Explain empirical duration and why it may diverge from analytical duration
The idea
Empirical duration is estimated by regressing observed bond price changes on observed benchmark yield changes, rather than computed from a model. It matters because for lower-quality credit the two diverge sharply: when government yields fall in a flight to quality, credit spreads usually widen at the same time, so a high-yield bond's price rises less than its analytical duration predicts, and sometimes falls. Empirical duration captures that correlation between the benchmark and the spread, which analytical duration assumes away by holding the spread constant.
A practice question
How is empirical duration obtained?
- From the weighted average time to receipt of cash flowsWhy not: That is Macaulay duration, computed from the bond's terms rather than measured from its behaviour.
- By regressing observed price changes on observed benchmark yield changes — correct
- From the bond's stated maturity and couponWhy not: Those are inputs to the analytic measures. Empirical duration uses observed prices and yields instead.
It is measured from market behaviour rather than computed from a model, which is why it can reflect relationships the model does not contain.
A harder one
For which bonds does empirical duration most differ from analytical duration?
- Short-dated government bondsWhy not: Almost no spread to move, and little duration to be wrong about. The two measures agree closely.
- Zero-coupon government bondsWhy not: The analytic case works perfectly here — one cash flow, no credit spread, duration equal to maturity.
- Lower-quality corporate bonds — correct
The divergence comes from the correlation between benchmark yields and credit spreads, which is strong for weak credits and negligible for sovereigns.
The hardest tier
Government yields fall sharply in a flight to quality. Why might a high-yield bond fall in price?
- Because credit spreads widen at the same time — correct
- Because high-yield bonds have negative analytical durationWhy not: Analytical duration is positive for any ordinary bond. What is negative is the *empirical* relationship in this episode, and only because the spread moved.
- Because coupon payments are suspended in market stressWhy not: Coupons are contractual. Suspension would be a default, not a normal feature of a risk-off move.
For a weak credit the spread widening can exceed the benchmark fall. The same event that lowers the risk-free rate raises the perceived risk of the issuer, and analytical duration holds the spread constant, so it predicts a gain that does not occur.
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