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SnapFi

Bonds

25 formulas

Fixed-income essentials: pricing coupon and zero-coupon bonds, computing yields (current, YTM, YTC, YTW, BEY), measuring rate risk via Macaulay/modified/effective duration, convexity, and DV01, decomposing total return through holding-period return, and connecting nominal yields to credit spreads, default probabilities, forward rates, real yields, and breakeven inflation.

Pricing

Bond Price

Present value of a coupon bond: the sum of discounted periodic coupons plus the discounted face value at maturity. Discount rate is the yield to maturity, applied periodically as y/m.

P=F⋅CR/m⋅[1−(1+y/m)−Nm]y/m+F(1+y/m)NmP = \frac{F \cdot CR / m \cdot \left[1 - (1 + y/m)^{-Nm}\right]}{y/m} + \frac{F}{(1 + y/m)^{Nm}}

Zero-Coupon Bond Price

Present value of a zero-coupon bond — a single discounted cash flow equal to the face value at maturity, with no intermediate coupons. The simplest case of bond pricing and the building block for spot-rate analysis.

P=F(1+y/m)NmP = \frac{F}{(1 + y/m)^{Nm}}

Accrued Interest

Coupon income earned but not yet paid, computed as the periodic coupon scaled by the fraction of the coupon period that has elapsed. Bridges the quoted clean price and the actual settlement (dirty) price.

Accrued=Coupon×AccruedDaysPeriodDays\text{Accrued} = \text{Coupon} \times \frac{\text{AccruedDays}}{\text{PeriodDays}}

Present Value of Coupons

The value today of a bond's coupon stream alone: an ordinary annuity of F × CR / m per period, discounted at the periodic yield y / m over N × m periods. Together with the present value of the face, it makes up the bond price.

PVcoupons=F⋅CRm×1−(1+y/m)−Nmy/mPV_{\text{coupons}} = \frac{F \cdot CR}{m} \times \frac{1 - (1 + y/m)^{-Nm}}{y/m}

Present Value of Face Value

The value today of the principal a bond returns at maturity: the face value discounted at the periodic yield over N × m periods. This is also the price of a zero-coupon bond with the same face and maturity.

PVface=F(1+y/m)NmPV_{\text{face}} = \frac{F}{(1 + y/m)^{Nm}}

Yield Measures

Current Yield

Annual coupon income divided by current market price. Ignores any capital gain or loss to maturity, so it understates total return on a discount bond and overstates it on a premium bond.

CurrentYield=F⋅CRP\text{CurrentYield} = \frac{F \cdot CR}{P}

Yield to Maturity (YTM)

The internal rate of return that equates the present value of a bond's remaining cash flows (coupons + face) to its current market price. Solved iteratively because no closed-form exists for a coupon bond.

P=∑t=1NmF⋅CR/m(1+y/m)t+F(1+y/m)Nm,solve for yP = \sum_{t=1}^{Nm} \frac{F \cdot CR / m}{(1 + y/m)^{t}} + \frac{F}{(1 + y/m)^{Nm}}, \quad \text{solve for } y

Yield to Call (YTC)

The IRR that equates a callable bond's price to the present value of cash flows assuming the bond is called on the first call date at the call (redemption) price. Solved iteratively, like YTM, but the terminal payment and term are different.

P=∑t=1TcmF⋅CR/m(1+y/m)t+Redemption(1+y/m)Tcm,solve for yP = \sum_{t=1}^{T_c m} \frac{F \cdot CR / m}{(1 + y/m)^{t}} + \frac{\text{Redemption}}{(1 + y/m)^{T_c m}}, \quad \text{solve for } y

Yield to Worst (YTW)

Minimum of yield to maturity and the lowest yield to call across all applicable call dates supplied by the user. It is a contractual yield scenario, not a guaranteed realized-return floor.

YTW=min⁡(YTM,YTC)\text{YTW} = \min(\text{YTM}, \text{YTC})

Bond-Equivalent Yield (BEY)

Annualized yield computed by doubling the semi-annual periodic yield. The US Treasury convention for quoting yields on coupon bonds and (with adjustments) for converting other yield bases to a comparable Treasury basis.

BEY=2×YSemi\text{BEY} = 2 \times \text{YSemi}

Holding Period Return (HPR)

Total return realized over a finite holding period, including capital gain/loss and coupon income, expressed as a single percentage. Unlike YTM, makes no assumption about hold-to-maturity or coupon reinvestment.

HPR=P1+TotalCoupons−P0P0\text{HPR} = \frac{P_1 + \text{TotalCoupons} - P_0}{P_0}

Effective Annual Yield

The annually compounded yield equivalent to a nominal yield compounded m times per year. A bond quoted at a 6% semi-annual yield actually earns 6.09% a year.

EAY=(1+ym)m−1EAY = \left(1 + \frac{y}{m}\right)^m - 1

Duration & Convexity

Macaulay Duration

Weighted-average time to a bond's cash flows, where weights are the present values of those cash flows divided by the bond price. Expressed in years; the older of the two duration measures.

DMac=1P∑t=1Nmt/m⋅CFt(1+y/m)tD_{Mac} = \frac{1}{P} \sum_{t=1}^{Nm} \frac{t/m \cdot CF_t}{(1 + y/m)^{t}}

Modified Duration

Macaulay duration scaled by 1/(1 + y/m). Equals the negative percentage price change per unit yield change — the standard first-order rate-sensitivity coefficient for a bond.

DMod=DMac1+y/mD_{Mod} = \frac{D_{Mac}}{1 + y/m}

Convexity

Second-order rate sensitivity: the curvature in the price-yield relationship. Captures the fact that bond prices rise more on a yield drop than they fall on an equal yield rise. Always positive for option-free bonds.

C=1P∑t=1Nm(t/m)(t/m+1/m)⋅CFt(1+y/m)t+2C = \frac{1}{P} \sum_{t=1}^{Nm} \frac{(t/m)(t/m + 1/m) \cdot CF_t}{(1 + y/m)^{t+2}}

DV01 (PVBP)

Dollar Value of an 01 — the dollar price change of a bond for a one-basis-point change in yield. Approximates the bond's rate exposure in dollars: the standard risk unit on every fixed-income trading desk.

DV01≈DMod×P×0.0001\text{DV01} \approx D_{Mod} \times P \times 0.0001

Price-Change Approximation

Second-order Taylor approximation of bond price change: −ModDur × Δy captures the linear (duration) effect, and 0.5 × Conv × Δy² adds the curvature (convexity) refinement. Gives accurate price-change estimates for moderate yield shifts without recomputing the full bond price.

ΔPP≈−DMod⋅Δy+12⋅C⋅(Δy)2\frac{\Delta P}{P} \approx -D_{Mod} \cdot \Delta y + \frac{1}{2} \cdot C \cdot (\Delta y)^{2}

Effective Duration

Empirical duration computed from a parallel yield shift: the average price response per unit yield change, derived from a full bond repricing rather than analytical differentiation. The right duration measure for bonds with embedded options.

DEff=P−−P+2⋅P0⋅ΔyD_{Eff} = \frac{P_{-} - P_{+}}{2 \cdot P_{0} \cdot \Delta y}

Dollar Duration

Modified duration times price: the dollar change in a bond's price for a one-unit (100 percentage point) change in yield. Scale by the actual yield move to get the price move; DV01 is this divided by 10,000.

$D=Dmod×P\$D = D_{\text{mod}} \times P

Spread & Term Structure

Yield Spread

Difference between a bond's yield and a benchmark (typically a comparable-maturity Treasury). The simplest and most-cited measure of credit, liquidity, or sector premium over the risk-free rate.

Spread=BondYield−TreasuryYield\text{Spread} = \text{BondYield} - \text{TreasuryYield}

Spot Rate from Zero-Coupon Bond

Inverts the zero-coupon bond price formula to recover the annualized spot rate. Each spot rate is a single-cash-flow discount rate; the curve of spot rates is the term structure of interest rates.

Spot=m⋅[(FP)1/(Nm)−1]\text{Spot} = m \cdot \left[ \left(\frac{F}{P}\right)^{1/(Nm)} - 1 \right]

Forward Rate

Implied future interest rate for the period between two tenors, derived from spot rates via no-arbitrage. The forward rate is what makes "invest at the near spot then roll" produce the same return as "invest at the far spot directly." Both spot inputs and the forward output share the same compounding convention (m), so semi-annual Treasury spots produce a semi-annual forward, annually-compounded spots produce an annual forward, and so on.

(1+Forwardm)(Years2−Years1)m=(1+Spot2/m)Years2⋅m(1+Spot1/m)Years1⋅m\left(1 + \frac{\text{Forward}}{m}\right)^{(\text{Years}_2 - \text{Years}_1) m} = \frac{(1 + \text{Spot}_2/m)^{\text{Years}_2 \cdot m}}{(1 + \text{Spot}_1/m)^{\text{Years}_1 \cdot m}}

Implied Default Probability

Approximate annual default probability implied by a credit spread, using the standard reduced-form approximation: spread ≈ default probability × loss given default. Inverting gives the probability the market is pricing in.

DefaultProb≈Spread1−Recovery\text{DefaultProb} \approx \frac{\text{Spread}}{1 - \text{Recovery}}

Inflation

Fisher Real Yield

Inflation-adjusted real yield via the Fisher equation: (1 + nominal) = (1 + real)(1 + inflation). The exact relationship between stated nominal yields and inflation-adjusted purchasing-power yields.

Real=1+Nominal1+Inflation−1\text{Real} = \frac{1 + \text{Nominal}}{1 + \text{Inflation}} - 1

Breakeven Inflation

Market-implied inflation expectation: the inflation rate at which the nominal Treasury and the comparable-maturity TIPS would deliver the same real return. Approximated by the nominal-minus-real yield difference.

Breakeven=1+TreasuryYield1+TIPSYield−1≈TreasuryYield−TIPSYield\text{Breakeven} = \frac{1 + \text{TreasuryYield}}{1 + \text{TIPSYield}} - 1 \approx \text{TreasuryYield} - \text{TIPSYield}