Valuing debt securities is relatively straightforward compared with, say, valuing equity securities (see the Equity Securities chapter) because bonds typically have a finite life and predictable cash flows. The value of a debt security is usually estimated by using a discounted cash flow (DCF) approach. The DCF valuation approach is a valuation approach that takes into account the time value of money. Recall from the discussion of the time value of money in the Quantitative Concepts chapter that the timing of a cash flow affects the cash flow’s value. The DCF valuation approach estimates the value of a security as the present value of all future cash flows that the investor expects to receive from the security.
The cash flows for a debt security are typically the future coupon payments and the final principal payment. The value of a bond is the present value of the future coupon payments and the final principal payment expected from the bond. This valuation approach relies on an analysis of the investment fundamentals and characteristics of the issuer. The analysis includes an estimate of the probability of receiving the promised cash flows and an establishment of the appropriate discount rate. Once an estimate of the value of a bond is calculated, it can be compared with the current price of the bond to determine whether the bond is overvalued, undervalued, or fairly valued.
7.1 Current Yield
A bond’s current yield is calculated as the annual coupon payment divided by the current market price. This measure is simple to calculate and is often quoted. A bond’s current yield provides bondholders with an estimate of the annualised return from coupon income only, without concern for the effect of any capital gain or loss resulting from changes in the bond’s value over time. The current yield should not be confused with the discount rate used to calculate the value of the bond.
7.2 Valuation of Fixed-Rate and Zero-Coupon Bonds
For fixed-rate bonds and zero-coupon bonds, the timing and promised amount of the interest payments and final principal payment are known. Thus, the value of a fixed-rate bond or zero-coupon bond can be expressed as
where V0 is the current value of the bond, CFt is the bond’s cash flow (coupon payments and/or par value) at time t, r is the discount rate, and n is the number of periods until the maturity date. The bond’s cash flows and the timing of the cash flows are defined in the bond contract, but the discount rate reflects market conditions as well as the riskiness of the borrower. As always, you are not responsible for calculations, but the presentation of formulas and illustrative calculations may enhance your understanding.
It is important to note that the expected payments may not occur if the issuer defaults. Therefore, when estimating the value of a debt security using the DCF approach, an analyst or investor must estimate and use an appropriate discount rate (r) that reflects the riskiness of the bond’s cash flows. This discount rate represents the investor’s required rate of return on the bond given its riskiness. The expected cash flows of bonds with higher credit risk should be discounted at relatively higher discount rates, which results in lower estimates of value.
Although you are not responsible for calculating a bond’s value, Example 5 illustrates how to do so and the effect of using different discount rates. This example also serves to illustrate the effect of a change in discount rates on a bond. A change in discount rates may be the result of a change in interest rates in the market or a change in credit risk of the bond issuer.
EXAMPLE 5. BOND VALUATION USING DIFFERENT DISCOUNT RATES
Consider a three-year fixed-rate bond with a par value of $1,000 and a coupon rate of 6%, with coupon payments made semiannually. The bond will make six coupon payments of $30 (one coupon payment every six months over the life of the bond) and a final principal payment of $1,000 on the maturity date. The value of the bond can be estimated by discounting the bond’s promised payments using an appropriate discount rate that reflects the riskiness of the cash flows. If an investor determines that a discount rate of 7% per year, or 3.5% semiannually, is appropriate for this bond given its risk, the value of the bond is $973.36, calculated as
Example 5 also illustrates how the relationship between the coupon rate and the discount rate (required rate of return) affects the bond’s value relative to the par value. To explain this relationship further,
■ if the bond’s coupon rate and the required rate of return are the same, the bond’s value is its par value. Thus, the bond should trade at par value.
■ if the bond’s coupon rate is lower than the required rate of return, the bond’s value is less than its par value. Thus, the bond should trade at a discount (trade at less than par value).
■ if the bond’s coupon rate is higher than the required rate of return, the bond’s value is greater than its par value. Thus, the bond should trade at a premium (trade at more than par value).
In the case of a zero-coupon bond, the only promised payment is the par value on the maturity date. To estimate the value of a zero-coupon bond, the single promised payment equal to the bond’s par value is discounted to its present value by using an appropriate discount rate that reflects the riskiness of the bond.
7.3 Yield to Maturity
Investors can also use the DCF approach to estimate the discount rate implied by a bond’s market price. The discount rate that equates the present value of a bond’s promised cash flows to its market price is the bond’s yield to maturity, or yield. An investor can compare this yield to maturity with the required rate of return on the bond given its riskiness to decide whether to purchase it.
A bond’s yield to maturity can be expressed as
where P0 represents the current market price of the bond, and rytm represents the bond’s yield to maturity.
Many investors use a bond’s yield to maturity to approximate the annualised return from buying the bond at the current market price and holding it until maturity, assuming that all promised payments are made on time and in full. When a bond’s payments are known, as in the case of fixed-rate bonds and zero-coupon bonds, the yield to maturity can be inferred by using the current market price. Example 6 shows the calculation of yield to maturity. Again, you are not responsible for knowing how to do the calculation.
EXAMPLE 6. YIELD TO MATURITY
It is important to understand that bond prices and bond yields to maturity are inversely related. That is, as bond prices fall, their yields to maturity increase, and as bond prices rise, their yields to maturity decrease.
7.4 Yield Curve
When investors try to determine the appropriate discount rate (yield to maturity or required rate of return) for a particular bond, they often begin by looking at the yields to maturity offered by government bonds. The term structure of interest rates, often referred to simply as the term structure, shows how interest rates on government bonds vary with maturity. The term structure is often presented in graphical form, referred to as the yield curve. The yield curve graphs the yield to maturity of government bonds (y-axis) against the maturity of these bonds (x-axis). It is important when developing a yield curve to ensure that bonds have identical features other than their maturity, such as identical coupon rates. In other words, the bonds considered should only differ in maturity.
A yield curve applied by investors to US debt securities is the US Treasury yield curve, which graphs yields on US government bonds by maturity. Exhibit 2 illustrates the US Treasury yield curve as of 22 April 2014. In this case, the yield curve is upward sloping, indicating that longer-maturity bonds offer higher yields to maturity than shorter-maturity bonds. For example, the yield to maturity on a 30-year Treasury bond is 3.50%, but the yield to maturity on a 1-year Treasury bill is only 0.11%.
Although an upward-sloping curve is the norm, there are times when the yield curve may be flat, meaning that the yield to maturity of US Treasury bonds is the same no matter what the maturity date is. There are also times when the yield curve is downward sloping, or inverted, which can happen if interest rates are expected to decline in the future.
The term structure for government bonds, such as Treasury bonds, provides investors with a base yield to maturity, which serves as a reference to compare yields to maturity offered by riskier bonds. Relative to Treasury bonds, riskier bonds should offer higher yields to maturity to compensate investors for the higher credit or default risk.