CFA Level I
Fixed Income
Formula Reference Sheet
Bond Pricing
Full Price of a Bond
PV × (1 + YTMn)t/T
Full price = flat price + accrued interest
Flat Price of a Bond (quoted price)
Full price − Accrued interest
= Full price − Coupon × tT
= Full price − Coupon × tT
Current Yield
CPNFlat Price
Bond Price Using Spot Rates
CPN(1+r)n
+ ··· +
FV + CPN(1 + r + volatility spread)n
Yields & Rates
Effective Yield
EFF = (1 + r)n − 1
Periodic Rate
r = YTMn
Periodicity Conversion
(1 + YTMmm)m
=
(1 + YTMnn)n
Floating Rate Bond Coupon
Coupon Rate = Reference Rate + Quoted Margin
Money Market Instruments
Money Market Yield
FV − PVPV
× 360n
Bond Equivalent Yield (add-on)
FV − PVPV
× 365n
Discount Yield
FV − PVFV
× 360n
Discount Rate
YearsDays
×
FV − PVFV
Yield Spreads
Yield on Bond with Embedded Option
Yield on Gov’t Bondsbenchmark
+
OASoption-adj spread
+
Option Valuecost of option
OAS = Z-spread − Option value ·
Z-spread = comparable gov’t bond + discount
Floating Rate PV (Index-Based)
PV =
Index + QMm · FV
1 + Index + DMm
QM = quoted margin · DM = discount margin
Periodic Bond Price (Par-Based)
Pmkt = Index + DMm · FV
r = Index + DM / m
Mortgage-Backed Securities
Single Monthly Mortality (SMM)
Prepayment for the month
Mortgage bal. at start − Scheduled principal
Loan-to-Value (LTV) Ratio
LTV = Loan AmountMarket Value of Collateral
Debt-Service Coverage (DSCR)
DSCR = Net Operating IncomeDebt Service
MBS portfolio weights = par value weights (not market price weights)
Duration
Modified Duration
Macaulay Duration1 + YTMn
Macaulay Duration = weighted average term to maturity of cash flows from bond
% Change in Bond Price
[− Annual Modified Duration] × ΔYTM
Approximate Modified Duration
V− − V+2 × V0 × ΔYTM
Effective Duration (embedded options)
V− − V+2 × V0 × ΔCurve
Money Duration
Annual Modified Duration × Full Price of Bond
Price Value of a Basis Point (PVBP)
PVBP = V− − V+2
V± = price of bond if yield changes by 1 basis point
Duration Gap
Macaulay Duration − Investment Horizon
Convexity
Approximate Convexity (non-embedded)
V− + V+ − 2V0
(ΔYTM)2 × V0
Effective Convexity (embedded options)
V− + V+ − 2V0
(ΔCurve)2 × V0
Full Change in Bond Price (Duration + Convexity)
[− Annual Modified Duration] × ΔYTM
← Duration effect (can also use Money Duration)
+
← Duration effect (can also use Money Duration)
½ × Annual Convexity × (ΔYTM)2
← Convexity adjustment
← Convexity adjustment
Investment Horizon & Risk
Duration Gap vs. Risk Type
| Horizon vs. Macaulay Duration | Dominant Risk |
|---|---|
| Horizon = Macaulay Duration | Reinvestment risk = Price risk |
| Horizon > Macaulay Duration | Greater reinvestment risk |
| Horizon < Macaulay Duration | Greater price risk |
Credit & Default
Expected Market Loss
Market Loss = Par Value × [P(default) × (1 − Recovery Rate)]