Solutions to Chapter Valuing Bonds Note: Unless otherwise stated, assume all bonds have $1,000 face (par) value a The coupon payments are fixed at $60 per year Coupon rate = coupon payment/par value = 60/1000 = 6%, which remains unchanged b When the market yield increases, the bond price will fall The cash flows are discounted at a higher rate c At a lower price, the bond’s yield to maturity will be higher The higher yield to maturity on the bond is commensurate with the higher yields available in the rest of the bond market d Current yield = coupon payment/bond price As coupon payment remains the same and the bond price decreases, the current yield increases When the bond is selling at a discount, $970 in this case, the yield to maturity is greater than 6% We know that if the discount rate were 6%, the bond would sell at par At a price below par, the YTM must exceed the coupon rate Current yield equals coupon payment/bond price, in this case, 60/970 So current yield is also greater than 6% Coupon payment = 08 x 1000 = $80 Current yield = 80/bond price = 075 Therefore, bond price = 80/.075 = $1,066.67 Par value is $1000 by assumption Coupon rate = $75/$1000 = 075 = 7.5% Current yield = $75/$950 = 0789 = 7.89% Yield to maturity = 8.6% [n = 6; PV= (-)950; FV = 1000; PMT = 75) To sell at par, the coupon rate must equal yield to maturity Since Circular bonds yield 8.6%, this must be the coupon rate 5-1 Copyright © 2006 McGraw-Hill Ryerson Limited a Current yield = annual coupon/price = $80/1050 = 0762 = 7.62% b YTM = 7.2789% On the calculator, enter PV = (-)1050, FV = 1000, n = 10, PMT = 80, compute i When the bond is selling at par, its yield to maturity equals its coupon rate This firm’s bonds are selling at a yield to maturity of 9.25% So the coupon rate on the new bonds must be 9.25% if they are to sell at par The current bid yield on the bond was 4.60% To buy the bond, investors pay the ask price The investor would pay 102.52 percent of par value With $1,000 par value, this means paying $1,025.2 to buy a bond Coupon payment = interest = 05 × 1000 = 50 Capital gain = 1100 – 1000 = 100 Rate of return = = = 15 = 15% 10 Tax on interest received = tax rate × interest = × 50 = 15 After-tax interest received = interest – tax = 50 – 15 = 35 Fast way to calculate: After-tax interest received = (1 – tax rate) × interest = (1 – 3)× 50 = 35 Tax on capital gain = × × 100 = 15 After-tax capital gain = 100 – 15 = 85 Fast way to calculate: After-tax capital gain = (1 – tax rate) × capital gain = (1 – 5×.3)×100 = 85 After-tax rate of return = = = 12 = 12% 11 Bond year 1: PMT = 80, FV = 1000, i = 10%, n = 10; Compute PV0 = $877.11 year 2: PMT = 80, FV = l000, i = 10%, n = 9; Compute PV1 = $884.82 Rate of return = = 10 = 10% Bond year 1: PMT = 120, FV = 1000, i = 10%, n = 10; Compute PV0 = $1122.89 year 2: PMT = 120, FV = l000, i = 10%, n = 9; Compute PV1 =$1115.18 5-2 Copyright © 2006 McGraw-Hill Ryerson Limited Rate of return = = 10 = 10% Both bonds provide the same rate of return 12 Accrued interest= Coupon payment × = 22.5 × = $16.63 Dirty bond price= clean bond price + accrued interest = $990+ $16.63= $1006.63 The quoted clean price is $990 The bond pays semi-annual interest The last $22.5 coupon was paid on March 1, 2011, and the next coupon will be paid on September 1, 2011 The number of days from the last coupon payment to the purchase date is 136 (from March to July 15) and the total number of days in the coupon period is 184 (from March to September 1) The accrued interest is $16.63, and the total cost of buying one bond is $1006.63 13 a b If YTM = 8%, price will be $1000 Rate of return = = = 0286 = 2.86% c Real return = – = 14 15 1.0286 – = –.001359, or about – 136% 1.03 a With a par value of $1000 and a coupon rate of 8%, the bondholder receives payments of $40 per year, for a total of $80 per year b Assume it is 9%, compounded semi-annually Per period rate is 9%/2, or 4.5% Price = 40 × annuity factor(4.5%, 18 years) + 1000/1.04518 = $939.20 c If the yield to maturity is 7%, compounded semi-annually, the bond will sell above par, specifically for $1,065.95: Per period rate is 7%/2 = 3.5% Price = 40 × annuity factor(3.5%, 18 years) + 1000/1.03518 = $1,065.95 On your calculator, set n = 30, FV =1000, PMT = 97.5 a b Set PV = (-)900 and compute the interest rate to find that YTM = 10.89% Set PV = (-)1000 and compute the interest rate to find that YTM = 9.75% 5-3 Copyright © 2006 McGraw-Hill Ryerson Limited c 16 17 Set PV = (-)1100 and compute the interest rate to find that YTM = 8.794% On your calculator, set n=60, FV=1000, PMT=48.75 a Set PV = (-)900 and compute the interest rate to find that the (semiannual) YTM =5.443% The bond equivalent yield to maturity is therefore 5.443 × = 10.886% b Set PV = (-)1000 and compute the interest rate to find that YTM = 4.875% The annualized bond equivalent yield to maturity is therefore × 2= 9.75% c Set PV = (-)1100 and compute the interest rate to find that YTM = 4.399% The annualized bond equivalent yield to maturity is therefore 4.399 × = 8.798% In each case we solve this equation for the missing variable: Price= 1000/(1 + YTM)maturity Price 300 300 385.54 Maturity (years) 30.0 15.64 10.0 YTM 4.095% 8.0% 10.0% Alternatively the problem can be solved using a financial calculator: Solving the first question: PV = (-)300, PMT = 0, n = 30, FV = 1000, and compute i 18 PV of perpetuity = coupon payment/rate of return PV = C/r = 60/.06 = $1000 If the required rate of return is 10%, the bond sells for: PV = C/r = 60/.1 = $600 19 Because current yield = 098375, bond price can be solved from: 90/Price = 098375, which implies that price = $914.87 On your calculator, you can now enter: i = 10; PV = (-)914.87; FV = 1000; PMT = 90, and solve for n to find that n =20 years 20 Assume that the yield to maturity is a stated rate Thus the per period rate is 7%/2 or 3.5% We must solve the following equation: 5-4 Copyright â 2006 McGraw-Hill Ryerson Limited PMT ì annuity factor(3.5%, 18 periods) + 1000/(1.035)18 = $1065.95 To solve, use a calculator to find the PMT that makes the PV of the bond cash flows equal to $1065.95 You should find PMT = $40 The coupon rate is 2×40/1000 = 8% 21 NOTE: Typo in the text! The yield to maturity on the bond at issue should be 6.5%, not 65%!! Also, the solution at the end of textbook does not match this question In fact, it is the solution for the case where the bond’s yield to maturity at issue was 8% See below to get the solution in the back of the textbook.a Assume that the bonds were issued at par value With a yield to maturity of 6.5% at issue, the coupon rate must be 6.5% The semi-annual coupon payment is 0.065/2 × $1,000 = $32.50 Now, the price is 32.50 × Annuity factor(7%, 16 periods) + 1000/1.0716 = $645.75 b The investors pay $645.75 for the bond They expect to receive the promised coupons plus $800 at maturity We calculate the yield to maturity based on these expectations: 32.50 × Annuity factor(i, 16 periods) + 800/(1 + i)16 = $645.75 which can be solved on the calculator to show that i =5.97% On an annual basis, this 2×5.97% or 11.94% [n = 16; PV = (-)645.75; FV = 800; PMT = 32.50] ALTERNATE SOLUTION: If the yield to maturity at issue was 8%, then you get the following answers (this corresponds to the solution found in Appendix B at the back of the book) a Assume that the bonds were issued at par value With a yield to maturity of 8% at issue, the coupon rate must be 8% The semi-annual coupon payment is 0.08/2 × $1,000 = $40 Now, the price is 40 × Annuity factor(7%, 16 periods) + 1000/1.0716 = $716.60 b The investors pay $716.60 for the bond They expect to receive the promised coupons plus $800 at maturity We calculate the yield to maturity based on these expectations: 49 × Annuity factor(i, 16 periods) + 800/(1 + i)16 = $716.60 which can be solved on the calculator to show that i =6.03% On an annual basis, this 2×6.03% or 12.06% [n = 16; PV = (-)716.60; FV = 800; PMT = 40] 22 a Today, at a price of 980 and maturity of 10 years, the bond’s yield to maturity is 8.3% (n = 10, PV = (-) 980, PMT = 80, FV = 1000) 5-5 Copyright © 2006 McGraw-Hill Ryerson Limited In one year, at a price of 1050 and remaining maturity of years, the bond’s yield to maturity is 7.23% (n = 9, PV = (-) 1050, PMT = 80, FV = 1000) b 23 Rate of return = = 15.31% Assume the bond pays an annual coupon The answer is: PV0 = $935.82 (n = 10, PMT = 80, FV = 1000, i = 9) PV1 = $884.82 (n = 9, PMT = 80, FV = 1000, i = 10) Rate of return = 80 + 884.82 − 935.82 = 3.10% 935.82 If the bond pays coupons semi-annually, the solution becomes more complex First, decide if the yields are effective annual rates or APRs Second, make an assumption regarding the rate at which the first (mid-year) coupon payment is reinvested for the second half of the year Your assumptions will affect the calculated rate of return on the investment Here is one possible solution: Assume that the yields are APR and the yield changes from 9% to 10% at the end of the year The bond prices today and one year from today are: PV0 = $934.96 (n = × 10 = 20, PMT = 80/2 = 40, FV = 1000, i = 9/2 = 4.5) PV1 = $883.10 (n = × = 18, PMT = 80/2 = 40, FV = 1000, i = 10/2 = 5) Assuming that the yield doesn’t increase to 10% until the end of year, the $40 midyear coupon payment is reinvested for half a year at 9%, compounded monthly Its future value at the end of the year is: $40 × (1.045) = $41.80 and the rate of return on the bond investment is: Rate of return = = 3.20% 24 The price of the bond at the end of the year depends on the interest rate at that time With one year until maturity, the bond price will be $ 1065/(1 + r) a Price = 1065/1.06 = $1004.72 Return = [65 + (1004.72 – 1000)]/1000 = 06972 = 6.972% b Price = 1065/1.08 = $986.11 Return = [65+ (986.11 – 1000)]/1000 = 05111 = 5.111% c Price = 1065/1.10 = $968.18 Return = [65 + (968.18 – 1000)]/1000 = 0332 = 3.32% 5-6 Copyright © 2006 McGraw-Hill Ryerson Limited 25 The bond price is originally $549.69 (On your calculator, input n = 30, PMT = 40, FV =1000, and i = 8%.) After one year, the maturity of the bond will be 29 years and its price will be $490.09 (On your calculator, input n = 29, PMT = 40, FV = 1000, and i = 9%.) The rate of return is therefore [40 + (490.09 – 549.69)]/549.69 = –.0357 = –3.57% 26 a Annual coupon = 08 × 1000 = $80 Total coupons received after years = × 80 = $400 Total cash flows, after years = 400 + 1000 = $1400 Rate of return = b () 1/5 – = 075 = 7.5% Future value of coupons after years = 80 × future value factor(1%, years) = 408.08 Total cash flows, after years = 408.08 + 1000 = $1408.8 Rate of return = c () 1/5 – = 0763 = 7.63% Future value of coupons after years = 80 × future value factor(8.64%, years) = 475.35 Total cash flows, after years = 475.35 + 1000 = $1475.35 Rate of return = 27 () 1/5 – = 0864 = 8.64% To solve for the rate of return using the YTM method, find the discount rate that makes the original price equal to the present value of the bond’s cash flows: 975 = 80 × annuity factor( YTM, years ) + 1000/(1 + YTM)5 Using the calculator, enter PV = (-)975, n = 5, PMT = 80, FV = 1000 and compute i You will find i = 8.64%, the same answer we found in 26 (c) 28 a False Since a bond's coupon payments and principal are fixed, as interest rates rise, the present value of the bond's future cash flow falls Hence, the bond price falls Example: Two-year bond 3% coupon, paid annual Current YTM = 6% Price = 30 × annuity factor(6%, 2) + 1000/(1 + 06)2 = 945 If rate rises to 7%, the new price is: 5-7 Copyright © 2006 McGraw-Hill Ryerson Limited Price = 30 × annuity factor(7%, 2) + 1000/(1 + 07)2 = 927.68 29 b False If the bond's YMT is greater than its coupon rate, the bond must sell at a discount to make up for the lower coupon rate For an example, see the bond in a In both cases, the bond's coupon rate of 3% is less than its YTM and the bond sells for less than its $1,000 par value c False With a higher coupon rate, everything else equal, the bond pays more future cash flow and will sell for a higher price Consider a bond identical to the one in a but with a 6% coupon rate With the YTM equal to 6%, the bond will sell for par value, $1,000 This is greater the $945 price of the otherwise identical bond with a 3% coupon rate d False Compare the 3% coupon bond in a with the 6% coupon bond in c When YTM rises from 6% to 7%, the 3% coupon bond's price falls from $945 to $927.68, a -1.8328% decrease (= (927.68 - 945)/945) The otherwise identical 6% bonds price falls to 981.92 (= 60 × annuity factor(7%, 2) + 1000/(1 + 07)2) when the YTM increases to 7% This is a -1.808% decrease (= 981.92 1000/1000), which is slightly smaller The prices of bonds with lower coupon rates are more sensitivity to changes in interest rates than bonds with higher coupon rates e False As interest rates rise, the value of bonds fall A 10 percent, year Canada bond pays $50 of interest semi-annually (= 10/2 × $1,000) If the interest rate is assumed to be compounded semi-annually, the per period rate of 2% (= 4%/2) rises to 2.5% (=5%/2) The bond price changes from: Price = 50 × annuity factor(2%, 2×5) + 1000/(1 + 02)10 = $1,269.48 to: Price = 50 × annuity factor(2.5%, 2×5) + 1000/(1 + 025)10 = $1,218.80 The wealth of the investor falls 4% (=$1,218.80 - $1,269.48/$1,269.48) Internet: Using historical yield-to-maturity data from Bank of Canada Tips: Students will need to read the instructions on how to put the data into a spreadsheet They will want to save the data in CSV format so that it will be easily moved into the spreadsheet The data will be automatically put into Excel if you access the website with Internet Explorer Watch that the headings for the columns of data in your spreadsheet aren’t out of line (we found that the Government of Canada bond yield heading took two columns, displacing the other two headings – the data itself were in the correct columns) Expected results: Long-term Government of Canada bonds have the lowest yield, followed by the yields for the provincial long bonds and then for the corporate bonds The graph of the yields clearly shows the consistent spreads but also how the level of interest rates varies over time For an even clearer picture, have the students pick data from 1990 onward 5-8 Copyright © 2006 McGraw-Hill Ryerson Limited Time Series: Low/High/Average (Accessed November 22, 2008) Date Range: 2002/07 – 2007/06 'V122544=Government of Canada benchmark bond yields - long-term 'V122517=Average weighted bond yields (Scotia Capital Inc.) - Provincial long-term 'V122518=Average weighted bond yields (Scotia Capital Inc.) - All corporates - long-term Yield spread Yield Spread (Provincial (Corporate vs Date V122544 V122517 V122518 vs Canada) Canada) 2002/0 5.73 6.13 7.19 0.4 1.46 2002/08 5.58 6.99 0.42 1.41 2002/09 5.43 5.83 6.84 0.4 1.41 2002/10 5.63 6.05 7.17 0.42 1.54 2002/11 5.58 5.99 6.96 0.41 1.38 2002/12 5.42 5.81 6.73 0.39 1.31 2003/01 5.49 5.92 6.85 0.43 1.36 2003/02 5.46 5.88 6.81 0.42 1.35 2003/03 5.58 6.02 7.06 0.44 1.48 2003/0 5.41 5.82 6.7 0.41 1.29 2003/0 5.12 5.52 6.35 0.4 1.23 2003/0 5.03 5.41 6.22 0.38 1.19 2003/0 5.4 5.7 6.48 0.3 1.08 2003/08 5.44 5.79 6.54 0.35 1.1 2003/09 5.23 5.57 6.29 0.34 1.06 2003/10 5.38 5.73 6.39 0.35 1.01 2003/11 5.29 5.63 6.27 0.34 0.98 2003/12 5.2 5.52 6.07 0.32 0.87 2004/0 5.23 5.5 6.03 0.27 0.8 2004/0 5.09 5.37 5.87 0.28 0.78 2004/0 5.04 5.38 5.85 0.34 0.81 2004/0 5.31 5.66 6.15 0.35 0.84 2004/0 5.32 5.71 6.25 0.39 0.93 2004/0 5.33 5.78 6.36 0.45 1.03 2004/0 5.29 5.76 6.34 0.47 1.05 2004/0 5.15 5.58 6.17 0.43 1.02 2004/0 5.04 5.44 6.05 0.4 1.01 5-9 Copyright © 2006 McGraw-Hill Ryerson Limited 2004/1 2004/11 2004/12 2005/0 2005/0 2005/0 2005/0 2005/0 2005/0 2005/0 2005/0 2005/0 2005/1 2005/11 2005/12 2006/0 2006/0 2006/0 2006/0 2006/0 2006/0 2006/0 2006/0 2006/0 2006/1 2006/11 2006/12 2007/0 2007/0 2007/0 2007/0 4.9 4.92 5.39 5.29 5.3 5.99 5.88 5.82 0.39 0.39 0.38 0.99 0.98 0.9 4.74 5.14 5.66 0.4 0.92 4.76 5.11 5.62 0.35 0.86 4.77 5.21 5.73 0.44 0.96 4.59 5.04 5.58 0.45 0.99 4.46 4.89 5.46 0.43 4.29 4.69 5.2 0.4 0.91 4.31 4.72 5.25 0.41 0.94 4.12 4.52 5.04 0.4 0.92 4.21 4.64 5.15 0.43 0.94 4.37 4.18 4.02 4.82 4.67 4.54 5.34 5.24 5.09 0.45 0.49 0.52 0.97 1.06 1.07 4.2 4.71 5.3 0.51 1.1 4.15 4.67 5.27 0.52 1.12 4.23 4.78 5.37 0.55 1.14 4.57 5.07 5.67 0.5 1.1 4.5 5.01 5.6 0.51 1.1 4.67 5.18 5.81 0.51 1.14 4.45 4.96 5.6 0.51 1.15 4.2 4.69 5.33 0.49 1.13 4.07 4.55 5.18 0.48 1.11 4.24 4.02 4.1 4.7 4.47 4.56 5.33 5.11 5.18 0.46 0.45 0.46 1.09 1.09 1.08 4.22 4.66 5.28 0.44 1.06 4.09 4.53 5.15 0.44 1.06 4.21 4.2 4.64 4.64 5.27 5.38 0.43 0.44 1.06 1.18 5-10 Copyright © 2006 McGraw-Hill Ryerson Limited 2007/0 2007/0 4.39 4.84 5.63 0.45 1.24 4.56 5.07 5.82 0.51 1.26 Average Yield Spread of the provincial bonds over the Canada bonds：0.42% Average Yield Spread of the corporate bonds over the Canada bonds： 1.09% We can see that long-term Government of Canada bonds have the lowest yield over time, followed by the yields for long-term provincial long bonds and then for the corporate bonds The graph of the yields clearly shows the consistent spreads but also how the level of interest rates varies over time The result makes sense because YTM of long-term Canada bonds has the lowest risk premium of the three, followed by YTM of the provincial bonds YTM of long-term corporate bonds has larger spreads over Canada bonds because it has much higher default and liquidity risk than Canada Bonds 30 a Strips pay no interest, only principal Assume each bond pays $100 principal on the maturity date Bond June 2010 June 2012 June 2015 June 2019 June 2025 Time to Maturity (Years) 1.583 3.583 6.583 10.583 16.583 5-11 Copyright © 2006 McGraw-Hill Ryerson Limited YTM = (100/Price)1/time to maturity - = (100/96.94)1/1.583 - = 01926 = (100/91.04)1/3.583 - = 02655 = (100/80.58)1/6.583 - = 03334 = (100/65.43)1/10.583 - = 04090 = (100/45.75)1/16.583 - = 04828 b The term structure (yield curve) is upward sloping 31 Price of bond today = 40 × PVIFA(5%, 3) + 50 × PVIFA(5%,3) × PVIF(5%,3) + 60 × PVIFA(5%,3)×PVIF(5%,6) + 1000 × PVIF(5%, 9) = 108.93 + 117.62 + 121.93 + 644.61 = $993.09 32 a., b Price of each bond at different yields to maturity Maturity of bond years years Yield (%) 1124.09 1000.00 897.26 Difference between prices (YTM=7% vs YTM=9%) 226.83 c 33 1033.87 1059.71 1000.00 1000.00 967.60 944.65 66.27 115.06 30 years The table shows that prices of longer-term bonds respond with more sensitivity to changes in interest rates This can be illustrated in a variety of ways In the table we compare the prices of the bonds at percent and percent yields When the yield falls from to 7%, the price of the 30-year bond increases $226.83 but the price of the 4-year bond only increases $66.27 Another way to compare the bonds’ sensitivity to changes in the yield is to look at the percentage change in the prices For example, with an increase in the yield from to 9%, the price of the 4-year bond falls (967.6/1000) –1, or 3.24% but the 30-year bond price falls (897.26/1000) – 1, or 10.27% The bond’s yield to maturity will increase from 8.5%, effective annual interest (EAR) to 8.8%, EAR, when the perceived default risk increases month interest rate equivalent to 8.5% EAR = (1.085)1/2 – = 041633 month interest rate equivalent to 8.8% EAR = (1.088)1/2 – = 043072 Price at AA rating = $978.12 (n = 2×10 = 20, PMT = 80/2 = 40, FV =1000, i = 4.1633) 5-12 Copyright © 2006 McGraw-Hill Ryerson Limited Price at A rating = $959.36 (n = 2×10 = 20, PMT = 80/2 = 40, FV =1000, i = 4.3072) The price falls by $18.76 dollars due to the drop in the bond rating and the increase in the required rate of return 34 Internet: Credit spreads on corporate bonds At www.bondsonline.com/Todays_Market/Corporate_Bond_Spreads.php, the spread for a 10 year A2/A-rated bond was reported to be “95”, meaning 95 basis points (bp) or 95% The spread for a 10 year B2/B-rated bond was 265 bp or 2.65% As of March 9, 2009 the yield to maturity on a 10 year US Treasury bond was 2.88% The estimated required rate of return on each corporate bond is: Required rate of return = US treasury bond yield to maturity + credit spread 10 year A2/A-rated bond required rate of return = 2.88% + 95% = 3.83% 10 year B2/B-rated bond required rate of return = 2.88% + 2.65% = 5.53% 35 Internet: Canadian corporate bond yields Tips: If you click on “Bond Type” it will sort the bonds by type, making it easier to find a set of corporate bonds Alternatively, the data in the table can be copied and pasted into Excel and sorted there If you sort by type and maturity, it is easier to get a group of corporate bonds with similar maturity dates At www.dbrs.com, type the company name into the search box If the company is rated, it will be listed Click on the name and pick the rating of the subordinated debt (or just the lowest rating) Find a Government of Canada bond (CANADA FEDGOV) with a similar maturity date in the bond list Calculate the yield spread: corporate bond yield – government bond yield and compare the yields and spread with the different ratings Here’s sample of data taken from the globeinvestor.com bond table and assembled into a table in Excel “Spread” in the final column is calculated as the difference between the corporate bond yield and the corresponding Government of Canada bond Federal government bonds could not be found with exactly the same maturity date for all corporate bonds So the Federal bond with closest maturity date was chosen The BBB rate bonds have the largest spread, between 410 and 466 basis points The Canadian Tire A low bond spread, 6.47 – 2.06 = 4.06 is a bit lower than the BBB spread but not by much By contrast the two AA corporate 5-13 Copyright © 2006 McGraw-Hill Ryerson Limited bonds (Bank of Montreal and Bank of Nova Scotia) are substantially smaller, only about 260 basis points DBRS Rating CANADIAN NATURAL RESOURCES METRO INC LOBLAWS CANADIAN TIRE BANK OF MTL BANK OF NOVA SCOTIA 36 CORPORATE BONDS Coupon Coupon Rate Freq Maturity BBB BBB BBB A low AA low AA low 4.95 4.98 7.1 4.95 4.55 4.15 S S S S S S CANADA FEDGOV Price Yield Maturity 6/1/2015 91.09 10/15/2015 92.08 6/1/2016 104.02 6/1/2015 93.99 8/1/2017 96.52 10/27/2017 98.97 6.72 6.47 6.39 6.12 5.06 5.17 Yield Spread 2.06 2.06 2.29 2.06 2.51 2.51 4.66 4.41 4.10 4.06 2.55 2.66 6/1/2015 6/1/2015 6/1/2016 6/1/2015 6/1/2017 6/1/2017 YTM = 4% Real interest rate = + nominal interest rate = 1.04 - = 0196, or 1.96% + expected rate of inflation 1.02 Real interest rate ≈ nominal interest rate - expected inflation rate = 4% - 2% = 2% 37 The nominal return is 1060/1000, or 6% The real return is 1.06/(1 + inflation) – a b c d 1.06/1.02 – = 0392 = 3.92% 1.06/1.04 – = 0192 = 1.92% 1.06/1.06 – = 0% 1.06/1.08 – = – 0185 = –1.85% 38 The principal value of the bond will increase by the inflation rate, and since the coupon is 4% of the principal, it too will rise along with the general level of prices The total cash flow provided by the bond will be 1000 × (1 + inflation rate) + coupon rate × 1000 × (1 + inflation rate) Since the bond is purchased for par value, or $1000, total dollar nominal return is therefore the increase in the principal due to the inflation indexing, plus coupon income: Income = 1000 × inflation rate + coupon rate × 1000 × (1 + inflation rate) Finally, the nominal rate of return = income/1000 a Nominal return = = 0608 Real return = – = 04 b Nominal return = = 0816 Real return = – = 04 c Nominal return = = 1024 Real return = – = 04 5-14 Copyright © 2006 McGraw-Hill Ryerson Limited d Nominal return = = 1232 a b c d First year income 40x1.02=$40.80 40x1.04=$41.60 40x1.06=$42.40 40x1.08=$43.20 39 40 Real return = – = 04 Second year income 1040 x 1.022 = $1082.02 1040 x 1.042 = $1124.86 1040 x 1.062 = $1168.54 1040 x 1.082 = $1213.06 a YTM = 5.76% (n=15, PV = (-)1048, PMT=62.5, FV=1000) b YTC = 6.33% (n=10, PV = (-)1048, PMT=62.5, FV=1100) 41 a Current price = 1,112.38 (n=6, i=4.8%, PMT=70, FV=1000) b Current call price = 1,137.35 (n=6, i=4.35%, PMT=70, FV=1000) 42 a YTM on ABC bond at issue = 5.5% (since sold at par, coupon rate = required rate of return) 10-year Gov't of Canada bond yield at issue = ABC bond YTM - credit spread = 5.5% - 25% = 5.25% Required yield to meet Canada call: = 10-year Gov't of Canada bond yield + 15% = 5.25 + 15% = 5.4% Call price at issue = 1,007.57 (n=10, i=5.4%, PMT=55, FV=1000) b Required yield to call bond = 4.9% + 15% = 5.05% Call price now, years later = 1,019.46 (n=5, i=5.05%, PMT=55, FV=1000) c Based on new interest rates, the bond price is: Price now, years later = 1,021.65 (n=5, i=5%, PMT=55, FV=1000) Now the current price is greater than the call price The company can call bonds and reduce its cost of debt 43 The coupon bond will fall from an initial price of $1000 (when yield to maturity = 8%) to a new price of $897.26 when YTM immediately rises to 9% This is a 10.27% decline in the bond price The zero coupon bond will fall from an initial price of = $99.38 to a new price of = $75.37 This is a price decline of 24.16%, far greater than that of 5-15 Copyright © 2006 McGraw-Hill Ryerson Limited the coupon bond The price of the coupon bond is much less sensitive to the change in yield It seems to act like a shorter maturity bond This makes sense: the 8% bond makes many coupon payments, most of which come years before the bond’s maturity date Each payment may be considered to have its own “maturity date” which suggests that the effective maturity of the bond should be measured as some sort of average of the maturities of all the cash flows paid out by the bond The zero–coupon bond, by contrast, makes only one payment at the final maturity date 44 a Annual after-tax coupon = (1 - 35) × 08 × 1000 = $52 Total coupons received after years = × 52 = $260 Capital gains tax = × 35 × (1000 – 975) = 4.375 After-tax capital gains = 1000 – 975 – 4.375 = 20.625 Total cash flows, after years = 260 + 1000 – 4.375 = $ 1255.625 Rate of return = () 1/5 – = 05189, or 5.189% Note: This can also be answered by first calculating the five-year rate of return and then converting it into a one-year rate of return This way students can continue to use the coupons + capital gains/original investment approach: Five-year rate of return = = = 28782 The one-year rate of return equivalent to the five-year rate of return is: (1 + 28782) 1/5 – = 05189, or 5.189% b Future value of coupons after years = (1 – 35) × 80 × future value factor((1–.35)×1%, years) = 263.4 Total cash flows, after years = 263.4 + 1000 – 4.375 = $1259.025 Rate of return = c () 1/5 – = 0525 = 5.246% Future value of coupons after years = (1 – 35) × 80 × future value factor((1–.35)×8.64%, years) = 290.89 Total cash flows, after years = 290.89 + 1000 – 4.375 = $1286.5 Rate of return = () 1/5 – = 057 = 5.7% 5-16 Copyright © 2006 McGraw-Hill Ryerson Limited 45 The new bonds must be priced to have a yield to maturity of 5% + 1.5% = 6.5% To sell at par, the coupon rate on the new bonds must be set at 6.5% 46 Standard & Poor's Expected results: Students should be able to see some evidence supporting the difference in the bond ratings of these two companies BCE, Inc provides wire line and wireless communications services, Internet access, data services, and video services in Canada BCE has S&P rating of BBAgrium, Inc produces and markets agricultural nutrients, industrial products, and specialty products worldwide The company has S&P Issuer credit rating: BBB BCE: Times interest earned= = 6.7 BCE: Debt/Equity = 64.2% AGU: Times interest earned== 23.5 AGU: Debt/Equity= 61.5% Agrium has a higher times interest earned ratio of 23.5 while BCE’s times interest earned is 6.7 Thus, Agrium has greater ability to make its interest payment than BCE BCE’s indebtedness is higher than AGU because it has higher debt to equity ratio than AGU Both ratios justify AGU’s higher credit rating 5-17 Copyright © 2006 McGraw-Hill Ryerson Limited ... 1 .05 2004/0 5.15 5.58 6.17 0.43 1.02 2004/0 5.04 5.44 6 .05 0.4 1.01 5-9 Copyright © 2006 McGraw-Hill Ryerson Limited 2004/1 2004/11 2004/12 2 005/ 0 2 005/ 0 2 005/ 0 2 005/ 0 2 005/ 0 2 005/ 0 2 005/ 0 2 005/ 0... Yield Spread of the provincial bonds over the Canada bonds：0.42% Average Yield Spread of the corporate bonds over the Canada bonds： 1.09% We can see that long-term Government of Canada bonds have... because YTM of long-term Canada bonds has the lowest risk premium of the three, followed by YTM of the provincial bonds YTM of long-term corporate bonds has larger spreads over Canada bonds because