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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1 NOTES E DEBT INSTRUMENTS Debt instrument is defined as a particular type of security that requires the borrower to pay the lender certain fixed dollar amounts at regular intervals until a specified time is reached. I. Simple Loan Contract Under the terms of a simple loan contract, the borrower (contract issuer) receives from the lender (contract holder) a specified amount of funds (the principal) for a specified period of time (the maturity). The borrower agrees that, at the end of this period of time -- referred to as the maturity date -- the borrower will repay the principal to the lender together with an additional payment referred to as the interest payment. Borrower Receives: Principal START MATURITY DATE Lender Principal Receives: + Interest Payment The annual borrowing fee for a simple loan with a principal P, an interest payment I, and a maturity of N years is measured by the SIMPLE interest rate given by I divided by [P times N]. NOTE: Your textbook implicitly assumes that the maturity N on simple loans is one year (N=1). As will be clarified further below -- "for simple loans, the simple interest rate equals the yield to maturity" -- is only true if N 1 is assumed for the simple loans and the yield to maturity is calculated as an annual rate. Example of a Simple Loan Contract: A borrower receives a loan on January 1, 1999, in amount $500, and agrees to pay the lender $550 on January 1, Thus, the principal is $500, the maturity is two years, the maturity date is January 1, 2001, and the interest payment is $50. The SIMPLE (annual) interest rate for this loan is then $50/[$500*2] =.05, or 5 percent. II. Fixed-Payment Loan Contracts Under the terms of a fixed-payment loan contract, the borrower (contract issuer) receives from the lender (contract holder) a specified amount of funds -- the loan value -- and, in return, makes periodic fixed payments to the lender until a specified maturity date. These periodic fixed payments include both principal (loan value) and interest, so at maturity there is no lump sum repayment of principal. Borrower Receives: Loan Value LV MATURITY START DATE Lender Fixed Fixed Fixed Receives: Payment FP Payment FP Payment FP Example of a Fixed-Payment Loan Contract: Joe arranges a 15-year installment loan with a finance company to help pay for a new car. Under the terms of this loan, Joe receives $20,000 now to finance the purchase of a new car but must make payments of $2000 every year for the next 15 years to the finance company. III. Coupon Bond Note: For simplicity, the case of a newly issued coupon bond is considered below, so that the seller is the borrower (issuer of the bond) and the buyer is the initial lender. Coupon bonds can also be re-sold in secondary markets, in which case the seller is not the borrower and the buyer is not the original lender. Under the terms of a coupon bond, the borrower (bond issuer) agrees to pay the lender (bond purchaser) a fixed amount of funds (the coupon payment) on a periodic basis until a specified maturity date, at which time the borrower must also pay the lender the face value (or "par value") of the bond. The coupon rate of a coupon bond is, by definition, the amount of the coupon payment divided by the face value of the bond. As will be clarified in the next section, below, the purchase price of a coupon bond depends on the "present value" of the stream of anticipated coupon payments plus the final face value payment promised by the bond. Coupon bonds that sell above their face value are said to sell at a premium, and those that sell below their face value, at a discount. Receives: Price Pb MATURITY START /\/\/\ DATE Lender Coupon Coupon... Coupon Receives: Payment C Payment C Payment C + Face Value F Example of a Coupon Bond: Suppose a coupon bond has a face value of $1000, a maturity of five years, and an annual coupon payment of $60. Then, at the end of each year for the next five years, the borrower (bond issuer) must pay the lender (bond holder) a coupon payment of $60. In addition, at the end of five years (the maturity date), the borrower must pay the lender the face value of the bond, $1000. The coupon rate for this coupon bond is $60/$1000 =.06, or 6 percent. 9

2 IV. Discount Bond (or Zero-Coupon Bond): Note: For simplicity, the case of a newly issued discount bond is considered below, so that the seller is the borrower (issuer of the bond) and the buyer is the initial lender. Discount bonds can also be re-sold in secondary markets, in which case the seller is not the borrower and the buyer is not the original lender. Under the terms of a discount bond, the borrower (bond issuer) immediately receives from the lender (bond holder) the purchase price Pd of the bond, which is typically less than the face value F of the bond. In return, the borrower promises that, at the bond's maturity date, he will pay the lender the face value F of the bond. Receives: Price Pd START MATURITY DATE Lender Face Value FV Receives: Important Cautionary Note: The above definition of a discount bond is not unique. Some authors refer to zero-coupon bonds as PURE discount bonds, labeling as a "discount bond" any bond that sells at a discount in the sense that its market price is less than its face value. Also, while people often say that discount bonds make no interest payments (this is literally true, in the sense that only a face value payment is made), it is NOT true that the interest RATE on discount bonds is zero. Indeed, as will be seen below, the most basic measure of interest rates in use today is the annual "yield to maturity" i. For a one-year discount bond, the formula for calculating i reduces to i = [FV-Pd]/Pd, hence i is only zero in the highly unlikely event that FV=Pd. Discount Bond Example: On June 1, 2004, a borrower gives a lender a discount bond with a face value of $100 and a maturity of ½ of a year, and the lender gives $95 to the borrower. The borrower must then pay the lender $100 on January 1, The yield to maturity on this contract is equal to (FV-Pd)/(Pd*N)=(100-95)/(95*½) =10%. (recall that yield to maturity = simple int. rate if N 1). V. The Concept of Present Value Suppose someone promises to pay you $100 in some future period T. This amount of money actually has two different values: a nominal value of $100, which is simply a measure of the number of dollars that you will receive in period T; and a present value (sometimes referred to as a present discounted value), roughly defined to be the minimum number of dollars that you would have to give up today in return for receiving $100 in period T. Stated somewhat differently, the present value of the future $100 payment is the value of this future $100 payment measured in terms of current (or present) dollars. The concept of present value permits debt instruments with different associated payment streams to be compared with each other by calculating their values in terms of a single common unit: namely, current dollars. Specific formulas for the calculation of present value for future payments will now be developed and applied to the determination of present value for debt instruments with various types of payment streams. Present Value of Payments One Period into the Future Notation: Let V(N) be amount of money paid N years into the future. If you save $1 today for a period of one year at an annual interest rate i, the nominal value of your savings after one year will be (1) V(1) = (1+i)*$1, where the asterisk "*" denotes multiplication. On the other hand, proceeding in the reverse direction from the future to the present, the present value of the future dollar amount V(1) = (1+i)*$1 is equal to $1. That is, the amount you would have to save today in order to receive back V(1)=(1+i)*$1 in one year's time is $1. Notice that this calculation of $1 as the present value of V(1)=(1+i)*$1 satisfies the following formula: V(1) (2) Present Value = of V(1) (1+i) Indeed, given any fixed annual interest rate i, and any payment V(1) to be received one year from today, the present value of V(1) is given by formula (2). In effect, then, the payment V(1) to be received one year from now has been discounted back to the present using the annual interest rate i, so that the value of V(1) is now expressed in current dollars. Pres.Value of Payments Multiple Periods Into the Future If you save $1 today at a fixed annual interest rate i, what will be the value of your savings in one year's time? In two year's time? In n year's time? If you save $1 at a fixed annual interest rate i, the nominal value of your savings in one year's time will be V(1)=(1+i)*$1. If you then put aside V(1) as savings for an additional year rather than spend it, the nominal value of your savings at the end of the second year will be (3) V(2) = (1+i)*V(1) = (1+i)*(1+i)*$1 = (1+i)2*$1. And so forth for any number of years n. (4) START /\/\/\ >YEAR 1 2 n Nominal Value of $1 (1+i)*$1 (1+i) 2 *$1 (1+i) n * $1 Savings: 10

3 NOTES F Now consider the present value of V(n) = (1+i) n *$1 for any year n. By construction, V(n) is the nominal value obtained after n years when a single dollar is saved for n successive years at the fixed annual interest rate i. Consequently, the present value of V(n) is simply equal to $1, regardless of the value of n. Notice, however, that the present value of V(n) -- namely, $1 -- can be obtained from the following formula: V(n) (5) Present Value = of V(n) (1+i) n Indeed, given any fixed annual interest rate i, and any nominal amount V(n) to be received n years from today, the present value of V(n) can be calculated by using formula (5). Present Value of Any Arbitrary Payment Stream Now suppose you will be receiving a sequence of three payments over the next three years. The nominal value of the first payment is $100, to be received at the end of the first year; the nominal value of the second payment is $150, to be received at the end of the second year; and the nominal value of the third payment is $200, to be received at the end of the third year. Given a fixed annual interest rate i, what is the present value of the payment stream ($100,$150,$200) consisting of the three separate payments $100, $150, and $200 to be received over the next three years? To calculate the present value of the payment stream ($100,$150,$200), use the following two steps: Step 1: Use formula (5) to separately calculate the present value of each of the individual payments in the payment stream, taking care to note how many years into the future each payment is going to be received. Step 2: Sum the separate present value calculations obtained in Step 1 to obtain the present value of the payment stream as a whole. Carrying out Step 1, it follows from formula (5) that the present value of the $100 payment to be received at the end of the first year is $100/(1+i). Similarly, it follows from formula (5) that the present value of the $150 payment to be received at the end of the second year is $150 (6) (1+i) 2 Finally, it follows from formula (3) that the present value of the $200 payment to be received at the end of the third year is $ (7) (1+i) 3 Consequently, adding together these three separate present value calculations in accordance with Step 2, the present value PV(i) of the payment stream ($100,$150,$200) is given by $100 $150 $200 (8) PV(i) = (1 + i) (1 + i) 2 (1 + i) 3 More generally, given any fixed annual interest rate i, and given any payment stream (V1,V2,V3,...,VN) consisting of individual payments to be received over the next N years, the present value of this payment stream can be found by following the two steps outlined above. In particular, then, given any fixed annual interest rate, and given any debt instrument with an associated payment stream paid out on a yearly basis to the lender (debt instrument holder), the present (current dollar) value of this debt instrument is found by calculating the present value of its associated payment stream in accordance with Steps 1 and 2 outlined above. Consequently, regardless how different the payment streams associated with two such debt instruments may be, one can calculate the present values for these debt instruments in current dollar terms and hence have a way to compare them. Technical Note: The above procedure for calculating the present value of a debt instrument whose payments are paid out on an annual basis to a lender can be generalized to debt instruments whose payments are paid out at arbitrary times to lenders. To do this, one needs to transform the annual interest rate used to discount each payment so that its period matches that of the payment. For example, to convert an annual interest rate to a monthly interest rate, you divide the annual interest rate by 12 (the number of months in a year). Thus, for example, if the annual interest rate is.12 (i.e., 12 percent), this is equivalent to a monthly interest rate of.01 (i.e., 1 percent). Consequently, if a payment V is to be received at the end of the next month, and the annual interest rate is.12, then the present value of this payment V is V/(1+.01). Similarly, to convert an annual interest rate to a quarterly interest rate, you divide the annual interest rate by 4. Thus, a 12 percent annual interest rate is equivalent to a 3 percent quarterly interest rate. VI. Measuring Interest Rates by Yield to Maturity By definition, the current yield to maturity for a marketed debt instrument is the particular fixed annual interest rate i which, when used to calculate the present value of the debt instrument's future stream of payments to the instrument's holder, yields a present value equal to the current market value of the instrument. Yield to Maturity for Simple Loans: Recall from previous discussion the general form of a simple loan contract: 11

4 Borrower Receives: Principal START MATURITY DATE Lender Principal Receives: + Interest Payment Let N be the maturity in years. If N 1 we can use the simple interest year formula. Thus i= I / [P * N]. If N>1, we need to re-examine the definition of yield to maturity to get the answer. Recall that Principal must equal to present value of future payment of [P+I], thus from equation (5) we can solve for (1+i) N = (P+I) / P. Therefore i = (1+ I / P) 1/N -1 Example of a Simple Loan Contract: A borrower receives a loan on January 1, 2000, in amount $100, and agrees to pay the lender $144 on January 1, Thus, the principal is $100, the maturity is two years, and the interest payment is $44. The yield to maturity for this loan is then i = [$144 / $100] ½ -1 = 0.2, or 20%. Yield to Maturity for Discount Bonds: Recall that discount bond has the same characteristics as the simple loan Receives: Price Pd START MATURITY DATE Lender Receives: Face Value FV Thus the yield to maturity is computed similarly: If N 1 then i = [FV-Pd] / [Pd * N] = ( [FV-Pd] / Pd )* 365 / #of days to mat. (and if N>1 then i = (FV/Pd) 1/N -1 ) Yield to Maturity for Fixed-Payment Loan Contracts: Recall from previous discussion the general form of a fixed-payment loan contract: Borrower Receives: Loan Value LV MATURITY START DATE Lender Fixed Fixed Fixed Receives: Payment FP Payment FP Payment FP Consider a particular fixed-payment loan contract with a loan value LV = $20,000, annual fixed payments FP = $2,000, and a maturity of N = 15 years. What is the yield to maturity for this loan contract? By definition, then, the yield to maturity of this fixed-payment loan contract is the particular fixed annual interest rate i which, when used to calculate the present value of the loan contract, results in a present value that is exactly equal to $20,000, the current value of the loan contract. The payment stream to the lender generated by this loan contract consists of twenty successive yearly fixed payments, each having the nominal value FP=$2,000. Using formula (3), given any year k, k = 1,...,15, and any fixed annual interest rate i, the present value of the particular fixed payment FP = $2,000 received at the end of year n is FP / (1+i) n Given any fixed annual interest rate i, the present value PV( i ) of the loan contract is then given by the sum of all of these separate present value calculations for the fixed payments FP received by the lender (debt instrument holder) at the end of years 1 through 15, i.e., (9) PV(i) = FP/(1+i) + FP/(1+i) FP/(1+i) 15. Since the right hand side is a geometric sequence, it can be simplified as follows: (10) PV(i) = (FP / i) * (1-1/(1+i) n ) Since the current value of the loan contract is $20,000, the desired yield to maturity is then found by solving the following equation for i: $20,000 = PV( i ). Because the present value PV( i ) depends in a rather complicated way on i, the determination of i from formula (10) is not straightforward. To make life easier, tables have been published that can be used to determine yields to maturity for various types of fixed-payment loan contracts once the current value and fixed payments of the loan are known. For example, using such tables, it can be shown that the solution for i is approximately i = 5,5%. That is, the yield to maturity i for a fixed-payment loan contract with a current value of $20000, with annual fixed payments of $2000, and with a maturity of fifteen years, is approximately 5.5%. Yield to Maturity for Coupon Bonds: Recall from previous discussion the basic contractual terms of a coupon bond: Receives: Price Pb MATURITY START /\/\/\ DATE Coupon Coupon... Coupon Lender Payment C Payment C Payment C Receives: + Face Value FV Consider a coupon bond whose purchase price is Pb=$94, whose face value is FV = $100, whose coupon payment is C = $10, and whose maturity is 10 years. By definition, the coupon rate for this bond is equal to C/FV = $10/$100 =.10 (i.e., 10 percent). 12

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