Question

A bond has a yield to maturity of 5 percent. It matures in 12 years. Its coupon rate is 7 percent. What is its modified duration? The bond pays coupons twice a year

Answer 1

YTM= 5%

Time to maturity= 12 years

Coupon rate =7%

Pays Coupon semi-annually,therefore, bond will pay 3.5 every 6 months if it has face value of 100 say

To calculate Modified duration, we first need to calculate Macualay duration

And to calculate Macualay duration, we need to know how much and when each cash flow is paid by the bond

This is simple, we know that the bond will pay coupon of 3.5 every half year and then will make a principle payment of 100 at maurity i.e. at t=12

We then have to calulate present value of each of these cash flows using discount rate as yield to maturity

For example for coupon payment of 3.5 at t=0.5, we will discount the value using discount rate i.e. YTM of 5% i.e. 3.5/(1+5%)^0.5------this is simple since we have discounted using YTM and time period as 0.5

Once you have got the present value of all the bonds' cash flows, add them up to get the total present value

you can also get the total present value using your financial calculator, just entrer the value of PMT-3.5, N=24, I/Y=5%, FV=100, then click on CPT PV to get total present value as 117.88

As a next step , divide each cash flows' present value with the total present value of the bond

This will give you weighted cash flows at each period i.e. percent of total cash flow occuring at a particular time period

Then, multiply this weighted cash flows with the corresponding time period to get the value of Macaulay duration as 8.63( i.e. the cash flow which is occuring at t=0.5, multiply the present value fo that cash flow with 0.5)

Modified duration = Macaulay duration/ (1+YTM/no of paymets per year)= Macaulay duration/ (1+5%/2)= 8.63/(1+2.5%)=8.42