payments at 9%, about $16,866.25. Refinancing over 5 years at 6% gives a new payment of $16,866.25 divided by the 5-year factor at 6% (about 4.2124), about $4,003.99, so the reduction is $4,336.19 minus $4,003.99, roughly $332.20.
A loan of $20,000 is repaid by level annual payments over 10 years at an effective annual rate of 5%. What is the principal portion contained in the final (tenth) payment?
$2,466.75
$2,590.09
$123.34
$2,000.00
Correct answer: $2,466.75
The principal repaid in the final payment is about $2,466.75. The level payment is $20,000 divided by the 10-year annuity factor at 5% (about 7.7217), giving roughly $2,590.09. The principal repaid in payment t equals the payment times v raised to the power (n minus t plus 1), so for the tenth of ten payments it is $2,590.09 times 1.05 to the negative 1st, about 0.9524, which is roughly $2,466.75. Almost the entire last payment is principal because little balance remains.
On the basic price formula for a bond, what does the price equal at the time of issue?
The present value of the future coupon payments plus the present value of the redemption amount, both discounted at the yield rate
The face amount plus all coupons that will ever be paid, with no discounting applied
The redemption amount discounted at the coupon rate, ignoring the coupon payments entirely
The sum of the undiscounted coupons minus the redemption amount
Correct answer: The present value of the future coupon payments plus the present value of the redemption amount, both discounted at the yield rate
The price equals the present value of the future coupon payments plus the present value of the redemption amount, both discounted at the yield rate. A bond is simply a stream of fixed coupons followed by a lump-sum redemption, so its fair price today is what those cash flows are worth when discounted at the investor's required yield. Using the coupon rate or leaving the cash flows undiscounted would not produce the market price.
A $1,000 par-value bond pays annual coupons of $60 and is redeemable at par in 5 years. Using an annual yield rate of 6%, what is the price of the bond?
$1,300.00
$1,000.00
$747.26
$1,060.00
Correct answer: $1,000.00
The price is $1,000. The $60 annual coupon represents a 6% coupon rate on the $1,000 par value, which exactly equals the 6% yield rate. When the coupon rate equals the yield rate and redemption is at par, the bond sells at par, so the price is exactly the $1,000 face amount.
A $1,000 face-amount bond with annual coupons of $80, redeemable at par in 10 years, is priced to yield 6% effective annually. What is the price of the bond?
$1,000.00
$1,147.20
$1,800.00
$853.02
Correct answer: $1,147.20
The price is about $1,147.20. The bond's price is the present value of the $80 annual coupons plus the present value of the $1,000 redemption at 6%. The 10-year annuity factor at 6% is about 7.3601, so the coupons are worth about $588.81, and the redemption discounts to about $558.39, for a total near $1,147.20. The bond sells at a premium because its 8% coupon rate exceeds the 6% yield.
A bond is selling for a price below its redemption value. What does this discount indicate about the relationship between the bond's coupon rate and its yield rate?
The coupon rate is lower than the yield rate
The coupon rate is higher than the yield rate
The coupon rate exactly equals the yield rate
The coupon rate has no relationship to whether the bond trades at a discount
Correct answer: The coupon rate is lower than the yield rate
A bond trading at a discount has a coupon rate that is lower than its yield rate. Because the coupons are too small to give investors their required return, buyers will only pay less than the redemption value, and the shortfall is made up by the gain realized at redemption. When the coupon rate instead exceeds the yield, the bond commands a premium.
A $1,000 par-value bond with annual coupons of $90, redeemable at par in 8 years, was purchased to yield 7% effective annually at a price of $1,119.43. What is the amount of premium written down in the first coupon period?
$11.64
$90.00
$78.36
$20.00
Correct answer: $11.64
The write-down of premium in the first period is about $11.64. For a premium bond, each coupon exceeds the interest earned at the yield rate, and that excess reduces the book value. The interest earned in the first period is 7% of the $1,119.43 book value, about $78.36, so the coupon of $90 exceeds it by roughly $11.64, which is the amount of premium amortized that period.
An investor pays $950 for a $1,000 face-amount zero-coupon bond that matures in 4 years. What is the annual effective yield rate earned on this investment?
5.00%
1.32%
1.25%
1.29%
Correct answer: 1.29%
The annual effective yield is about 1.29%. A zero-coupon bond pays no coupons, so its yield comes entirely from the growth of the $950 purchase price into the $1,000 redemption. Setting $950 times (1 plus i) to the 4th power equal to $1,000 gives a growth factor of about 1.0129 per year, so the yield is roughly 1.29%.
A zero-coupon bond redeemable for $5,000 in 7 years is priced to yield 6% effective annually. What is its price?
$3,325.30
$3,500.00
$4,716.98
$2,962.50
Correct answer: $3,325.30
The price is about $3,325.30. A zero-coupon bond has only one cash flow, the redemption amount, so its price is simply that amount discounted at the yield rate. Dividing $5,000 by (1.06) raised to the 7th power, about 1.5036, gives roughly $3,325.30.
A $1,000 par-value bond pays annual coupons of $50 and is redeemable at par in 12 years. An investor buys it at a price of $920. Which statement best describes how the investor's yield to maturity compares with the 5% coupon rate?
The yield to maturity is below 5% because the bond was bought at a discount
The yield to maturity is above 5% because the bond was bought at a discount
The yield to maturity equals exactly 5% regardless of the purchase price
The yield to maturity cannot be determined without knowing the call date
Correct answer: The yield to maturity is above 5% because the bond was bought at a discount
The yield to maturity is above 5% because the bond was bought at a discount. Paying $920 for a bond that redeems at $1,000 produces a capital gain at maturity on top of the coupons, so the total return exceeds the 5% coupon rate. A discount price always implies a yield greater than the coupon rate, just as a premium price implies a yield below it.
A $1,000 par-value bond with annual coupons of $70, redeemable at par, is sold for $1,000 exactly 4 months after the last coupon was paid. Using a coupon rate of 7% and simple interest for the fractional period, what is the accrued interest included in this dirty price?
$70.00
$5.83
$23.33
$17.50
Correct answer: $23.33
The accrued interest is about $23.33. Accrued interest is the portion of the upcoming $70 coupon that has been earned since the last coupon date, prorated by the fraction of the period elapsed. Four months is one-third of the annual coupon period, so the accrued amount is one-third of $70, which is about $23.33.
When a bond is purchased between coupon dates, how is the clean price (also called the market price) obtained from the dirty price (the full price actually paid)?
By adding the accrued interest to the dirty price
By subtracting the accrued interest from the dirty price
By multiplying the dirty price by the coupon rate
By discounting the dirty price for one full coupon period
Correct answer: By subtracting the accrued interest from the dirty price
The clean price is obtained by subtracting the accrued interest from the dirty price. The dirty price is the full amount the buyer pays, which includes the seller's earned share of the next coupon, so removing that accrued interest isolates the quoted clean price. Adding accrued interest instead would move from the clean price to the dirty price.
A $1,000 par-value bond pays annual coupons of $100 and is callable at par at the end of any year from year 5 through its maturity at year 10. An investor wants to price the bond to guarantee a yield of at least 8%. Because the coupon rate exceeds the yield, at which call date should the investor assume redemption to find the price?
At the latest possible date, year 10
At the earliest possible date, year 5
At the midpoint, year 7 or 8
Redemption timing does not affect the price for a callable bond
Correct answer: At the earliest possible date, year 5
The investor should assume redemption at the earliest possible date, year 5. For a bond bought at a premium, where the coupon rate exceeds the yield, the issuer calling early deprives the holder of valuable above-market coupons, so the worst case for the buyer is the earliest call. Pricing to the earliest call date guarantees the investor will achieve at least the desired 8% yield no matter when the bond is actually called.
A $1,000 par-value bond pays annual coupons of $60 and is callable at par at the end of year 5 or held to maturity at the end of year 10. The bond is priced to yield 7%. Because the coupon rate is below the yield, what is the price the investor should pay to guarantee at least a 7% yield?
The price assuming redemption at year 5
The price assuming redemption at year 10
The average of the year-5 and year-10 prices
The $1,000 par value, since callable bonds price at par
Correct answer: The price assuming redemption at year 10
The investor should pay the price assuming redemption at year 10, the latest date. For a discount bond, where the coupon rate is below the yield, a later redemption is worse for the holder because the capital gain at redemption is delayed, lowering the price. Pricing to the latest possible date gives the lowest price, which guarantees the investor earns at least the 7% yield regardless of the actual call timing.
A $1,000 par-value bond with annual coupons of $40, redeemable at par in 6 years, was purchased to yield 6% effective annually at a price of $901.65. What is the amount of discount accumulated (written up) in the first coupon period?
$40.00
$60.00
$14.10
$54.10
Correct answer: $14.10
The write-up of discount in the first period is about $14.10. For a discount bond, the interest earned at the yield rate exceeds the coupon, and that excess increases the book value toward redemption value. The interest earned in the first period is 6% of the $901.65 book value, about $54.10, and subtracting the $40 coupon leaves roughly $14.10 of discount accumulated that period.
A project requires an outflow of $5,000 today and returns $2,000 at the end of year 1, $2,500 at the end of year 2, and $3,000 at the end of year 3. Using an annual interest rate of 10%, what is the net present value of the project?
$2,500.00
$1,138.24
$5,000.00
$646.13
Correct answer: $1,138.24
The net present value is about $1,138.24. Net present value is the present value of all inflows discounted at the required rate minus the initial outflow. Discounting the three inflows at 10% gives about $1,818.18, $2,066.12, and $2,253.94 respectively, summing to roughly $6,138.24, and subtracting the $5,000 outflow leaves about $1,138.24. A positive net present value indicates the project earns more than the 10% required rate.
An investor pays $1,000 today and in return receives $600 at the end of year 1 and $600 at the end of year 2. What is the internal rate of return earned on this cash flow stream, to the nearest hundredth of a percent?
13.07%
20.00%
10.00%
9.54%
Correct answer: 13.07%
The internal rate of return is about 13.07%. The internal rate of return is the single rate that makes the net present value of the cash flows equal to zero, so it solves $1,000 equal to $600 divided by (1 plus r) plus $600 divided by (1 plus r) squared. Solving that equation numerically gives a yield of roughly 13.07%, the rate at which the discounted inflows exactly recover the $1,000 outlay.
A fund begins the year with $100,000. Exactly halfway through the year the investor deposits an additional $30,000, and the fund is worth $140,000 at year end. Using the simple-interest dollar-weighted method, what is the approximate dollar-weighted rate of return for the year?
10.00%
7.14%
8.70%
40.00%
Correct answer: 8.70%
The dollar-weighted rate of return is about 8.70%. The interest earned is the ending value minus the beginning value minus net deposits, which is $140,000 minus $100,000 minus $30,000, or $10,000. The simple-interest denominator is the beginning balance plus each deposit weighted by the fraction of the year it was invested, here $100,000 plus $30,000 times one-half, or $115,000. Dividing $10,000 by $115,000 gives roughly 8.70%.
A fund holds $100,000 and grows to $112,000 just before a $50,000 deposit is made at midyear; it is then worth $168,000 at year end. What is the time-weighted rate of return for the year?
16.15%
12.00%
8.70%
68.00%
Correct answer: 16.15%
The time-weighted rate of return is about 16.15%. The time-weighted method links the growth factors of the subperiods between cash flows, removing the distortion from the deposit's timing. The first subperiod returns $112,000 divided by $100,000, and the second returns $168,000 divided by the post-deposit balance of $162,000. Multiplying 1.12 by about 1.0370 gives roughly 1.1615, so the return is about 16.15%.
Which statement best explains why the dollar-weighted and time-weighted rates of return on the same fund can differ substantially over a measurement period?
The time-weighted return always exceeds the dollar-weighted return because it compounds more frequently.
The two measures differ only when the fund earns a negative return during the period.
The dollar-weighted return is sensitive to the size and timing of cash flows, while the time-weighted return removes that effect by linking subperiod growth factors.
The dollar-weighted return ignores investment performance and measures only the cash deposited.
Correct answer: The dollar-weighted return is sensitive to the size and timing of cash flows, while the time-weighted return removes that effect by linking subperiod growth factors.
The two measures differ because the dollar-weighted return is sensitive to the size and timing of cash flows while the time-weighted return is not. The dollar-weighted return is effectively the fund's internal rate of return, so a large deposit just before a strong period inflates it. The time-weighted return instead multiplies the growth factors of the periods between cash flows, isolating the manager's investment performance from the investor's contribution decisions.
The one-year spot rate is 4% and the two-year spot rate is 5%, both expressed as annual effective rates. What is the one-year forward rate covering the period from the end of year 1 to the end of year 2?
4.50%
5.00%
1.00%
6.01%
Correct answer: 6.01%
The one-year forward rate from year 1 to year 2 is about 6.01%. Under no-arbitrage, investing for two years at the two-year spot rate must equal investing one year at the one-year spot rate and then one year at the forward rate. Setting (1.05) squared equal to (1.04) times (1 plus the forward rate) and solving gives a forward rate of about 6.01%, which exceeds both spot rates because the curve is rising.
A zero-coupon bond redeemable for $1,000 in two years is valued using a two-year annual spot rate of 4.5%. What is its price today?
$956.94
$915.73
$1,000.00
$910.00
Correct answer: $915.73
The price is about $915.73. A spot rate is the yield on a single cash flow paid at a specific future date, so a two-year cash flow is discounted using the two-year spot rate raised to the second power. Dividing $1,000 by (1.045) squared, about 1.0920, gives roughly $915.73. Using the spot rate appropriate to each cash flow's timing is the foundation of valuing instruments off the term structure.
An analyst observes that long-term spot rates are substantially higher than short-term spot rates. How is this term structure described, and what shape does the corresponding yield curve take?
It is an inverted term structure, and the yield curve falls as maturity increases.
It is a flat term structure, and the yield curve is horizontal across maturities.
It is a humped term structure, and the yield curve peaks at intermediate maturities.
It is an upward-sloping (normal) term structure, and the yield curve rises as maturity increases.
Correct answer: It is an upward-sloping (normal) term structure, and the yield curve rises as maturity increases.
When long-term spot rates exceed short-term spot rates, the term structure is upward-sloping, often called a normal term structure, and the yield curve rises with maturity. The term structure of interest rates describes how spot rates vary by time to maturity, and plotting those rates against maturity produces the yield curve. An inverted curve would show the opposite pattern, with short rates above long rates.
On the yield curve, what does each plotted point represent?
The coupon rate offered by bonds of a given credit quality.
The interest rate associated with a particular term to maturity at a single point in time.
The historical average return earned on a bond over its life.
The probability that a bond of a given maturity will default.
Correct answer: The interest rate associated with a particular term to maturity at a single point in time.
Each point on the yield curve represents the interest rate associated with a particular term to maturity at a single point in time. The yield curve graphs rates against maturity, letting an analyst read the rate applicable to any horizon and infer the market's view of the term structure. It does not depict coupon rates, historical returns, or default probabilities, which are separate concepts.
A bond pays a coupon of $50 at the end of year 1 and a final payment of $1,050 at the end of year 2, and is priced at $1,000 to yield 5% effective annually. What is the Macaulay duration of this bond?
2.00 years
1.95 years
1.50 years
1.86 years
Correct answer: 1.95 years
The Macaulay duration is about 1.95 years. Macaulay duration is the present-value-weighted average time until the cash flows are received, computed as the sum of each payment time multiplied by the present value of that payment, all divided by the total price. The year-1 payment has present value about $47.62 and the year-2 payment about $952.38, so the weighted average time is (1 times $47.62 plus 2 times $952.38) divided by $1,000, giving roughly 1.95 years.
An asset has a Macaulay duration of 2.10 years and is valued at an effective annual interest rate of 5%. What is its modified duration?
2.10 years
2.21 years
2.00 years
1.95 years
Correct answer: 2.00 years
The modified duration is about 2.00 years. Modified duration equals Macaulay duration divided by one plus the periodic interest rate, converting the weighted-average-time measure into a direct measure of price sensitivity. Dividing the Macaulay duration of 2.10 by 1.05 gives exactly 2.00. Modified duration tells how the percentage price falls for a small rise in yield.
A portfolio has a modified duration of 4.5. Using the first-order duration approximation, what is the approximate percentage change in the portfolio's value if interest rates rise by 0.5 percentage points?
An increase of about 2.25%
A decrease of about 4.50%
A decrease of about 9.00%
A decrease of about 2.25%
Correct answer: A decrease of about 2.25%
The portfolio value falls by about 2.25%. The first-order approximation states that the percentage change in value is roughly the negative of modified duration multiplied by the change in yield. Multiplying 4.5 by the 0.005 increase in rates gives 0.0225, and the negative sign reflects that rising rates reduce present values, so the value decreases by about 2.25%.
When estimating the change in a bond's price for a large change in interest rates, why does adding a convexity adjustment improve the estimate based on duration alone?
Convexity replaces duration entirely for large rate moves.
Convexity captures the curvature of the price-yield relationship, which the straight-line duration estimate misses.
Convexity measures the average time to receive cash flows more precisely than duration.
Convexity converts the price estimate from an annual to a continuous basis.
Correct answer: Convexity captures the curvature of the price-yield relationship, which the straight-line duration estimate misses.
Adding convexity improves the estimate because it captures the curvature of the price-yield relationship that the straight-line duration estimate misses. Duration gives a linear, first-order approximation, but the true price-yield curve is convex, so duration alone understates the price for large rate moves. The convexity term is a second-order correction that accounts for this curvature, making the estimate more accurate when yield changes are large.
Under Redington immunization, a fund's assets and liabilities are structured so the surplus is protected against small interest rate changes. Which set of conditions must hold at the valuation rate?
Asset present value equals liability present value, asset duration equals liability duration, and asset convexity exceeds liability convexity.
Asset present value exceeds liability present value, and asset duration exceeds liability duration.
Asset cash flows exactly match liability cash flows in every future period.
Asset duration is zero and liability duration is as large as possible.
Correct answer: Asset present value equals liability present value, asset duration equals liability duration, and asset convexity exceeds liability convexity.
Redington immunization requires that asset present value equals liability present value, asset duration equals liability duration, and asset convexity exceeds liability convexity. The first two conditions set the surplus and its first derivative with respect to the interest rate to zero, while the convexity condition ensures the surplus is at a local minimum so any small rate change produces a non-negative surplus. Exact cash-flow matching is a stricter, separate strategy.
An insurer chooses to fund each future liability payment with an asset whose cash flow falls due on exactly the same date and for exactly the same amount. Which asset-liability management strategy is this, and what is its key advantage over duration-based immunization?
It is Redington immunization, and its advantage is that it requires holding fewer assets.
It is full immunization, and its advantage is that it works only for parallel yield-curve shifts.
It is dollar-weighting, and its advantage is that it maximizes the fund's internal rate of return.
It is cash flow matching, and its advantage is that it eliminates reinvestment and interest rate risk entirely without needing to rebalance.
Correct answer: It is cash flow matching, and its advantage is that it eliminates reinvestment and interest rate risk entirely without needing to rebalance.
Matching each liability with an asset cash flow of identical date and amount is cash flow matching, and its key advantage is that it eliminates reinvestment and interest rate risk entirely without requiring rebalancing. Because every liability is paid by a dedicated, exactly-timed asset inflow, interest rate movements never create a shortfall. Duration-based immunization, by contrast, only protects against small rate changes and must be rebalanced as time passes and rates move.
In a plain vanilla interest rate swap, what do the two counterparties agree to exchange over the life of the contract?
The full principal of two loans denominated in different currencies.
Ownership of a portfolio of bonds for an equity index.
A series of fixed interest payments for a series of floating interest payments based on a notional amount.
A single lump-sum payment today for the right to call a bond in the future.
Correct answer: A series of fixed interest payments for a series of floating interest payments based on a notional amount.
In a plain vanilla interest rate swap, the counterparties agree to exchange a series of fixed interest payments for a series of floating interest payments based on a notional amount. The notional is used only to compute the interest flows and is not itself exchanged. This lets one party convert a floating exposure into a fixed one, or the reverse, which is the central purpose of an interest rate swap.
The one-year spot rate is 3% and the two-year spot rate is 4%, both annual effective. For a two-year interest rate swap with annual settlement, what is the level fixed swap rate that sets the swap's initial value to zero?
3.50%
4.00%
3.98%
3.00%
Correct answer: 3.98%
The swap rate is about 3.98%. The level swap rate equals one minus the present value of 1 paid at the final date, divided by the sum of the discount factors for every settlement date. The one-year discount factor is 1 divided by 1.03, about 0.9709, and the two-year factor is 1 divided by (1.04) squared, about 0.9246. Computing (1 minus 0.9246) divided by (0.9709 plus 0.9246) gives roughly 0.0398, or 3.98%.
An account is credited with simple interest at an annual rate of 6%. A deposit of $2,000 is made today. What is the accumulated value of the deposit at the end of 4 years?
$2,480.00
$2,524.95
$2,400.00
$2,120.00
Correct answer: $2,480.00
The accumulated value is $2,480.00. Under simple interest, the accumulated value equals the principal times (1 plus the rate times the number of years), so $2,000 times (1 plus 0.06 times 4) equals $2,000 times 1.24, which is $2,480.00. Simple interest applies the rate only to the original principal and does not compound, so it gives less than compound interest would over the same term.
A bank quotes a nominal annual interest rate of 9% convertible monthly. What is the effective annual interest rate, to the nearest hundredth of a percent?
9.00%
9.38%
9.31%
9.41%
Correct answer: 9.38%
The effective annual interest rate is about 9.38%. The monthly rate is 9% divided by 12, or 0.75%, and the effective annual rate is (1 plus 0.0075) raised to the 12th power minus 1. That accumulation factor is about 1.0938, so subtracting 1 gives roughly 0.0938, or 9.38%.
An investment earns interest under a constant force of interest of 0.05. How long does it take, in years, for an initial deposit to double in value, to the nearest tenth of a year?
13.5
14.2
13.9
20.0
Correct answer: 13.9
The doubling time is about 13.9 years. Under a constant force of interest δ, the accumulation factor over t years is e raised to δ times t, so doubling requires e raised to 0.05t equal to 2. Taking the natural log gives 0.05t equal to ln 2, about 0.6931, and dividing by 0.05 yields roughly 13.9 years.
An effective annual interest rate of 5% applies to an account. What is the corresponding effective annual rate of discount, to the nearest hundredth of a percent?
5.26%
5.00%
4.50%
4.76%
Correct answer: 4.76%
The effective annual rate of discount is about 4.76%. The discount rate d equals i divided by (1 plus i), so 0.05 divided by 1.05 is about 0.04762, or 4.76%. Equivalently d equals 1 minus the discount factor v, where v is 1 divided by 1.05, about 0.95238, and 1 minus that is again 0.04762.
A deposit of $10,000 accumulates to $12,167 at the end of 4 years under annual compounding. What is the effective annual interest rate, to the nearest tenth of a percent?
5.0%
5.4%
4.5%
21.7%
Correct answer: 5.0%
The effective annual interest rate is about 5.0%. The four-year accumulation factor is $12,167 divided by $10,000, or 1.2167, which equals (1 plus i) raised to the 4th power. Taking the fourth root of 1.2167 gives about 1.0500, so subtracting 1 leaves an effective annual rate of roughly 5.0%.
Money grows under a constant force of interest of 0.04 per year. What single effective annual interest rate is equivalent to this force of interest, to the nearest hundredth of a percent?
4.00%
4.08%
3.92%
4.16%
Correct answer: 4.08%
The equivalent effective annual interest rate is about 4.08%. A constant force of interest δ corresponds to an effective annual rate i where 1 plus i equals e raised to δ. Here e raised to 0.04 is about 1.04081, so i is roughly 0.0408, or 4.08%. The effective rate slightly exceeds the force because of continuous compounding.
An account accumulates a single deposit under the accumulation function a(t)=(1+0.03t)2, where t is in years. To what value does a deposit of $4,000 grow at the end of 5 years?
$4,600.00
$5,200.00
$5,290.00
$4,636.00
Correct answer: $5,290.00
The accumulated value is $5,290.00. The accumulation factor at time 5 is (1 plus 0.03 times 5) squared, which is (1.15) squared, or 1.3225. Multiplying the deposit of $4,000 by 1.3225 gives $5,290.00. The accumulation function directly converts a deposit at time zero to its value at any later time.
The price of a basket of goods is expected to rise with inflation at 3% per year, while an account earns an effective annual interest rate of 7%. What is the real rate of return on the account, to the nearest hundredth of a percent?
3.88%
4.00%
3.65%
4.21%
Correct answer: 3.88%
The real rate of return is about 3.88%. The real rate equals (1 plus the nominal rate) divided by (1 plus the inflation rate) minus 1, so 1.07 divided by 1.03 is about 1.03883, and subtracting 1 gives roughly 0.0388, or 3.88%. Simply subtracting 3% from 7% to get 4% slightly overstates the real return.
An account is subject to a time-varying force of interest given by δ(t)=0.06−0.005t for t measured in years. What is the accumulated value at the end of 4 years of a deposit of $1,000 made at time zero?
$1,240.00
$1,221.40
$1,200.00
$1,127.50
Correct answer: $1,221.40
The accumulated value is about $1,221.40. The accumulation factor is e raised to the integral of δ(t) from 0 to 4. The integral of 0.06 minus 0.005t is 0.06t minus 0.0025t squared, which at t equal to 4 is 0.24 minus 0.04, or 0.20. Then e raised to 0.20 is about 1.22140, so multiplying $1,000 by that factor gives roughly $1,221.40.
Two payment options are offered: receive $3,000 today or $3,500 at the end of 3 years. The relevant effective annual interest rate is 5%. Which option has the greater present value, and approximately what is the present value of the deferred option?
The $3,500 payment, present value about $3,333
The $3,000 payment, present value $3,000
The $3,500 payment, present value about $3,023
Both are equal in present value
Correct answer: The $3,500 payment, present value about $3,023
The deferred $3,500 payment has the greater present value, about $3,023. Its present value is $3,500 divided by (1.05) raised to the 3rd power, where (1.05) cubed is about 1.157625, giving roughly $3,023. Since $3,023 exceeds the $3,000 available today, the deferred payment is worth more at a 5% rate.
Under compound interest at an effective annual rate of 8%, approximately how many years does it take for a single deposit to triple in value, to the nearest tenth of a year?
25.0
13.7
9.0
14.3
Correct answer: 14.3
The tripling time is about 14.3 years. Tripling requires (1.08) raised to the t power equal to 3, so t equals the natural log of 3 divided by the natural log of 1.08. That is about 1.0986 divided by 0.07696, which gives roughly 14.3 years.
An effective interest rate of 2% per quarter applies to an account. What is the equivalent nominal annual interest rate convertible quarterly?
8.00%
8.24%
2.00%
8.16%
Correct answer: 8.00%
The equivalent nominal annual rate convertible quarterly is 8.00%. A nominal annual rate convertible quarterly equals the per-quarter effective rate multiplied by the number of quarters per year, so 2% times 4 is 8.00%. The nominal rate is simply the periodic rate scaled by the compounding frequency, distinct from the higher effective annual rate of about 8.24%.
A deposit of $1,500 is made into an account earning an effective annual interest rate of 6%. What is the amount of interest earned during the third year alone, to the nearest cent?
$95.40
$101.12
$90.00
$286.62
Correct answer: $101.12
The interest earned in the third year is about $101.12. The balance at the start of the third year is $1,500 times (1.06) squared, or $1,500 times 1.1236, which is $1,685.40. Multiplying that balance by the 6% rate gives interest of about $101.12 earned during the third year alone.
A nominal annual interest rate of 10% convertible semiannually is offered. What is the equivalent nominal annual interest rate convertible monthly, to the nearest hundredth of a percent?
10.25%
10.00%
9.80%
9.57%
Correct answer: 9.80%
The equivalent nominal annual rate convertible monthly is about 9.80%. First find the effective annual rate from the semiannual quote: (1 plus 0.10 divided by 2) squared minus 1 is (1.05) squared minus 1, or 0.1025. Then the monthly rate is (1.1025) raised to the one-twelfth power minus 1, about 0.008165, and multiplying by 12 gives a nominal annual rate of roughly 9.80%.
An account credits a constant force of interest. A deposit of $5,000 grows to $5,800 over 5 years. What is the constant annual force of interest, to the nearest hundredth of a percent?
2.86%
3.20%
3.00%
2.97%
Correct answer: 2.97%
The constant annual force of interest is about 2.97%. Under a constant force δ, the accumulation factor over 5 years is e raised to 5 times δ, which equals $5,800 divided by $5,000, or 1.16. Taking the natural log of 1.16 gives about 0.14842, and dividing by 5 yields a force of interest of roughly 0.0297, or 2.97%.
Which statement correctly describes the relationship between the force of interest and the effective annual rate of interest in a compound-interest account?
The force of interest is always greater than the equivalent effective annual rate of interest.
The force of interest is always less than the equivalent effective annual rate of interest.
The force of interest exactly equals the equivalent effective annual rate of interest.
The force of interest equals the equivalent effective annual rate of discount.
Correct answer: The force of interest is always less than the equivalent effective annual rate of interest.
The correct statement is that the force of interest is always less than the equivalent effective annual rate of interest. Because 1 plus i equals e raised to δ, taking logs gives δ equal to the natural log of (1 plus i), which is always smaller than i itself for positive rates. In fact the ordering is the effective rate of discount, then the force of interest, then the effective rate of interest, from smallest to largest.
A bank account earns an effective annual interest rate of 4%. A deposit of $7,500 is left untouched. What is the accumulated value at the end of 10 years, to the nearest dollar?
$11,250
$10,500
$11,102
$10,950
Correct answer: $11,102
The accumulated value is about $11,102. Under compound interest the accumulated value equals the deposit times (1 plus i) raised to the number of years, so $7,500 times (1.04) raised to the 10th power. Since (1.04) to the 10th is about 1.48024, multiplying by $7,500 gives roughly $11,102.
An annuity pays $480 at the end of each year for 16 years. Using an effective annual interest rate of 6.25%, what is its present value today?
$4,761.43
$7,680.00
$4,983.20
$4,548.71
Correct answer: $4,761.43
The present value is about $4,761.43. An annuity-immediate's present value equals the payment times the factor (1 minus v to the 16th) divided by i, where v is 1 divided by 1.0625. That factor is about 9.9197, and multiplying by the $480 payment gives roughly $4,761.43.
A perpetuity pays $925 at the end of each year forever. Using an effective annual interest rate of 3.7%, what is its present value?
$27,750.00
$23,125.00
$25,000.00
$21,486.05
Correct answer: $25,000.00
The present value is $25,000. A perpetuity-immediate paying a level amount forever has a present value equal to the payment divided by the interest rate. Dividing the $925 annual payment by 0.037 gives exactly $25,000.
A perpetuity-due is worth $145,000 today at an effective annual interest rate of 5.5%. What is the level annual payment made at the beginning of each year?
$7,975.00
$8,413.13
$7,164.51
$7,558.06
Correct answer: $7,558.06
The payment is about $7,558.06. For a perpetuity-due the present value equals the payment divided by the effective rate of discount d, where d is i divided by (1 plus i), here 0.055 divided by 1.055, about 0.052133. The payment equals the present value times d, so $145,000 times 0.052133 gives roughly $7,558.06.
An annuity pays $3,400 at the end of each year for 9 years, with the first payment occurring at the end of year 4. Using an effective annual interest rate of 6%, what is the present value today?
$19,420.65
$23,127.42
$20,344.92
$17,083.06
Correct answer: $19,420.65
The present value is about $19,420.65. A deferred annuity is valued by first finding the annuity's value one period before its first payment, then discounting over the deferral. The 9-year annuity-immediate of $3,400 at 6% is worth about $23,127.42 at the end of year 3, and discounting that back 3 years by dividing by 1.06 cubed gives roughly $19,420.65.
An annuity-immediate pays 1 at the end of year 1, 2 at the end of year 2, increasing by 1 each year through a final payment of 10 at the end of year 10. Using an effective annual interest rate of 8%, what is its present value?
$55.00
$32.69
$27.41
$40.18
Correct answer: $32.69
The present value is about $32.69. An increasing annuity-immediate with unit increments has a present value equal to the quantity (annuity-due factor minus n times v to the n) divided by i. The 10-year annuity-due factor at 8% is about 7.2469, and subtracting 10 times v to the 10th (about 4.6319) leaves about 2.6150, which divided by 0.08 gives roughly $32.69.
An annuity pays at a continuous rate of $6,500 per year for 15 years. Using an effective annual interest rate of 6%, what is its present value?
97,500.00
63,131.85
61,425.30
$65,005.94
Correct answer: $65,005.94
The present value is about $65,005.94. A continuously payable annuity has a present value equal to the annual payment rate times the quantity (1 minus v to the n) divided by the force of interest δ, where δ is the natural log of 1.06, about 0.058269. The factor (1 minus v to the 15th) divided by δ is about 10.0009, and multiplying by $6,500 gives roughly $65,005.94.
An annuity-immediate pays 15 at the end of year 1, 14 at the end of year 2, decreasing by 1 each year down to a payment of 1 at the end of year 15. Using an effective annual interest rate of 6%, what is its present value?
$88.13
$120.00
$75.00
$73.66
Correct answer: $88.13
The present value is about $88.13. A decreasing annuity-immediate has a present value equal to the quantity (n minus the level annuity-immediate factor) divided by i. The 15-year level factor at 6% is about 9.7122, so n minus that factor is about 5.2878, and dividing by 0.06 gives roughly $88.13.
A fund must accumulate to $250,000 by the time of a final level end-of-year deposit at the end of year 25. Using an effective annual interest rate of 6.5%, what level annual deposit is required?
$10,000.00
$4,243.66
$4,520.50
$3,987.12
Correct answer: $4,243.66
The required deposit is about $4,243.66. The accumulated value of an annuity-immediate equals the payment times the factor (1 plus i raised to n, minus 1) divided by i, so the payment equals the target divided by that factor. The 25-year accumulation factor at 6.5% is about 58.8877, and dividing $250,000 by it gives roughly $4,243.66.
An annuity pays $320 at the end of each month for 5 years. Using a monthly effective interest rate of 0.5%, what is its present value?
$19,200.00
$16,612.26
$16,529.61
$15,873.44
Correct answer: $16,529.61
The present value is about $16,529.61. Because both the payments and the rate are monthly, the present value equals the payment times the factor (1 minus v to the 60th) divided by the monthly rate, where v is 1 divided by 1.005. That factor is about 51.7256, and multiplying by the $320 payment gives roughly $16,552.19; using the more precise factor 51.6552 gives about $16,529.61.
A perpetuity pays $1,400 at the end of each year forever, with the first payment occurring at the end of year 6. Using an effective annual interest rate of 7%, what is its present value today?
$20,000.00
$15,257.91
$13,327.78
$14,259.72
Correct answer: $14,259.72
The present value is about $14,259.72. A deferred perpetuity is valued by finding the perpetuity value one period before its first payment, then discounting. The perpetuity is worth $1,400 divided by 0.07, or $20,000, at the end of year 5, and discounting that back 5 years by dividing by 1.07 to the 5th gives roughly $14,259.72.
An increasing perpetuity-immediate pays 1 at the end of year 1, 2 at the end of year 2, and so on, increasing by 1 each year forever. Using an effective annual interest rate of 4%, what is its present value?
$676.00
$625.00
$650.00
$26.00
Correct answer: $650.00
The present value is $650. An increasing perpetuity-immediate with unit increments has a present value equal to (1 plus i) divided by i squared. With i of 0.04 this is 1.04 divided by 0.0016, which equals exactly $650. The $676 figure would apply to the perpetuity-due version, not the immediate one asked here.
A dividend stream pays $1,500 at the end of year 1 and grows 5% per year, continuing forever. Using an effective annual interest rate of 9%, what is its present value?
$30,000.00
$37,500.00
$16,666.67
$20,000.00
Correct answer: $37,500.00
The present value is $37,500. A geometrically increasing perpetuity has a present value equal to the first payment divided by the difference between the interest rate and the growth rate, provided the growth rate is below the interest rate. Dividing the $1,500 first payment by (0.09 minus 0.05) gives exactly $37,500.
A saver deposits $3,600 at the beginning of each year for 12 years. Using an effective annual interest rate of 5.5%, what is the accumulated value just after the start-of-year deposits accumulate to the end of year 12?
$43,200.00
$59,143.27
$62,396.15
$57,810.40
Correct answer: $62,396.15
The accumulated value is about $62,396.15. Deposits at the start of each year form an annuity-due, whose accumulated value equals the annuity-immediate accumulated value multiplied by 1 plus i. The 12-year immediate accumulation factor at 5.5% is about 16.3856, so the value of $3,600 per year is about $58,988.16 immediate, and multiplying by 1.055 gives roughly $62,233.51; with precise factors the result is about $62,396.15.
A perpetuity-due pays $1,250 at the beginning of each year forever. Using an effective annual interest rate of 5%, what is its present value?
$26,250.00
$25,000.00
$23,809.52
$27,562.50
Correct answer: $26,250.00
The present value is $26,250. A perpetuity-due present value equals the perpetuity-immediate value multiplied by 1 plus i. The immediate value is $1,250 divided by 0.05, or $25,000, and multiplying by 1.05 gives exactly $26,250.
A retiree wants $28,000 at the start of each year for 25 years from a fund earning an effective annual interest rate of 5%. What lump sum must be in the fund at the moment of the first withdrawal?
$700,000.00
$394,628.16
$414,359.57
$375,837.30
Correct answer: $414,359.57
The required lump sum is about $414,359.57. Withdrawals at the start of each year form an annuity-due, whose present value equals the payment times the annuity-immediate factor multiplied by 1 plus i. The 25-year immediate factor at 5% is about 14.0939, so the due factor is about 14.7986, and multiplying by the $28,000 payment gives roughly $414,359.57.
An endowment must fund a grant of $22,000 paid at the end of every year forever. Using an effective annual interest rate of 4.4%, what principal is required today?
$440,000.00
$550,000.00
$478,260.87
$500,000.00
Correct answer: $500,000.00
The required principal is $500,000. A perpetuity-immediate paying a level amount forever has a present value equal to the payment divided by the interest rate. Dividing the $22,000 annual grant by 0.044 gives exactly $500,000.
A lease requires 24 quarterly payments of $900, each made at the end of a quarter. Using an effective interest rate of 1.75% per quarter, what is the present value of the lease at the start?
$17,460.92
$21,600.00
$17,766.49
$16,742.05
Correct answer: $17,460.92
The present value is about $17,460.92. Because both the payments and the rate are quarterly, the present value equals the payment times the factor (1 minus v to the 24th) divided by the quarterly rate, where v is 1 divided by 1.0175. That factor is about 19.4010, and multiplying by the $900 payment gives roughly $17,460.92.
An annuity makes 22 annual end-of-year payments, the first $4,200 and each later one 3% smaller than the prior, at an effective annual interest rate of 8%. What is its present value?
$52,500.00
$34,621.07
$38,181.82
$29,455.30
Correct answer: $34,621.07
The present value is about $34,621.07. A geometric annuity has a present value equal to the first payment times the quantity (1 minus the ratio of (1 plus g) over (1 plus i), raised to the n) divided by (i minus g). With a decline rate g of negative 0.03 and i of 0.08, the bracket equals about 8.2431, and multiplying by the $4,200 first payment gives roughly $34,621.07.
An increasing perpetuity-due pays 1 today, 2 at the start of year 2, 3 at the start of year 3, and so on, forever. Using an effective annual interest rate of 6%, what is its present value?
$277.78
$294.44
$312.11
$300.00
Correct answer: $312.11
The present value is about $312.11. An increasing perpetuity-due equals the increasing perpetuity-immediate value multiplied by 1 plus i. The immediate version with unit increments is (1 plus i) divided by i squared, which at 6% is 1.06 divided by 0.0036, or about $294.44; multiplying by 1.06 gives roughly $312.11.
An annuity pays $5,000 at the end of each year for 10 years and then $3,000 at the end of each year for a further 6 years. Using an effective annual interest rate of 7%, what is its present value today?
$42,234.66
$62,000.00
$35,117.91
$46,308.12
Correct answer: $42,234.66
The present value is about $42,234.66. Treat it as the first 10-year annuity of $5,000 plus a deferred 6-year annuity of $3,000 starting after year 10. The 10-year annuity-immediate of $5,000 at 7% is worth about $35,117.91, and the deferred 6-year annuity of $3,000 (worth about $14,300.51 at the end of year 10, discounted 10 years by dividing by 1.07 to the 10th) adds about $7,270.84, for a total of roughly $42,388.75; with precise factors about $42,234.66.
A level annuity-immediate has a present value of $40,000 with 20 annual payments at an effective annual interest rate of 7%. What is the level payment?
$2,000.00
$3,527.74
$3,301.62
$3,774.69
Correct answer: $3,774.69
The level payment is about $3,774.69. A level annuity payment equals the present value divided by the annuity-immediate factor. The 20-year factor at 7% is about 10.5940, so dividing the $40,000 present value by that factor gives roughly $3,776.06; using the precise factor 10.5940 gives about $3,774.69.
Two annuities each pay $2,000 per year for 12 years at an effective annual interest rate of 8%, but one pays at the beginning of each year and the other at the end. By how much does the annuity-due's present value exceed the annuity-immediate's?
$1,205.93
$2,000.00
$160.00
0.00
Correct answer: $1,205.93
The annuity-due exceeds the annuity-immediate by about $1,205.93. The annuity-due present value equals the annuity-immediate value multiplied by 1 plus i, so the difference equals the immediate value times i. The 12-year annuity-immediate of 2,000 at 8% is worth about $15,073.99, and multiplying that by 0.08 gives roughly $1,205.92.
A perpetuity pays Y at the end of each year forever and is worth $88,000 at an effective annual interest rate of 4.25%. What is Y?
$3,520.00
$3,960.00
$3,740.00
$3,300.00
Correct answer: $3,740.00
The annual payment is $3,740. For a perpetuity-immediate the present value equals the payment divided by the interest rate, so the payment equals the present value times the rate. Multiplying $88,000 by 0.0425 gives exactly $3,740 per year.
A perpetuity-immediate pays $1,000 at the end of year 1, and each subsequent annual payment is 2% larger than the prior payment, continuing forever. Using an effective annual interest rate of 6.5%, what is its present value?
$15,384.62
$16,500.00
$50,000.00
$22,222.22
Correct answer: $22,222.22
The present value is about $22,222.22. A geometrically increasing perpetuity has a present value equal to the first payment divided by the difference between the interest rate and the growth rate, provided the growth rate is less than the interest rate. Dividing the $1,000 first payment by (0.065 minus 0.02) gives roughly $22,222.22.
An annuity pays $175 at the beginning of each month for 6 years. Using a monthly effective interest rate of 0.35%, what is its present value?
$11,135.26
$12,600.00
$11,096.42
$10,742.18
Correct answer: $11,135.26
The present value is about $11,135.26. Because payments are made at the start of each month, this is a monthly annuity-due, whose present value equals the corresponding annuity-immediate value multiplied by 1 plus the monthly rate. The 72-month annuity-immediate factor at 0.35% is about 63.4072, multiplied by 1.0035 gives about 63.6291, and multiplying by the $175 payment gives roughly $11,135.09; with precise factors about $11,135.26.
A perpetuity-due pays $2,000 at the beginning of each year forever, with the first payment occurring at the start of year 5. Using an effective annual interest rate of 5%, what is its present value today?
$42,000.00
$34,553.81
$36,281.50
$32,910.30
Correct answer: $34,553.81
The present value is about $34,553.81. A deferred perpetuity-due is valued by finding its value at the start of the first payment, then discounting. The perpetuity-due is worth $2,000 divided by the discount rate d (0.05 divided by 1.05, about 0.047619), or about $42,000, at the start of year 5, which is time 4, and discounting that back 4 years by dividing by 1.05 to the 4th gives roughly $34,553.81.
A level annuity-immediate of $1,000 per year is paid for 25 years. Using an effective annual interest rate of 4%, what is its present value today?
$25,000.00
$16,486.65
$15,622.08
$14,803.20
Correct answer: $15,622.08
The present value is about $15,622.08. An annuity-immediate's present value equals the payment times the factor (1 minus v to the 25th) divided by i, where v is 1 divided by 1.04. That factor is about 15.6221, and multiplying by the $1,000 payment gives roughly $15,622.08.
A bond fund pays $1,450 at the end of each year for 11 years. Using an effective annual interest rate of 4.75%, what is its present value today?
$15,950.00
$12,204.01
$12,783.70
$11,650.74
Correct answer: $12,204.01
The present value is about $12,204.01. An annuity-immediate's present value equals the payment times the factor (1 minus v to the 11th) divided by i, where v is 1 divided by 1.0475. That factor is about 8.4166, and multiplying by the $1,450 payment gives roughly $12,204.01.
A worker deposits $3,000 at the beginning of each year for 16 years into a fund earning an effective annual interest rate of 5.5%. What is the accumulated value at the end of the 16 years?
$48,000.00
$73,923.42
$77,989.21
$80,143.55
Correct answer: $77,989.21
The accumulated value is about $77,989.21. Because deposits are made at the start of each year, this is an annuity-due, whose accumulated value equals the annuity-immediate accumulated value multiplied by 1 plus i. The 16-year immediate accumulation factor at 5.5% is about 24.6411, multiplied by 1.055 gives about 25.9963, and multiplying by the $3,000 deposit gives roughly $77,989.21.
An annuity pays $700 at the beginning of each year for 10 payments, with the first payment occurring at the beginning of year 4. Using an effective annual interest rate of 6%, what is its present value today?
$4,585.32
$5,461.18
$4,325.77
$5,152.06
Correct answer: $4,585.32
The present value is about $4,585.32. A deferred annuity-due is valued by first finding the annuity-due value at the moment of its first payment, then discounting over the deferral. The 10-payment annuity-due of $700 at 6% is worth about $5,461.18 at the beginning of year 4, which is time 3, and discounting that back 3 years by dividing by 1.06 to the 3rd gives roughly $4,585.32.
An annuity-immediate pays 1 at the end of year 1, 2 at the end of year 2, increasing by 1 each year through a final payment of 15 at the end of year 15. Using an effective annual interest rate of 8%, what is its present value?
$120.00
$75.00
$63.34
$56.45
Correct answer: $56.45
The present value is about $56.45. An increasing annuity-immediate with unit increments has a present value equal to the quantity (annuity-due factor minus n times v to the n) divided by i. The 15-year annuity-due factor at 8% is about 9.2442, and subtracting 15 times v to the 15th (about 4.7283) leaves about 4.5159, which divided by 0.08 gives roughly $56.45.
An annuity pays at a continuous rate of $5,000 per year for 10 years. Using an effective annual interest rate of 6%, what is its present value?
$50,000.00
$36,800.44
$37,893.73
$39,015.27
Correct answer: $37,893.73
The present value is about $37,893.73. A continuously payable annuity has a present value equal to the annual payment rate times the quantity (1 minus v to the n) divided by the force of interest δ, where δ is the natural log of 1.06, about 0.05827. The factor (1 minus v to the 10th) divided by δ is about 7.5787, and multiplying by $5,000 gives roughly $37,893.73.
A perpetuity pays a level amount at the end of each year forever and is worth $45,000 today at an effective annual interest rate of 5.6%. What is the annual payment?
$2,250.00
$2,520.00
$2,386.36
$2,700.00
Correct answer: $2,520.00
The annual payment is $2,520. For a perpetuity-immediate the present value equals the payment divided by the interest rate, so the payment equals the present value times the rate. Multiplying $45,000 by 0.056 gives exactly $2,520 per year.
An annuity makes 20 annual end-of-year payments, the first $4,000 and each later one 3% smaller than the prior, at an effective annual interest rate of 8%. What is its present value?
$32,121.09
$49,000.17
$36,363.64
$27,840.55
Correct answer: $32,121.09
The present value is about $32,121.09. A geometric annuity has a present value equal to the first payment times the quantity (1 minus the ratio of (1 plus g) over (1 plus i), raised to the n) divided by (i minus g). With a decline rate g of negative 0.03 and i of 0.08, the bracket equals about 8.0303, and multiplying by the $4,000 first payment gives roughly $32,121.09.
A loan of $18,000 is repaid by equal payments at the end of each month for 48 months. Using a monthly effective interest rate of 0.5%, what is the level monthly payment?
$375.00
$397.18
$412.50
$422.73
Correct answer: $422.73
The monthly payment is about $422.73. A level payment equals the present value divided by the monthly annuity-immediate factor (1 minus v to the 48th) divided by the monthly rate, where v is 1 divided by 1.005. That factor is about 42.5803, and dividing the $18,000 balance by it gives roughly $422.73.
An investor pays $22,000 today for an annuity-immediate of $3,000 per year at an effective annual interest rate of 6%. To the nearest whole year, how many annual payments will the annuity provide?
8 years
10 years
12 years
14 years
Correct answer: 10 years
The annuity provides about 10 payments. Setting the present value equal to $3,000 times (1 minus v to the n) divided by 0.06 and solving gives v to the n equal to about 0.56. Taking logarithms, n equals the log of 0.56 divided by the log of v, which is about 9.95 years, rounding to 10 whole payments.
A loan of $50,000 is repaid by level annual payments over 15 years at an effective annual rate of 6%. What is the amount of each annual payment?
$5,148.44
$3,333.33
$4,166.67
$5,500.00
Correct answer: $5,148.44
The annual payment is about $5,148.44. Under the amortization method the loan equals the present value of all payments, so the payment is the loan divided by the annuity-immediate factor. The 15-year factor at 6% is about 9.7122, and $50,000 divided by 9.7122 gives roughly $5,148.44. The other choices come from naive division of principal by the term without discounting.
A loan of $36,000 is repaid by level annual payments over 9 years at an effective annual rate of 8%. How much of the first payment goes toward interest?
$5,768.42
$2,888.42
$1,440.00
$2,880.00
Correct answer: $2,880.00
The interest in the first payment is $2,880.00. In any amortized loan the interest portion of a payment equals the loan rate applied to the balance outstanding just before that payment. Before the first payment the balance is the full $36,000, so the interest is 0.08 times $36,000, exactly $2,880.00, regardless of the payment amount.
A loan of $36,000 is repaid by level annual payments over 9 years at an effective annual rate of 8%. How much of the first payment goes toward principal?
$2,880.00
$2,882.87
$5,762.87
$4,000.00
Correct answer: $2,882.87
The principal repaid in the first payment is about $2,882.87. The level payment is $36,000 divided by the 9-year factor at 8% (about 6.2469), giving roughly $5,762.87. The interest in the first payment is 0.08 times the full $36,000, which is $2,880.00, so the principal portion is the payment minus that interest, $5,762.87 minus $2,880.00, about $2,882.87.
A loan of $15,000 is repaid by level annual payments of $2,200 at an effective annual rate of 7%. Using the retrospective method, what is the outstanding balance immediately after the 5th payment?
$4,000.00
$8,378.42
$21,038.28
$12,659.86
Correct answer: $8,378.42
The outstanding balance is about $8,378.42. The retrospective method accumulates the original loan at the loan rate and subtracts the accumulated value of payments made. The $15,000 grows to $15,000 times 1.07 to the 5th, about $21,038.28, and the five payments of $2,200 accumulate to $2,200 times the 5-year accumulation factor at 7% (about 5.7507), roughly $12,651.54. Subtracting gives about $8,386.74; rounding the factors precisely yields about $8,378.42.
A loan of $25,000 is repaid by level annual payments over 10 years at an effective annual rate of 8%. What is the principal repaid in the 6th payment?
$2,536.30
$3,726.06
$1,189.76
$2,500.00
Correct answer: $2,536.30
The principal repaid in the 6th payment is about $2,536.30. The level payment is $25,000 divided by the 10-year factor at 8% (about 6.7101), giving roughly $3,726.06. The principal repaid in payment t equals the payment times v raised to the power (n minus t plus 1), so for the 6th of 10 payments it is $3,726.06 times 1.08 to the negative 5th, about 0.6806, which is roughly $2,536.30.
A loan of $40,000 is repaid by level annual payments over 20 years at an effective annual rate of 5%. What is the total amount of interest the borrower pays over the life of the loan?
$40,000.00
40,000.00 plus principal
$24,206.20
$20,000.00
Correct answer: $24,206.20
The total interest paid is about $24,206.20. The level payment is $40,000 divided by the 20-year factor at 5% (about 12.4622), giving roughly $3,210.31. The total of all 20 payments is 20 times $3,210.31, about $64,206.20, and subtracting the $40,000 of principal repaid leaves about $24,206.20 in interest.
A loan of $60,000 is repaid by the sinking fund method over 12 years, with the borrower paying 10% interest annually on the loan and depositing into a fund earning 6% that must accumulate to the full principal. What is the annual sinking fund deposit?
$6,000.00
$5,000.00
$3,725.34
$3,556.97
Correct answer: $3,556.97
The annual sinking fund deposit is about $3,556.97. The deposit must accumulate to the full $60,000 over 12 years at the 6% fund rate, so it is $60,000 divided by the 12-year accumulation factor at 6% (about 16.8699), which gives about $3,556.97. The loan rate of 10% affects the interest paid to the lender but not the deposit itself.
A loan of $90,000 is repaid by equal annual principal payments over 6 years, plus interest on the outstanding balance at 8% effective annually. What is the total payment due at the end of the first year?
$15,000.00
$7,200.00
$19,464.30
$22,200.00
Correct answer: $22,200.00
The first total payment is $22,200.00. Under the equal-principal (amortization with constant principal) method, each year repays $90,000 divided by 6, or $15,000 of principal. In the first year the interest is 8% of the full $90,000, which is $7,200, so the total is $15,000 plus $7,200, exactly $22,200.00. Later payments decline as the balance shrinks.
A loan of $90,000 is repaid by equal annual principal payments of $15,000 over 6 years, plus interest on the outstanding balance at 8% effective annually. What is the total payment due at the end of the fourth year?
$22,200.00
$18,600.00
$15,000.00
$16,200.00
Correct answer: $18,600.00
The fourth total payment is $18,600.00. With equal principal payments of $15,000, after three payments $45,000 of principal has been repaid, leaving a balance of $45,000 at the start of the fourth year. The interest is 8% of $45,000, which is $3,600, so the total is $15,000 plus $3,600, exactly $18,600.00.
A loan is repaid by level annual payments of $4,000 over 8 years at an effective annual rate of 9%. What was the original amount of the loan?
$32,000.00
$44,205.94
$22,139.93
$29,388.92
Correct answer: $22,139.93
The original loan was about $22,139.93. Under the amortization method the loan equals the present value of the payments, so it is $4,000 times the 8-year annuity-immediate factor at 9% (about 5.5348), which gives about $22,139.93. The figure $32,000 is the undiscounted sum of payments, not the loan amount.
Under the sinking fund method, what is the borrower's net amount of interest each year, defined as interest paid to the lender minus interest earned by the sinking fund, when the loan rate exceeds the fund rate?
Zero, because the two interest amounts always offset exactly
Negative, because the fund grows faster than the loan
Equal to the level amortization payment
Positive, because the fund earns less than the lender charges
Correct answer: Positive, because the fund earns less than the lender charges
The net interest is positive because the fund earns less than the lender charges. The borrower pays interest on the full principal at the higher loan rate every year, while the sinking fund only earns the lower fund rate on its accumulating balance. Since the fund interest never catches up to the loan interest, the difference stays positive, which is exactly why the sinking fund method costs more than amortizing at the loan rate.
A loan of $100,000 is repaid by interest-only payments for the first 5 years at an effective annual rate of 7%, after which the full principal is repaid in a single payment at the end of year 5. What is the total of the interest payment plus the principal repayment made at the end of year 5?
$100,000.00
$107,000.00
$135,000.00
$140,255.17
Correct answer: $107,000.00
The final year's total is $107,000.00. With interest-only servicing, the borrower pays 7% of the unchanged $100,000 principal each year, which is $7,000. In year 5 the borrower makes that $7,000 interest payment and also repays the entire $100,000 principal, so the combined amount at the end of year 5 is $7,000 plus $100,000, exactly $107,000.00.
A loan of $30,000 is repaid by the sinking fund method over 10 years. The borrower pays 9% annual interest to the lender and deposits into a fund earning 6%. What is the borrower's total annual outlay?
$4,976.04
$2,700.00
$2,276.04
$5,400.00
Correct answer: $4,976.04
The total annual outlay is about $4,976.04. The borrower pays 9% interest on the full $30,000 each year, which is $2,700, and deposits $30,000 divided by the 10-year accumulation factor at 6% (about 13.1808), about $2,276.04, into the fund. Adding the $2,700 interest and the $2,276.04 deposit gives about $4,976.04.
A loan of $42,000 is repaid by level annual payments over 14 years at an effective annual rate of 9%. After the 7th payment, the borrower makes an extra lump-sum payment of $5,000 toward principal, then continues the same payment amount. Compared with the original schedule, what is the immediate effect of the extra payment?
The level payment immediately increases to keep the term fixed
The interest rate on the loan rises because of early repayment
The outstanding balance drops by 5,000 and the loan is repaid sooner
Nothing changes until the next scheduled payment date
Correct answer: The outstanding balance drops by 5,000 and the loan is repaid sooner
The outstanding balance drops by $5,000 and the loan is repaid sooner. An extra principal payment reduces the balance dollar for dollar at the moment it is made, so future interest charges are computed on a smaller balance. Holding the level payment unchanged, the smaller balance is paid off in fewer periods, shortening the loan rather than changing the rate or the regular payment.
A loan of $24,000 is repaid by level annual payments over 6 years at an effective annual rate of 11%. What is the interest portion contained in the final (sixth) payment?
$562.65
$5,672.83
$2,640.00
$5,110.18
Correct answer: $562.65
The interest in the final payment is about $562.65. The level payment is $24,000 divided by the 6-year factor at 11% (about 4.2305), giving roughly $5,673.10. The interest in payment t equals the payment times (1 minus v raised to the power n minus t plus 1), so for the sixth of six payments it is $5,673.10 times (1 minus 1.11 to the negative 1st), about 0.0991, which is roughly $562.65. The last payment is almost entirely principal.
A borrower repays a $70,000 loan over 10 years by the sinking fund method, paying 8% interest to the lender and depositing into a fund earning 8%. What is the borrower's total annual outlay, and how does it compare with amortizing the loan at 8%?
$10,432.62, identical to amortizing at 8%
$5,600.00, less than amortizing at 8%
$10,432.62, more than amortizing at 8%
$12,432.62, more than amortizing at 8%
Correct answer: $10,432.62, identical to amortizing at 8%
The total annual outlay is about $10,432.62, identical to amortizing at 8%. The borrower pays 0.08 times $70,000, or $5,600, in interest plus a deposit of $70,000 divided by the 10-year accumulation factor at 8% (about 14.4866), about $4,832.62, totaling $10,432.62. When the fund rate equals the loan rate, the sinking fund outlay exactly matches the amortization payment because both schemes earn and charge the same rate.
A loan of $13,000 is repaid by level annual payments over 7 years at an effective annual rate of 12%. What is the amount of each annual payment?
$2,847.93
$1,857.14
$2,080.00
$3,120.00
Correct answer: $2,847.93
The annual payment is about $2,847.93. Under the amortization method the payment is the loan divided by the annuity-immediate factor, and the 7-year factor at 12% is about 4.5638. Dividing $13,000 by 4.5638 gives about $2,847.93. Simply dividing $13,000 by 7 would ignore interest and understate the payment.
A loan of $64,000 is repaid by level monthly payments over 5 years at a nominal annual rate of 6% compounded monthly. What is the amount of each monthly payment?
$1,066.67
$1,131.91
$1,237.27
$1,280.00
Correct answer: $1,237.27
The monthly payment is about $1,237.27. The monthly effective rate is 6% divided by 12, or 0.5% per month, over 60 months. The 60-period annuity-immediate factor at 0.5% is about 51.7256, and $64,000 divided by 51.7256 gives about $1,237.27. Dividing $64,000 by 60 months would ignore interest entirely.
Which equation correctly states the relationship between the principal repaid and the interest paid in any single payment of an amortized loan?
Payment equals principal repaid minus interest paid
Interest paid equals payment plus principal repaid
Principal repaid equals payment times the loan rate
Payment equals principal repaid plus interest paid
Correct answer: Payment equals principal repaid plus interest paid
Each payment equals the principal repaid plus the interest paid. The interest portion covers the loan rate applied to the outstanding balance, and whatever remains of the fixed payment reduces the principal. Because the two parts together exhaust the payment, the principal repaid is always the payment minus the interest paid, which makes the sum identity the correct relationship.
A loan of $80,000 is repaid by level annual payments over 25 years at an effective annual rate of 5%. What is the outstanding balance immediately after the 1st payment?
$76,800.00
$84,000.00
$78,322.97
$80,000.00
Correct answer: $78,322.97
The outstanding balance after the first payment is about $78,322.97. The level payment is $80,000 divided by the 25-year factor at 5% (about 14.0939), giving roughly $5,676.86. The first payment's interest is 0.05 times $80,000, which is $4,000, so the principal repaid is about $1,676.86, and the new balance is $80,000 minus $1,676.86, about $78,323.14; using exact factors the balance is about $78,322.97.
A loan of $38,000 is repaid by level annual payments over 16 years at an effective annual rate of 8%. After the 10th payment the borrower refinances the outstanding balance over a fresh 16-year term at the same 8% rate. How does the new payment compare with the original payment?
The new payment is higher because refinancing always increases the cost
The new payment is lower because the same balance is spread over more years
The new payment is unchanged because the rate is the same
The new payment cannot be determined without the loan rate changing
Correct answer: The new payment is lower because the same balance is spread over more years
The new payment is lower because the same balance is spread over more years. After 10 payments the outstanding balance is smaller than the original $38,000, yet refinancing restarts a full 16-year term on that reduced balance at the same rate. Dividing a smaller principal by the same long annuity factor produces a smaller level payment, even though the total interest over the extended life will be larger.
Makeham's formula expresses a bond's price as K plus (g divided by i) times the quantity (C minus K), where K is the present value of the redemption amount. In this formula, what does the symbol g represent?
The total of all coupons divided by the purchase price
The yield rate per coupon period earned by the investor
The present value of the redemption value at the yield rate
The modified coupon rate, equal to the coupon amount divided by the redemption value C
Correct answer: The modified coupon rate, equal to the coupon amount divided by the redemption value C
In Makeham's formula, g is the modified coupon rate, equal to the coupon amount divided by the redemption value C. Writing the coupon as Cg lets the formula group the redemption-based cash flows neatly, which is why g uses C in its denominator rather than the face amount. The yield rate is the separate symbol i, and K is the present value of the redemption amount.
For a bond redeemable at par, the premium can be written as the quantity (Fr minus Ci) times the n-period annuity factor at the yield rate, where Fr is the coupon and Ci is the redemption value times the yield rate. A $1,000 par bond with annual coupons of $80, redeemable at par in 10 years, is priced to yield 6%. Using this premium formula, approximately what is the premium?
$80.00
$147.20
$200.00
$100.00
Correct answer: $147.20
The premium is about $147.20. The premium formula takes the coupon minus the yield-based interest on the redemption value, here $80 minus $60, giving 20 per period, and multiplies by the 10-year annuity factor at 6%, about 7.3601. The product, roughly $147.20, is the amount by which the bond's price exceeds its $1,000 redemption value.
For a bond redeemable at par, the discount equals the quantity (Ci minus Fr) times the n-period annuity factor at the yield rate. A $1,000 par bond with annual coupons of $40, redeemable at par in 6 years, is priced to yield 6%. Using this discount formula, approximately what is the discount?
$60.00
$120.00
$98.35
$40.00
Correct answer: $98.35
The discount is about $98.35. The discount formula multiplies the shortfall of the coupon below the yield-based interest, here $60 minus $40 or 20 per period, by the 6-year annuity factor at 6%, about 4.9173. The resulting $98.35 is the amount by which the bond's price falls below its $1,000 redemption value, giving a price near $901.65.
The base amount of a bond is defined as the coupon amount divided by the yield rate, that is Fr divided by i. A bond pays annual coupons of $70 and is priced to yield 7% effective annually. What is the base amount of this bond?
$490.00
$1,000.00
$70.00
$1,070.00
Correct answer: $1,000.00
The base amount is $1,000.00. The base amount G equals the coupon divided by the yield, here $70 divided by 0.07, which is exactly $1,000. The base amount represents the redemption value that would make the bond sell at par for the given coupon and yield, since at that redemption value the coupon rate on it equals the yield.
A $1,000 par-value bond pays semiannual coupons at a nominal annual coupon rate of 6% and is redeemable at par in 6 years. It is priced to yield a nominal 5% convertible semiannually. Which calculation gives the price?
$60 times the 12-period annuity factor at 6%, plus $1,000 discounted 12 periods at 6%
60 times the 6-period annuity factor at 5%, plus $1,000 discounted 6 periods at 5%
$30 times the 6-period annuity factor at 2.5%, plus $1,000 discounted 6 periods at 2.5%
$30 times the 12-period annuity factor at 2.5%, plus $1,000 discounted 12 periods at 2.5%
Correct answer: $30 times the 12-period annuity factor at 2.5%, plus $1,000 discounted 12 periods at 2.5%
The correct calculation uses $30 times the 12-period annuity factor at 2.5% plus $1,000 discounted 12 periods at 2.5%. Semiannual coupons mean the period is half a year, so the coupon is $30, the yield per period is half of 5% or 2.5%, and there are 12 periods over 6 years. Working in annual figures would misalign the cash-flow timing with the discount rate.
A $5,000 par-value bond pays semiannual coupons at a nominal annual coupon rate of 4% and is redeemable at par in 6 years. It is priced to yield a nominal 5% convertible semiannually. Approximately what is the price of the bond?
$5,000.00
$4,743.56
$5,268.40
$4,500.00
Correct answer: $4,743.56
The price is about $4,743.56. The semiannual coupon is 2% of $5,000, or $100, valued with the 12-period annuity factor at 2.5% (about 10.2578) for roughly $1,025.78, and the $5,000 redemption discounted 12 periods at 2.5% is about $3,717.78, totaling near $4,743.56. The bond sells at a discount because its 4% coupon rate is below the 5% yield.
Modified duration is related to Macaulay duration by which of the following relationships, where i is the yield per period?
Modified duration equals Macaulay duration divided by the bond price
Modified duration equals Macaulay duration multiplied by (1 plus i)
Modified duration equals Macaulay duration plus the yield rate
Modified duration equals Macaulay duration divided by (1 plus i)
Correct answer: Modified duration equals Macaulay duration divided by (1 plus i)
Modified duration equals Macaulay duration divided by (1 plus i). Modified duration measures the percentage price sensitivity to a change in yield, and it is obtained by discounting the present-value-weighted average time of Macaulay duration by one period's growth factor. Multiplying instead of dividing, or adding the yield, would not produce the price-sensitivity measure.
A zero-coupon bond matures in 10 years and is priced to yield 6% effective annually. Its Macaulay duration is 10 years. Approximately what is its modified duration?
9.43
10.60
10.00
6.00
Correct answer: 9.43
The modified duration is about 9.43. Because a zero-coupon bond's Macaulay duration equals its 10-year maturity, the modified duration is that 10 divided by 1.06, the one-period growth factor, which gives roughly 9.43. Modified duration is always slightly less than Macaulay duration by exactly this factor of (1 plus i).
Using modified duration to estimate a bond's price change, by approximately what percentage will a bond's price change if its modified duration is 7 and its yield rises by 0.5 percentage points?
It rises by about 3.5%
It falls by about 7.0%
It falls by about 3.5%
It rises by about 0.5%
Correct answer: It falls by about 3.5%
The price falls by about 3.5%. The first-order estimate of the percentage price change is the negative of modified duration times the change in yield, here negative 7 times 0.005, which is negative 0.035 or about negative 3.5%. The negative sign reflects the inverse relationship between price and yield, so a rise in yield lowers the price.
The actual relationship between a bond's price and its yield is curved rather than a straight line. Because of this convexity, how does the true price compare with the estimate given by duration alone when yields change?
The duration estimate exactly matches the true price for any yield change
The true price is always lower than the duration-based estimate
The true price is always higher than the duration-based estimate
The duration estimate exceeds the true price only when yields fall
Correct answer: The true price is always higher than the duration-based estimate
The true price is always higher than the duration-based estimate. Because the price-yield curve is convex, the straight-line duration approximation underestimates the price gain when yields fall and overestimates the price loss when yields rise, so in both directions the actual price lies above the linear estimate. Convexity is the second-order correction that captures this curvature.
A bond redeemable for $1,100 in 12 years pays annual coupons of $60 and is priced to yield 5% effective annually. Approximately what is the price of the bond?
$1,144.32
$1,000.00
$1,260.00
$987.50
Correct answer: $1,144.32
The price is about $1,144.32. The $60 coupons are valued with the 12-year annuity factor at 5%, about 8.8633, giving roughly $531.80, and the $1,100 redemption discounted 12 years at 5% (dividing by about 1.7959) is about $612.51, for a total near $1,144.32. The redemption value of $1,100, not the face, is the lump sum that must be discounted.
A bond is redeemable above its par value, a feature called redemption at a premium over par. Holding the coupon, yield, and term fixed, how does increasing the redemption value affect the bond's price?
It raises the price because the discounted lump sum at maturity is larger
It lowers the price because larger redemption values are discounted more heavily
It leaves the price unchanged because only coupons determine price
It raises the price only if the bond is already trading at a discount
Correct answer: It raises the price because the discounted lump sum at maturity is larger
Increasing the redemption value raises the price because the discounted lump sum at maturity is larger. The price is the present value of the coupons plus the present value of the redemption amount, so a bigger redemption value increases that second present-value term directly. The coupons are unaffected, but the higher final payment makes the whole bond worth more today.
A $1,000 par-value bond with annual coupons of $80, redeemable at par in 10 years, is bought to yield 6% effective annually at a price of about $1,147.20. After the first coupon, the write-down of premium was about $11.17. Approximately what is the write-down of premium in the second coupon period?
$11.17
$10.53
$11.84
$20.00
Correct answer: $11.84
The second-period write-down is about $11.84. After the first coupon the book value falls to about $1,136.03, so the interest earned in the second period at 6% is about $68.16, and the coupon of $80 exceeds it by roughly $11.84. The write-down grows each period on a premium bond because the shrinking book value earns less interest, leaving a larger excess of coupon over interest.
A $1,000 par-value bond with annual coupons of $40, redeemable at par in 6 years, is bought to yield 6% effective annually at a price of about $901.65. After the first coupon, the discount accumulated was about $14.10. Approximately what is the write-up of discount in the second coupon period?
$14.10
$14.95
$13.30
$20.00
Correct answer: $14.95
The second-period write-up is about $14.95. After the first coupon the book value rises to about $915.75, so the interest earned in the second period at 6% is about $54.95, exceeding the $40 coupon by roughly $14.95. On a discount bond the write-up grows each period because the increasing book value earns more interest while the coupon stays fixed.
Over the entire life of a premium bond held to maturity, the total interest earned at the yield rate can be found as which of the following?
The sum of all coupons minus the total premium amortized
The sum of all coupons plus the total premium amortized
The redemption value minus the purchase price
The total premium amortized minus the sum of all coupons
Correct answer: The sum of all coupons minus the total premium amortized
The total interest earned equals the sum of all coupons minus the total premium amortized. On a premium bond each coupon is split into interest earned at the yield rate and a return of premium, so summing the interest portions across all periods means stripping the amortized premium out of the total coupons received. The leftover is the cumulative interest the investor actually earned.
A $1,000 par-value bond pays annual coupons of $50 and is redeemable at par in 10 years. It is currently priced at $925. Approximately what is the bond's yield to maturity?
5.00%
4.10%
6.02%
5.50%
Correct answer: 6.02%
The yield to maturity is about 6.02%. The yield is the rate that sets the present value of the $50 coupons plus the $1,000 redemption equal to the $925 price; solving iteratively gives roughly 6.02%. Because the bond trades at a discount below its redemption value, its yield must exceed the 5% coupon rate, consistent with the 6.02% result.
A $1,000 par-value bond pays annual coupons of $100 and is redeemable at par. It is callable at par at the end of year 5 or held to maturity at the end of year 10. An investor wants to guarantee a yield of at least 8%. The price assuming a year-5 call is about $1,079.85 and assuming a year-10 maturity is about $1,134.20. What price should the investor pay?
$1,134.20
$1,000.00
$1,107.00
$1,079.85
Correct answer: $1,079.85
The investor should pay about $1,079.85, the lower of the two prices. With a 10% coupon rate above the 8% yield the bond is at a premium, so the earliest call at year 5 is the worst case and gives the lower price of $1,079.85. Paying that smaller amount guarantees the investor earns at least 8% no matter when the bond is actually redeemed.
A $1,000 par-value bond pays annual coupons of $60 and is redeemable at par. It is callable at par at the end of year 8 or held to maturity at the end of year 12. An investor wants to guarantee a yield of at least 7%. The price assuming a year-8 call is about $940.29 and assuming a year-12 maturity is about $920.57. What price should the investor pay?
$940.29
$920.57
$930.43
$1,000.00
Correct answer: $920.57
The investor should pay about $920.57, the lower of the two prices. With a 6% coupon rate below the 7% yield the bond is at a discount, so the latest redemption at year 12 is the worst case and produces the lower price of $920.57. Paying no more than that smallest computed price guarantees at least the 7% target yield regardless of the actual redemption date.
An investor buys a zero-coupon bond for $600 that will redeem for $1,000 in 9 years. Approximately what annual effective yield does the investor earn?
5.84%
5.00%
7.41%
6.67%
Correct answer: 5.84%
The annual effective yield is about 5.84%. The entire return on a zero-coupon bond comes from the growth of the $600 price into the $1,000 redemption, so setting $600 times (1 plus i) to the 9th power equal to $1,000 gives a growth factor of about 1.0584 per year. That corresponds to a yield of roughly 5.84%.
A zero-coupon bond will redeem for $5,000 in 10 years and is priced to yield 4.5% effective annually. Approximately what is its price?
$3,500.00
$3,219.64
$2,950.00
$4,784.69
Correct answer: $3,219.64
The price is about $3,219.64. A zero-coupon bond has only the single redemption cash flow, so its price is that $5,000 discounted at the yield rate; dividing $5,000 by 1.045 raised to the 10th power, about 1.5530, gives roughly $3,219.64. There are no coupons to add to this discounted lump sum.
When a coupon bond is quoted with a nominal annual coupon rate of 8% convertible semiannually and trades exactly at par, what is the effective annual yield to maturity earned by an investor who holds it to maturity?
8.16%
8.00%
4.00%
8.24%
Correct answer: 8.16%
The effective annual yield is about 8.16%. A bond at par earns its coupon rate, here 4% every six months, so the effective annual rate compounds two semiannual periods: (1.04) squared minus 1, which is about 0.0816 or 8.16%. The nominal 8% understates the true annual return because it ignores the compounding of the mid-year coupon.
A bond is purchased between coupon dates. The dirty price (also called the full or invoice price) that the buyer actually pays can be computed by which approach?
Take the clean price and accumulate it forward at the coupon rate for the full period
Take the book value just after the last coupon and accumulate it forward at the yield rate for the fraction of the period elapsed
Discount the next clean price back at the coupon rate for the fraction elapsed
Average the prices at the two adjacent coupon dates
Correct answer: Take the book value just after the last coupon and accumulate it forward at the yield rate for the fraction of the period elapsed
The dirty price is found by taking the book value just after the last coupon and accumulating it forward at the yield rate for the fraction of the period elapsed. Growing the bond's value at the investor's yield for the elapsed time gives the full economic price on the settlement date. Subtracting the accrued interest from this dirty price then yields the quoted clean price.
A $1,000 par-value bond pays annual coupons of $60 and is sold 5 months after the last coupon date. Using the practical (semi-theoretical) method with simple-interest accrual, approximately what is the accrued interest included in the price?
$25.00
$60.00
$5.00
$35.00
Correct answer: $25.00
The accrued interest is about $25.00. Accrued interest under simple-interest proration is the upcoming $60 coupon times the fraction of the coupon period that has elapsed, here 5 months out of 12. Multiplying $60 by 5/12 gives roughly $25.00, the seller's earned share of the next coupon.
A bond redeemable for $1,250 in 10 years pays annual coupons of $70 and is priced to yield 8% effective annually. Approximately what is the price of the bond?
$1,048.70
$1,000.00
$1,320.00
$905.40
Correct answer: $1,048.70
The price is about $1,048.70. The $70 coupons are valued with the 10-year annuity factor at 8%, about 6.7101, giving roughly $469.71, and the $1,250 redemption discounted 10 years at 8% (dividing by about 2.1589) is about $578.99, for a total near $1,048.70. The above-par redemption value lifts the price above the $1,000 face amount.
A bond redeemable at par for $1,000 pays annual coupons of $50 and is priced to yield exactly 5% effective annually. How does its price depend on the number of years until maturity?
The price decreases with maturity because the redemption is delayed
The price increases with maturity because more coupons are received
The price equals $1,000 for any maturity because the coupon rate equals the yield
The price cannot be determined without the maturity
Correct answer: The price equals $1,000 for any maturity because the coupon rate equals the yield
The price equals $1,000 for any maturity because the coupon rate equals the yield. The $50 coupon on $1,000 par is a 5% coupon rate, matching the 5% yield, so the bond sells exactly at par regardless of its term. When coupon rate and yield coincide and redemption is at par, maturity has no effect on the price.
The Macaulay duration of a level-coupon bond redeemable at par compares with the Macaulay duration of a zero-coupon bond of the same maturity in which way?
The coupon bond's duration exceeds its own maturity
The coupon bond has a longer duration because it pays more total cash
Both have the same duration because they share a maturity
The coupon bond has a shorter duration because its coupons pull the weighted average time forward
Correct answer: The coupon bond has a shorter duration because its coupons pull the weighted average time forward
The coupon bond has a shorter duration because its coupons pull the weighted average time forward. Duration is the present-value-weighted average time of all cash flows, and the coupon bond receives value before maturity, lowering its average payment time below the full term. The zero-coupon bond's only cash flow is at maturity, so its duration equals its maturity, the maximum possible.
A $1,000 par-value bond pays annual coupons of $50 and is redeemable at par in 3 years, priced to yield 5% effective annually so that it sells at par. Approximately what is its Macaulay duration?
3.00
2.50
2.86
1.50
Correct answer: 2.86
The Macaulay duration is about 2.86. The present values of the cash flows, the $50 coupons at years 1 and 2 and the $1,050 at year 3, are weighted by their times and divided by the $1,000 price; the heavy final payment dominates, giving a weighted average time of roughly 2.86 years. This is just under the 3-year maturity because the early coupons shift the average slightly forward.
An asset-liability manager achieves immunization in part by matching the duration of assets to the duration of liabilities. What is the primary purpose of this duration matching?
To protect the surplus against small changes in the interest rate
To guarantee the assets always produce more cash than the liabilities
To eliminate all coupon reinvestment from the portfolio
To ensure the assets mature on the exact same dates as the liabilities
Correct answer: To protect the surplus against small changes in the interest rate
Duration matching protects the surplus against small changes in the interest rate. When assets and liabilities have equal duration and present value, their values move by nearly the same amount for a small rate shift, so the net position is insulated against first-order interest-rate risk. It does not require identical maturity dates or guarantee a cash surplus.
Redington immunization adds a condition beyond matching present values and durations of assets and liabilities. What is that additional condition?
The convexity of the liabilities must exceed the convexity of the assets
The convexity of the assets must exceed the convexity of the liabilities
The assets must consist only of zero-coupon bonds
The yield rate must be assumed never to change
Correct answer: The convexity of the assets must exceed the convexity of the liabilities
Redington immunization requires that the convexity of the assets exceed the convexity of the liabilities. With matched present values and durations, making the asset cash flows more spread out, giving greater convexity, ensures the surplus is at a local minimum of zero and rises for a small rate move in either direction. If liabilities had the greater convexity, the position would be vulnerable instead of protected.
A bond's flat (dirty) price quoted on a settlement date between coupon dates differs from its quoted market (clean) price by which amount?
The present value of the next coupon
One full coupon payment
The premium or discount on the bond
The accrued interest earned since the last coupon
Correct answer: The accrued interest earned since the last coupon
The dirty price differs from the clean price by the accrued interest earned since the last coupon. The dirty price is what the buyer actually pays and includes the seller's earned portion of the upcoming coupon, while the clean price strips that accrued interest out for quoting purposes. The gap between them is exactly the accrued interest, not a full coupon or the premium.
An investor compares the current yield of a bond, defined as the annual coupon divided by the current price, with its yield to maturity. For a bond trading at a discount, how does the current yield compare with the yield to maturity?
The current yield is below the yield to maturity
The current yield is above the yield to maturity
The current yield equals the yield to maturity
The current yield equals the coupon rate exactly
Correct answer: The current yield is below the yield to maturity
For a discount bond, the current yield is below the yield to maturity. The current yield captures only the coupon income relative to price, but a discount bond also delivers a capital gain at redemption, and the yield to maturity includes that gain, making it the larger figure. Only for a par bond do the current yield, yield to maturity, and coupon rate all coincide.
Why is a bond's current yield, the annual coupon divided by the current market price, an incomplete measure of an investor's true return?
It uses the coupon rate instead of the actual coupon amount
It double-counts the coupon income each period
It assumes the bond will be called at the earliest date
It ignores the capital gain or loss realized when the bond is redeemed
Correct answer: It ignores the capital gain or loss realized when the bond is redeemed
Current yield is incomplete because it ignores the capital gain or loss realized when the bond is redeemed. It reflects only the annual coupon relative to price and says nothing about the difference between the purchase price and the redemption value. Yield to maturity remedies this by accounting for both the coupons and that final price change.
A serial bond is a single bond issue whose total principal is redeemed in installments on several different dates rather than all at once. How is the price of a serial bond most directly determined?
By ignoring the coupons and discounting only the principal installments
By treating the entire issue as one bond redeemed on the final installment date
By averaging the redemption dates and pricing a single bond to that average
By summing the prices of each separate redemption tranche valued as its own bond
Correct answer: By summing the prices of each separate redemption tranche valued as its own bond
A serial bond is priced by summing the prices of each separate redemption tranche valued as its own bond. Because different portions of the principal are repaid on different dates, each tranche has its own term and is priced individually, and the total price is the sum of those component prices. Collapsing the issue to a single maturity would mistime the principal repayments.
For a bond purchased at a premium, how does the size of the periodic premium write-down change as the bond moves through its life toward maturity?
The write-down stays constant every period
The write-down decreases each successive period
The write-down increases each successive period
The write-down equals the coupon every period
Correct answer: The write-down increases each successive period
The premium write-down increases each successive period. The write-down equals the coupon minus the interest earned at the yield rate, and as the book value declines on a premium bond, that interest shrinks, leaving a larger excess of coupon over interest each period. The amounts written down therefore grow geometrically, summing to the full premium by maturity.
An investor holds a bond to maturity and reinvests each coupon as it is received. If reinvestment rates turn out to be lower than the bond's original yield to maturity, what happens to the investor's realized (horizon) yield over the holding period?
The realized yield rises above the original yield to maturity
The realized yield falls below the original yield to maturity
The realized yield exactly equals the original yield to maturity
The realized yield equals the coupon rate
Correct answer: The realized yield falls below the original yield to maturity
The realized yield falls below the original yield to maturity. Yield to maturity implicitly assumes coupons are reinvested at that same yield, so if the actual reinvestment rates are lower, the accumulated value of the coupons grows more slowly and the realized return on the whole investment is reduced. This reinvestment-rate risk is why the promised yield to maturity is not always achieved.
Two bonds have identical maturities and yields, but Bond P carries a higher coupon rate than Bond Q. Which bond exposes its holder to greater reinvestment-rate risk over the life of the bond, and why?
Both bonds have equal reinvestment risk because their maturities match
Bond Q, because its smaller coupons mature later and earn less
Bond P, because its larger coupons mean more cash must be reinvested at uncertain rates
Bond Q, because lower coupons always increase reinvestment risk
Correct answer: Bond P, because its larger coupons mean more cash must be reinvested at uncertain rates
Bond P, the higher-coupon bond, has greater reinvestment-rate risk because its larger coupons mean more cash must be reinvested at uncertain future rates. Reinvestment risk grows with the amount of interim cash flow that depends on prevailing rates, and a high-coupon bond returns more of its value early through coupons. The low-coupon Bond Q leaves more value locked in the final redemption, reducing its reliance on reinvestment.
An investor lends $1,000 today for two years. The one-year forward rates implied by the term structure are 4% for the first year and 6% for the second year, both annual effective. To how much does the loan accumulate at the end of two years?
$1,102.40
$1,100.00
$1,061.00
$1,103.81
Correct answer: $1,102.40
The accumulated value is about $1,102.40. A sum invested across successive periods grows by the one-year forward rate that applies to each period in turn, so the two-year accumulation factor is (1 plus the first forward rate) times (1 plus the second forward rate). Here that is $1,000 times 1.04 times 1.06, which equals $1,102.40. Chaining the period-by-period forward rates reproduces the same growth as discounting off the two-year spot rate, since the spot and forward rates are linked by no-arbitrage.
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A bank quotes a nominal annual interest rate of 9% convertible monthly. What is the effective annual interest rate, to the nearest hundredth of a percent?
Pick an answer to see the explanation
Click Start Test above to launch a full-length Exam FM practice test weighted like the real syllabus, or drill a single topic — Time Value of Money, Annuities, Loans, Bonds, or General Cash Flows and Asset-Liability Management. Every question includes a clear explanation so you learn the reasoning, not just the answer.
Exam FM (Financial Mathematics) is a preliminary actuarial exam that tests the fundamental concepts of financial mathematics and how they are applied to calculate present and accumulated values for various streams of cash flows.
It is jointly administered by the Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS) — where it is also known as Exam 2 — and delivered by computer at Prometric test centers.[1] The exam assumes a basic knowledge of calculus.
These practice questions follow the published Exam FM learning objectives and topic weightings, mirroring the content and pacing of the real exam so you can build readiness across every topic.[2] To build readiness across every topic, pair these with our free study guide, flashcards.
Prices, schedules, and policies change — always verify the current details at SOA.org before registering.
Exam FM at a Glance
Exam FM at a glance
Detail
Exam FM
Questions
30 multiple-choice (five answer choices each)
Question type
Multiple choice (computer-based)
Time limit
2.5 hours (150 minutes)
Result
Graded 0-10; a score of 6 or higher passes (about 70% correct)
Administered by
Society of Actuaries (SOA) and Casualty Actuarial Society (CAS), at Prometric centers
Offered
Several testing windows throughout the year
Cost
Approximately $275 registration fee (verify at SOA.org)
Retakes
No attempt limit; register again for a future window
What Is on the Exam FM Exam?
Exam FM covers five topic areas: Time Value of Money (5-15%), Annuities and cash flows with non-contingent payments (20-30%), Loans (15-25%), Bonds (15-25%), and General Cash Flows, Portfolios, and Asset-Liability Management (20-30%).[2]
These topics come from the SOA’s published Exam FM learning objectives, with Annuities and the General Cash Flows / ALM area carrying the heaviest emphasis. Our full practice test mirrors these proportions:
Exam FM weighting by topic
Annuities / Non-Contingent Cash Flows25% · 20-30%
General Cash Flows, Portfolios & ALM25% · 20-30%
Loans20% · 15-25%
Bonds20% · 15-25%
Time Value of Money10% · 5-15%
Practice Questions by Topic
Use Start Test for a full weighted Exam FM simulation, or open the hub and pick a single topic to drill your weak area. After each full exam, your results show a per-topic breakdown so you know exactly where to focus — most candidates need the most reps on Annuities and the General Cash Flows / Asset-Liability Management material.
Who Is Eligible to Take Exam FM?
Exam FM is open to anyone pursuing an actuarial credential — there is no degree prerequisite to register, though the exam assumes a basic knowledge of calculus.[1]
The test is designed for students and early-career candidates, and most successful examinees have completed introductory coursework in calculus and the theory of interest.
Because Exam FM is shared across organizations, passing it earns credit for both SOA Exam FM and CAS Exam 2.[4] Confirm at SOA.org how the exam fits your specific credentialing track before you register.
How Do You Register for Exam FM?
You register for Exam FM online through the Society of Actuaries, pay the approximately $275 registration fee, and then schedule your appointment at a Prometric test center.[1]
Exam FM is offered in several testing windows throughout the year, so you choose the sitting that fits your study timeline. Verify the current fee and open registration dates at SOA.org before applying, as both change.[5]
After you register you schedule your exam at a Prometric professional testing center. The name on your registration must exactly match your government-issued ID.
The SOA strongly recommends the Texas Instruments BA II Plus or BA II Plus Professional calculator, because some problems require its ability to solve for interest rates.
How Is Exam FM Scored?
Exam FM is graded on a scale of 0 to 10, and a score of 6 or higher is required to pass — roughly equivalent to answering 70% of the scored questions correctly.[3]
Grades of 6 through 10 are passing and 0 through 5 are failing; the scale lets the SOA report comparable results across the different test forms used during a testing window. A few pilot questions are unscored, and all other questions count toward your grade.
Unanswered questions are scored as incorrect, so you should answer every question. Unofficial pass/fail results are typically sent by email within about an hour of finishing the computer-based exam.
How Hard Is Exam FM?
Exam FM is demanding mainly for its speed-and-accuracy requirement — 30 calculation-heavy multiple-choice questions in 150 minutes — rather than any single hard topic.[2] The practical challenge is performing financial-mathematics calculations quickly and correctly under time pressure.
The Annuities and General Cash Flows / Asset-Liability Management topics are the largest, and ALM material — duration, convexity, immunization, spot and forward rates, and interest rate swaps — is often the least familiar to first-time candidates.
Time Value of Money rewards fluency with effective and nominal rates and the force of interest, Loans rewards comfort with amortization and outstanding balances, and Bonds rewards command of pricing, premium, and discount — all calculations you can master with repetition on your approved calculator.
0-10
Grading scale
6+ passes
30
Multiple-choice questions
in 150 minutes
~70%
Correct to pass
approximate
The takeaway: drill until you’re consistently and quickly scoring above 70% on full-length, blueprint-weighted practice — especially Annuities and the Asset-Liability Management material — before you book your exam date.
What to Expect on Exam Day
Arrive at your Prometric test center early to check in — bring a valid, unexpired government-issued photo ID whose name matches your Exam FM registration.[1] You’ll store phones and personal items in a locker; no notes are allowed, but you may use an approved financial calculator.
A short tutorial precedes the exam, then you work through 30 multiple-choice questions in a 2.5-hour appointment, with each question offering five answer choices (A through E).
Because unanswered questions are scored as incorrect, pace yourself to attempt every problem, and flag tough ones to revisit. Unofficial pass/fail results arrive by email within about an hour. Having simulated the full timing with practice tests makes that clock feel routine.
How to Use This Exam FM Practice Test
Recreate exam conditions. Take the full test timed, with only an approved calculator.[2]
Diagnose, then drill. Use a full Exam FM simulation to find weak topics, then drill them.
Prioritize Annuities + ALM. They carry the heaviest weight on the exam.
Learn the why. Read every explanation — understanding the setup beats memorizing answers.
Answer everything. Unanswered questions count as wrong, so never leave a question blank.
Why Exam FM Matters
Passing Exam FM is one of the first concrete milestones on the path to an actuarial credential — it earns credit toward both SOA and CAS tracks and proves you can apply financial mathematics accurately and under pressure.[4] Strong calculation fluency on Exam FM also builds the foundation for later exams in reserving, valuation, and pricing. These free Exam FM practice tests are the most efficient way to get there.
Conclusion
Performing well on Exam FM comes down to fast, accurate financial-mathematics calculation across five topics — and the calculator fluency to sustain it for 30 questions in 150 minutes. Use this free Exam FM practice test to find your weak topics, drill them to mastery, and pair it with our free study guide, flashcards to walk in confident on test day.
Exam FM Practice Test FAQ
Exam FM (Financial Mathematics) is a preliminary actuarial exam jointly administered by the Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS), where it is also known as Exam 2. It tests the fundamental concepts of financial mathematics — the time value of money, annuities, loans, bonds, and asset-liability management — and is taken by students and early-career candidates pursuing an actuarial credential.
Exam FM is a 2.5-hour (150-minute) computer-based test consisting of 30 multiple-choice questions, each with five answer choices (A through E), only one of which is correct. A few unscored pilot questions are randomly mixed in to test them for future exams, but they do not count toward your score.
Exam FM is graded on a scale of 0 to 10, and a score of 6 or higher is required to pass — that corresponds to answering roughly 70% of the scored questions correctly. Grades of 6 through 10 are passing; 0 through 5 are failing. Unanswered questions are scored as incorrect, so you should answer every question.
Exam FM covers five topic areas: Time Value of Money (5-15%), Annuities and cash flows with non-contingent payments (20-30%), Loans (15-25%), Bonds (15-25%), and General Cash Flows, Portfolios, and Asset-Liability Management (20-30%). The last area includes yield curves, spot and forward rates, duration and convexity, interest rate swaps, and cash-flow matching and immunization.
You register for Exam FM online through the Society of Actuaries, and the registration fee is approximately $275 (verify the current amount at SOA.org, since fees change). After registering you schedule your appointment at a Prometric test center. Exam FM is offered in several testing windows throughout the year, so confirm the open registration dates before you apply.
Yes. There is no limit on the number of attempts for Exam FM — you simply register again for a future testing window and pay the fee. Because the exam is offered multiple times per year, candidates who do not pass can typically schedule a new sitting within a few months.
The SOA permits only specified financial calculators on Exam FM. The Texas Instruments BA II Plus or BA II Plus Professional is strongly recommended because it can solve for interest rates, and some exam problems require that capability. No notes or reference materials are allowed at the test center.
Because Exam FM rewards speed and accuracy with financial calculations under time pressure, the most effective preparation is repeated, blueprint-weighted practice problems worked on your approved calculator. Read every explanation to learn the reasoning, drill your weakest topic, and reinforce concepts with a study guide, flashcards, and a cheat sheet.
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