## IntroductionConsider a zero-coupon bond, which is purchased at a discount to its face value and can be redeemed for its full face value at maturity. For example, a zero-coupon bond with a face value of $10,000 that earns 5% annual interest and matures in 10 years would have a present value of:
For a zero-coupon bond, the bond duration is simply the number of years until maturity; in this example, the duration would be 10 years. There is however another definition of bond duration, which is an estimate of the percentage change in the price of the bond for a 1% change in interest rates. Recall that a bond's price moves in the opposite direction to its yield. In the example above, if interest rates went down, a buyer would be willing to pay more for the bond at present because the buyer would be receiving less interest between the present time and the maturity date. Let's now look at how much the price of our example bond would change if interest rates went down 1% (in this case, from 5% to 4%).
The price of the bond would therefore increase by $617 if interest rates went down 1%. On a percentage basis, the price of the bond would increase by $617/$6139 or approximately 10%. Hold that thought for a moment. Let's look at a different example, this time a zero-coupon bond with a face value of $10,000 that earns 8% annual interest and matures in 20 years. If interest rates fell 1% (that is, to 7%), the percentage change in price would be:
Notice that in the first example, the bond had a duration of 10 years and its price changed approximately 10% in response to a 1% change in the interest rate. In the second example, the bond had a duration of 20 years and its price changed approximately 20% in response to a 1% change in the interest rate. What we're seeing, therefore, is that a zero-coupon bond's duration is BOTH the number of years until maturity AND an estimate of the percentage change in the price of the bond in response to a 1% change in the interest rate! This latter result isn't intuitive. For example, notice the assertion that the price change depends only on the duration and not on the absolute interest rate. We used the absolute interest rates in these example calculations, but if this assertion holds true, the absolute interest rate isn't required to estimate the percentage price change in response to a 1% change in the interest rate. Let's explore the mathematics here to see if we can understand why this relationship would hold true. ## Framing the problemLet x be the interest rate, where 0<x<1.Let the price at maturity be the unit price of 1. Let n be the duration, which is the number of years until maturity. Let P be the present value of the zero-coupon bond. Now, we want to see if we can show that for a 1% change in the interest rate, the price changes by approximately ## Solving the problemWe can frame P as a function of x as follows:
Or, equivalently:
Since we're interested in how P changes with respect to x, we can start by computing the slope of the P vs x curve using calculus:
Now, for a small Δx (in our case, 0.01):
Now, let's compute the percentage change in price:
Finally, for small x (e.g. x < 0.1), 1+x can be approximated as 1, so the % change can be approximated as n. |