Are flashcards effective for Exam FM?

Yes. Flashcards use active recall — retrieving an answer from memory — which research shows is one of the most effective study methods, especially in short, spaced sessions. Because Exam FM rewards instant recall of formulas and annuity symbols, the cards are an efficient way to make that knowledge automatic before you reach for the calculator.

What topics do the Exam FM flashcards cover?

All six syllabus topics: interest and discount and the force of interest (Time Value of Money); annuity-immediate, annuity-due, perpetuities and varying annuities; loan amortization, balances and sinking funds; bond pricing, premium/discount and callable bonds; NPV, IRR, duration, convexity and immunization; and spot/forward rates, the yield curve and interest rate swaps.

How should I study with these Exam FM flashcards?

Lead with the heaviest topics — Annuities and Cash Flows and General Cash Flows/ALM each carry the most weight — and make sure you also drill the newer Swaps topic. Mix the modes: flip to learn the formulas, type to test recall, match for speed, and quiz to check yourself before working full practice questions on the calculator.

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Yes — 100% free, all four study modes, no paywall.

Do these flashcards match the current Exam FM syllabus?

Yes. The cards are organized to the current SOA/CAS Exam FM (Financial Mathematics) syllabus, including the General Cash Flows/Portfolios/ALM topic (duration, convexity, immunization) and the Term Structure & Interest Rate Swaps topic. The notation matches the actuarial symbols you'll see on exam day.

Effective rate of interest (i)

The rate earned over one period on the balance at the START of the period: 1 grows to $ 1 + i $ after one period.

Effective rate of discount (d)

The rate applied to the END-of-period balance: $ d = \dfrac{i}{1+i} = 1 - v = iv $.

The present value of 1 due in one period: $ v = \dfrac{1}{1+i} = 1 - d $.

Accumulated value of 1 in n periods

$ (1+i)^n $ under compound interest.

Simple interest accumulation

$ a(t) = 1 + it $: interest is earned only on the original principal, never on prior interest.

Compound interest accumulation

$ a(t) = (1+i)^t $: interest earns interest. Exam FM is compound unless told otherwise.

Accumulation function a(t)

The accumulated value at time t of 1 invested at time 0. Under compound interest $ a(t) = (1+i)^t $.

Nominal rate of interest i⁽ᵐ⁾

An annual rate convertible m times a year; each subperiod earns $ i^{(m)}/m $. Effective annual rate $ = (1 + i^{(m)}/m)^m - 1 $.

Nominal rate of discount d⁽ᵐ⁾

Convertible m times a year; $ 1 + i = \left(1 - \dfrac{d^{(m)}}{m}\right)^{-m} $.

Effective annual rate from i⁽ᵐ⁾

$ i = \left(1 + \dfrac{i^{(m)}}{m}\right)^{m} - 1 $. More frequent compounding raises the effective rate.

$ e^{\delta} = 1 + i $, so $ v = e^{-\delta} $ and $ \delta = \ln(1+i) $.

Varying force of interest accumulation

$ a(t) = \exp\!\left(\int_0^{t} \delta(s)\,ds\right) $.

Force of interest from a(t)

$ \delta(t) = \dfrac{a'(t)}{a(t)} = \dfrac{d}{dt}\ln a(t) $.

Set the value of all inflows equal to the value of all outflows at a single chosen comparison date, after discounting/accumulating each cash flow to it.

Discounting vs accumulating

Discount a future payment back by multiplying by $ v^t $; accumulate a deposit forward by multiplying by $ (1+i)^t $.

Ordering: i, i⁽ᵐ⁾, δ, d⁽ᵐ⁾, d

For a fixed effective rate: $ d < d^{(m)} < \delta < i^{(m)} < i $. The force δ sits between the nominal discount and nominal interest rates.

$ v = 1 - d $; the discount factor is one minus the discount rate.

Doubling time (constant force)

Solve $ e^{\delta t} = 2 $, so $ t = \dfrac{\ln 2}{\delta} $.

Rule of 72 (intuition)

A balance roughly doubles in $ 72/(100i) $ years; an approximation, exact doubling uses $ \ln 2 \div \ln(1+i) $.

Effective rate over a non-unit period

Over period [a,b], the effective rate is $ \dfrac{a(b) - a(a)}{a(a)} $.

The value today of one or more future cash flows, found by discounting each at the appropriate rate.

The value at a future date of one or more cash flows, found by accumulating each forward at the appropriate rate.

Interest in the t-th period a(t) − a(t−1)

The amount of interest earned in period t on an investment of 1 is $ a(t) - a(t-1) $.

Convert force δ to discount factor

$ v = e^{-\delta} $; to discount n years, multiply by $ e^{-\delta n} $.

Continuous compounding limit

As m → ∞, $ \left(1 + \dfrac{i^{(m)}}{m}\right)^{m} \to e^{\delta} = 1+i $.

Why d < i for the same effective rate

Discount is taken off the larger end-of-period amount, so a smaller rate d achieves the same effect as interest rate i on the smaller starting amount.

Effective vs nominal: which is bigger?

For a positive nominal rate, the effective annual rate exceeds the nominal rate whenever m > 1 (more than one compounding per year).

Present value of a single payment C in n periods

$ PV = C\,v^n = \dfrac{C}{(1+i)^n} $.

Level payments at the END of each period. Present value $ a_{\overline{n}|} = \dfrac{1 - v^n}{i} $.

Level payments at the START of each period. Present value $ \ddot{a}_{\overline{n}|} = \dfrac{1 - v^n}{d} = a_{\overline{n}|}(1+i) $.

Relationship between ä and a

$ \ddot{a}_{\overline{n}|} = a_{\overline{n}|}\,(1+i) $: each due payment lands one period earlier.

Accumulated value, annuity-immediate

$ s_{\overline{n}|} = \dfrac{(1+i)^n - 1}{i} $.

Accumulated value, annuity-due

$ \ddot{s}_{\overline{n}|} = s_{\overline{n}|}(1+i) = \dfrac{(1+i)^n - 1}{d} $.

$ s_{\overline{n}|} = a_{\overline{n}|}\,(1+i)^n $: the accumulated value is the present value rolled forward n periods.

Pays 1 at the end of each period forever; present value $ = \dfrac{1}{i} $.

Pays 1 at the start of each period forever; present value $ = \dfrac{1}{d} $.

Value the annuity one period before its first payment, then discount over the deferral: $ {}_{k|}a_{\overline{n}|} = v^k\,a_{\overline{n}|} $.

Increasing annuity-immediate (Ia)ₙ

Payments 1, 2, …, n at period ends: $ (Ia)_{\overline{n}|} = \dfrac{\ddot{a}_{\overline{n}|} - n v^n}{i} $.

Decreasing annuity-immediate (Da)ₙ

Payments n, n−1, …, 1 at period ends: $ (Da)_{\overline{n}|} = \dfrac{n - a_{\overline{n}|}}{i} $.

Increasing perpetuity-immediate

Payments 1, 2, 3, … forever: present value $ = \dfrac{1+i}{i^2} $. It also equals $ \dfrac{1}{i} + \dfrac{1}{i^2} $.

Geometric (growing) annuity

First payment P, growing at g per period, n payments: $ PV = P\cdot\dfrac{1 - \left(\frac{1+g}{1+i}\right)^n}{i - g} $ (for $ g \neq i $).

First payment P, growing at g forever (g < i): $ PV = \dfrac{P}{i - g} $.

Paid continuously at rate 1 per period: $ \bar{a}_{\overline{n}|} = \dfrac{1 - v^n}{\delta} $.

Your FREE Exam FM Flashcards 2026 – 200+ Cards

Realistic Exam FM flashcards across all six Financial Mathematics topics — flip, match, type, and quiz yourself on interest theory, annuities, loans, bonds, and swaps.

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Exam FM Flashcards

Effective rate of interest (i): The rate earned over one period on the balance at the START of the period: 1 grows to $1 + i$ after one period.
Effective rate of discount (d): The rate applied to the END-of-period balance: $d = \dfrac{i}{1+i} = 1 - v = iv$ .
Discount factor (v): The present value of 1 due in one period: $v = \dfrac{1}{1+i} = 1 - d$ .
Present value of 1 in n periods: $v^n = (1+i)^{-n}$ .
Accumulated value of 1 in n periods: $(1+i)^n$ under compound interest.
Simple interest accumulation: $a(t) = 1 + it$ : interest is earned only on the original principal, never on prior interest.
Compound interest accumulation: $a(t) = (1+i)^t$ : interest earns interest. Exam FM is compound unless told otherwise.
Accumulation function a(t): The accumulated value at time t of 1 invested at time 0. Under compound interest $a(t) = (1+i)^t$ .
Nominal rate of interest i⁽ᵐ⁾: An annual rate convertible m times a year; each subperiod earns $i^{(m)}/m$ . Effective annual rate $= (1 + i^{(m)}/m)^m - 1$ .
Nominal rate of discount d⁽ᵐ⁾: Convertible m times a year; $1 + i = \left(1 - \dfrac{d^{(m)}}{m}\right)^{-m}$ .
Effective annual rate from i⁽ᵐ⁾: $i = \left(1 + \dfrac{i^{(m)}}{m}\right)^{m} - 1$ . More frequent compounding raises the effective rate.
Force of interest (δ): The instantaneous, continuously compounded rate. For a constant force, accumulation over t years is $e^{\delta t}$ , and $\delta = \ln(1+i)$ .
Relationship e^δ: $e^{\delta} = 1 + i$ , so $v = e^{-\delta}$ and $\delta = \ln(1+i)$ .
Varying force of interest accumulation: $a(t) = \exp\!\left(\int_0^{t} \delta(s)\,ds\right)$ .
Force of interest from a(t): $\delta(t) = \dfrac{a'(t)}{a(t)} = \dfrac{d}{dt}\ln a(t)$ .
Equation of value: Set the value of all inflows equal to the value of all outflows at a single chosen comparison date, after discounting/accumulating each cash flow to it.
Discounting vs accumulating: Discount a future payment back by multiplying by $v^t$ ; accumulate a deposit forward by multiplying by $(1+i)^t$ .
Ordering: i, i⁽ᵐ⁾, δ, d⁽ᵐ⁾, d: For a fixed effective rate: $d < d^{(m)} < \delta < i^{(m)} < i$ . The force δ sits between the nominal discount and nominal interest rates.
v and d relationship: $v = 1 - d$ ; the discount factor is one minus the discount rate.
Doubling time (constant force): Solve $e^{\delta t} = 2$ , so $t = \dfrac{\ln 2}{\delta}$ .
Rule of 72 (intuition): A balance roughly doubles in $72/(100i)$ years; an approximation, exact doubling uses $\ln 2 \div \ln(1+i)$ .
Effective rate over a non-unit period: Over period [a,b], the effective rate is $\dfrac{a(b) - a(a)}{a(a)}$ .
Present value: The value today of one or more future cash flows, found by discounting each at the appropriate rate.
Accumulated value: The value at a future date of one or more cash flows, found by accumulating each forward at the appropriate rate.
Interest in the t-th period a(t) − a(t−1): The amount of interest earned in period t on an investment of 1 is $a(t) - a(t-1)$ .
Convert force δ to discount factor: $v = e^{-\delta}$ ; to discount n years, multiply by $e^{-\delta n}$ .
Continuous compounding limit: As m → ∞, $\left(1 + \dfrac{i^{(m)}}{m}\right)^{m} \to e^{\delta} = 1+i$ .
Why d < i for the same effective rate: Discount is taken off the larger end-of-period amount, so a smaller rate d achieves the same effect as interest rate i on the smaller starting amount.
Effective vs nominal: which is bigger?: For a positive nominal rate, the effective annual rate exceeds the nominal rate whenever m > 1 (more than one compounding per year).
Present value of a single payment C in n periods: $PV = C\,v^n = \dfrac{C}{(1+i)^n}$ .
Annuity-immediate: Level payments at the END of each period. Present value $a_{\overline{n}|} = \dfrac{1 - v^n}{i}$ .
Annuity-due: Level payments at the START of each period. Present value $\ddot{a}_{\overline{n}|} = \dfrac{1 - v^n}{d} = a_{\overline{n}|}(1+i)$ .
Relationship between ä and a: $\ddot{a}_{\overline{n}|} = a_{\overline{n}|}\,(1+i)$ : each due payment lands one period earlier.
Accumulated value, annuity-immediate: $s_{\overline{n}|} = \dfrac{(1+i)^n - 1}{i}$ .
Accumulated value, annuity-due: $\ddot{s}_{\overline{n}|} = s_{\overline{n}|}(1+i) = \dfrac{(1+i)^n - 1}{d}$ .
PV–AV relationship: $s_{\overline{n}|} = a_{\overline{n}|}\,(1+i)^n$ : the accumulated value is the present value rolled forward n periods.
Perpetuity-immediate: Pays 1 at the end of each period forever; present value $= \dfrac{1}{i}$ .
Perpetuity-due: Pays 1 at the start of each period forever; present value $= \dfrac{1}{d}$ .
Deferred annuity: Value the annuity one period before its first payment, then discount over the deferral: ${}_{k|}a_{\overline{n}|} = v^k\,a_{\overline{n}|}$ .
Increasing annuity-immediate (Ia)ₙ: Payments 1, 2, …, n at period ends: $(Ia)_{\overline{n}|} = \dfrac{\ddot{a}_{\overline{n}|} - n v^n}{i}$ .
Decreasing annuity-immediate (Da)ₙ: Payments n, n−1, …, 1 at period ends: $(Da)_{\overline{n}|} = \dfrac{n - a_{\overline{n}|}}{i}$ .
Increasing perpetuity-immediate: Payments 1, 2, 3, … forever: present value $= \dfrac{1+i}{i^2}$ . It also equals $\dfrac{1}{i} + \dfrac{1}{i^2}$ .
Geometric (growing) annuity: First payment P, growing at g per period, n payments: $PV = P\cdot\dfrac{1 - \left(\frac{1+g}{1+i}\right)^n}{i - g}$ (for $g \neq i$ ).
Growing perpetuity: First payment P, growing at g forever (g < i): $PV = \dfrac{P}{i - g}$ .
Continuous annuity: Paid continuously at rate 1 per period: $\bar{a}_{\overline{n}|} = \dfrac{1 - v^n}{\delta}$ .
Continuously increasing annuity $(\bar{I}\bar{a})_{\overline{n}|}$: $(\bar{I}\bar{a})_{\overline{n}|} = \dfrac{\bar{a}_{\overline{n}|} - n v^n}{\delta}$ .
m-thly payable annuity-immediate $a^{(m)}_{\overline{n}|}$: Payments of $1/m$ at the end of each m-thly subperiod: $a^{(m)}_{\overline{n}|} = \dfrac{1 - v^n}{i^{(m)}}$ .
Finding the number of payments n: From $PV = P\,a_{\overline{n}|}$ , solve $v^n = 1 - \dfrac{i \cdot PV}{P}$ , then $n = \dfrac{\ln(v^n)}{\ln v}$ .
Finding the unknown rate i: An annuity equation in i has no closed form; solve numerically (calculator I/Y key) or by interpolation.
Level payment from a present value: $P = \dfrac{PV}{a_{\overline{n}|}}$ : divide the present value by the annuity-immediate factor.
Difference: due value minus immediate value: $\ddot{a}_{\overline{n}|} - a_{\overline{n}|} = a_{\overline{n}|}\cdot i$ (in present-value terms, the immediate value times i).
Annuity-immediate vs annuity-due timing: Immediate: first payment at time 1, last at time n. Due: first payment at time 0, last at time n−1.
Perpetuity with first payment deferred: Value the perpetuity one period before its first payment, then discount: e.g. first payment at time k+1 → $v^k \cdot \dfrac{1}{i}$ .
Annuity payable less frequently than interest converts: Adjust by converting the interest rate to the payment period's effective rate first, then apply the standard annuity factor.
Accumulated value of a deferred annuity: The deferral does not change the accumulated value at the end of the payment stream; only the present value is discounted by the deferral.
Outstanding stream as an annuity: Any level future cash-flow stream of n payments is valued with $a_{\overline{n}|}$ — the engine behind loans and bonds.
$(Is)_{\overline{n}|}$ increasing accumulated annuity: $(Is)_{\overline{n}|} = \dfrac{\ddot{s}_{\overline{n}|} - n}{i}$ .
Sum check: aₙ + level payment recovers PV: If you discount each of the n level payments by $v^t$ and add, you recover $P\,a_{\overline{n}|}$ — the annuity factor is just that sum.
Payment timing keyword 'first payment now': Signals an annuity-DUE; the first payment is at time 0.
Payment timing keyword 'first payment in one year': Signals an annuity-IMMEDIATE; the first payment is at time 1.
Loan amortization method: Each level payment covers interest on the balance first, then repays principal. $L = P\,a_{\overline{n}|}$ .
Level loan payment: $P = \dfrac{L}{a_{\overline{n}|}}$ for a loan L repaid with n level payments at rate i.
Interest portion of payment t: $I_t = i\,B_{t-1} = P\left(1 - v^{\,n-t+1}\right)$ , where $B_{t-1}$ is the prior balance.
Principal portion of payment t: $P_t = P\,v^{\,n-t+1}$ . It grows geometrically: $P_{t+1} = P_t(1+i)$ .
Outstanding balance — prospective: Present value of the remaining payments: $B_t = P\,a_{\overline{n-t}|}$ .
Outstanding balance — retrospective: Loan accumulated minus payments accumulated: $B_t = L(1+i)^t - P\,s_{\overline{t}|}$ .
Prospective = retrospective: For a level-payment loan the two methods give the same outstanding balance at every time t.
Sinking-fund method: Borrower pays the lender interest $iL$ each period and deposits into a fund (rate j) that accumulates to L. Total outlay $= iL + \dfrac{L}{s_{\overline{n}|\,j}}$ .
Sinking-fund deposit: $D = \dfrac{L}{s_{\overline{n}|\,j}}$ , where j is the sinking-fund rate.
Sinking-fund balance after k deposits: $D\,s_{\overline{k}|\,j}$ : the deposits accumulated at the fund rate j.
Total interest paid over a loan's life: Sum of all payments minus the original principal: $nP - L$ .
Principal repaid grows by (1+i): In a level-payment loan, $P_{t+1} = P_t(1+i)$ ; the final payment is almost all principal.
Interest in the first payment: $I_1 = iL$ : interest on the full original balance.
Refinancing a loan: Find the outstanding balance at the refinance date (PV of remaining payments at the old rate), then amortize that balance over the new term at the new rate.
Amortization schedule columns: Payment, interest paid ( $iB_{t-1}$ ), principal repaid (payment − interest), and new balance ( $B_{t-1} - P_t$ ).
When does the sinking-fund method cost more?: When the sinking-fund rate j is below the loan rate i, because the fund earns less than the interest charged on the full balance.
Balance just after vs just before a payment: Just before payment t the balance is $B_{t-1}(1+i)$ ; just after it is $B_t = B_{t-1}(1+i) - P$ .
Combined principal in payments t and t+1: $P_t + P_{t+1} = P\left(v^{\,n-t+1} + v^{\,n-t}\right)$ ; use the geometric growth of principal.
Loan with non-level payments: Use the retrospective method (or first principles), since the prospective annuity factor assumes level payments.
Final payment / balloon adjustment: If level payments don't exactly clear the loan, a smaller (or drop) final payment settles the residual balance.
Bond price formula (basic): $P = Fr\,a_{\overline{n}|} + C v^n$ : present value of coupons plus present value of redemption, at the yield rate.
Bond price (premium/discount form): $P = C + (Fr - Ci)\,a_{\overline{n}|}$ : redemption value plus the present value of the coupon-vs-yield difference.
Coupon amount: $Fr$ : the face amount F times the coupon rate r. The coupon is fixed for the bond's life.
Premium bond: Coupon rate > yield rate → price > redemption value. The premium $P - C$ is written down each period.
Discount bond: Coupon rate < yield rate → price < redemption value. The discount $C - P$ is accumulated (written up) each period.
Par bond: Coupon rate = yield rate → price equals the redemption value exactly.
Book value: The present value of a bond's remaining cash flows at the ORIGINAL yield rate; it glides to the redemption value C at maturity.
Write-down of premium in period t: Coupon minus yield-rate interest: $Fr - i\,B_{t-1}$ . It reduces the book value toward C.
Write-up of discount in period t: Yield-rate interest minus the coupon: $i\,B_{t-1} - Fr$ . It increases the book value toward C.
Interest earned in period t (bond): $i\,B_{t-1}$ : the yield rate applied to the start-of-period book value.
Zero-coupon bond price: Only one cash flow: $P = C v^n = \dfrac{C}{(1+i)^n}$ . All return comes from the price-to-redemption growth.
Yield to maturity (YTM): The single yield rate i that equates the bond's price to the present value of its coupons plus redemption.
Discount price ⇒ yield vs coupon: A bond bought below redemption value (discount) has a yield ABOVE its coupon rate; a premium price gives a yield below the coupon.
Accrued interest: The seller's earned share of the next coupon when sold between coupon dates: $(\text{fraction of period elapsed}) \times Fr$ .
Clean (market) price: The quoted price: $\text{clean} = \text{dirty} - \text{accrued interest}$ .
Dirty (full) price: The cash actually paid: $\text{dirty} = \text{clean} + \text{accrued interest}$ .
Callable bond — premium: Coupon > yield: price to the EARLIEST call date (worst for the holder) to guarantee at least the desired yield.
Callable bond — discount: Coupon < yield: price to the LATEST possible redemption date (lowest price) to guarantee at least the desired yield.
Makeham's formula: $P = K + \dfrac{g}{i}(C - K)$ , where $K = C v^n$ is the PV of redemption and $g = Fr/C$ is the modified coupon rate.
Modified coupon rate g: $g = \dfrac{Fr}{C}$ : the coupon as a fraction of the redemption value, used in Makeham's formula.
Redemption value C: The amount paid at maturity. If redeemable at par, $C = F$ ; otherwise C may differ from the face amount.
Face (par) value F: The amount used with the coupon rate to compute the coupon $Fr$ . It need not equal the price or the redemption value.
Why a premium amortizes down: Each coupon exceeds the yield-rate interest, so the surplus reduces book value until it reaches C at maturity.
Salvage / redemption at a premium: If C > F (redeemable above par), the redemption term $C v^n$ in the price formula uses C, not F.
Bond sold before maturity: Its sale price is the present value of the remaining coupons and redemption at the buyer's required yield at that time.
Total accumulated premium amortized: Over the bond's life the premium written down sums to $P - C$ ; the discount written up sums to $C - P$ .
Coupon vs yield: price direction: Higher yield → lower price (inverse relationship), holding coupons and term fixed.
Semiannual coupons: Use the per-period (e.g. semiannual) coupon, the per-period yield, and the number of half-years as n in the price formula.
Bond amortization schedule: Tracks coupon, interest earned ( $iB_{t-1}$ ), premium/discount adjustment, and the new book value each period.
Current yield vs YTM: Current yield = annual coupon ÷ price (ignores capital gain/loss); YTM accounts for the price-to-redemption change too.
Net present value (NPV): Sum of all cash flows discounted at the required rate: $NPV = \sum_t CF_t\, v^t$ . Accept the project if NPV > 0.
Internal rate of return (IRR): The single rate that makes NPV = 0: $\sum_t CF_t (1+r)^{-t} = 0$ .
NPV vs IRR decision rule: $NPV > 0 \iff IRR >$ required rate. Both accept the same projects for a simple cash-flow sign pattern.
Multiple IRRs: A cash-flow stream whose sign changes more than once can have more than one IRR; rely on NPV in that case.
Dollar-weighted rate of return: The fund's IRR — sensitive to the size and timing of deposits/withdrawals. Simple-interest approx: interest ÷ exposure-weighted balance.
Time-weighted rate of return: Product of subperiod growth factors between cash flows, minus 1: $\prod_k (1 + j_k) - 1$ . Removes timing effects.
Dollar- vs time-weighted: which for a manager?: Use TIME-weighted to judge the manager's skill (timing-neutral); use dollar-weighted for the investor's actual experience.
Simple-interest dollar-weighted approximation: $i \approx \dfrac{I}{A + \sum_t C_t (1 - t)}$ : interest over the exposure-weighted average balance.
Macaulay duration: PV-weighted average time of cash flows: $D_{\text{Mac}} = \dfrac{\sum_t t\,v^t\,CF_t}{\sum_t v^t\,CF_t}$ .
Modified duration: Price sensitivity to yield: $D_{\text{mod}} = \dfrac{D_{\text{Mac}}}{1+i}$ . Then $\dfrac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta i$ .
Duration of a zero-coupon bond: Equals its time to maturity n, because there is a single cash flow at time n.
Macaulay duration formula via aₙ (level): For a level annuity, duration is a PV-weighted average of payment times; longer streams and lower rates raise duration.
First-order price approximation: $P(i + \Delta i) \approx P(i)\,(1 - D_{\text{mod}}\,\Delta i)$ . Linear; understates the price for large moves.
Convexity: The second-order measure of curvature of the price-yield curve; adding it corrects the duration estimate for large rate changes.
Second-order price approximation: $\dfrac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta i + \tfrac{1}{2}\,C_{\text{onv}}\,(\Delta i)^2$ .
Why duration matters: Higher duration = greater interest-rate risk; a small yield change moves the price more for a high-duration asset.
Redington immunization: Three conditions at the valuation rate: $PV_A = PV_L$ , $D_A = D_L$ , and $C_A > C_L$ (asset convexity greater).
Redington — meaning of each condition: Equal PVs ⇒ surplus zero; equal durations ⇒ first derivative of surplus zero; greater asset convexity ⇒ surplus at a local minimum.
Full immunization: A liability is bracketed by asset cash flows before and after it, protecting against any single shift (not just small ones).
Cash-flow matching: Fund each liability with an asset cash flow of identical date and amount; eliminates reinvestment and rate risk with no rebalancing.
Immunization vs cash-flow matching: Immunization protects against SMALL rate moves and needs rebalancing; cash-flow matching handles any move but is harder to implement.
Surplus: Assets minus liabilities, $PV_A - PV_L$ . Immunization aims to keep surplus ≥ 0 as rates move.
Reinvestment risk: The risk that coupons/cash flows must be reinvested at lower-than-expected rates; immunization balances it against price risk.
Price risk vs reinvestment risk: Rising rates lower prices (price risk) but raise reinvestment income; at the duration horizon the two offset — the basis of immunization.
Portfolio yield rate: A single equivalent rate solving the portfolio's equation of value; the portfolio's overall IRR.
Reinvestment of annuity payments: If payments earn a different reinvestment rate, accumulate them at that rate separately, then combine with the rest of the stream.
Duration of a portfolio: The market-value-weighted average of the durations of its components.
Convexity is always positive for option-free bonds: Standard fixed cash-flow bonds have positive convexity, so duration always understates the price after a yield move (a favorable bias).
Effect of higher coupon on duration: Higher coupons shorten duration (more value arrives early); lower coupons and longer maturities lengthen it.
Discounted payback / project measures: Beyond NPV and IRR, exams may ask for the discounted payback period — the time for discounted inflows to recover the outlay.
Spot rate: Today's annual yield on a single cash flow paid at one future date t; discount by $(1+s_t)^{-t}$ .
Forward rate: An interest rate agreed today for a future period, derived from spot rates by no-arbitrage.
Forward rate from spot rates: $(1+s_2)^2 = (1+s_1)(1+f_{1,2})$ , so $f_{1,2} = \dfrac{(1+s_2)^2}{1+s_1} - 1$ .
General forward-rate relationship: $(1+s_n)^n = (1+s_m)^m\,(1+f_{m,n})^{\,n-m}$ for the forward rate covering periods m to n.
Yield curve: A plot of spot rates against time to maturity at a single point in time.
Normal (upward-sloping) term structure: Long-term spot rates exceed short-term rates; the yield curve rises with maturity.
Inverted term structure: Short-term spot rates exceed long-term rates; the yield curve falls with maturity.
Discount factor from spot rate: $P_t = (1 + s_t)^{-t}$ : the price today of 1 paid at time t.
Pricing a cash flow off the term structure: Discount each future cash flow by its OWN spot-rate discount factor $P_t$ , not a single flat rate.
Interest rate swap: A contract to exchange a stream of FIXED interest payments for a stream of FLOATING payments on a notional amount; the notional is never exchanged.
Notional amount: The principal used only to compute the swap's interest payments; it is not itself paid between counterparties.
Swap rate: The level fixed rate setting the swap's initial value to zero: $R = \dfrac{1 - P_n}{\sum_{t=1}^{n} P_t}$ .
Two-year swap rate example: With $P_1 = 1.03^{-1}, P_2 = 1.04^{-2}$ : $R = \dfrac{1 - P_2}{P_1 + P_2} \approx 3.98\%$ .
Purpose of a swap: To convert floating-rate exposure into fixed (or vice versa) without trading the underlying debt.
Deferred swap: A swap whose settlement payments start at a future date; the swap rate uses only the discount factors for the active settlement dates.
Swap value after inception: Re-value the fixed and floating legs off the current term structure; their difference is the swap's current value.
Components of an interest rate: A quoted rate = real risk-free rate + inflation premium + default-risk premium + liquidity premium + maturity/term premium.
Real vs nominal interest rate: The nominal rate includes expected inflation; the real rate strips it out: roughly $\text{real} \approx \text{nominal} - \text{inflation}$ .
Default-risk premium: Extra yield demanded for the chance a borrower fails to pay; higher for riskier issuers.
Liquidity premium: Extra yield for holding an asset that is harder to sell quickly without a price concession.
Term (maturity) premium: Extra yield typically demanded for longer maturities, contributing to an upward-sloping yield curve.
Supply and demand for loanable funds: Equilibrium interest rates are set where the supply of savings meets the demand for borrowing; shifts move rates.
Central-bank policy and rates: Monetary policy (e.g. setting short-term target rates) shapes the short end of the yield curve and influences the whole term structure.
Each point on the yield curve: Represents the interest rate for a particular term to maturity at a single moment in time.
No-arbitrage principle: Two strategies with identical cash flows must have the same price today; it pins down forward and swap rates from spot rates.
Bootstrapping spot rates: Deriving spot rates sequentially from the prices of coupon bonds or par yields, shortest maturity first.
Par yield: The coupon rate at which a bond prices at par given the current spot curve; closely related to the swap rate.
Floating leg of a swap: Payments tied to a reference rate that resets each period; its value at inception equals the notional minus the final discount factor times notional.
Fixed leg of a swap: Level payments at the swap rate R; valued as $R \times \sum_t P_t$ times the notional.
Inflation premium: The portion of a nominal rate compensating lenders for expected loss of purchasing power.
Comparison date (focal date): The single point in time to which all cash flows are moved when writing an equation of value; the answer is the same whichever date you choose.
Equivalent rates: Two rates are equivalent if they produce the same accumulated value over the same period, e.g. $i$ , $i^{(m)}$ , $d$ , and $\delta$ describing one investment.
Accumulation vs amount function: a(t) is the accumulation of 1; the amount function $A(t) = k\,a(t)$ accumulates an initial investment k.
Effective rate in period n: $i_n = \dfrac{a(n) - a(n-1)}{a(n-1)}$ : interest earned in period n relative to the start-of-period balance.
Sinking fund vs amortization equivalence: If the sinking-fund rate equals the loan rate, the sinking-fund method costs exactly the same as amortization.
Outstanding balance recursion: $B_t = B_{t-1}(1+i) - P$ : grow the balance by interest, then subtract the payment.
Coupon-bond price between dates: Accumulate the previous coupon-date price forward by the partial-period interest, then subtract any coupon just paid (for the dirty price).
Effective yield on a bond bought at a discount: Exceeds the coupon rate because the investor also earns the capital gain from price to redemption value.
Annuity-immediate factor as a geometric sum: $a_{\overline{n}|} = v + v^2 + \cdots + v^n = \dfrac{v(1 - v^n)}{1 - v} = \dfrac{1 - v^n}{i}$ .
Annuity-due factor as a geometric sum: $\ddot{a}_{\overline{n}|} = 1 + v + \cdots + v^{n-1} = \dfrac{1 - v^n}{1 - v} = \dfrac{1 - v^n}{d}$ .
Convexity sign for fixed cash flows: Positive — the price-yield curve is convex, so prices fall less and rise more than a linear duration estimate predicts.
Choosing assets to immunize: Pick assets whose duration matches the liability duration while keeping greater convexity, after matching present values.
Yield rate of a cash-flow stream: The IRR — the rate equating the present value of inflows to the present value of outflows.
Net cash flow: Inflows minus outflows in a period; the sign of the net cash flow sequence affects whether a unique IRR exists.
Reinvested coupons accumulation: If reinvested at rate j ≠ i, accumulate coupons with $s_{\overline{n}|\,j}$ and add the redemption to get the total accumulated value.
Price of 1 paid at time t (Pₜ): The discount factor from the spot curve: $P_t = (1+s_t)^{-t}$ ; the building block for swap and bond valuation.
Swap as a series of forwards: A swap can be decomposed into a portfolio of forward rate agreements; its fair fixed rate is a discount-factor-weighted average of forward rates.
Level vs varying interest over time: When rates vary by period, discount each cash flow with the product of the relevant one-period factors rather than a single $v^t$ .
Accumulated value with reinvestment at a second rate: Split into the stream's own growth plus the separately-accumulated reinvested interest, then sum at the horizon.
Loan interest savings from extra payments: An extra principal payment reduces all future interest because subsequent interest is charged on a smaller balance.
Bond yield approximation: When no closed form exists, estimate the yield by interpolation between two trial rates that bracket the price.
Continuous force vs effective rate: $\delta = \ln(1+i) < i$ for $i > 0$ : the force of interest is always below the effective annual rate.
Annuity symbol $a_{\overline{n}|i}$ meaning: Present value, one period before the first of n level payments of 1, valued at rate i.
Sinking-fund total cost vs amortization: Equal when fund rate = loan rate; the sinking-fund method costs more when the fund earns less than the loan rate.
Book value at issue equals price: At purchase the book value equals the purchase price; it then amortizes toward the redemption value C.
Duration as a hedge target: Setting asset duration equal to the investment horizon (or liability duration) neutralizes first-order interest-rate risk.
Forward rate vs expected future spot rate: Under pure expectations they coincide; term and liquidity premiums make forward rates exceed expected future spot rates.
Why answer every question on Exam FM: There is no guessing penalty, so an unanswered question is a guaranteed miss; always fill in an answer.
Calculator TVM keys (BA II Plus): N (periods), I/Y (rate per period), PV, PMT, FV — enter four and solve for the fifth to handle most annuity and loan problems.
Cash-flow worksheet (BA II Plus): CF/NPV/IRR keys evaluate uneven cash-flow streams — the fast route to NPV and IRR questions.

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By Career EmployerUpdated Sep 12, 2026

200+ free Exam FM flashcards — 4 ways to study

Exam FM Flashcard of the DayNew daily

The classic card. Do you know this one?

Click Study Flashcards above to open the flashcard hub — over 200 Exam FM cards you can flip, match, type, or quiz yourself on. Every card is drawn from the SOA/CAS Financial Mathematics syllabus, so you study exactly what the exam measures.^[2] Pair them with our free practice test and study guide.

Exam FM Flashcard Study Modes

Flip mode is for first passes, where you read a front like Makeham’s formula and check yourself against the back. Type mode hides the term and asks you to produce it from the definition, so a prompt describing continuously compounded growth has to come back as Force of interest (δ). Match turns term-to-definition pairing into a timed game, and Quiz builds multiple-choice questions straight from the same 200 cards.

Free Exam FM flashcards from Career Employer — active recall for the SOA/CAS Financial Mathematics exam

Why Flashcards Work for Exam FM

Time Value of Money is the largest block at 37 cards, and it drills the machinery every later topic assumes. You get Present value, Accumulated value, and the Discount factor (v) as separate cards so the symbols stay distinct, while Force of interest (δ), Equivalent rates, and the Relationship e^δ push you to state conversions precisely instead of just recognizing them. Equation of value and the v and d relationship anchor the setup habits you will reuse constantly.

General Cash Flows, Portfolios & ALM carries 36 cards covering measurement and risk language. Macaulay duration, Modified duration, and Convexity appear as distinct fronts so you stop blurring them, and Full immunization, Surplus, and Reinvestment risk cover the asset-liability vocabulary. Net cash flow and Multiple IRRs keep the yield-measurement cautions in view.

Annuities & Cash Flows holds 35 cards built around timing distinctions: Annuity-immediate versus Annuity-due, then Perpetuity-immediate versus Perpetuity-due, with Deferred annuity, Continuous annuity, and Growing perpetuity extending the same pattern. The PV–AV relationship ties the block back to the discounting cards. Bonds matches it at 35 cards, separating Par bond, Premium bond, and Discount bond, then layering on Book value, Coupon amount, Accrued interest, Redemption value C, and Makeham’s formula.

Term Structure & Interest Rate Swaps runs 33 cards on rate curves and swap mechanics, including Spot rate, Forward rate, and Par yield alongside Swap rate, Notional amount, Deferred swap, and the card asking about the Purpose of a swap. Loans closes the deck with 24 cards on repayment structure: the Loan amortization method, the Sinking-fund method and Sinking-fund deposit, Level loan payment, Outstanding balance recursion, Prospective = retrospective, and Refinancing a loan.

Exam FM rewards instant recall of the annuity factors, the interest/discount conversions, the loan and bond formulas, and the duration/immunization rules.^[2] Spaced flashcards are the most efficient way to make that knowledge automatic, so you spend your exam time on the calculator, not on remembering which formula to use. Used alongside our practice test and study guide, they turn review time into measurable progress.

Exam FM Flashcards by Topic

The cards are organized by the six Financial Mathematics topics. Drill the heaviest ones first — Annuities and Cash Flows and General Cash Flows/ALM each carry the most weight — and make sure you also cover the newer Term Structure & Swaps topic:^[2]

Exam FM flashcards by syllabus topic
Topic	What the cards cover
Time Value of Money	Interest, discount, v, nominal rates, and the force of interest
Annuities & Cash Flows	Immediate vs due, perpetuities, deferred, and varying annuities
Loans	Amortization, the payment split, outstanding balance, and sinking funds
Bonds	Pricing, premium/discount, book value, callable bonds, and clean/dirty price
General Cash Flows, Portfolios & ALM	NPV, IRR, return measures, duration, convexity, and immunization
Term Structure & Interest Rate Swaps	Spot/forward rates, the yield curve, and swap rates

How to Get the Most Out of These Flashcards

Start with the biggest block. Time Value of Money leads the deck at 37 cards and its symbols reappear everywhere, so nail Present value and Discount factor (v) before touching bonds or swaps.
Type-drill the ones you paraphrase. Force of interest (δ) and Macaulay duration are easy to half-know, and typing the exact term from the definition exposes that gap fast.
Use Match on near-twins. The annuity timing pairs are ideal here, since sorting Annuity-due against Annuity-immediate under time pressure forces the distinction to become automatic.
Move to the practice test once Quiz feels flat. When multiple choice on Bonds and Loans stops surprising you, shift to full questions and use the study guide for the derivations behind the cards.
Rotate rather than binge. Two domains per sitting across 200 cards keeps Term Structure & Interest Rate Swaps and Loans from always landing last and staying weakest.

Exam FM Flashcards FAQ

Over 200 free Exam FM flashcards, organized across all six SOA/CAS Financial Mathematics topics — Time Value of Money, Annuities and Cash Flows, Loans, Bonds, General Cash Flows/Portfolios/Asset-Liability Management, and Term Structure & Interest Rate Swaps. They're free with no account required.

References

1.Society of Actuaries. “Exam FM: Financial Mathematics.” Society of Actuaries. ↑
2.Society of Actuaries. “Exam FM (Financial Mathematics) Syllabus.” Society of Actuaries. ↑
3.Casualty Actuarial Society. “Exams & Admissions (Financial Mathematics, jointly administered).” Casualty Actuarial Society. ↑

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