Last Updated on 15 October 2020 by F.R.Costa

Duration measures the sensitivity of a bond to interest rate changes. But how yield, coupon, and maturity interact with duration?

The concept of duration is of key importance for someone willing to invest in bonds, as it summarises, in a single number, the exposure of a portfolio to interest rate risk.

Duration is a quantification of the sensitivity of a bond price to interest rate changes, many times measured as the weighted average maturity of bond payments. The larger the number, the greater the exposure to interest rate risk. Someone willing to speculate on interest rates should seek for higher duration portfolios, while someone concerned with interest rate risk should seek for lower durations.

Additionally, there are a few important properties that are worth taking a look at.

## P1 – For a zero-coupon bond, duration is equal to time to maturity

The weighted average maturity of payments is lower for a coupon bond than for a zero-coupon bond, because the first disburses money period after period while the second has just a single cash flow at maturity date. Thus, the duration for a zero-coupon bond is equal to its time to maturity. The consequence is simple: someone willing to reduce exposure to interest rate risk should opt for a coupon-paying bond when deciding between a coupon and a zero-coupon bond (all else equal).

## P2 – For a perpetuity, duration is equal to (1+YTM)/YTM

For example, the duration of a perpetuity trading with a 10% yield is (1+0.10) / 0.10, or 11 years. For a 5% yield perpetuity, duration is 21. The lower the yield, the higher the duration. The following chart reflects this relation:

## P3 – Duration is inversely related with coupon rate

The higher the coupon rate, the higher the weights on the early payments and thus the lower is the weighted average maturity of payments. Someone looking for less exposure to interest rate risk should then prefer a bond with higher coupon rate (all else equal).

## P4 – Duration is positively related with time to maturity

For almost every bond traded, larger maturities are associated with higher durations (with the exception of some bonds trading at deep discounts). Being this true, one could very well ask the question: if maturity and duration increase at the same time, why not just use time-to-maturity as an interest rate risk indicator? The answer is simple: time-to-maturity doesn’t take into account coupon payments. The larger the coupon payments, the lower the weighted average maturity of payments, as the repayment of par value has a lower relative weight. The conclusion is that duration is then a better measure of interest rate risk as it accounts for coupon payments. Still, for two bonds with the same coupon rate, duration is higher for the bond with longer maturity.

## P5 – Duration is inversely related with yield to maturity

In P2 it became already clear that the higher the yield the lower the duration. While a higher yield bond reduces the present value of all the bond’s cash flows, it reduces the value of more-distant payments like par value repayment, by a greater proportional amount. The consequence is then a higher weight given to coupon payments than to the par value repayment, which corresponds to a lower weighted average maturity of payments, or in other words, a lower duration.

## Final Words

With the above five bond properties in mind, and considering the actual level of interest rates, investors can better match their bond holdings to their key investment goals. There is a high likelihood of interest rate hikes, which is the source of risk evaluated by duration. More conservative investors, may then opt for bonds with higher yields, higher coupon rates and/or lower maturities, as they come with lower duration. A speculator willing to maximise a bet on an interest rate rise could short sell a zero-coupon bond with a larger maturity. Each case is a different case but, at least, duration helps summarise interest rate risk in a single number, which allows for comparison across portfolios. 