Foundations of Financial Management: An Analysis (part 32)

“Finance is not merely about making money. It’s about achieving our deep goals and protecting the fruits of our labor. It’s about stewardship and, therefore, about achieving the good society.” 

– Robert J. Shiller

Valuation and Rates of Return (part B)

by

Charles Lamson

Concept of Yield to Maturity

 In the previous example presented in the last post, the yield-to-maturity that was used at the as the discount rate was 10 percent. The yield-to-maturity, or discount rate, is the rate of return required by bondholders. The bondholder, or any investor for that matter, will allow three factors to influence his or her required rate of return.

1. The required real rate of return---This is the rate of return the investor demands for giving up the current use of the funds on an inflation-adjusted basis. It is the financial rent the investor charges for using his or her funds for one year, five years, or in a given period. Although it varies from time to time, historically the real rate of return demanded by investors has been about 2 to 3%.

2. Inflation premium---In addition to the real rate of return discussed above, the investor requires a premium to compensate for the eroding effect of inflation on the value of the dollar. It would hardly satisfy an investor to have a 3 percent total rate of return in a 5 percent inflationary economy. Under such circumstances, the lender (investor) would be paying the borrower 2 percent in purchasing power for use of the funds. This would represent an irrational action. No one wishes to pay another party to use his or her funds. The inflation premium added to the real rate of return ensures that this will not happen. The size of the inflation premium will be based on the investors expectations about future inflation. In the last three and a half decades, the inflation premium has been 2 to 4 percent.

If one combines the real rate of return (part 1) and the inflation premium (part 2), the risk-free rate of return is determined. This is the rate that compensates the investor for the current use of his or her funds and for the loss in purchasing power due to inflation, but not for taking risks. As an example, if the real rate of return were 3 percent and the inflation premium were 4 percent, we would say the risk-free rate of return is 7 percent.

3. Risk premium---We must now add the risk premium to the risk-free rate of return. This is a premium associated with the special risks of a given investment. Of primary interest to us are two types of risk: business risk and financial risk. Business risk relates to the inability of the firm to hold its competitive position and maintain stability and growth in its earnings. Financial risk relates to the inability of the firm to meet its debt obligations as they come due. In addition to the two forms of risk mentioned above, the risk premium will be greater or less for different types of Investments. For example, because bonds possess a contractual obligation for the firm to pay interest to bondholders, they are considered less risky than common stock where no such obligation exists.

The risk premium of an investment may range from as low as zero on a very short term U.S. government backed security to 10 to 15 percent on a gold mining expedition. The typical risk premium is 2 to 6 percent. Just as the required real rate of return and the inflation premium change over time, so does the risk premium. For example high-risk corporate bonds (sometimes referred to as junk bonds) normally require a risk premium of about 5 percentage points over the risk-free rate. As is emphasized in many parts of this analysis, there is a strong correlation between the risk the investor is taking and the return the investor demands. If you want a higher return you must take a greater risk.

We shall assume that in the investment we are examining the risk premium is 3 percent. If we add this risk premium to the two components of the risk-free rate of return developed in parts 1 and 2, we arrive at an overall required rate of return of 10 percent.

In this instance, we assume we are evaluating the required return on a bond issued by a firm. If the security had been the common stock of the same firm, the risk premium might be 5 to 6 percent and the required rate of return 12 to 13 percent.

Finally, in concluding this section you should recall that the required rate of return on a bond is effectively the same concept as required yield to maturity. 

Changing the Yield to Maturity and the Impact on Bond Valuation

In the earlier bond value calculation, we assumed the interest rate was 10 percent ($100 annual interest on a $1,000 par value bond) and the yield to maturity was also 10 percent. Under those circumstances, the price of the bond was basically equal to par value. Now let's assume conditions in the market caused the yield to maturity to change.

Increase in Inflation Premium For example, assume the inflation premium goes up from 4 to 6 percent. All else remains constant. The required rate of return would now be 12 percent.

With the required rate of return, or yield to maturity, now at 12 percent, the price of the bond will change. A bond that pays only 10 percent interest when the required rate of return (yield to maturity) is 12 percent will fall below its current value of approximately $1,000. The new price of the bond, as computed below, is $850.90.

Present Value of Interest Payments---We take the present value of a $100 annuity for 20 years. The discount rate is 12 percent. Using Table 1 from last post:

Present Value of Principal Payment at Maturity---We take the present value of $1,000 after 20 years. The discount rate is 12%. Using Table 2 from last post:

Total Present Value---

In this example we assumed increasing inflation caused the required rate of return (yield to maturity) to go up and the bond price to all by approximately $150. The same effect would occur if the business risk increased or the demanded level for the real rate of return became higher.

Decrease in Inflation Premium The opposite effect would happen if the required rate of return went down because of lower inflation, less risk, or other factors. Let's assume the inflation premium declines and the required rate of return (yield to maturity) goes down to 8 percent.

The 20-year bond with a 10% interest rate would now sell for $1,196.80. 

Present Value of Interest Payments---

Present Value of Principal Payment at Maturity---

Total Present Value---

The bond is now trading at $196.80 over par value. This is certainly the expected result because the bond is paying 10 percent interest when the yield and the market is only 8 percent. The two percentage point differential on a $1,000 par value bond represents $20 per year. The investor will receive the differential for the next two years. The present value of $20 for the next 20 years at the current market rate of interest of 8 percent is approximately $196.80. This explains why the bond is trading at $196.80 over its stated, or par, value.

The further the yield mature to maturity on a bond changes from the stated interest rate on the bond, the greater the price change effect will be. This is illustrated in Table 3 for the 10 percent interest rate, 20 year bonds discussed in the next several posts.

Table 3 Bond price table

We clearly see the impact that different yields to maturity have on the price of a bond. 

*MAIN SOURCE: BLOCK & HIRT, 2005, FOUNDATIONS OF FINANCIAL MANAGEMENT, 11TH ED., PP. 271-274*

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