# Duration and coupons

So duration is lower for bonds with higher coupons, book says because you get your payments more rapidly than a lower coupon. OK, can someone explain how does that make the price of a bond more sensitive? So what if you get your payments more rapidly?

Compared two bonds with same YTM and expires at same time, e.g., 8% coupon vs. zero semi-annual coupon both with 20 years and 8% YTM, the one with 8% coupon rate will have lower duration. You can see it from two perspectives: 1. Duration as weighted average maturity (in terms of years) of the payments It is attributable to the impact of early coupon payments on the average maturity of a bond’s payments. The higher these coupons, the higher the weights on the early payments and the lower is the weighted average maturity of the payments 2. Duration as price sensitivity. difficult to explain so I give you an example. For example above, the YTM goes from 8% to 9%, zero coupon bond goes from 208.29 to 171.93 gives a 17.46% price change while 8% coupon bond goes from 1000 to 907.99 or 9.2% price change (less relative price change). Notice here that we are talking about RELATIVE price change not ABSOLUTE price change. Hope it helps.

If you think of interest rates as the discount factor applied to bond coupons and principal payments in order to get today’s price, then the effect of coupons is easier to understand. Let’s imagine that you have a \$1000 bond that pays 5% once annually and matures in 2 years. So there is just a single coupon payment of \$50 in one year, plus a second coupon of \$50 and the principal payment of \$1000 in two years (\$1050 total). The price of the bond should be = PV(\$50 in one year) + PV(\$1050 in two years). To make it easy to calculate, let’s suppose that interest rates for similar bonds is 10%. Therefore the bond’s value is: \$50*(0.90) + \$1050*(0.90)^2 = \$895.50 [0.90 is the discount factor for a 10% interest rate = (1-10%)] What’s important here is the (0.90)^2 vs the (0.90). The payments that come later down the line are much more sensitive to interest rate changes because the discount factor compounds over time. Their contribution to the price of the bond today is much more sensitive to interest rate changes than the payments that come earlier, where the compounding is less. The coupon payments thus have a smaller duration, because they are less sensitive to the price. A bond’s duration can be thought of a the weighted average of the durations of all the payments. Since the payments that come earlier have smaller durations, bonds with more coupons (and larger coupons) will tend to have smaller durations.

bchadwick, the price of the bond should be 50/(1.10) + 1050/(1.10)^2 where are you getting PV(\$50) + PV(\$1050 in two years) ?

50/1.1 = PV(50 in 1 year)… isn’t it the same thing?

Oops, I took a computational shortcut that isn’t quite right (0.9 is approximately 1/1.1, but not quite), but the fundamental message of the explanation is the same. Remeber that the price of a bond is the PV of all payments you can expect for owning the bond (discounted further for any risks that you won’t get paid).