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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.