A type of bond called a consol has no maturity date. The issuer is not obligated to ever repay the principal but makes coupon payments each year forever. Thus, if I buy a con-sol today, I am entitled to receive the coupon payment forever but never to be repaid the principal. After some mathematical manipulation and simplifi cation of Equation (5- 6), found in endnote 13, such characteristics imply the following:13
i = C/P
The yield to maturity, or interest rate, i, on a consol is equal to the coupon pay-ment, C, divided by the price of the bond, P. Suppose a new $1,000 face value consol is issued today and promises to pay $50 in interest each year. This is the coupon payment each year. Assuming the price of the new consol is $1,000, the $50 divided by the price shows that the consol yields 5 percent ($50/$1,000 = .05).
Now assume that a year later another $1,000 consol is issued by the same com-pany. Suppose the prevailing level of interest rates in the economy has risen so that the new consol will have to pay $60 a year in interest to be competitive. Clearly, the new consol is a better investment than the one- year- old 5 percent consol.
Suppose that some unforeseen fi nancial problems lead the own er of the old 5 per-cent consol to sell it. Who would be willing to purchase the old 5 perper-cent consol, given that they could instead purchase a new 6 percent consol? The answer is nobody, at least not yet. The older consol will have to yield 6 percent to be sold, and it will sell if it can somehow be made to yield 6 percent.
The older consol cannot change the fact that it pays $50 a year in interest. How-ever, the old consol can sell for a lower price. If the price drops to $833.33, then $50 a year interest would represent a yield of 6 percent ($50/$833.33 = .06). In fact, this is exactly what will happen. The own er of the old consol will offer the bond for sale at $1,000— the original price. Because no buyers appear, the market maker handling the transaction will Consol
A perpetual bond with no maturity date; the issuer is never obliged to repay the principal but makes coupon payments each year, forever.
Suggested Readings
The article “The Cyclical Behavior of Interest Rates,”
Journal of Finance 52 (September 1997): 1519– 42, offers an advanced discussion of interest rates.
A classic work dealing with interest rate determination is Irving Fisher, The Theory of Interest (New York: Macmillan, 1930). A more recent article, “The Fisher Hypothesis Re-visited: New Evidence,” fi nds that nominal interest rates are directly related to expected infl ation rates and govern-ment borrowing. It can be found in Applied Economics 29 (August 1997): 1055– 59.
For a basic discussion of the “Fisher Effect,” see the Inter-net site http:// en .wikipedia .org/ wiki/ Fisher _hypothesis . For an interesting article on the relationship between in-terest rates and infl ation, see Edward Renshaw, “Infl ation
and the Search for a Neutral Rate of Return on T-Bills,”
Challenge (November- December 1994): 58– 61.
For an analysis of the relationship between interest rates and bond prices, see Dale Bremmer, “The Relationship Between Interest Rates and Bond Prices,” American Econo-mist 36 (Spring 1992): 85– 86.
For information on estimating yields on Trea sury securities, see http:// www .newyorkfed .org/ aboutthefed/ fedpoint/
fed28 .html .
Information on interest rate calculations can be found on-line at http:// www .newyorkfed .org/ education/ interest _ rates .html .
lower the price. The price cutting will continue until buyers appear; this will occur when the price falls to $833.33, because at this point the yield on the old consol is competitive with the yield on new consols. Finally, what if the interest rate on new bonds falls to 4 percent? What will happen to the price of the old consol and why?14
Endnotes
1. If Joe received the interest earned on the loan after one year but left the principal, the total return over two years would be $120, or $60 each year. The average annual rate of return would be 6 percent [.06 = ($120/$1,000) ÷ 2], which is simple interest. If, as in the example, no interest payment is made after one year— the funds being, in effect, re- lent or reinvested—
the total return is $123.60, and the compound annual return is 6.18 percent [.0618 = ($123.60/$1,000) ÷ 2]. The compound rate will always be higher than the simple rate due to the interest earned on interest.
2. For those who would like to work through all the steps, start with V2 = V0 + iV0 + i(V0 + iV0)
= V0 + iV0 + iV0 + i2V0 = V0 + 2iV0 + i2V0 = V0(1 + 2i + i2) = V0(1 + i)(1 + i) = V0(1 + i)2.
3. We are also assuming that she expects the interest rate to be 6 percent for the next fi ve years.
4. The present value of $7,500,000 to be received in fi ve years, assuming an interest rate of 4 per-cent, is $6,164,453.30. This is obviously more than $6,000,000. Put another way, if the actress took the $6,000,000 today and lent it at 4 percent for fi ve years, she would have only $7,299,917.41 at the end of the period rather than $7,500,000. The $6,164,453.30 is what she would need to lend today at 4 percent for fi ve years to have $7,500,000 at the end of the period. In sum, the actress should take the $7,500,000 in fi ve years rather than the $6,000,000 today.
5. Are you puzzled by the fact that the price of the bond in the marketplace equals the present value of the bond? If so, think of what happens in any market when a product is selling for less or more than buyers and sellers think it’s really worth. If it is selling for less, quantity demanded will be greater than quantity supplied, and the price will rise in response. If it is selling for more, quantity demanded will be less than quantity supplied, and the price will fall in response. Equilibrium is reached when the prevailing price in the market is such that quantity demanded equals quantity supplied. So, too, in fi nancial markets.
6. Technically, the time to maturity is now one year less a day, but to simplify, we ignore the one day.
7. In Chapter 23, we will see that the managers could also use fi nancial futures markets to re-duce the risk of losses from changes in interest rates.
8. The federal government’s demand for loanable funds is less sensitive to changes in the level of interest rates than are the other sectors. The federal government does not necessarily bor-row less at higher interest rates than at lower interest rates. Ceteris paribus, as interest rates rise, the government may actually borrow more because of higher interest payments on the outstanding national debt.
9. See any principles of economics text for a more detailed defi nition of GDP.
10. Later in the text, we shall see that continuous increases in the growth rate of the money sup-ply can lead to infl ation, changes in infl ationary expectations, and possible increases in inter-est rates.
11. This equation is a reduced- form equation resulting from simultaneously solving a demand and supply equation for loanable funds.
12. The correlation between the business cycle and interest rates is far from perfect. For exam-ple, during the expansion that began in the early 1990s and ended in the early 2000s, interest rates did not behave in the same manner described.
13. From Equation (5- 6), the price of a consol is equal to P = C/(1 + i)1 + C/(1 + i)2 + C/(1 + i)3 + C/(1 + i)4 + . . . = C [1/(1 + i) + 1/(1 + i)2 + 1/(1 + i)3 + 1/(1 + i)4 + . . . ] = C(1 /i) = C/i. Therefore, i = C/P.
14. The old consol represents a future stream of income of $50 per year forever. At an interest rate of 4 percent, the price rises to $1,250 ($50/.04 = $1,250), and the lucky own er makes a capital gain of $250.
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6
Learning Objectives
After reading this chapter, you should know:
What a yield curve is
What the expectations theory is
How expectations infl uence interest rates What determines expectations
How term to maturity, credit risk, liquidity, and tax treatment affect interest rates
C H A P T E R S I X
Time gives good advice.
—Maltese proverb