# NOTES E. Borrower Receives: Loan Value LV MATURITY START DATE. Lender Fixed Fixed Fixed Receives: Payment FP Payment FP Payment FP

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5 The payment stream to the lender generated by this coupon bond NOTES G is given by (11) ( \$10,\$10,\$10,\$10,\$10,\$10,\$10,\$10,\$10,[\$10+\$100] ). For any given fixed annual interest rate i, the present value PV(i) of the payment stream (11) is given by the sum of the separate present value calculations for each of the payments in this payment stream as determined by formula (5). That is, (12) PV(i) = \$10/(1+i) + \$10/(1+i) \$10/(1+i) \$100/(1+i) 10. The current value of the coupon bond is its current purchase price Pb = \$94. It then follows by definition that the yield to maturity for this coupon bond is found by solving the following equation for i: (13) Pb = PV(i) = (C / i) * (1-1/(1+i) n ) + FV / 1+i) n. The calculation of the yield to maturity i from formula (13) can be difficult, but tables have been published that permit one to read off the yield to maturity i for a coupon bond once the purchase price, the face value, the coupon rate, and the maturity are known. For example, using such tables, it can be shown that the yield to maturity i for the coupon bond currently under consideration, which has a purchase price of \$94 per \$100 of face value, a coupon rate of 10 percent, and a maturity of 10 years, is approximately equal to 11 percent. Yield to Maturity for a Consol: A consol is a coupon bond that has an infinite maturity and hence never repays its principal. Rather, the holder of a consol receives a coupon payment C in perpetuity -- that is, in each future payment period without end -- implying that the payment stream to the holder takes the special form (C,C,C,...). Let Pc be a price of a consol. The formula Pc = PV( i ) in equation (13) for determining the yield to maturity i for a consol reduces to C C (14) Pc = ---, which implies that i = --- i Pc Some Final Important Observations on Yield to Maturity: For any coupon bond with a fixed coupon payment C and a fixed face value F, the purchase price Pb of the bond is equal to the face value F if and only if the yield to maturity i for the bond is equal to the coupon rate C/F. This observation follows directly from the structure of a coupon bond. When the purchase price equals the face value, the coupon bond essentially functions as a bank deposit account into which a principal amount (the face value) is deposited by a lender, earns a fixed annual interest rate (the coupon rate) for some number of years, and is then recovered by the lender. Illustration for a One-Period Coupon Bond: For a one-period coupon bond with coupon payment C, face value FV, and purchase price Pb, the formula Pb = PV(i) for determining the yield to maturity i can be written as F + C (15) Pb = (1+i) Dividing each side of formula (15) by the face value F, one obtains 1 + C/FV (16) Pb/FV = (1+i) Given C and F, formula (16) implies that Pb equals FV (i.e., the left-hand side equals 1) if and only if i equals C/F (i.e., the right-hand side equals 1). More generally, given any coupon bond with a fixed coupon payment C and a fixed face value F, the purchase price Pb of the bond is lower (higher) than FV if and only if the yield to maturity i is higher (lower) than the coupon rate C/FV. This follows directly from formula (14) for determination of the yield to maturity, using the previously noted fact that the purchase price Pb is equal to FV if and only if the yield to maturity i is equal to the coupon rate C/FV. Moreover, for any given coupon bond with given C and FV, the yield to maturity i of the bond is inversely related to the purchase price Pb of the bond. That is, the higher the yield to maturity i, the lower the purchase price Pb, and conversely. This inverse relationship also follows directly from formula (13). This inverse relationship between the yield to maturity of a debt instrument and its purchase price actually holds in general. For any debt instrument with any given payment stream, when the yield to maturity for the debt instrument rises, the purchase price of the debt instrument must fall, and vice versa. This follows directly from the general definition for the yield to maturity, applicable to all debt instruments. VII. Basic Concepts, Key Issues and Practice Questions Simple loan contract Principal Maturity and maturity date Interest payment Simple interest rate Fixed-payment loan contract Coupon bond, Consol Face value Coupon payment Coupon rate Discount bond (or zero-coupon bond) Nominal value Present value (or present discounted value) Yield to maturity Diagrammatic representation of loan contracts Present value of a future payment and of a stream of future payments General formula for determining the yield to maturity for any bond Calculating the yield to maturity for a simple loan and for a discount bond Inverse relationship between the price of a bond and its yield to maturity Relationship between the purchase price of a coupon bond, its face value, its yield to maturity, and its coupon rate 13

6 VIII. Other Measures of Interest Rates The yield to maturity is the most accurate measure of interest rates and we will henceforth use the terms "interest rate" and "yield to maturity" interchangeably throughout the remainder of his text. Nevertheless, since the yield to maturity can be difficult to calculate, other less accurate measures of interest rates are commonly used in the financial pages of newspapers and elsewhere to report the properties of debt instruments. We will discusse two such measures at some length: "current yield" and "discount yield." Current Yield The current yield is an approximation to the yield to maturity for coupon bonds. More precisely, letting Pb denote the purchase price of a coupon bond, and C denote its coupon payment, the current yield, denoted below by ic, is given by: C (17) ic = Pb In general, for most coupon bonds, the current yield will differ in value from the yield to maturity. However, it can be shown that the current yield equals the yield to maturity for a special type of coupon bond, called a consol. Comparing (17) and (14), it follows that -- for a consol -- the current yield ic equals the yield to maturity i because both are equal to C/Pb. For coupon bonds with less than infinite maturities, the current yield ic no longer coincides with the yield to maturity i. However, the current yield becomes an increasingly better approximation for the yield to maturity as the maturity of a coupon bond becomes longer and longer (hence closer and closer to the infinite maturity of a consol). That is, all else remaining the same, ic provides an increasingly accurate approximation to i as one considers coupon bonds with successively longer maturities N. For fixed C, FV, and Pb: implies (18) Maturity N increases > ic approaches i Another aspect of a coupon bond that determines how accurate an approximation ic provides to the yield to maturity i is the difference between the bond's purchase price Pb and its face (or par) value FV. Given C, FV, and N: implies (19) Pb approaches F > ic approaches i. Finally, it follows directly from definition for the current yield: Discount Yield U.S. Treasury bills are an example of a discount bond. For ease of calculation, interest rates on many discount bonds such as Treasury bills and commercial paper are quoted on a 360-day "discount yield" basis (or "bank discount basis") rather than on a yield-to-maturity basis, as follows. Let FV denote the face value of a discount bond, and let Pd denote the purchase price of the discount bond. Then the discount yield, denoted below by idb, is given by: F - Pd 360 (21) idb = * F Days to Maturity Let us see how idb compares, for example, to the yield to maturity i for a one-year discount bond. In the case of a one-year discount bond the usual formula Pd = PV(i) for determining the yield to maturity takes the form F - Pd (22) i = Pd Comparing (7) with (6) for the special case of a discount bond with a one year maturity (i.e., days to maturity = 365), it follows that F 365 (23) i = idb * * Pd 360 Consequently, recalling that discount bonds are priced at a discount (Pd < F), it follows that the yield to maturity i for a discount bond with a one-year maturity is definitely greater than the discount yield idb. Also, or any given discount bond with a fixed face value F and a fixed maturity N, the discount yield idb is inversely related to the price Pd of the discount bond -- that is, when idb increases, Pd decreases, and vice versa. Recall from previous notes that the yield to maturity i on a discount bond is also inversely related to the purchase price Pd. This follows directly from the general formula Pd=PV(i) used to determine i for discount bonds -- see, for example, relation (22), which is what the general formula Pd=PV(i) reduces to when the discount bond has a one-year maturity. Consequently, as for the current yield, one obtains the following important observation: For any given discount bond with a fixed face value F and a fixed maturity, the discount yield idb and the yield to maturity i always move together in response to changes in the purchase price Pd. As for the current yield, this positive co-movement between idb and i holds even if idb is a bad approximation to i in level terms in the sense that the difference between idb and i is large. if and only if (20) ic increases < > Pb decreases.. 14

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