Callable Bond Duration Under Exercise Scenarios

# Macaulay Duration of a Callable Bond


# Macaulay Duration of a Callable Bond












I could not find any formula of Macaulay duration for a callable bond in the literature. Can anybody show how to derive it or give a reference where it is already obtained.

EDIT My goal is to find a closed form formula for callable bond. This paper derives modified duration for callable bond (see Appendix for derivation). The derivation is not clear to me, as the author does not specify what kind of yield, $Y$, is used to take the partial derivative of prices (YTM, YTC, YTW). Also, it is not Macaulay duration, but modified duration.

## Answer by Dimitri Vulis (score 2, accepted)

https://quant.stackexchange.com/a/77838

There are two commonly used approaches, which are OK for for duration/dv01, key rate duration / dv01 by tenor bucket, and other risk measures in which the risk scenario perturbs the market data only by a small amount. Neither should be used for stress scenarios, such as in VaR calculations.

1 find the yield to worst. Assume that the bond issuer would exercise to seek the worst yield. (Note also that for putable bonds, which are much less common that callable, we'd assume that the bond holder would seek the best yield.) Assume that a small bump to the market data would not affect the moneyness of the option. Calculate the risk measures under these assumptions.

2 making some assumptions about the volatilities of the market data, for each exercise scenario, calculate the probability of the scenario being realized, and the risk measures assuming this scenario, then use the probability-wwighted sum of the risk measures.

Numerical example. A bond matures in 3 years, but has Bermudan calls in 1 and 1 years. Label the scenarios:

$s=1$ the issuer calls the bond in 1 year

$s=2$ the issuer calls the bond in 2 years

$s=3$ the issuer doesn't call the bond, but lets it mature in 3 years

Approach 1: making up some numbers, the yields under each scenario are $y_s$, the YTW is realized if the bond is called in 1 year. Assume that this will happen, ignore the possibility of the bond not being called, use the risk measures of the 1-year bond.

Approach 2: making some assumptions about the volatilities of the interest rates and the bond issuer's credit spread, and probably using a trer, calculate that the probabilities of being called in 1 or 2 years are $p_1$ and $p_2$, resulting in durations $d_1$ and $d_2$ respectively, and the probability of letting the bind mature is $p_3$, use $d=\sum_{s=1}^3p_sd_s$.