# YTM and Reinvestment Risk

Yield to maturity calculates the total return an investor would earn by holding the bond till maturity. This includes both the coupon income and the capital gains from the bond. It also considers reinvestment income, however, with the assumption that coupon payments can be invested at the same rate as YTM. This means that you will actually earn a yield equal to YTM, even if you hold the bond to maturity, only if you are able to reinvest all the coupon payment at the YTM until the maturity of the bond. This is very unlikely and that's why we have reinvestment risk.

This assumption can be misleading because we have a term structure of interest rates, which can result in different yields at different points in time.

Let’s demonstrate this with the help of an example.

Let’s say that an investor has purchased a 15-year 8% semi-annual coupon bond at par. We assume that he purchased the bond at par for simplicity. Since there is no capital gain/loss, the bond’s YTM will also be 8%.

We can calculate the total future value of all cash flows from this bond as follows:

Future dollar value = $100 x (1.04)30 =$324.34

This value is made up of $100 of principal return and$224.34 of the total dollar return from coupon and reinvestment income. In fact if there was no reinvestment income the coupon income would only be $120. Remember that there is no capital gain, so the balance ($224.34 - $120) =$104.35 is earned from reinvestment income. This is the money which is earned by investing the periodic coupon payments at YTM of 8%.

If the coupons are not reinvested at 8%, the investor will not be able to realize the YTM of 8%.

How much reinvestment risk is present in a bond depends on several factors such as coupon rate and bond’s maturity. Here are some observations.

• A bond that has high coupon is more dependent on reinvestment income because more money needs to be reinvested at the YTM to maintain the YTM.
• For a bond selling at premium, there is capital loss, which further increases the dependency on reinvestment income.
• For a bond selling at a discount, there is capital gain, which reduces the dependency on reinvestment income.
• The longer the bond’s maturity, the higher is the reinvestment risk.