Fixed Income — Master Formula Sheet

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1. Bond Pricing

Present Value of a Bond:

$PV={\sum }_{t=1}^n\frac{PMT}{(1+r{)}^t}+\frac{FV}{(1+r{)}^n}$

where $PMT$ = coupon payment, $FV$ = face value, $r$ = market discount rate per period, $n$ = number of periods.

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2. Current Yield

$\text{Current Yield}=\frac{\text{Annual Coupon}}{\text{Bond Price}}$

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3. Accrued Interest

$AI=PMT\times \frac{t}{T}$

where $t$ = days since last coupon, $T$ = days in coupon period.

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4. Full Price and Flat Price

$\text{Full Price}=\text{Flat Price}+AI$

Full price between coupon dates:

$P{V}^{\text{Full}}=PV\times (1+r{)}^{t/T}$

where $PV$ = value at last coupon date, $t/T$ = fraction of period elapsed.

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5. Effective Annual Rate (EAR)

$EAR={\left(1+\frac{APR}{n}\right)}^n-1$

where $n$ = compounding periods per year.

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6. APR Conversion Between Periodicities

${\left(1+\frac{AP{R}_m}{m}\right)}^m={\left(1+\frac{AP{R}_n}{n}\right)}^n$

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7. Yield Spreads

G-Spread:

$G\text{-spread}=YT{M}_{\text{bond}}-YT{M}_{\text{govt}}$

Z-Spread (zero-volatility spread):

$PV={\sum }_{t=1}^n\frac{C{F}_t}{(1+S_t+Z{)}^t}$

where $S_t$ = spot rate for period $t$, $Z$ = Z-spread (solve for $Z$).

OAS (Option-Adjusted Spread):

$OAS=Z\text{-spread}-\text{Option Value (in spread terms)}$

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8. Forward Rates

Forward rate from spot rates:

$(1+S_n{)}^n=(1+S_1)(1+f_{1,1})(1+f_{2,1})\cdots (1+f_{n-1,1})$

General relationship:

$(1+S_{A+B}{)}^{A+B}=(1+S_A{)}^A\times (1+f_{A,B}{)}^B$

Solving for a forward rate:

$f_{A,B}={\left[\frac{(1+S_{A+B}{)}^{A+B}}{(1+S_A{)}^A}\right]}^{1/B}-1$

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9. Macaulay Duration

$\text{MacDur}=\frac{{\sum }_{t=1}^{n}t\times PV(C{F}_t)}{PV}$

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10. Modified Duration

$\text{ModDur}=\frac{\text{MacDur}}{1+r}$

Price change approximation:

$\%\Delta PV\approx -\text{ModDur}\times \Delta \text{Yield}$

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11. Approximate Modified Duration

$\text{ApproxModDur}=\frac{P{V}_{-}-P{V}_{+}}{2\times P{V}_0\times \Delta \text{Yield}}$

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12. Money Duration (Dollar Duration)

$\text{MoneyDur}=\text{ModDur}\times P{V}^{\text{Full}}$

$\Delta PV\approx -\text{MoneyDur}\times \Delta \text{Yield}$

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13. PVBP (Price Value of a Basis Point)

$PVBP=\frac{P{V}_{-}-P{V}_{+}}{2}$

where $P{V}_{\pm }$ = prices after ±1 bp yield change.

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14. Approximate Convexity

$\text{ApproxCon}=\frac{P{V}_{-}+P{V}_{+}-2\times P{V}_0}{\Delta {\text{Yield}}^2\times P{V}_0}$

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15. Price Change with Duration and Convexity

$\%\Delta PV\approx (-\text{ModDur}\times \Delta \text{Yield})+\left(\frac{1}{2}\times \text{Convexity}\times \Delta {\text{Yield}}^2\right)$

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16. Effective Duration

$\text{EffDur}=\frac{P{V}_{-}-P{V}_{+}}{2\times P{V}_0\times \Delta \text{Curve}}$

For bonds with embedded options; uses benchmark curve shift.

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17. Effective Convexity

$\text{EffCon}=\frac{P{V}_{-}+P{V}_{+}-2\times P{V}_0}{\Delta {\text{Curve}}^2\times P{V}_0}$

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18. Key Rate Duration

${\text{KeyRateDur}}_k=\frac{-1}{PV}\times \frac{\Delta PV}{\Delta r_k}$

where $\Delta r_k$ = change in the spot rate at maturity $k$ only.

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19. Credit Risk

Expected Loss:

$\text{Expected Loss}=POD\times LGD\times \text{Exposure}$

Loss Given Default:

$LGD=1-\text{Recovery Rate}$

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20. FRN Discount Margin

The discount margin (DM) is the spread over the reference rate such that the FRN’s price equals its market value:

$PV=\frac{(QM+R)\times FV/m}{(1+\frac{R+DM}{m}{)}^1}+\frac{(QM+R)\times FV/m}{(1+\frac{R+DM}{m}{)}^2}+\cdots +\frac{(QM+R)\times FV/m+FV}{(1+\frac{R+DM}{m}{)}^N}$

where $QM$ = quoted margin, $R$ = reference rate, $DM$ = discount margin, $m$ = periods per year.

• If $DM>QM$: FRN trades at a discount
• If $DM<QM$: FRN trades at a premium
• If $DM=QM$: FRN trades at par

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21. Money Market Yields

Bank Discount Yield:

$r_{BD}=\frac{D}{FV}\times \frac{360}{\text{Days}}$

Money Market Yield:

$r_{MM}=\frac{D}{PV}\times \frac{360}{\text{Days}}$

Bond Equivalent Yield:

$r_{BEY}=\frac{D}{PV}\times \frac{365}{\text{Days}}$

where $D=FV-PV$.

Conversion:

$r_{MM}=r_{BD}\times \frac{FV}{PV}=\frac{r_{BD}}{1-r_{BD}\times \frac{\text{Days}}{360}}$

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22. Portfolio Duration

$D_{\text{portfolio}}={\sum }_{i=1}^n{w}_i\times D_i$

where $w_i$ = market value weight of bond $i$.

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Related Pages

• Bond Valuation
• Duration
• Convexity
• Yield Measures
• Yield Spreads
• Term Structure
• Credit Risk