8.3 Coupon Bonds, Current yield, and Yield to Maturity

Transcription

1 inancial Economics, Spring Coupon Bonds, Current yield, and Yield to Maturity 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 may not be 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. rom zero rates to YM. 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 bond. 4. Sum their PV's. 5. Set it equal to price. 6. or a given price, find internal rate of return for that bond. his IRR is "yield to maturity." Example: 3-year Bond with 0% Coupon Suppose that, in the bond market, we observe the following set of spot rate, i.e., zero rates; numerical example in 8.2. We apply annual compounding. Pure Discount Bond Maturity price per $ of face value spot rate as APR year years years able. Suppose also that there is a 3-year bond with 0% coupon. We like to find its theoretical price and its yield to maturity. ace value is $,000. Coupon is paid once a year. Each coupon is $00. We apply annual compounding. Price is equal to sum of PV' s of coupons and face value here are three cash inflows from the coupon bond. time of receipt year 2 year 3 year dollar amount We evaluate these cash flows as pure discount bonds. If we apply zero rates, then we calculate the following equation; 00 PV Ch08b.nb

2 inancial Economics, Spring 20 In[950]:= 00 ClearPV; PV ; Print "PV", PV PV Alternatively, we can take a short cut. able implies the following relationships. price of pure discount bond and zero rate We calculate the following; PV = ( ) 0.8. In[952]:= ClearPV; Print"PV ", PV 063. We calculate the following; PV = ( ) 0.8. hen, PV= 063. Due to the rounding errors, figures do not coincide. Interest rates should be more precise. Let's use $063 in this example. Set price equal to PV. Bond price is $, dollars. In[953]:= Let's check precise interest rates Clearx, x2, x3 NSolve0.95 x,x NSolve0.88,x2 x2 2 NSolve0.80,x3 x3 3 Out[954]= x Out[955]= x , x Out[956]= x , x , x Yield to Maturity YM is the value of single discount rate y which makes NPV equal to zero. In other words, YM is a solution for the following equation. Let y be APR for annual compounding. price 00 y 00 y y Ch08b.nb 2

3 inancial Economics, Spring 20 In[957]:= Clearprice, y, ans; price 063; ans NSolveprice 00 y ,y; y 2 y 3 y y. ans3; Print"Yield to maturity is equal to ", 00 y, "." Yield to maturity is equal to Current Yield Example 0% coupon bond with one year remaining An example given on page 228 of the text book: ace value is $, 000. Coupon rate 0 %. Coupon is paid once a year. We apply annual compounding. One year zero rate is 5 %. 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. here is another kind of "yield", called current yield. Its definition is given by current yield Current yield of the above example is coupon current bond price 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 $00. It is smaller than you paid. Your yield to maturity of one year investment is given by coupon ace value price YM 0.05 price It is 5%. Here, = is capital loss Bond Pricing Principle Depending on the relationships between price and face value, bonds are categorized into three groups; par, premium and discount bonds.. par bond : P= ; 2. P>: premium bond; 3. P<: discount bond Each group of bonds has specific relationship between coupon rate and yield to maturity as follows; Principle : P= fi c = y If a bond' s price equals its face value, then its yield to maturity equals its coupon rate. Coupon bonds with a market price equal to their face value are called par bonds. So YM of par bond equal to its coupon rate. 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. 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> fl c > y. Principle 3 says P< fl c < y 20-05Ch08b.nb 3

4 inancial Economics, Spring 20 Principle 2: or premium bonds, YM < coupon rate Principle 3: or discount bonds, YM > coupon rate 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? Principle P= ïc=y his can be shown using the formula of sum of geometric sequence. P=ì c=y his can be shown using graph of value of cash flow as a function of YM. Principle 2 and 3 follow from the above proof. Graphical Presentation of Bond Pricing Principles Let g(y) be PV of bond as a function of YM y. part Shape of function for y>- If y > - then, the first derivative is always negative and that the second derivative is positive. or y > -. g'(y)<0 and g''(y)>0. In addition, as yø -, g(y) goes to infinity. And as y Ø +, g(y) Ø 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 > -. Also there exits only one solution because of the shape of g(y). part 2 c=y fl P= If YM equals to coupon rate, then PV equals to bond price. his can be proved using a formula of sum of geometric sequences. his means that g(c)= where c is coupon rate. his equality always holds for any value of c. Result of part tells that solution is unique. So the coupon rate c is the only solution for equation g(y)=. It implies that if bond price equals to, then YM equals to coupon rate. Graphically, the above results tell that. g(y) intersects with horizontal line of when y=c. Par bond 2. or y > c, then g(y) < Discount bond 3. or y < c, then g(y)> Premium bond 20-05Ch08b.nb 4

5 inancial Economics, Spring 20 Example: coupon rate 6%, 5 years to maturity In[96]:= Clearg, x, c,,, y; c 0.06; 00; 5; gy_ : c y t y t Plotgy, y, 0.05,, ImageSize 250, PlotLabel "Shape of gy where g'y0 and g''y0", AxesLabel y, present value Shape of gy where g'y<0 and g''y>0 present value 50 Out[963]= y Principle holds for any remaining years. Let's see examples for = 3, 5, 7, In[964]:= Clear, graph; 3; graph3 Plotgy, y, 0.055, 0.065, ImageSize 200, PlotLabel "3", PlotStyle Dotted; Clear; 5; graph5 Plotgy, y, 0.055, 0.065, ImageSize 200, PlotLabel "5"; Clear; 7; graph7 Plotgy, y, 0.055, 0.065, ImageSize 200, PlotLabel "7", PlotStyle hick; Showgraph3, graph5, graph7, PlotLabel 3, 5, 7, AxesLabel y, P P , 5, 7 Out[970]= y 8.5 Why Yields for the Same Maturity May Differ Often bonds with the same maturity have different YM. here are following reasons.. effect of coupon rate 2. effect of default risk 3. callability and convertibility 20-05Ch08b.nb 5

6 inancial Economics, Spring 20 Effect of Coupon Rates In the following, we show that two bonds with the same maturity but with different coupon rate have the different YM. Such a situation happens, when zero rate yield curve is not flat. wo bonds have the same maturity but different coupon rates. he same maturity means the same YM? Let S t be spot rates. Bond prices are determined in the following way: p t c S t t S t... () p2 t c 2 S t t S t... (2) YM is IRR which satisfies p t c y t y... (3) p2 t c2 y2 t y2... (4) Question: "Is y equal to y2"? Answer: hey are different except for by chance, unless spot rate yield curve is flat. hat spot rate yield curve is flat means that St are the same for all t. If spot rates are the same for all maturities, then equations () and (3) have the same answers. Also (2) and (4) have the same result. Because spot rates are the same for () and (2), y and y2 are equal. IRR is a kind of weighted average of "rates of return per period". YM is can be interpreted such a king of weighted average of spot rates. Different coupon rates mean different weights. Weighted average takes different value. Example Annual compounding is assumed. 6 years to maturity. Coupon rate are c = 0.03 and c2 = year spot rate in APR In[97]:= Clearp, p2,, S, c, c2, 6; c 0.03; c2 0.06; 00; z "year" "spot rate in APR" ; ablesk z2, k, k,, 6; 20-05Ch08b.nb 6

7 inancial Economics, Spring 20 In[975]:= c p St t t ; S c2 p2 St t t ; Print"p ", p, " and p2 ", p2 S p and p In[977]:= Cleary, ans c ans NSolvep y t t, y; y y y. ans6 Out[979]= In[980]:= Cleary2, ans2 ans2 NSolvep2 c2 y2 t t, y2; y2 y2 y2. ans26; Print"y", y, " and y2 ", y2 y and y hus, two bonds have the same maturity but have the different yield to maturity. Callable Bond and Convertible Bond Callability: he issuer of the bond has the right to redeem the bond before the final maturity date. 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. Homework No.4, due on May 8, 20 Q: Problem 8. Hint: Biannual compounding apply. Each coupon is half of c. Q2: Draw the same graph as igure 8.4 A on page 236, assuming 20-year 6% coupon and 5-year 6% coupon bonds and also 20 year zero coupon. Apply annual compounding. Q3: Problem 8.7 Correction of problem: 2 year price is $9.4834, not $ Q4: Problem 8.8 Q5. Problem 8.9 Hint about semiannual coupons: Japanese and US government bonds pay coupons every 6 months. In this case, we use semiannual compounding. Amount of each coupon is c ; half of annual payment. Let x be yield to maturity per 6 2 months. hen x satisfies the following. 0 b P t where b = c x t x 0 2 YM as annual percentage rate is 2x per year; y=2 x Ch08b.nb 7

8 inancial Economics, Spring 20 How to Draw ables by Mathematica Showing able Step. Make a table in "input mode" cell. Step 2. Copy the table and paste it in "text mode" cell Detail of process. Choose or create Input mode cell. 2. Input - Create Matrix/able/Palette 3. Make- table, Specify the number of rows and columns. Check "draw lines and frames" boxes. 4. Copy the table and paste it in the text mode cell Ch08b.nb 8