Question

Suppose that a zero coupon bond selling at $1,000 par has a duration of four years. If interest rates increase from 6 percent to 7 percent annually, the value of the bond will fall by what amount using Equation 6.14? Use semiannual compounding. Then, use the PV formula to determine the actual price of the bond at 7 percent. What is the difference? Why is there a difference?

Answer 1

For a zero coupon bond, duration is equal to maturity. Given that the bond has duration of 4 years so maturity is also 4 years.

Interest rate=6% originally.

So, volatility=4/1.06=3.77 which means bond price will change by 3.77% if interest rate changes by 1%.

Thus, when interest rate is 7%, bond price will fall by 3.77*1=3.77%.

Par value=$1,000

At 7% interest, price fall=$1,000*0.0377=$37.7

But according to PV formula,

price=1,000/(1.06)⁴=$792.09 when interest rate is 6% and it is 1,000/(1.07)⁴=$762.895 when interest rate is 7%.

So, fall in price=$(792.09-762.895)=$29.198

This is less than the fall calculated with the help of duration, because duration assumes a linear relationship between interest rate and price which is not actually true because the curve is convex and not linear in nature.