Library/CFA (Chartered Financial Analyst)/JuiceNotes 2024 Fixed Income/Curve-Based and Empirical Fixed-Income Risk Measures

Curve-Based and Empirical Fixed-Income Risk Measures

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Effective Duration and Convexity10 min

Traditional modified duration assumes cash flows are fixed. However, for bonds with embedded options like callable or putable bonds, cash flows change based on interest rate movements. Effective duration is the appropriate measure here, calculating price sensitivity to changes in the benchmark yield curve rather than the bond's own yield. A callable bond's value is the non-callable bond value minus the call option value. Because the call option becomes more valuable to the issuer as rates fall (capping the bond price), callable bonds exhibit negative convexity in low-rate environments and have lower effective duration than straight bonds. Putable bonds, where the investor holds the option to sell back, always have positive convexity and lower duration in high-rate environments.

Key Points

Effective duration measures sensitivity to benchmark yield curve changes.
Required for bonds with embedded options (uncertain cash flows).
Callable Bond Price = Non-Callable Price - Call Option Value.
Putable Bond Price = Non-Putable Price + Put Option Value.
Callable bonds can have negative convexity; putable bonds always have positive convexity.

Calculating Risk Measures10 min

Effective duration and convexity are estimated by shifting the benchmark yield curve up and down by a small amount (e.g., 25 basis points) and observing the new prices (V- and V+). The effective duration formula utilizes the difference between the higher price (V-) and lower price (V+) divided by the initial price and the total shift. Effective convexity captures the curvature or the second-order effect, using the sum of the shifted prices minus twice the initial price. These measures are then used to predict percentage price changes for a given change in yield.

Key Points

Effective Duration = (V_minus - V_plus) / (2 * V0 * Change_in_Curve).
Effective Convexity = (V_minus + V_plus - 2 * V0) / (V0 * Change_in_Curve^2).
Total Price Change = (-Effective Duration * Change_in_Yield) + (0.5 * Effective Convexity * Change_in_Yield^2).

Key Rate and Empirical Duration10 min

Key Rate Duration (KRD) measures sensitivity to shifts at specific maturity points on the yield curve, identifying 'shaping risk' or non-parallel shifts. The sum of KRDs equals the effective duration. Empirical duration uses historical statistical data to estimate sensitivity, differing from analytical duration which uses formulas. Analytical duration often assumes benchmark yields and credit spreads are uncorrelated. However, for corporate bonds, they are often negatively correlated (spreads widen when yields fall), causing empirical duration to be lower than analytical duration.

Key Points

Key Rate Duration measures sensitivity to non-parallel curve shifts (shaping risk).
Sum of Key Rate Durations = Effective Duration.
Analytical duration assumes uncorrelated spreads and yields.
Empirical duration reflects the reality that corporate spreads and government yields are often negatively correlated.
For corporate bonds, Empirical Duration is typically lower than Analytical Duration.

Questions

Question 1

Why are Macaulay and modified durations considered inappropriate for bonds with embedded options?

Curve-Based and Empirical Fixed-Income Risk Measures

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