Your FREE Exam FM Flashcards 2026 – 200+ Cards

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

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: $v = \dfrac{1}{1+i} = 1 - d$.

$v^n = (1+i)^{-n}$.

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

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

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

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

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

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

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

The instantaneous, continuously compounded rate. For a constant force, accumulation over t years is $e^{\delta t}$, and $\delta = \ln(1+i)$.

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

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

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

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.

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

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.

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

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

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.

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

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

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

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.

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

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

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.

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.

a(t) is the accumulation of 1; the amount function $A(t) = k\,a(t)$ accumulates an initial investment k.

$i_n = \dfrac{a(n) - a(n-1)}{a(n-1)}$: interest earned in period n relative to the start-of-period balance.

When rates vary by period, discount each cash flow with the product of the relevant one-period factors rather than a single $v^t$.

$\delta = \ln(1+i) < i$ for $i > 0$: the force of interest is always below the effective annual rate.

There is no guessing penalty, so an unanswered question is a guaranteed miss; always fill in an answer.

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

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

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

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

$s_{\overline{n}|} = a_{\overline{n}|}\,(1+i)^n$: 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: ${}_{k|}a_{\overline{n}|} = v^k\,a_{\overline{n}|}$.

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

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

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

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: $\bar{a}_{\overline{n}|} = \dfrac{1 - v^n}{\delta}$.

$(\bar{I}\bar{a})_{\overline{n}|} = \dfrac{\bar{a}_{\overline{n}|} - n v^n}{\delta}$.

Payments of $1/m$ at the end of each m-thly subperiod: $a^{(m)}_{\overline{n}|} = \dfrac{1 - v^n}{i^{(m)}}$.

From $PV = P\,a_{\overline{n}|}$, solve $v^n = 1 - \dfrac{i \cdot PV}{P}$, then $n = \dfrac{\ln(v^n)}{\ln v}$.

An annuity equation in i has no closed form; solve numerically (calculator I/Y key) or by interpolation.

$P = \dfrac{PV}{a_{\overline{n}|}}$: divide the present value by the annuity-immediate factor.

$\ddot{a}_{\overline{n}|} - a_{\overline{n}|} = a_{\overline{n}|}\cdot i$ (in present-value terms, the immediate value times i).

Immediate: first payment at time 1, last at time n. Due: first payment at time 0, last at time n−1.

Value the perpetuity one period before its first payment, then discount: e.g. first payment at time k+1 → $v^k \cdot \dfrac{1}{i}$.

Adjust by converting the interest rate to the payment period's effective rate first, then apply the standard annuity factor.

The deferral does not change the accumulated value at the end of the payment stream; only the present value is discounted by the deferral.

Any level future cash-flow stream of n payments is valued with $a_{\overline{n}|}$ — the engine behind loans and bonds.

$(Is)_{\overline{n}|} = \dfrac{\ddot{s}_{\overline{n}|} - n}{i}$.

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.

Signals an annuity-DUE; the first payment is at time 0.

Signals an annuity-IMMEDIATE; the first payment is at time 1.

$a_{\overline{n}|} = v + v^2 + \cdots + v^n = \dfrac{v(1 - v^n)}{1 - v} = \dfrac{1 - v^n}{i}$.

$\ddot{a}_{\overline{n}|} = 1 + v + \cdots + v^{n-1} = \dfrac{1 - v^n}{1 - v} = \dfrac{1 - v^n}{d}$.

Split into the stream's own growth plus the separately-accumulated reinvested interest, then sum at the horizon.

Present value, one period before the first of n level payments of 1, valued at rate i.

N (periods), I/Y (rate per period), PV, PMT, FV — enter four and solve for the fifth to handle most annuity and loan problems.

Each level payment covers interest on the balance first, then repays principal. $L = P\,a_{\overline{n}|}$.

$P = \dfrac{L}{a_{\overline{n}|}}$ for a loan L repaid with n level payments at rate i.

$I_t = i\,B_{t-1} = P\left(1 - v^{\,n-t+1}\right)$, where $B_{t-1}$ is the prior balance.

$P_t = P\,v^{\,n-t+1}$. It grows geometrically: $P_{t+1} = P_t(1+i)$.

Present value of the remaining payments: $B_t = P\,a_{\overline{n-t}|}$.

Loan accumulated minus payments accumulated: $B_t = L(1+i)^t - P\,s_{\overline{t}|}$.

For a level-payment loan the two methods give the same outstanding balance at every time t.

Borrower pays the lender interest $iL$ each period and deposits into a fund (rate j) that accumulates to L. Total outlay $= iL + \dfrac{L}{s_{\overline{n}|\,j}}$.

$D = \dfrac{L}{s_{\overline{n}|\,j}}$, where j is the sinking-fund rate.

$D\,s_{\overline{k}|\,j}$: the deposits accumulated at the fund rate j.

Sum of all payments minus the original principal: $nP - L$.

In a level-payment loan, $P_{t+1} = P_t(1+i)$; the final payment is almost all principal.

$I_1 = iL$: interest on the full original balance.

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.

Payment, interest paid ($iB_{t-1}$), principal repaid (payment − interest), and new balance ($B_{t-1} - P_t$).

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.

Just before payment t the balance is $B_{t-1}(1+i)$; just after it is $B_t = B_{t-1}(1+i) - P$.

$P_t + P_{t+1} = P\left(v^{\,n-t+1} + v^{\,n-t}\right)$; use the geometric growth of principal.

Use the retrospective method (or first principles), since the prospective annuity factor assumes level payments.

If level payments don't exactly clear the loan, a smaller (or drop) final payment settles the residual balance.

If the sinking-fund rate equals the loan rate, the sinking-fund method costs exactly the same as amortization.

$B_t = B_{t-1}(1+i) - P$: grow the balance by interest, then subtract the payment.

An extra principal payment reduces all future interest because subsequent interest is charged on a smaller balance.

Equal when fund rate = loan rate; the sinking-fund method costs more when the fund earns less than the loan rate.

$P = Fr\,a_{\overline{n}|} + C v^n$: present value of coupons plus present value of redemption, at the yield rate.

$P = C + (Fr - Ci)\,a_{\overline{n}|}$: redemption value plus the present value of the coupon-vs-yield difference.

$Fr$: the face amount F times the coupon rate r. The coupon is fixed for the bond's life.

Coupon rate > yield rate → price > redemption value. The premium $P - C$ is written down each period.

Coupon rate < yield rate → price < redemption value. The discount $C - P$ is accumulated (written up) each period.

Coupon rate = yield rate → price equals the redemption value exactly.

The present value of a bond's remaining cash flows at the ORIGINAL yield rate; it glides to the redemption value C at maturity.

Coupon minus yield-rate interest: $Fr - i\,B_{t-1}$. It reduces the book value toward C.

Yield-rate interest minus the coupon: $i\,B_{t-1} - Fr$. It increases the book value toward C.

$i\,B_{t-1}$: the yield rate applied to the start-of-period book value.

Only one cash flow: $P = C v^n = \dfrac{C}{(1+i)^n}$. All return comes from the price-to-redemption growth.

The single yield rate i that equates the bond's price to the present value of its coupons plus redemption.

A bond bought below redemption value (discount) has a yield ABOVE its coupon rate; a premium price gives a yield below the coupon.

The seller's earned share of the next coupon when sold between coupon dates: $(\text{fraction of period elapsed}) \times Fr$.

The quoted price: $\text{clean} = \text{dirty} - \text{accrued interest}$.

The cash actually paid: $\text{dirty} = \text{clean} + \text{accrued interest}$.

Coupon > yield: price to the EARLIEST call date (worst for the holder) to guarantee at least the desired yield.

Coupon < yield: price to the LATEST possible redemption date (lowest price) to guarantee at least the desired yield.

$P = K + \dfrac{g}{i}(C - K)$, where $K = C v^n$ is the PV of redemption and $g = Fr/C$ is the modified coupon rate.

$g = \dfrac{Fr}{C}$: the coupon as a fraction of the redemption value, used in Makeham's formula.

The amount paid at maturity. If redeemable at par, $C = F$; otherwise C may differ from the face amount.

The amount used with the coupon rate to compute the coupon $Fr$. It need not equal the price or the redemption value.

Each coupon exceeds the yield-rate interest, so the surplus reduces book value until it reaches C at maturity.

If C > F (redeemable above par), the redemption term $C v^n$ in the price formula uses C, not F.

Its sale price is the present value of the remaining coupons and redemption at the buyer's required yield at that time.

Over the bond's life the premium written down sums to $P - C$; the discount written up sums to $C - P$.

Higher yield → lower price (inverse relationship), holding coupons and term fixed.

Use the per-period (e.g. semiannual) coupon, the per-period yield, and the number of half-years as n in the price formula.

Tracks coupon, interest earned ($iB_{t-1}$), premium/discount adjustment, and the new book value each period.

Current yield = annual coupon ÷ price (ignores capital gain/loss); YTM accounts for the price-to-redemption change too.

Accumulate the previous coupon-date price forward by the partial-period interest, then subtract any coupon just paid (for the dirty price).

Exceeds the coupon rate because the investor also earns the capital gain from price to redemption value.

If reinvested at rate j ≠ i, accumulate coupons with $s_{\overline{n}|\,j}$ and add the redemption to get the total accumulated value.

When no closed form exists, estimate the yield by interpolation between two trial rates that bracket the price.

At purchase the book value equals the purchase price; it then amortizes toward the redemption value C.

Sum of all cash flows discounted at the required rate: $NPV = \sum_t CF_t\, v^t$. Accept the project if NPV > 0.

The single rate that makes NPV = 0: $\sum_t CF_t (1+r)^{-t} = 0$.

$NPV > 0 \iff IRR >$ required rate. Both accept the same projects for a simple cash-flow sign pattern.

A cash-flow stream whose sign changes more than once can have more than one IRR; rely on NPV in that case.

The fund's IRR — sensitive to the size and timing of deposits/withdrawals. Simple-interest approx: interest ÷ exposure-weighted balance.

Product of subperiod growth factors between cash flows, minus 1: $\prod_k (1 + j_k) - 1$. Removes timing effects.

Use TIME-weighted to judge the manager's skill (timing-neutral); use dollar-weighted for the investor's actual experience.

$i \approx \dfrac{I}{A + \sum_t C_t (1 - t)}$: interest over the exposure-weighted average balance.

PV-weighted average time of cash flows: $D_{\text{Mac}} = \dfrac{\sum_t t\,v^t\,CF_t}{\sum_t v^t\,CF_t}$.

Price sensitivity to yield: $D_{\text{mod}} = \dfrac{D_{\text{Mac}}}{1+i}$. Then $\dfrac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta i$.

Equals its time to maturity n, because there is a single cash flow at time n.

For a level annuity, duration is a PV-weighted average of payment times; longer streams and lower rates raise duration.

$P(i + \Delta i) \approx P(i)\,(1 - D_{\text{mod}}\,\Delta i)$. Linear; understates the price for large moves.

The second-order measure of curvature of the price-yield curve; adding it corrects the duration estimate for large rate changes.

$\dfrac{\Delta P}{P} \approx -D_{\text{mod}}\,\Delta i + \tfrac{1}{2}\,C_{\text{onv}}\,(\Delta i)^2$.

Higher duration = greater interest-rate risk; a small yield change moves the price more for a high-duration asset.

Three conditions at the valuation rate: $PV_A = PV_L$, $D_A = D_L$, and $C_A > C_L$ (asset convexity greater).

Equal PVs ⇒ surplus zero; equal durations ⇒ first derivative of surplus zero; greater asset convexity ⇒ surplus at a local minimum.

A liability is bracketed by asset cash flows before and after it, protecting against any single shift (not just small ones).

Fund each liability with an asset cash flow of identical date and amount; eliminates reinvestment and rate risk with no rebalancing.

Immunization protects against SMALL rate moves and needs rebalancing; cash-flow matching handles any move but is harder to implement.

Assets minus liabilities, $PV_A - PV_L$. Immunization aims to keep surplus ≥ 0 as rates move.

The risk that coupons/cash flows must be reinvested at lower-than-expected rates; immunization balances it against price risk.

Rising rates lower prices (price risk) but raise reinvestment income; at the duration horizon the two offset — the basis of immunization.

A single equivalent rate solving the portfolio's equation of value; the portfolio's overall IRR.

If payments earn a different reinvestment rate, accumulate them at that rate separately, then combine with the rest of the stream.

The market-value-weighted average of the durations of its components.

Standard fixed cash-flow bonds have positive convexity, so duration always understates the price after a yield move (a favorable bias).

Higher coupons shorten duration (more value arrives early); lower coupons and longer maturities lengthen it.

Beyond NPV and IRR, exams may ask for the discounted payback period — the time for discounted inflows to recover the outlay.

Positive — the price-yield curve is convex, so prices fall less and rise more than a linear duration estimate predicts.

Pick assets whose duration matches the liability duration while keeping greater convexity, after matching present values.

The IRR — the rate equating the present value of inflows to the present value of outflows.

Inflows minus outflows in a period; the sign of the net cash flow sequence affects whether a unique IRR exists.

Setting asset duration equal to the investment horizon (or liability duration) neutralizes first-order interest-rate risk.

CF/NPV/IRR keys evaluate uneven cash-flow streams — the fast route to NPV and IRR questions.

Today's annual yield on a single cash flow paid at one future date t; discount by $(1+s_t)^{-t}$.

An interest rate agreed today for a future period, derived from spot rates by no-arbitrage.

$(1+s_2)^2 = (1+s_1)(1+f_{1,2})$, so $f_{1,2} = \dfrac{(1+s_2)^2}{1+s_1} - 1$.

$(1+s_n)^n = (1+s_m)^m\,(1+f_{m,n})^{\,n-m}$ for the forward rate covering periods m to n.

A plot of spot rates against time to maturity at a single point in time.

Long-term spot rates exceed short-term rates; the yield curve rises with maturity.

Short-term spot rates exceed long-term rates; the yield curve falls with maturity.

$P_t = (1 + s_t)^{-t}$: the price today of 1 paid at time t.

Discount each future cash flow by its OWN spot-rate discount factor $P_t$, not a single flat rate.

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.

The principal used only to compute the swap's interest payments; it is not itself paid between counterparties.

The level fixed rate setting the swap's initial value to zero: $R = \dfrac{1 - P_n}{\sum_{t=1}^{n} P_t}$.

With $P_1 = 1.03^{-1}, P_2 = 1.04^{-2}$: $R = \dfrac{1 - P_2}{P_1 + P_2} \approx 3.98\%$.

To convert floating-rate exposure into fixed (or vice versa) without trading the underlying debt.

A swap whose settlement payments start at a future date; the swap rate uses only the discount factors for the active settlement dates.

Re-value the fixed and floating legs off the current term structure; their difference is the swap's current value.

A quoted rate = real risk-free rate + inflation premium + default-risk premium + liquidity premium + maturity/term premium.

The nominal rate includes expected inflation; the real rate strips it out: roughly $\text{real} \approx \text{nominal} - \text{inflation}$.

Extra yield demanded for the chance a borrower fails to pay; higher for riskier issuers.

Extra yield for holding an asset that is harder to sell quickly without a price concession.

Extra yield typically demanded for longer maturities, contributing to an upward-sloping yield curve.

Equilibrium interest rates are set where the supply of savings meets the demand for borrowing; shifts move rates.

Monetary policy (e.g. setting short-term target rates) shapes the short end of the yield curve and influences the whole term structure.

Represents the interest rate for a particular term to maturity at a single moment in time.

Two strategies with identical cash flows must have the same price today; it pins down forward and swap rates from spot rates.

Deriving spot rates sequentially from the prices of coupon bonds or par yields, shortest maturity first.

The coupon rate at which a bond prices at par given the current spot curve; closely related to the swap rate.

Payments tied to a reference rate that resets each period; its value at inception equals the notional minus the final discount factor times notional.

Level payments at the swap rate R; valued as $R \times \sum_t P_t$ times the notional.

The portion of a nominal rate compensating lenders for expected loss of purchasing power.

The discount factor from the spot curve: $P_t = (1+s_t)^{-t}$; the building block for swap and bond valuation.

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.

Under pure expectations they coincide; term and liquidity premiums make forward rates exceed expected future spot rates.