VALUE %. $100, (=MATURITY

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1 NOTES H IX. How to Read Financial Bond Pages Understanding of the previously discussed interest rate measures will permit you to make sense out of the tables found in the financial sections of newspapers and magazines that report on U.S. Treasury debt instruments and corporate bonds traded on stock exchanges. He illustrates his discussion using the financial pages from the January 26, 2 issue of the Wall Street Journal. This section reviews the main points of textbook discussion, using the financial pages from the February 17, 1999 issue of the New York Times to illustrate and elaborate key points. As will be seen, the form in which financial information is reported in the New York Times is essentially the same as in the Wall Street Journal, with only minor notational differences. A. Treasury Bonds and Notes: Treasury bonds (T-bonds) are coupon bonds with a maturity greater than ten years, and Treasury notes (T-notes) are coupon bonds with a maturity of between one and ten years. As in the Wall Street Journal, the New York Times provides a single table reporting on T-bonds and T-notes because both have the same structure. Below is a sample listing from the T-bonds and T-notes table appearing in the Wall Street Journal (September 16, 22, handout) which reports information for the previous trading day, September 15, 22: Rate Maturity Bid Ask Chg Ask. Yld Aug 3 18:15 18: Aug 3 n 11:25 11: Notational Note: For expositional simplicity, T-bonds and T-notes will hereafter be lumped together and simply referred to as "bonds." The first column (Rate) identifies a bond's annual coupon rate, i.e., the annual coupon payment as a percentage of face value. Usually this annual coupon payment is paid in two equal semiannual installments. The second column (Month) of the sample listing identifies the month and year that a bond matures. A footnote may next be provided to indicate that a bond has some special feature. For example: the letter "n" denotes "T-note". The third and fourth columns (Bid, Ask) provide information about a bond's bid and asked prices, which by convention are quoted as a percentage per $1 of face value (so that 1 = face value) with fractions in 32s. Unlike the Wall Street Journal, which uses a colon to indicate fractional values in 32s (e.g., 12:8 = 12 8/32), the New York Times uses a decimal point (e.g., 12.8 = 12 8/32). More precisely, the bid price quoted for a bond is the approximate market price offered by prospective buyers of the bond on the trading day in question, so it indicates approximately how much you would have received if you had sold the bond on that day. In contrast, the asked price quoted for a bond is the approximate market price demanded by prospective sellers of the bond on the trading day in question, so it indicates approximately how much you would have had to pay to purchase the bond on that day. The asked price minus the bid price -- referred to as the bid-ask spread -- reflects the profit margin of the bond dealers who handle trades in this bond. Hence, for obvious reasons, the asked price always exceeds the bid price. The fifth column (Chg) indicates the change in the BID price from the previous trading day's quotation. The sixth and final column (Yld) provides the yield to maturity on the bond using the currently quoted ASKED price as the purchase price. The asked price is used because the yield to maturity is most relevant for a person who intends to purchase and hold the bond and thus earn the yield. IMPORTANT: Both bid and ask prices are clean prices. That is, they do not include the accrued interest. See the following example based on quotes above. This ant the following example are only for illustration, this will not be required in tests! Assume that the T-bond has COUPON RATE equal to %. Its FACE VALUE of $1, will be paid on August 13 th, 23 (=MATURITY DATE). Note that T-bonds have semi-annual coupons. 1/ What is accrued interest on September 15 th, 22 First notice that during the remaining 332 days this bond will pay total interest of %*$1,=$11,125 and the principal of $1,. There will be two payments. In 149 days (March 13 th, 23) owner of the bond will receive first of the two coupon payments of $5, days later (August 13 th, 23) she will receive $15,556.5 of second coupon and principal. On September 15 th, it was 33 days from the last payment of coupon on August 13 th. Thus, accrued interest is 33/(33+149) * $5,562.5=$1,8.6. 2/ How much would you pay on September 15 th, 22 to buy this bond You will have to pay both ask price and the accrued interest. Therefore you would pay $1, * (18+16/32) + $1,8.6 = $19, / What was this bond s ask yield to maturity on September 15 th, 22 Ask yield to maturity i, satisfies: Present Value of future payments = Purchasing price. That is, purchasing price of $19,58.6 equals to present value of 5, , i*(149/365) 1+i*(332/365) This is true only for i=.166=1.66%. Note: The treasury note with coupon rate has a maturity date August 29 th. It also pays interest twice a year (in 165 and 348 days). Thus, its price is $11,98.85 ($11,812.5 of clean price + $ of accrued interest), its ask yield to maturity is 1.7%. Note on Treasury Zero Coupon Issues: Coupon stripping is the act of removing the individual coupon payments from a coupon bond and treating each payment as a separate zero coupon bond. The remaining face-only bond is then also in effect a 15

2 zero coupon bond. For example, a 2-year bond with a face value of $1, and an annual coupon payment of $1, could be stripped into 21 separate zerocoupon instruments: namely, 2 "interest strips" consisting of the 2 annual coupon payments of $1,, each due on the specified annual coupon payment date; and one "principal strip" consisting of an instrument having a face value of $1, due in 2 years. Merrill Lynch began the market 1 1 for stripped securities in The U.S. Treasury introduced stripping of its coupon bond issues in February 1985, referring to the resulting zero-coupon securities as STRIPs (Separate Trading of Registered Interest and Principal of Securities). U.S. Treasury strips are mentioned in the explanatory notes for the Wall Street Journal bonds. B. Treasury bills: Treasury bills (T-bills) are discount bonds with a maturity of one year or less. Since a coupon rate is zero, they are identified solely by their maturity date. Below is a sample listing from the T-bills table appearing in the WSJ (September 16, 22) which reports information for the previous trading day, September 15, 22: Date Days to Mat. Bid Ask Chg Yield Jan Jan The first column (Date) gives the month, day, and year of the maturity date. The third column (Bid) gives the discount yield in percentage terms using as the purchase price Pd the BID price, i.e., the price offered by prospective buyers. The third column (Ask) gives the discount yield in percentage terms using as the purchase price Pd the ASKED price, i.e., the price demanded by prospective sellers. Recall that the discount yield varies inversely with the purchase price Pd. It follows that, for T-bill issues, the bid discount yield reported in the Bid column is always greater than the asked discount yield reported in the Ask column, indicating that the bid price is less than the asked price. The fourth column (Chg) reports the change in the asked discount yield from the previous trading day measured in terms of basis points, which are hundredths of a percentage point (e.g., -.4 means the asked discount yield has fallen in percentage terms by 4 basis points). The fifth and final column (Yield) provides the yield to maturity using the current ASKED price as the current value. Can you verify that the presented yields to maturity is computed correctly Assume that F=$1. In case of the first T-bill, idb=1.64% and there are 114 days to maturity. Thus from equation (6) we can find price of the bill Pd: $1 - Pd % = * > Pd = $99.48 $1 114 Yield to maturity equals to i, where $1 Pd= i*(114/365) thus, $1 - Pd 365 i = * =.167 = 1.67%. Pd 114 C. Corporate Bonds Traded on Exchanges: A majority of bonds, and all municipal or tax-exempt bonds, are not listed on exchanges; rather, they are traded over-the-counter. However, the New York Stock Exchange (NYSE), and to a much less extent the American Stock Exchange (AMEX), do list various coupon bonds issued by corporations with strong credit ratings. Below is a sample listing from the NYSE corporate bond table appearing in the New York Times (February 17, 1999, page C17) which reports information for the previous trading day, February 16, 1999: Company Coupon Mat. Cur.Yld. Vol. Price Chg. Rate ATT 5 1/ /8-1/8 ARetire 5 3/4 2 cv / /2 The first column (Company) shows the issuing company, the second column gives the original coupon rate, and the third column gives the last two digits of the maturity year. The fourth column reports the annual current yield (Cur. Yld.). In some cases, a footnote may instead be inserted to call attention to a special feature of the bond; for example, the letters "cv" in the above table denote "convertible into stock under special conditions". The remaining three columns report the number of bonds traded for the day measured in $1 face value (Vol.), the bond's closing price for the day expressed as a percentage of face value with 1 equaling face value (Price), and the difference between the current trading day's closing price and the previous trading day's closing price (Chg). X. Interest Rates vs. Return Rates Given any asset A held over any given time period T, the return to A over the holding period T is, by definition: the sum of all payments (rents, coupon payments, dividends, etc.) generated by A during period T, assumed paid out at the end of the period, 16

3 NOTES I P (1) P (1) R ate of R etu rn PLUS the capital gain (+) or loss (-) in the market value of A over period T, measured as the market value of A at the end of period T minus the market value of A at the beginning of period T. C C C P (2) C + P (2) Y ield to M a tu rity P V = P (1) The return rate on asset A over the holding period T is then defined to be the return on A over period T divided by the market value of A at the beginning of period T. More precisely, suppose that an asset A is held over a time period that starts at some time t and ends at time t+1. Let the market value of A at time t be denoted by P(t) and the market value of A at time t+1 be denoted by P(t+1). Finally, let V(t,t+1) denote the sum of all payments accruing to the holder of asset A from t to t+1, assumed to be paid out at time t+1. Then, by definition, the return rate on asset A from t to t+1 is given by the following formula: (24) Return Rate on V(t,t+1) + P(t+1) - P(t) Asset A From = time t to t+1 P(t) C V(t,t+1) P(t+1) - P(t) = P(t) P(t) = payments + Capital Gain (if +) received as or Loss (if -) as percentage percentage of P(t) of P(t) Formula (24) holds for any asset A, whether physical or financial. In particular, it holds for bonds. The question then arises: For bonds, what is the connection between C C + F V C + F V the return rate defined by formula (24) and the interest rate on the bond defined by yield to maturity, current yield, or discount yield The return rate on a bond is not necessarily equal to the interest rate on that bond, whether defined by yield to maturity, the current yield, or the discount yield. The reason for this is that the return rate calculated for a particular holding period takes into account any capital gains or losses that occur during this holding period, in addition to payments received during the holding period. In contrast, the current yield ignores capital gains and losses altogether, and the yield to maturity and the discount yield only take into account the overall anticipated capital gain or loss that is incurred when the bond is held to maturity (as measured by the difference between the final face value payment and the initial purchase price). EXAMPLE A: COUPON BONDS Suppose you purchase a coupon bond at time t at a price P(t) with coupon payment C and face value F, you receive a coupon payment C at time t+1, and you also sell the coupon bond in a secondary market at a price P(t+1) at time t+1. By definition, the current yield that you receive on this coupon bond during the holding period from t to t+1 is given by (25) ic(t) = C/ P(t) Also, the percentage capital gain or loss you incur on the coupon bond during the holding period from t to t+1, denoted by g(t,t+1), is given by P(t+1) - P(t) (26) g(t,t+1) = P(t) It then follows from definition (9) that the return rate on the coupon bond from t to t+1 can be expressed as C + P(t+1) - P(t) (27) = ic(t) + g(t,t+1). P(t) Clearly the return rate (24) coincides with the current yield ic(t) if (28) P(t) = P(t+1). Condition (28) implies that there are no capital gains or losses on the coupon bond during the holding period from t to t+1, i.e., g(t,t+1) =. Conversely, if condition (28) fails to hold, then the return rate (27) does not coincide with the current yield ic(t). Thus, condition (28) is both necessary and sufficient for the return rate (27) from t to t+1 to equal the current yield ic(t). That is: if and only if Return Rate = ic(t) < > P(t) = P(t+1) From t To t+1 Now suppose, instead, that Time Period from t to t+1 = Bond Maturity Period: if and only if Return Rate = i(t) < > Period From t to t+1 From t To t+1 = Maturity Period 17

4 EXAMPLE B: DISCOUNT BOND For a discount bond with a purchase price Pd and a face value F, the return rate (9) over any holding period t to t+1 reduces to P(t+1) - Pd (29) Pd Recalling definition for the discount yield idb, it is seen that the return rate will generally differ from idb except in the degenerate case Pd = P(t+1) = F when both are zero. In summary, then, only under special conditions will the return rate for a bond over a given holding period coincide with the yield to maturity, the current yield, or the discount yield. XI. Real vs. Nominal Interest Rates The interest rate measures examined to date have all been "nominal" in the sense that they have not been adjusted for expected changes in prices. What actually concerns a "rational" saver considering the purchase of a debt instrument is not the nominal payment stream he or she expects to earn in future periods but rather the command over purchasing power that this nominal payment stream is expected to entail. This purchasing power depends on the behavior of prices. Let infe(t) denote the expected inflation rate at time t, and let i(t) denote the (nominal) interest rate for some debt instrument at time t. Then the real interest rate associated with i(t) is defined by the following "Fisher equation:" (3) ir(t) = i(t) - inf e (t). That is, the real interest rate is the nominal interest rate minus the expected inflation rate. Note: As explained by Mishkin, the real interest rate defined by (16) is more precisely called the ex ante real interest rate because it adjusts for expected changes in the price level. If the expected inflation rate in (16) is replaced by the actual inflation rate, one obtains the ex post real interest rate. Real interest rates provide a more accurate measure of the true costs of borrowing and the true gains from lending than nominal interest rates, and hence provide a better indicator of the incentives to borrow and lend. In particular, for any given nominal interest rate i on a debt instrument D, the incentive to borrow (issue D) will be higher if the real interest rate associated with i is lower (i.e., the expected inflation rate is higher). This is so since a higher expected inflation rate means the borrower (issuer of D) can expect to pay off his future nominal debt obligations using cheaper dollars than he borrowed. For this same reason, the incentive to lend (purchase D) will be lower if the real interest rate associated with i is lower. A similar distinction is made between the (nominal) return rate defined by (9), which has not been adjusted for expected changes in prices, and the "real return rate" which is subject to such adjustment. More precisely, the real return rate on any asset A over any holding period from t to t+1 is defined to be the (nominal) return rate (9) minus the expected inflation rate inf e (t). TABLE One-Year Returns on Bonds When Interest Rates Rise from 1% to (1) (2) (3) (4) (5) (6) Years to Initial Initial Price Rate of Rate of maturity current price next capital Return when bond yield year gain (2+5) is purchased (%) ($) ($) (%) (%) XII. Interest Rate Risk As previously seen, at any time t, the yield to maturity i(t) for a bond with a maturity date greater than t moves inversely with its current acquisition price P(t) -- that is, if $ 1, $ 1, infinite % $ 1 + $1, 1% P(2) $ 1 + P(2) $1 one goes up, the other goes down. $ 1 + $ 1, $ 1, $ P(1) 1, $ 1 + P(2) A fall in the price of an already held bond signals a capital loss to the holder. Consequently, the net effect of an increase in the yield to maturity for an already held bond can be a decrease in the return rate to its holder. P(2) 1% 1% $ 1 + P(2) P(2) $ 1 + $ 1, $1 $1 $ 1 + $ 1 18

5 NOTES J The uncertainty regarding return rate that bond holders face due to possible changes in yield to maturity is called interest rate risk. Table illustrates interest rate risk for bonds of different maturities, each with a coupon payment of $1 and a face value of $1. This illustration is worth reviewing with some care. First note that, for each bond in the Table, the initial yield to maturity i(1) for year 1 ("this year") is equal to the initial current yield ic(1) = 1 percent for year 1 because the initial price of the bond is set at its face value of $1. However, by assumption, the yield to maturity i(2) for year 2 ("next year") increases to 2 percent. The coupon bond listed in the Table with a 1-year maturity has a price P(2) in year 2 that is fixed (by contract) at the bond's face value, $1. For all other listed coupon bonds, however, their maturities exceed one year. Consequently, when i(2) increases, their price P(2) in year 2 decreases to some value smaller than their original price P(1) = $1 in year 1 and hence smaller than the bond's face value of $1. For example, for the coupon bond with a 3-year maturity, P(2) = $53. The return rate from year 1 to year 2 for each of the coupon bonds in Table 2 is given by the sum of the current yield ic(1) = C/P(1) for year 1 and the capital loss g(1,2) = [P(2)-P(1)]/P(1) from year 1 to year 2. Consequently, except for the bond with a one-year maturity, these return rates are smaller than they would have been without the increase in the yield to maturity in year 2. Indeed, for the coupon bond with a 3-year maturity, the capital loss g(1,2) is so large (-49.7 percent) that it overwhelms the 1 percent initial current yield ic(1) = 1 percent, resulting in a negative return rate of percent from year t=1 to t=2. More precisely, examining the return rates in column (6) of the Table as the maturity is decreased from 3 years to 1 year, it is seen that the coupon bonds with longer maturities experience a greater decline in their return rates when the yield to maturity i(2) increases. This is due to the fact that this increase in i(2) results in a smaller decline in the price P(2) for coupon bonds with smaller maturities and hence a smaller capital loss. [To see why, consider the formula Pb = PV(i) from which the yield to maturity i is determined.] Indeed, for coupon bonds with a one-year maturity, P(2) remains fixed at the face value $1. Increase in yield to maturity from year 1 to year 2 / \ i(1)=ic(1) i(2) (3) /\/\/\-- Year 1 2 N P(1) P(2) Maturity Date (N > 2) \ / Capital loss from t=1 to t=2 An important implication of this illustration is that the return rates of bonds with longer-term maturities respond more dramatically to changes in the yield to maturity than bonds with shorter-term maturities. That is, longer-term bonds are more subject to interest rate risk. This is one reason why investment in longer-term bonds is considered more risky than investment in shorter-term bonds. XIII. More Basic Concepts and Key Issues Current yield Discount yield (Nominal) return rate Real interest rate Real return rate Consol bond (perpentuity) Capital gain or loss Interest rate risk Expected inflation rate Calculating the current yield for a consol bond For a coupon bond, how does its maturity affect the relationship between its current yield and its yield to maturity What is the relationship between its current yield, its coupon rate, its purchase price, and its face value What is the relationship between its current yield, its yield to maturity, its purchase price, and its face value What is the relationship between its current yield and its purchase price, given any fixed level for its coupon payment For a discount bond, all else remaining the same, why do its discount yield and its yield to maturity always move together How to read financial bond pages for information on Treasury bonds and notes, Treasury bills, and corporate bonds traded on stock exchanges. Why is the return rate on a bond not necessarily equal to its interest rate How does the maturity of a bond affect its interest rate risk Why do bonds with long maturities expose bond holders to greater interest rate risk than bonds with shorter maturities What is the relationship between real and nominal interest rates Why do real interest rates provide a more accurate measure of the true costs of borrowing and the true gains from lending than nominal interest rates Why do real return rates provide a more accurate measure of the true gains or losses from holding an asset than nominal return rates 19

6 XIV. Practice Questions Answers for questions below: 1D 2A 3B 4 22% 5 6D 7B 8C 9C 1 C 11D 12D Q1. Which of the following statements is/are true in general for FIXED PAYMENT loans A. At maturity the borrower makes one fixed payment, equal to face value. B. The borrower makes only one fixed payment, at maturity, and this payment combines interest and principal repayment. C. The borrower makes the same fixed payment in every payment period until maturity, where the payments consist entirely of principal repayments. D. The borrower makes the same fixed payment in every payment period until maturity, where the payments consist of both interest and principal. Q2. Which of the following statements is/are true in general for COUPON BONDS A. The issuer makes a fixed coupon payment in every payment period during the life of the bond, plus a face value payment at maturity. B. The issuer makes a fixed coupon payment in every payment period during the life of the bond, where the present value of these cumulated payments equals the face value of the bond. C. Treasury bills are examples of coupon bonds. D. Only A and C of the above E. Only B and C of the above Q3. The COUPON RATE on a coupon bond with a purchase price of $25, a $3 face value, annual coupon payments of $125, and a 4-year maturity is A. the coupon payment $125 divided by the purchase price $25. B. the coupon payment $125 divided by the face value $3. C. the average coupon payment per year, which here is $125. D. total coupon payments ($5) divided by the purchase price $25. Q4. Assume the following simple loan contract: Principal=5, Interest payment = 22, Maturity is 2 years (maturity date is 2 years from today). What is the SIMPLE (annual) INTEREST RATE Q5. In the example from question Q4, what is the yield to maturity i A. 1% B. 11% C. D.22% Q6. Letting "*" denote multiplication, if the annual interest rate is 5 percent, then the PRESENT VALUE of a payment stream ($5,$,$,$7) with $5 to be received at the end of the FIRST year, $ to be received at the end of the SECOND and THIRD years, and $7 to be received at the end of the FOURTH year is given by A. $5/(1.5) + $7/(1.2) B. $5*(1.5) + $7*(1.5) 4 C. [$5 + $7]/(1.2) D. $5/(1.5) + $7/(1.5) 4 Q7. The (ANNUAL) YIELD TO MATURITY i on a coupon bond with a purchase price $65, a face value $75, a 2-year coupon payment stream ($5,$5), and a 2-year maturity is calculated as follows: i equals the annual interest rate that, when used to calculate the present value of the income stream, results in a present value equal to. A. ($5,$5), $65. B. ($5,$8), $65. C. ($5,$5), $75. D. ($5,$8), $75. Q8. Which of the following $6 face-value securities has the HIGHEST yield to maturity A coupon bond with a coupon rate of percent that sells for. A. 5 ; $6, B. 1 ; $6, C. 15 ; $6, D. 15 ; $6,2 Q9. The current yield on a coupon bond with a $7 face value, a 5 percent coupon rate, an 8 year maturity, and a current purchase price of $35 is A. 5 % B. 8 % C. 1 % D. 16 % E. 2 % Q1. Which of the following statements is/are FALSE for the current yield ic of a coupon bond with coupon payment C, face value F, and maturity N. A. For a consol bond, the ic equals the yield to maturity. B. For fixed C and F, the ic is a better approximation for the yield to maturity the greater is the bond's time to maturity N. C. For fixed C, F, and N, the ic is a better approximation for the yield to maturity the more the bond's purchase price exceeds the face value F. D. all of the above are false statements E. only A and B are false statements Q11. Consider a coupon bond that has an annual coupon payment C=$1, a face value F=$3,, and a maturity date January 1, 28. Suppose you BUY this bond on January 1, 23 for Pb=$25 and you SELL it on January 1, 24 for $2. Which of the following statements is/are TRUE for this bond: A. Your (annual) current yield on this bond from 1/1/23 to 1/1/24 is equal to C=$1 divided by the purchase price Pb=$25. B. Your return rate on this bond from 1/1/23 to 1/1/24 can be expressed as the sum of the current yield and the rate of your capital gain or loss. C. Your return rate on this bond from 1/1/23 to 1/1/24 is LESS than the current yield on the bond. D. All of the above are true. E. Only A and B are true. Q12 Suppose a consol bond pays $1 at 11:59 P.M. on December 31 of each year. Suppose you purchased the consol bond for $1 at midnight on December 31, 2, and you sold it for $19 at midnight on December 31, 21. Suppose the inflation rate during 21 was 3 percent. Then your NOMINAL return rate on the consol bond for 21 was and your REAL return rate on the consol bond for 21 was. A. 1 %; -2 % B. 1 %; 4 % C. 9 %; 6 % D. 1 %; 7 %t E. 1 %; 13 % 2

7 NOTES K Summary of major interest rates on the market Government securities Municipal Issues (notice that some are tax free and so their yield is adjusted by 31% tax bracket) Private Bonds 9