8.3 Coupon Bonds, Current yield, and Yield to Maturity

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1 8.3 Coupon Bonds, Current yield, and Yield to Maturity 8.3.a Coupon Bonds Coupon bond: It makes periodic payments of interest. Coupon payments:periodic payments of interest are called coupons. Coupon rate: $amount of coupon payment is determined as follows. coupon per year = coupon rate Semiannual Coupon Payments face value Japanese and US government bonds pay coupons every 6 months. Let F be face value, c be coupon rate. hen each coupon payment is equal to c 2 F. You receive this amount every 6 months until maturity. On maturity you receive coupon payment plus face value of the bond. Cash inflow will be as follows; bf, bf,... bf + F where b = c 2 coupon rate. Yield to Maturity, YM half the he relation between prices and yields on coupon bonds is more complicated than for pure discount bonds. YM is IRR, internal rate of return for a bond which makes PV of cash inflows from the bond equal to its price. YM is not the same as zero rate. What are the differences? Assume annual compounding. Suppose we have 5-year coupon bond and that its coupon rate is c per year. Coupon is paid once a year. Let P be price of bond. Let y be its YM. It satisfies the following equation; P 5 cf F 1 y y 5 Japanese and US government bonds pay coupons every 6 months. In this case, we use semiannual compounding. Amount of each coupon is c 2 the following. F. Let x be yield to maturity per 6 months. hen x satisfies P 10 bf F 1 x x 10 YM as annual percentage rate is 2x per year; y=2 x. Example: Value of Coupon Bond Suppose you have a coupon bond. Coupon rate is 3%. Maturity is 5 years. Coupon is paid every 6 months. So we apply semiannual compounding. All the interest rates are expressed as APR. Coupon is paid twice a year. Coupon rate is expressed as per year. Each coupon is = 1.5 Q1. What is the value of this bond if yield to maturity, i.e., IRR, is 2%. If face value is not specified, it is 100 dollars or yen. Q2. Draw a graph of value of bond as a function of yield to maturity. Yield to maturity is annual rate Ch08b.nb 1

2 Q1. value of coupon bond Clear v ; Clear c, b, F, v c c 0.03; b 2 ; F 100; y 0.02; x y 2 ; v 10 bf 1 x t F 1 x 10; Print "each coupon payment is ", bf, " dollars. and price of bond is equal to ", v each coupon payment is 1.5 dollars. and price of bond is equal to Value of bond as a function of YM per 6 month is given by the following; 10 bf F 1 x x 10 In principle, interest rate is expressed as annual rate. Let y be YM as annual rate. hen value of coupon bond is given by 10 bf 1 y t 2 F 1 y Q2. value of coupon bond as a function of YM Clear y ; f y : 10 bf Plot f x, x, 0, 0.2, 1 y 2 PlotLabel "value of bond", AxesLabel "YM", "$", ImageSize 220 t 1 F y 2 10 $ value of bond YM 8.3.b Relationships between zero rates, bond price and yield to maturity Yield to maturity, YM, is an internal rate of return, IRR for a bond. Internal rate of return is interest rate such that NPV becomes zero. YM is not equal to zero rate. Zero rate is interest rate which makes price of pure discount bond equal to PV of its face value. Spot rate is another name for zero rate. 8.3.b.1 Coupon bond as a set of pure discount bonds 1. Coupon bond has periodic coupon payments. 2. Each coupon can be considered as a zero coupon, i.e. pure discount bond. 3. Evaluate these coupons as pure discount bonds. 4. Sum their PV's. 5. Set PV equal to price Ch08b.nb 2

3 6. For a given price, find internal rate of return for that bond. his IRR is called "yield to maturity." 8.3.b.2. From zero rate to yield to maturity (1) finding zero rates prices of pure discount bonds zero rates Suppose we have pure discount bonds with maturities of 0.5, 1, 1.5,..., n years. heir face values are F 1, F 2,..., F 2 n. heir prices are P 1, P 2,..., P 2 n. We calculate zero rates, X 1, X 2,..., X 2 n such that P 1 = F1 1 x1, P 2= Here we apply semiannual compounding. (2) apply zero rates on cash flows of coupon bond F2 1 x2 2,..., P 2 n = F 2 n 1 x2n 2 n hold. Apply zero rate on each cash flow of coupon bond. hen we have price of coupon bond. We can interpret coupon bond as a set of pure discount bonds. heir maturities are 0.5, 1, 1.5,..., n years. heir face values are bf, bf,..., bf+f. We observe zero rates in the market. hen we apply these zero rates on the pure discount bond which constitute a coupon bond. Present values are given by the following; PV 1 = bf 1 x1, PV 2 = bf 1 x2 2,..., PV 2 n = bf F 1 x2 2 n Coupon bond is a set of pure discount bonds. Price of the coupon bond is equal to a sum of present values of pure discount bonds. price of coupon bond = bf 1 x1 + bf 1 x (3) finding IRR of coupon bond bf F 1 x2 2 n For a given price of coupon bond, find yield to maturity. Yield to maturity YM is interest rate y which makes net present value of the bond equal to zero; bf 1 y + bf + + bf F - price of coupon bond = 0 1 y 2 1 y 2 n Essence of the relationship is as follows; bf + bf x1 1 x2 2 bf F 1 x2 2 n = bf + bf + + bf F 1 y 1 y 2 1 y 2 n (4) annual percentage rate Government bonds pay coupon twice a year. So semiannual compounding is applied. In principle, interest rates are expressed as annual rates. If percentage is used, they are APR s. decimals and per 6 months. heir APR are 200 x 1..., 200 y. 8.3.b. 3. Current Yield here is another kind of yield, called current yield. Its definition is given by current yield annual coupon current bond price Example on p.228: 10% coupon bond with one year remaining x 1, x 2,..., y are in An example given on page 228 of the text book: Face value is $1, 000 with coupon rate 10 %. Coupon is paid once a year. We apply annual compounding. One year zero rate is 5 % Ch08b.nb 3

4 Since zero rate is 5%, bond price is given by the following; price Price is higher than its face value. Such a bond is premium bond. Current yield of the example on page 228 is his yield ignores capital loss. At maturity, you won't receive principal amount of your investment, which is $ You receive coupon plus face value which is $1100. It is smaller than you paid. Your yield to maturity of one year investment is given by YM coupon Face value price price It is 5%. Here, = is capital loss. 8.3.c. Bond Pricing Principle (1) Definitions: par, premium and discount bonds 0.05 Depending on the relationships between price and face value, bonds are categorized into three groups; par, premium and discount bonds. 1. P= F: par bond 2. P>F: premium bond; 3. P<F: discount bond Each group of bonds has specific relationship between coupon rate and yield to maturity shown as principle 1 to 3. If bond pays coupon twice a year, then variables b and y in the following are meant to be per 6 month value. (2) Principle 1: P=F b = y If a bond' s price equals its face value, then its yield to maturity equals its coupon rate. In other words, if bond is par, then its YM and coupon rate are equal. Although bond was issued at par, the level of interest rate may change later. So you often observe that the price of coupon bond is not equal to the face value. (3) Bond Pricing Principle 2 & 3 If bond's price is higher than its face value, such a bond is called premium bond. If price is lower than face value, such a bond is called discount bond. Principle 2 says P>F b > y. Principle 3 says P<F b < y Principle 2: For premium bonds, YM < coupon rate. Also the other direction holds. Principle 3: For discount bonds, YM > coupon rate. Also the other direction holds. If a coupon bond has a price higher than its face value, its yield to maturity is less than its coupon rate. And vice versa. Reason of these relationship? Ch08b.nb 4

5 Principle 1 P=F b=y his can be shown using the formula of a sum of geometric sequence. P=F b=y his can be shown using graph of PV of coupon bond as a function of YM. Principle 2 and 3 follow from the above proof. (4) Graphical Presentation of Bond Pricing Principles Let g(y) be PV of bond as a function of y, YM. g y bf F, 1 y y where is number of year or half year. If bond pays coupon twice a year and maturity is 5 years later, then =10. Let s see the shape of g(y). par Shape of function for y > -1 (1) If y > -1 then, the first derivative is always negative and that the second derivative is positive. For y > -1. g'(y)<0 and g''(y)>0. (2) In addition, as y -1, g(y) goes to infinity. And as y +, g(y) 0. Such a shape of function g(y) means that, for a given bond price p, an equation g(y) = p has always a solution in the range of y > -1. Also the solution is unique. We have only one solution. part 2 b=y P=F If YM equals to coupon rate, then PV equals to bond price. his can be proved using the formula for a sum of geometric sequences. his means that g(b)=f where c is coupon rate. his equality always holds for any value of b. he result in part1 tells that solution is unique. So the coupon rate b is the only solution for equation g(y)=f. It implies that if bond price equals to F, then YM equals to coupon rate. Graphically, the above results tell that 1. g(y) intersects with horizontal line of F when y=b. Par bond 2. For y > b, then g(y) < F Discount bond 3. For y < b, then g(y)> F Premium bond Example: coupon rate 6%, 5 years to maturity Suppose we have a coupon bond; 6% coupon rate and 5 years to maturity. Coupon is paid every 6 months. So c=0.06, F=100. Coupon is paid 10 times. Let s draw a graph of value of this bond as a function of YM. Let y be YM per 6 months. b is also per 6month; b= c Clear g, b, F,, y ; b ; F 100; 2 5; 2 bf F g y : 1 y y Ch08b.nb 5

6 Clear graph0, hline hline Plo00, y, 0.1, 0.6 ; graph0 Plot g y, y, 0.1, 0.6, ImageSize 220, PlotLabel "Graph0: Value of bond as function of YM", AxesLabel "y", "$" ; Show graph0, hline Value of bond as function of YM $ y Clear y, ans ans NSolve g y F, y ; y y. ans 10 ; Print "par bond yield per 6 months ", y par bond yield per 6 months 0.03 In the above example, principle 1 holds. Principle 1 holds for any remaining years. It means for any remaining year, when y=b, PV of this bond is equal to 100, which is face value. Let's see examples of = 6, 10, 14. Clear, graph1, graph2, graph3 ; 2 3; graph1 Plot g y, y, 0.05, 0.2, ImageSize 240, PlotLabel " 6", PlotStyle Dotted, hick ; Clear ; 2 5; graph2 Plot g y, y, 0.05, 0.2, ImageSize 240, PlotLabel " 10", PlotStyle Dashed ; Clear ; 2 7; graph3 Plot g y, y, 0.05, 0.2, ImageSize 240, PlotLabel " 14" ; Ch08b.nb 6

7 Show graph1, graph2, graph3, hline, PlotLabel " 6, 10, 14, c 0.06", AxesLabel "y", "PV" 160 PV 6, 10, 14, c y 8.5 Why Yields for the Same Maturity May Differ Often bonds with the same maturity have different YM. here are following reasons. 1. effect of coupon rate 2. effect of default risk 3. callability and convertibility 1.Effect of Coupon Rates In the following, we show that two bonds with the same maturity but with different coupon rates have the different YM. Such a situation happens, when zero rate yield curve is not flat. Suppose that you find two bonds with the same maturity. hey have different coupon rates. Does the same maturity means the same YM? Not necessarily. Let x t be zero rate / spot rate. Bond prices are determined in the following way: p1 p2 b 1 F F 1 x t x t... 1 b 2 F F 1 x t x t... 2 YMs are IRR which satisfy (3) and (4). p1 p2 b 1 F F 1 y 1 y b 2 F F 1 y t 2 1 y Question: Is y 1 equal to y 2? Answer: hey are different except for by chance. If zero rate yield curve is flat, they y1 and y2 become equal. hat zero rate yield curve is flat means that x t s are the same for all maturities. If x t s are the same for all t in (1), then value of y which is solution to equations (3) is the same as x t. (Graph0 above tells that it cannot have different solution.) Also (2) and (4) have the same result. hen, because zero rates are the same for (1) and (2), y 1 and y 2 are equal. If zero rate yield curve is flat, then YM of bonds of the same maturity are equal, even if coupon rates are different. Interpretation to convince yourself: Ch08b.nb 7

8 IRR is a kind of weighted average of "rates of return per period". YM can be interpreted such a kind of weighted average of zero rates; x 1,x 2,..., x. Different coupon rates mean different weights for them. Different weights result in different average values y 1 and y 2. Example of the same maturity with different YM Semiannual compounding is assumed. Suppose we have two coupon bonds with 3 years to maturity. Coupon rates are c1 = 0.06 and c2 = Zero rates are as shown below. year zero rate inapr Clear p1, p2,, b1, b2, F, z, x 6; b1 0.03; b2 0.06; F 100; z "period" "zero rate per 6m" ; x able z 2, k 1, k, 1, 6 ; PV as Sum of Pure Discount Bonds We evaluate each cash inflow from the coupon bond as a separate pure discount bond. o do so, we apply zero rate which corresponds to each maturity; xt, t=1,..., 6. p1 p2 b1f 1 x t b2f 1 x t t t F 1 x F 1 x ; ; Print "p1 ", p1, " and p2 ", p2 p and p Yield to Maturity Yield to maturity is internal rate of return. his is a single interest rate which makes NPV equal zero. Clear y1, ans1 ans1 NSolve y1 y1. ans1 6 ; b1f F p1, y1 ; 1 y1 y1 Clear y2, ans2 ans2 NSolve Print "y1 ", y1, " and b2f F p2, y2 ; y2 y2. ans2 6 ; 1 y2 y2 y2 ", y2 y and y hus, two bonds have the same maturity but have the different yield to maturity Ch08b.nb 8

2. Effect of Default Risk he riskier, the higher interest rate. 3. Callability and Convertibility 1. Callability: he issuer of the bond has the right to redeem the bond before the final maturity date.

9 2. Effect of Default Risk he riskier, the higher interest rate. 3. Callability and Convertibility 1. Callability: he issuer of the bond has the right to redeem the bond before the final maturity date. 2. Convertibility: he holder of a bond issued by a corporation has the right to convert the bond into a prespecified number of share of common stock. Do they increase or decrease bond prices? 1. Suppose that after the callable bond was issued, interest rates become lower. hen Issuer may want to pay back the original debt and borrow again but at lower rate. Holders of bond have to reinvest at lower rate if bond is paid out. Callability is advantageous to the issuer and disadvantageous to buyers of bond. Price of callable bond becomes lower than straight bond. 2. If you anticipate that stock price will rise, then you will be happy to buy convertible bond. Convertible bond price would be higher. Addendum to Chapter 8, Receipt Products Began in 1982 ime Slip You think you fell asleep after you study Financial Economics. Suppose thatyoufindyourselfintheearly1980's whenyouwakeup. Not knowing how to return to 2015, you look for a job. You start working in a security company. In early 1980 s interest rates are historically high. Investors wished such a high rate of return to continue many more years. Receipt Products known as receipt product he separation of the coupons from the bond principal began in Once separated, each coupon and principal payment becomes separate pure discount bond. Here comes a customer Here is a married couple who want to make sure their grand child can afford to study abroad in hey pick up a pamphlet in your office. You try to sell them 30-year pure discount bonds which your company creates. A man asks you many questions about risk etc. You notice they have the same last name as you. Anyway, you explain to them how good the bonds are. aking 5-year Bond as Example You take 1 to 2.5 year pure discount bonds as examples to convince him. Your company creates 1 to 2.5- year pure discount bonds from 2.5-year coupon bonds. he 2.5-year coupon bond pays 20% of face Ch08b.nb 9

10 value per year. Its face value is $100. So each coupon is $10. From a single 2.5-year coupon bond, 5 pure discount bonds are created. Your company calls these bonds "cat1", "cat2",... table A pure discount maturity bond from0.5to2.5years nameofbond cat1 cat2 cat3 cat4 cat5 face value $10 $10 $10 $10 $110 price p1 p2 p3 p4 p5 Your colleague calculated zero rates from various bonds traded in the market. Semiannual compounding is applied here. Zero rates are shown in table B. current situation maturity in6months table B zero rate per6month x1 x2 x3 x4 x5 x in decimal notation Question 1. Draw zero rate yield curve, which shows relationships between zero rate and time to maturity Question 2. What should be prices of cat1 to cat5? Question 3. What should be price of 2.5-year coupon bond with 20% coupons? Risk of Price Change he man who has the last name same as yours asks, ---"Is cat5 safe?" You answer to him. -- Yes. When we issue pure discount bonds, we buy 2.5-year reasury bond at the same time. When we receive coupon and principal of 2.5-year reasury bond, we pay you back cat5. Cash inflow we receive becomes payment to you. here is no possibility of default. --- hat's good. I won't loose. --- However it is possible to loose if you try to sell it before the maturity. If you try to sell cat5 before maturity, it is possible you loose. Question 4. Suppose that the man buys cat5 today. Also suppose that in the next year he tries to sell it. Zero rates are as shown in table C next year. Calculate his holding period rate of return over one year. Be aware that maturity of his bond becomes one year shorter Ch08b.nb 10

11 table C valuesforthenextyear maturity in6month zero rate x1 x2 x3 x4 x5 in decimal notation Question 5. Coupon rate of par bond Suppose we have 2.5-year coupon bond which pays coupon twice a year. Suppose zero rates are as shown in table C. What should be coupon rate if this 2.5-year bond is par. Question 6. Coupon rate of par bond; proof Let F be face value of coupon bond. Let c be coupon rate so that each coupon payment is cf. Prove that if yield to maturity is equal to coupon rate, then price P is equal to face value F. Hint: Use formula of a sum of geometric sequence. Homework No.6, due a pm on June 6th, Monday Please submit your answer to the box on the door of room 537 by 1 pm on June 6th. You can submit your answer before the weekend. Answers to Homework No.6 will be downloadable Monday afternoon. Q1. Problem 8.3, p240, Hint: annual compounding is applied. Q2. Problem 8.5, p240, Q3. Suppose you observe the following prices of pure discount bonds. a.what are zero rates? Biannual compounding apply. b.suppose that there is a 2-year bond with 10% coupon. Find its theoretical price and its yield to maturity. Maturity price face value year Q4: here is 3% coupon bond with 6 years to maturity. Coupon is paid every 6 months. Bond price is 100 dollars. Find YM. Q5. Problem 8.8, p241, Hint1: Definition of current yield is given on p Hint2: Consider relationship between price, F, c, YM. Assume F = 100. About Midterm Exam Midterm exam will be on June 8th Ch08b.nb 11

8. Valuation of Known Cash Flows: Bonds

8. Valuation of Known Cash Flows: Bonds 8. Using PV Factors to Value Known Cash Flows We use interest rate to value known cash flow. Which discount rate to use? In practice, you do not usually know which