How to Value Bonds
15 Feb 2025
A bond is a loan made to the issuer of the bond. When an investor buys a bond, they lend a principal amount to the issuer in exchange for periodic interest payments (coupons) and the repayment of the principal at maturity. Thus, bondholders are entitled to two types of cash flows: periodic interest payments and the repayment of the principal at maturity. We can calculate the present value of these cash flows to determine the value or price of the bond.
For example, consider a bond that has a face value of $1000, a coupon rate of 5%, paid semi-annually, and which matures in 2 years. The set of cash flows for this bond is:
Discounting these cash flows by the appropriate discount rate gives the present value of the bond. Let's assume that the rate of return (or yield) that investors expect from comparable bonds is 4%, or 2% semi-annually. The present values of these cash flows are thus:
Yielding a total present value of $1019.04. If investors expect the yield of comparable bonds to be 4%, then they would be willing to pay $1019.04 for this bond. Notice that the price of the bond is higher than its face value. This is because comparable bonds are yielding 4%, while this bond has a coupon rate of 5%. Thus, investors are willing to pay a premium for the higher coupon rate. This higher price, in turn, lowers the yield of the bond to 4%, bringing it in line with comparable bonds. Bonds selling above their face value are called premium bonds. The opposite occurs when yields on comparable bonds are higher than the coupon rate, in which case the price of the bond must fall in order to bring its yield in line with comparable bonds. These bonds are called discount bonds.
Yields are closely linked to bond prices. When yields rise, bond prices fall, and vice versa. However, different bonds react differently to changes in yields. A longer-term bond is more sensitive to changes in yields than a shorter-term bond. By longer-term, we refer to the duration of the bond, which is the weighted average of the times to each cash payment. For a given coupon rate, a bond that matures further into the future has a longer duration than one that matures sooner. For a given maturity, a bond with a lower coupon rate has a longer duration than one with a higher coupon rate, because a larger portion of the cash flows comes from the repayment of the principal at maturity.
Short- and long-term interest rates are not necessarily the same. The relationship between the two is called the term structure of interest rates. Typically, short-term interest rates are lower than long-term interest rates.
If long-term rates are higher than short-term rates, why would investors buy short-term bonds? One reason is that long-term bonds are exposed to more interest rate risk than short-term bonds. The prices of long-term bonds are more sensitive to changes in interest rates, which makes them riskier. Also, investors may worry about future inflation eroding the value of the fixed interest payments from long-term bonds, and therefore they may be uncomfortable locking in their money for a long time. A third reason is that investors may expect short-term rates to rise in the future, and they would prefer to invest their money in short-term bonds today, and reinvest their returns from these bonds at higher rates in the future.
Although investors can earn money through their investments, they also face the risk of their money losing value over time due to inflation. When prices rise, the purchasing power of money falls. Thus, in addition to nominal interest rates, investors are also concerned with real interest rates, which are nominal rates adjusted for inflation.
The real rate of return is:
1 + rreal = (1 + rnominal) / (1 + r)
where i is the inflation rate. For example, if the nominal rate is 10%, and inflation is 6%, then the real rate of return is:
( 1.10 / 1.06 ) - 1 = 1.0377 - 1 = 0.0377 = 3.77%
Thus, $100 invested at a 10% nominal rate will grow to $110, but the purchasing power of this $110 will only be $103.77 due to inflation. In other words, although you will have $110 in the future, you will only be able to buy $103.77 worth of goods in today's dollars.
Some bonds are indexed to inflation, meaning that the coupon payments and the principal are adjusted for inflation. With these bonds, investors can be sure that the returns from their investments can be used to buy the same amount of goods in the future as they can today.
Real interest rates are driven by the supply of and demand for capital. During times of economic growth, businesses will want to invest in new projects, which increases the demand for capital. This drives up real interest rates, which in turn attracts more savings from individuals. The opposite occurs during economic downturns.
Corporate bonds are riskier than government bonds. A corporation that issues a bond that is rated investment grade is considered to have a low risk of default. Issuers of junk bonds, on the other hand, have a higher risk of default. Although it is rare for a government to default on its debt, it can happen.