CFA Level I · CFA Level I Exam

Curve-Based and Empirical Fixed-Income Risk Measures: formula sheet

Full chapter guide

Key formulas

Bond price: PV = Σ [PMT ÷ (1 + r)^t] + FV ÷ (1 + r)^N; r is the yield per period, N the number of periods. Higher r gives lower PV.
Price and yield relationship: Yield ↑ ⇒ Price ↓; Yield ↓ ⇒ Price ↑; The relationship is inverse and convex for an option-free bond.
Coupon versus yield: Coupon = YTM: par; Coupon > YTM: premium; Coupon < YTM: discount; Uses the yield per period on the same compounding basis.
Sensitivity rules (other things equal): Longer maturity ⇒ more sensitive; Lower coupon ⇒ more sensitive; Lower yield ⇒ more sensitive; Compare one feature at a time. Do not state these as guaranteed if two features change together.
Percentage price change: % change = (New price − Old price) ÷ Old price; Use this to compare the size of gains and losses.
Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve); PV₋ is the price when the benchmark curve falls by ΔCurve. PV₊ is the price when it rises. ΔCurve is a decimal, so 25 bps = 0.0025.
Approximate price change: %ΔPV ≈ −EffDur × ΔCurve; Use for small parallel benchmark shifts. Add a convexity term for larger moves.
Modified duration from Macaulay: ModDur = MacDur ÷ (1 + YTM per period); Applies only to option-free bonds with fixed cash flows. Use periodic yield with the periodic compounding frequency.
Callable bond duration ranking: EffDur(callable) ≤ EffDur(option-free); The call option shortens expected cash flows when rates fall, so the callable has the lower effective duration.
Key rate duration: KRD_k = (PV₋ − PV₊) ÷ (2 × PV₀ × Δy_k); PV₋ and PV₊ are values after the k-th key rate falls and rises by Δy_k, with all other key rates unchanged. Δy_k is in decimals (10 bp = 0.0010).
Price change from one key rate: %ΔPV ≈ −KRD_k × Δy_k; For several key rate changes, add the individual effects: %ΔPV ≈ −Σ(KRD_k × Δy_k). This is a first-order estimate and ignores convexity.
Sum of key rate durations: Σ KRD_k ≈ effective duration; Holds when all key rates shift by the same amount (a parallel shift).
Portfolio key rate duration: KRD_p,k = Σ (w_i × KRD_i,k); w_i is the market value weight of bond i. Use market value weights, not par weights.
Active key rate duration: Active KRD_k = KRD_portfolio,k − KRD_benchmark,k; A positive value means the portfolio gains more than the benchmark if that key rate falls, and loses more if it rises.
Effective duration: EffDur = (PV₋ − PV₊) ÷ (2 × PV₀ × ΔCurve); PV₋ is the price when the benchmark curve falls by ΔCurve, and PV₊ is the price when it rises. ΔCurve is a decimal, so 25 bps = 0.0025.
Effective convexity: EffCon = (PV₋ + PV₊ − 2 × PV₀) ÷ [(ΔCurve)² × PV₀]; Prices come from a model that allows cash flows to change with the option. The result is negative when PV₋ + PV₊ is less than 2 × PV₀.
Approximate convexity: ApproxCon = (PV₋ + PV₊ − 2 × PV₀) ÷ [(ΔYTM)² × PV₀]; Uses the bond's own yield to maturity and fixed cash flows. Suitable for option-free bonds.
Percentage price change estimate: %ΔPV ≈ (−EffDur × ΔCurve) + ½ × EffCon × (ΔCurve)²; Use ΔCurve as a decimal. For option-free bonds, use modified duration and ΔYTM with approximate convexity.
Convexity of embedded-option bonds: Callable: positive at high yields, negative at low yields | Putable: positive; Putable convexity is at least that of a comparable option-free bond. Callable convexity is lower than that of a comparable option-free bond.
Empirical regression: %ΔPrice = a + b × ΔYield(benchmark) + error; Fit on historical data. b is the slope and is normally negative for a bond.
Empirical duration: Empirical duration = −b; The slope equals duration (with the sign flipped) only when %ΔPrice and ΔYield are in the same unit: both in decimals, or both in percent and percentage points. Mixing units, such as price in percent and yield in decimal, scales the slope by 100.
Yield beta: Yield beta = slope from regressing Δ(bond yield) on Δ(benchmark yield); A beta below 1 means the bond's yield moves less than the benchmark's.
Adjusted duration estimate: Empirical duration ≈ Analytical duration × Yield beta; Approximation for a bond whose yield moves with the benchmark by a stable beta.
Price change estimate: %ΔPrice ≈ −Duration × ΔYield; Use the benchmark yield change with empirical duration.

Quick revision

Bond price and yield move in opposite directions, and the relationship is convex.
Effective duration = (PV₋ − PV₊) ÷ (2 × ΔCurve × PV₀), with ΔCurve as a decimal.
Effective duration works for bonds with embedded options; modified duration assumes fixed cash flows.
Price change ≈ −EffDur × ΔCurve, then add the convexity term for a better estimate.
Convexity adjustment = ½ × EffConvexity × (ΔCurve)².
Effective convexity = (PV₋ + PV₊ − 2 × PV₀) ÷ (ΔCurve² × PV₀).
A callable bond can show negative convexity when yields fall enough that the call option is near or in the money, so its price rise is capped.
A putable bond has positive convexity, and the put option puts a floor under the price when yields rise, so its price falls less than that of an otherwise identical straight bond.
Key rate duration measures price sensitivity to a change at one maturity point, with other points unchanged.
The sum of a bond's key rate durations approximately equals its effective duration (the parallel-shift sensitivity).
Empirical duration is estimated from observed price and yield data and can differ from model duration.
Higher convexity is a benefit to the holder, all else equal.

Common mistakes

Thinking higher-coupon bonds are more sensitive to yield changes. Fix: Higher coupons return cash sooner, so the value is less tied to distant principal. Lower coupon means more sensitivity, other things equal.
Treating the price-yield curve as a straight line. Fix: The curve is convex for an option-free bond and lies above its tangent line. A duration-only estimate therefore understates the new price after a yield fall and after a yield rise. Add convexity to correct this.
Using modified duration for a callable or putable bond. Fix: If the bond has an embedded option, cash flows change with rates. Use the curve-shift formula with model prices.
Forgetting the 2 in the denominator. Fix: Write the denominator as 2 × PV₀ × ΔCurve every time. The total move across the two prices is twice the shift.
Forgetting the 2 in the denominator Fix: The 2 appears because you use both a down shift and an up shift. The price difference spans a 2 × Δy range. Always write (PV₋ − PV₊) ÷ (2 × PV₀ × Δy).
Using basis points as whole numbers Fix: Convert basis points to decimals before calculating. 1 bp = 0.0001.
Using approximate convexity or YTM changes for a bond with an embedded option. Fix: If the bond has a call or put, use effective measures based on benchmark curve shifts and model-based prices.
Saying callable bonds always have negative convexity. Fix: Say a callable bond has positive convexity at high yields and negative convexity at low yields.
Reporting the regression slope as duration without flipping the sign. Fix: Empirical duration is the negative of the slope when regressing price changes on yield changes.
Assuming empirical duration is always lower than analytical duration. Fix: It depends on the yield beta. A beta above 1 gives a larger empirical duration. Calculate rather than assume.

Exam tips

Settle the direction first. Many options fail on direction alone.
Learn the three sensitivity rules with their condition: other things equal.
For up versus down moves of equal size, choose the larger price gain from a yield fall.
In numerical questions, the answer must sit on the correct side of par: premium if the coupon exceeds the yield.
On the BA II Plus, use N, I/Y, PMT, FV, then CPT PV, and make sure PMT and FV carry opposite sign conventions to PV.
If a question mentions a call, put, or other embedded option, expect effective duration to be the right tool.
In conceptual items, rank callable, putable and option-free durations. Callable bonds lose duration as rates fall. Putable bonds lose duration as rates rise.
Watch units. Basis points must become decimals before you divide.