Fixed Income formula sheet.

Where Coupon rate = stated annual rate, Par value = face amount repaid at maturity, Payments per year = 1, 2, 4 or 12.

The coupon is a fixed amount of cash set by the par value, whatever price the bond trades at. Almost every pricing question starts by turning the annual coupon rate into the cash paid each period.

Exam tip A 6% semiannual bond pays 3 per 100 of par every six months; never apply the coupon rate to the market price.

Lesson: Coupons: fixed, floating and zero →All Fixed-Income Instrument Features formulas

Where $A$ = level payment per period, $r$ = market rate per period, $N$ = number of payments, Principal = amount borrowed.

A fully amortizing bond or loan repays part of the principal with every payment, so the balance reaches zero at maturity. Early payments are mostly interest and later ones mostly principal; the exam asks for the payment or for the interest and principal split in a given period.

Exam tip On the calculator set PV = principal, FV = 0, N and I/Y per period, then CPT PMT; for a monthly mortgage divide the annual rate by 12.

Lesson: Bullet, fully amortizing and partially amortizing bonds →All Fixed-Income Cash Flows and Types formulas

Where Purchase price$_0$ = cash lent at the start, Repo rate = annual simple rate, Days = repo term, Year = 360 or 365 by market convention.

A repo is a collateralized loan: the borrower sells a security for cash and agrees to buy it back at a higher price. The difference is the interest. The cash lent is the security's value less the haircut, which protects the lender if the collateral falls in value.

Exam tip The repo rate is a simple money market rate on the cash lent, not on the security's market value; a bigger haircut means less cash is lent.

Lesson: Repo mechanics: repo rate, initial margin, haircut and variation margin →All Fixed-Income Markets for Corporate Issuers formulas

Where $PMT$ = coupon per period, $FV$ = par value, $r$ = market discount rate (YTM) per period, $N$ = periods to maturity.

Price is the present value of the promised cash flows at one discount rate. Coupon rate above the yield gives a premium, below gives a discount, equal gives par. Solving the same equation for $r$ from an observed price gives the YTM, which assumes the bond is held to maturity, pays in full and coupons are reinvested at the YTM.

Exam tip For a semiannual bond halve the coupon and the yield and double the years; the rate the calculator returns is per period, so multiply it by 2 for the quoted YTM.

Lesson: Pricing a bond with a market discount rate →All Fixed-Income Bond Valuation: Prices and Yields formulas

Where $PV^{Full}$ = full (invoice, dirty) price, $PV^{Flat}$ = flat (quoted, clean) price, $AI$ = accrued interest owed to the seller.

Between coupon dates the buyer pays the quoted flat price plus the interest the seller has earned since the last coupon. Dealers quote flat prices so the quote does not jump on coupon dates; the cash that changes hands is the full price.

Exam tip Compute the full price first, then subtract accrued interest to get the flat price; giving the full price as the quoted price is the classic error.

Lesson: Between coupon dates: accrued interest, flat and full price →All Fixed-Income Bond Valuation: Prices and Yields formulas

Where $t$ = days from the last coupon to settlement, $T$ = days in the coupon period, $PMT$ = coupon per period.

Accrued interest is the seller's share of the next coupon, in proportion to the part of the period the seller held the bond. Government bonds usually use actual/actual and corporate bonds 30/360.

Exam tip Count $t$ and $T$ under the bond's day count convention (30/360 treats every month as 30 days) and use the coupon per period, not the annual coupon.

Lesson: Between coupon dates: accrued interest, flat and full price →All Fixed-Income Bond Valuation: Prices and Yields formulas

Where $PV$ = price at the last coupon date with whole periods, $r$ = yield per period, $t/T$ = fraction of the coupon period elapsed.

Price the bond as of the last coupon date, then grow that value at the per-period yield for the fraction of the period that has passed. It gives the same answer as discounting every cash flow with a fractional exponent, much faster.

Exam tip $PV$ here is not the flat price: the flat price is this full price minus accrued interest.

Lesson: Between coupon dates: accrued interest, flat and full price →All Fixed-Income Bond Valuation: Prices and Yields formulas

Where $n$ = maturity in years, $y_{short}$ and $y_{long}$ = average YTMs of comparable bonds at the bracketing maturities, $y_{target}$ = estimated yield.

Illiquid or not-yet-issued bonds have no market price, so estimate a yield from liquid bonds with similar credit quality and bracketing maturities, then discount the bond's cash flows at that yield. For a new issue the spread over the interpolated government yield is the key output.

Exam tip Average the comparable yields at each maturity first, then interpolate by maturity, then price the bond at the interpolated yield.

Lesson: Matrix pricing for illiquid and new bonds →All Fixed-Income Bond Valuation: Prices and Yields formulas

Where $APR_m$ = annual rate compounded $m$ times a year, $APR_n$ = equivalent annual rate compounded $n$ times a year.

Two yields can only be compared on the same periodicity. Use it to restate a semiannual bond yield on an annual or quarterly basis, or to put a money market rate on a bond basis.

Exam tip More frequent compounding means a lower stated rate for the same return: 6% semiannual equals 6.09% on an annual basis.

Lesson: Periodicity and converting annual yields →All Yield and Yield Spread Measures for Fixed-Rate Bonds formulas

Where $\text{YTC}_i$ = yield-to-call for each possible call date, YTM = yield-to-maturity.

The issuer calls a bond when that suits the issuer, so the investor should plan for the least favourable outcome. Yield-to-worst is the most conservative yield quoted on a callable bond.

Exam tip Each yield-to-call uses that date's call price in place of par and the number of periods to that date; then take the smallest of all the yields.

Lesson: Callable bonds: yield-to-call, yield-to-worst and option-adjusted yield →All Yield and Yield Spread Measures for Fixed-Rate Bonds formulas

Where $YTM_{\text{bond}}$ = yield of the bond, $YTM_{\text{government}}$ = yield of a government bond with the same maturity, interpolated if none matches.

The G-spread is the extra yield over a government bond of the same maturity, the compensation for credit, liquidity and tax differences. The I-spread is the same idea measured against the swap rate of the same tenor.

Exam tip Put both yields on the same periodicity and day count before subtracting, and quote the result in basis points (1% = 100 bp).

Lesson: Benchmark rates, yield spreads and the G-spread →All Yield and Yield Spread Measures for Fixed-Rate Bonds formulas

Where $z_1, \dots, z_N$ = benchmark spot rates per period, $Z$ = Z-spread per period (the same for every cash flow), $PV$ = bond price.

The Z-spread is the constant spread added to every point of the government spot curve that makes the discounted cash flows equal the price. It is more precise than the G-spread when the curve is not flat. For a bond with an embedded option, removing the option value gives the option-adjusted spread.

Exam tip OAS = Z-spread minus the option value, so a callable bond's OAS is below its Z-spread and a putable bond's OAS is above it.

Lesson: I-spread, Z-spread and option-adjusted spread →All Yield and Yield Spread Measures for Fixed-Rate Bonds formulas

Where $MRR$ = market reference rate, $QM$ = quoted margin set at issue, $DM$ = discount (required) margin, $m$ = periods per year, $N$ = periods to maturity, $FV$ = par.

Coupons use the quoted margin fixed at issue; the discount rate uses the margin the market now requires. If DM equals QM the FRN is priced at par on a reset date; if credit quality worsens, DM rises above QM and the note trades below par.

Exam tip Coupons use QM, discounting uses DM; to back out DM, solve for the periodic rate $r$ from the price and use $DM = r \times m - MRR$.

Lesson: Pricing an FRN and estimating its discount margin →All Yield and Yield Spread Measures for Floating-Rate Instruments formulas

Where $FV$ = face value paid at maturity, $DR$ = annual discount rate, Days = days to maturity, Year = 360 or 365 by convention.

Treasury bills and commercial paper are often quoted as discount rates: the rate is applied to face value, not to the amount invested, so it understates the investor's true return.

Exam tip A discount rate divides interest by FV, an add-on rate by PV; to compare them find PV first, then convert to a bond equivalent yield: $\frac{365}{Days} \times \frac{FV - PV}{PV}$.

Lesson: Money market yields and the discount rate basis →All Yield and Yield Spread Measures for Floating-Rate Instruments formulas

Where $FV$ = amount paid at maturity including interest, $PV$ = amount invested or current price, $AOR$ = annual add-on rate, Days and Year as quoted.

Certificates of deposit, repos and interbank deposits pay interest on top of the principal, so the add-on rate is interest divided by the amount invested. The bond equivalent yield is an add-on rate stated on a 365-day year.

Exam tip For a CD the maturity amount is principal $\times (1 + \frac{Days}{Year} \times AOR)$; do not deduct interest from face value as you would with a discount rate.

Lesson: Add-on rates and certificates of deposit →All Yield and Yield Spread Measures for Floating-Rate Instruments formulas

Where $Z_t$ = spot (zero-coupon) rate for period $t$, $PMT$ = coupon per period, $FV$ = par.

Each cash flow is discounted at the spot rate for its own date, which gives the no-arbitrage price. The YTM is then the single rate that reproduces this price, a kind of weighted average of the spot rates.

Exam tip With an upward-sloping spot curve, a coupon bond's YTM is below the spot rate at its maturity, because the earlier coupons are discounted at lower rates.

Lesson: Pricing a bond with spot rates →All The Term Structure of Interest Rates: Spot, Par, and Forward Curves formulas

Where $PMT$ = par coupon per 100 of par, $DF_t$ = discount factor for period $t$, $Z_t$ = spot rate for period $t$.

The par rate for maturity $N$ is the coupon rate that makes a bond priced off the spot curve worth exactly 100. The par curve lies close to, but not on, the spot curve; bootstrapping runs the other way, from par rates back to spot rates.

Exam tip Compute each discount factor once, add them up, then apply the formula; the answer is a coupon per 100, so divide by 100 for the rate.

Lesson: Par rates and the par curve →All The Term Structure of Interest Rates: Spot, Par, and Forward Curves formulas

Where $Z_A$, $Z_B$ = spot rates for $A$ and $B$ years ($A < B$), $IFR_{A,B-A}$ = forward rate starting in year $A$ for $B - A$ years.

Investing for $B$ years at the spot rate must earn the same as investing for $A$ years and then rolling over at the forward rate, so the forward rate is the break-even reinvestment rate. When the spot curve slopes up, forward rates lie above the spot rates.

Exam tip $IFR_{2,1}$ is the 2y1y rate: it starts in 2 years and lasts 1 year; take the $(B-A)$th root so a multi-year forward is an annual rate.

Lesson: Implied forward rates from spot rates →All The Term Structure of Interest Rates: Spot, Par, and Forward Curves formulas

Where $FV_{\text{coupons}}$ = coupons plus interest on interest at the horizon, $P_{\text{sale}}$ = sale price (or par at maturity), $P_{\text{purchase}}$ = price paid, $T$ = holding period in years.

The return actually earned combines coupons, interest on interest and the sale price. It equals the purchase YTM only if coupons are reinvested at that yield and the bond is either held to maturity without default or sold at a price on the constant-yield path.

Exam tip Use the price actually paid, not par, in the denominator; when rates rise, reinvestment income goes up but the sale price falls.

Lesson: Horizon yield: the realized return when rates move →All Interest Rate Risk and Return formulas

Where Macaulay duration and the investment horizon, both in years.

When the horizon equals Macaulay duration, price risk and reinvestment risk roughly offset for a one-time parallel shift in rates, so the horizon yield stays close to the purchase YTM. The sign of the gap tells you which risk dominates.

Exam tip Positive gap (duration longer than the horizon): price risk dominates and rising rates hurt; negative gap: reinvestment risk dominates and falling rates hurt.

Lesson: Reinvestment risk, price risk and the duration gap →All Interest Rate Risk and Return formulas

Where $k$ = time to receipt in periods, $CF_k$ = cash flow at $k$, $r$ = YTM per period, $PV$ = bond price, $w_k$ = share of the price from cash flow $k$.

Macaulay duration is the weighted average time to receive the bond's cash flows, using present values as weights. A lower coupon and a lower yield raise it, and so, in general, does a longer maturity; a zero-coupon bond's Macaulay duration equals its maturity.

Exam tip The result is in periods: divide by 2 for a semiannual bond to state it in years.

Lesson: Calculating Macaulay duration on a coupon date →All Interest Rate Risk and Return formulas

Where $MacDur$ = Macaulay duration, $r$ = yield per period (annual YTM divided by periods per year).

Modified duration is the approximate percentage price change for a 1% change in yield: $\%\Delta PV^{Full} \approx -AnnModDur \times \Delta Yield$. It is the main yield-based risk measure for option-free bonds, and the minus sign means price moves opposite to yield.

Exam tip Divide by 1 + yield per period: for a 6% semiannual bond use 1.03, not 1.06.

Lesson: From Macaulay duration to modified duration →All Yield-Based Bond Duration Measures and Properties formulas

Where $PV_{-}$, $PV_{+}$ = full prices after the yield falls and rises by $\Delta Yield$, $PV_0$ = current full price.

Reprice the bond with the yield shifted down and up by the same amount; the average price sensitivity gives modified duration without computing cash flow weights. Multiply by $1 + r$ to get approximate Macaulay duration.

Exam tip Enter $\Delta Yield$ as a decimal (10 bp = 0.0010); $PV_{-}$ is the higher price because the yield fell, so the numerator is positive.

Lesson: Approximate modified duration →All Yield-Based Bond Duration Measures and Properties formulas

Where $AnnModDur$ = annual modified duration, $PV^{Full}_{position}$ = full market value of the position, $PVBP$ = price value of a basis point.

Money duration turns the percentage sensitivity into currency: $\Delta PV^{Full} \approx -MoneyDur \times \Delta Yield$. PVBP is the change in value for a 1 bp move in yield, the figure traders use to size hedges.

Exam tip Use the full value of the holding (par held $\times$ full price $\div$ 100), not the par amount or the flat price.

Lesson: Integrated cases: duration of a bond bought mid-period →All Yield-Based Bond Duration Measures and Properties formulas

Where $PV_-$, $PV_+$ = full prices after the yield falls and rises by $\Delta y$, $PV_0$ = current full price, $\Delta y$ as a decimal.

Convexity measures the curvature that duration misses. Option-free bonds have positive convexity: the price rises more when yields fall than it drops when yields rise by the same amount. In general, longer maturity, lower coupon and lower yield mean more convexity.

Exam tip Use the same three prices as approximate duration and remember to square $\Delta y$ in the denominator.

Lesson: Convexity: measuring the curve →All Yield-Based Bond Convexity and Portfolio Properties formulas

Where $\Delta y$ = change in annual yield as a decimal, $\text{AnnModDur}$ = annual modified duration, $\text{AnnConvexity}$ = annual convexity.

For an option-free bond, duration alone gives a straight-line estimate that understates the price after a fall in yields and overstates the price after a rise. The convexity adjustment corrects for the curve and matters more the larger the yield change.

Exam tip Keep the sign of $\Delta y$ in the duration term; for an option-free bond the convexity term is always positive, so it adds to gains and shrinks losses.

Lesson: Estimating price changes with duration and convexity →All Yield-Based Bond Convexity and Portfolio Properties formulas

Where $PV_-$, $PV_+$ = model prices after the benchmark curve shifts down and up by $\Delta\text{Curve}$, $PV_0$ = current full price.

For bonds with embedded options, such as callables, putables and mortgage-backed securities, the cash flows change when rates change, so yield duration does not apply. Effective duration shifts the benchmark curve and reprices the bond with an option pricing model.

Exam tip A call option shortens effective duration when rates fall and a put option shortens it when rates rise; callable bonds can show negative effective convexity.

Lesson: Effective duration: why curve duration replaces yield duration →All Curve-Based and Empirical Fixed-Income Risk Measures formulas

Where POD = probability of default (usually annual), LGD = loss given default, in money or as a % of exposure.

Expected loss combines how likely default is with how much is lost if it happens. The credit spread is roughly POD $\times$ LGD, so comparing the spread with expected loss shows whether investors are fairly paid for credit risk.

Exam tip With LGD in money, EL is in money; with LGD as a % of principal, EL is a % per year that you can compare directly with the spread.

Lesson: Expected loss: POD × LGD →All Credit Risk formulas

Where EE = expected exposure at default, RR = recovery rate, $1 - \text{RR}$ = loss severity.

LGD is the part of the exposure that is not recovered. Seniority and collateral drive the recovery rate, so senior secured debt of an issuer has a lower LGD, and lower expected loss, than its subordinated debt even though the default probability is the same.

Exam tip Use $1 - \text{RR}$, not RR: a 40% recovery rate means a 60% loss severity.

Lesson: Expected loss: POD × LGD →All Credit Risk formulas

Where $\text{AnnModDur}$ = annual modified duration, $\Delta\text{Spread}$ = change in the credit spread as a decimal.

A change in the credit spread moves the price like a yield change of the same size, so modified duration converts spread widening into a loss. After a downgrade, longer-duration bonds lose more for the same widening.

Exam tip Narrowing is a negative $\Delta\text{Spread}$ and produces a gain; for large spread moves add $\tfrac{1}{2} \times$ convexity $\times (\Delta\text{Spread})^2$.

Lesson: How spread changes move bond prices →All Credit Risk formulas