V Pattabhi Ram When you buy a bond you are subject to interest rate risk. Interest rate risk is the risk associated with changes in market interest rates (a k a current yield) after the bond has been issued.

For instance, suppose after a bond has been issued at a coupon of 10%, the interest rates in the market go up to 11% or fall down to 9%. This causes risk to the investor. Here risk means that the terminal value of his investment undergoes a change

To understand this we must appreciate two facts.

One, the annual interest received will be reinvested at market interest rate. Two, when this rate increases the value of the bond will fall. However, the terminal value of the interest reinvested will go up because annual interest is now reinvested at higher rates. Similarly if the current yield falls, the value of the bond will go up but the terminal value of the interest reinvested will fall because they are now reinvested at lower rates. This changing value is what we call risk.

Let me explain with a couple of linked examples.

**Case 1:** You buy a 5 year bond with a face value of Rs 100, a coupon rate of 10% and redemption at par. This means that you will get Rs 10 every year for five years. At the end of the 5th year you will receive the redemption price of Rs 100

Now if the going yield is 10% it means that you can reinvest your interest amounts at 10%. If you do reinvest for the balance period of the bond you will end up having Rs 161.05 on maturity.

Suppose as soon as you bought the bond, the going yield drops to 9%. The maturity value of your investment will now be Rs 159.85. The drop in interest rate has hurt you. Alternatively if the going yield increased to 11%, the maturity value of your investment will be Rs 162.28. The increase in interest rate has helped you.

**Case 2**: Now let us say that when you bought the bond you decided you will keep it for 3 years only. That means that the first year interest will be invested for 2 years, the second year’s interest for 1 year etc. If the yield is 10%, the price of the bond at the end of the third year when you wish to sell would be Rs 99.95. Remember, the market price of a bond at any point in time is the present value of the future cash flows associated with the bond discounted at the desired yield.. Annual interest amounts would have been invested to mature at the end of the third year from the date of purchase of the bond. The total money, including reinvested interest, that you will receive at maturity (3rd year) will be Rs 133.05.

Suppose after you buy the yield falls to 9%. The price of the bond on sale after three years will fetch Rs 101.68. The total money including reinvested interest that you will receive on maturity is Rs 134.46. You have gained the difference between 134.46 vs. 133.05. Alternatively, if the yield after you bought went up to 11% the price of the bond on sale after three years will fetch Rs 98.33. The total money including reinvested interest that you will receive after three years is Rs 131.75. You have lost the difference between 133.05 vs. 131.75

**Computing duration **

Well, you have had a whole lot of numbers. So what are the conclusions?

One, when interest rate increases, the 5 year investor gains and the 3 year investor loses. There must therefore be some number of years at which the increase in interest rate neither leads to a win or a loss. Two, when interest rate falls, the 5 year investor loses and the 3 year investor wins. There must therefore be some number of years at which the fall in interest rate neither leads to a win nor a loss.

This some number of yeas is called the Duration of the bond. By buying bond for that duration you eliminate interest rate risk. i.e. there will be no change in terminal value based on interest rate movements.

Here’s the most critical question. How do we compute duration? Do we have to work in the elaborate manner shown above. Luckily no. A few quick steps and you get the duration

1. Compute the present value of the cash flows by discounting at current yield

2. Aggregate the result

3. Calculate the ratio of the present value of each year to the aggregated value.

4. Multiply this ratio with the number of years

5. Aggregate the result in (d)

In our example the duration is about 4.18 or 4 years. That is, in the given example, a 4-years investor is insulated from interest rate risk.

*Published in the HBL on 21st Sep, 2008*

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