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copyright © 2003 McGraw Hill Ryerson Limited 4-1 prepared by: Carol Edwards BA, MBA, CFA Instructor, Finance British Columbia Institute of Technology Fundamentals of Corporate Finance Second Canadian Edition

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copyright © 2003 McGraw Hill Ryerson Limited 4-2 Chapter 4 Valuing Bonds Chapter Outline Bond Characteristics Bond Prices and Yields

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copyright © 2003 McGraw Hill Ryerson Limited 4-3 Bond Characteristics Definitions Governments and corporations borrow money for the long term by issuing securities which are called bonds: They collect the money when they issue the bond, or sell it to the public. The money they collect is the amount of the loan. The amount of the loan may be known as the par value, face value, maturity value, or the principal. The date on which the loan will be paid off is the maturity date.

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copyright © 2003 McGraw Hill Ryerson Limited 4-4 Bond Characteristics Definitions The buyers of the bond are known as the bondholders. The bondholders are actually lending money when they purchase the bond. The seller of the bond is called the issuer. The issuer is actually borrowing money when it issues the bond. At maturity, the issuer repays the face value of the loan to the bondholders.

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copyright © 2003 McGraw Hill Ryerson Limited 4-5 Bond Characteristics Coupon Rate vs Discount Rate The issuer promises to make specified fixed income payments (interest) each year to the bondholders. These payments are known as the coupon. The coupon rate is the annual interest payment divided by the face value of the bond. The interest rate (or discount rate) is the rate at which the cash flows from the bond are discounted to determine its present value.

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copyright © 2003 McGraw Hill Ryerson Limited 4-6 Bond Characteristics Coupon Rate vs Discount Rate WARNING! The coupon rate, though a percent, is not the interest rate (or discount rate). The coupon rate tells us what cash flows a bond will produce. The coupon rate does not tell us the value of those cash flows. To determine the value of a cash flow, you must calculate its present value.

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copyright © 2003 McGraw Hill Ryerson Limited 4-7 Bond Prices and Yields What is a Bond Worth? What would you be willing to pay right now for a bond which: Pays a coupon of $65 per year for 3 years. Has a face value of $1,000. The cash flows on this bond would be: 0123 -Price ?$65 $65 + 1000 To determine the price, you must calculate the present value of these cash flows.

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copyright © 2003 McGraw Hill Ryerson Limited 4-8 Bond Prices and Yields What is a Bond Worth? The coupon rate on this bond is 6.5%: That is, you would receive $65 per year on a $1000 face value ($65/$1,000 = 6.5%). But, to determine the PV of the bond, you need to know the discount rate for the bond. The discount rate is the interest rate which investors are earning on a similar security. Assume that similar 3 year bonds offer a return of 5.1%. That is, 5.1% is the discount rate for this bond.

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copyright © 2003 McGraw Hill Ryerson Limited 4-9 Bond Prices and Yields What is a Bond Worth? To determine the price of the bond, you would calculate the PV of the cash flows using a 5.1% discount rate: $65/ (1.051) = $61.85 $65 / (1.051) 2 = $58.84 $1,038.05 BOND PRICE = PV today: 0123 -Price ? $65 $65 + 1000 $1065 / (1.051) 3 = $917.36

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copyright © 2003 McGraw Hill Ryerson Limited 4-10 Bond Prices and Yields What is a Bond Worth? Thus, if discount rates are 5.1%, you would be willing to pay $1,038.05 for a bond with a coupon rate of 6.5% and a remaining life of three years. But, suppose this were a 20 bond … what would it be worth? You could use the same procedure as you used for the first bond: Calculate the PV of each cash flow. Add up these present values. Can you see the problem with using this method?

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copyright © 2003 McGraw Hill Ryerson Limited 4-11 Multiple Cash Flows What is a Bond Worth? Calculating the PV this way would mean working out the PV for 20 separate cash flows... Yes! Look at the cash flows and see if you can identify what it would be … Is there an easier way?

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copyright © 2003 McGraw Hill Ryerson Limited 4-12 Bond Prices and Yields What is a Bond Worth? Did you notice that: The coupon payments on a bond are an annuity. In the last year of the bond’s life the holder receives a payment equal to the face value of the bond. Thus you can calculate the PV of the bond: PV = PV (coupons) + PV (face value) = (coupon x annuity factor) + (face value x discount factor) = $65 x [1/0.051–1/(0.051(1+0.051) 3 )]+1000x[1/(1+0.051) 3 ] = $176.68 + 861.37 = $1,038.05

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copyright © 2003 McGraw Hill Ryerson Limited 4-13 Bond Prices and Yields How Bond Prices Vary with Interest Rates When interest rates are 5.1%, a 3 year bond with a 6.5% coupon rate is worth $1,038.05. Critical question: What would happen to the value of this bond if interest rates were to change? You would have to recalculate the PV of the cash flows to determine its worth. Say interest rates rise to 6.5% and then to 15%. What is the value of this bond under each of these scenarios?

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copyright © 2003 McGraw Hill Ryerson Limited 4-14 Bond Prices and Yields How Bond Prices Vary with Interest Rates If the discount rate is 6.5% then the PV is: PV = PV (coupons) + PV (face value) = $65 x [1/0.15–1/(0.15(1+0.15) 3 )]+1000x[1/(1+0.15) 3 ] = $148.41 + 657.52 = $805.93 PV = PV (coupons) + PV (face value) = $65 x [1/0.065–1/(0.065(1+0.065) 3 )]+1000x[1/(1+0.065) 3 ] = $172.15 + 827.85 = $1,000.00 If the discount rate is 15% then the PV is:

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copyright © 2003 McGraw Hill Ryerson Limited 4-15 Bond Prices and Yields How Bond Prices Vary with Interest Rates Can you see the relationship between the coupon rate, the interest (discount) rate and the price of a bond? Check the next slide for a graph! Coupon RateInterest RatePrice of Bond 6.5% 5.1%$1,038.05 6.5% 6.5%$1,000.00 6.5% 15.0%$ 805.93

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copyright © 2003 McGraw Hill Ryerson Limited 4-16 Bond Prices and Yields How Bond Prices Vary with Interest Rates Can you see the relationship between the coupon rate, the interest (discount) rate and the price of a bond?

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copyright © 2003 McGraw Hill Ryerson Limited 4-17 Bond Prices and Yields How Bond Prices Vary with Interest Rates You should see that there is an inverse relationship between bond prices and the interest rate: When the market interest rate exceeds the coupon rate, bonds sell for less than face value. When the market interest rate equals the coupon rate, bonds sell for their face value. When the market interest rate is below the coupon rate, bonds sell for more than face value.

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copyright © 2003 McGraw Hill Ryerson Limited 4-18 Bond Prices and Yields Measuring the Returns on Bonds There are three methods used for measuring the return an investor would receive on an investment in bonds: Current Yield. Yield to Maturity. Rate of Return. Suppose you could buy a 3 year bond, with a 10% coupon rate. If you pay $1,136.16 for this bond, what is your return?

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copyright © 2003 McGraw Hill Ryerson Limited 4-19 Bond Prices and Yields Current Yield Current yield is calculated by dividing the coupon payments by the bond’s price: Current Yield = Coupon Payments Bond Price You are buying a bond with: Coupon Payments = $100 per year. Price of $1,136.16. Current Yield = $100 = 0.088 = 8.8% $1,136.16

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copyright © 2003 McGraw Hill Ryerson Limited 4-20 Bond Prices and Yields Current Yield The current yield on this bond is 8.8%. But, the current yield only takes into account the interest income earned on the bond. The current yield calculation does not include any capital gains or losses on your investment. If you pay $1,136.16 for this bond, will you have a capital gain, or a loss, when the bond matures?

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copyright © 2003 McGraw Hill Ryerson Limited 4-21 Bond Prices and Yields Current Yield In three years, when the bond matures, you will receive $1,000 for it. Thus, you will have a capital loss of $136.16 over the life of the bond. So, your actual return on this bond must be less than the 8.8% current yield. Conclusion: Current yield, because it ignores capital gains and losses, mismeasures the rate of return on a bond.

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copyright © 2003 McGraw Hill Ryerson Limited 4-22 Bond Prices and Yields Yield to Maturity We need a measure of return which takes account of both current yield (interest income) and changes in the bond’s value over its life (capital gains and losses). This measure is called Yield to Maturity. Yield to maturity is defined as the discount rate which makes the PV of the bond’s cash flows equal to its price.

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copyright © 2003 McGraw Hill Ryerson Limited 4-23 Bond Prices and Yields Yield to Maturity For the 3 year bond you are working with, you know the PV of the bond is $1,136.16. To calculate the yield to maturity, you must solve for “r” (the discount rate). In other words: PV BOND = PV (coupons) + PV (face value) $1,136.16 = $100 x [1/r–1/(r(1 + r) 3 )]+1000x[1/(1+ r) 3 ] Use your calculator to solve for “r”.

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copyright © 2003 McGraw Hill Ryerson Limited 4-24 Bond Prices and Yields Yield to Maturity (YTM) Did you calculate yield to maturity equals 5%? At a 5% discount rate, the PV of the bond’s cash flows is $1,136.16 (or the price of the bond). You can see a graphical representation of yield to maturity on the next slide.

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copyright © 2003 McGraw Hill Ryerson Limited 4-25 Bond Prices and Yields When Price = $1,136.16 the discount rate = 5% (ytm is 5%)

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copyright © 2003 McGraw Hill Ryerson Limited 4-26 Bond Prices and Yields Yield to Maturity Note that yield to maturity depends on the coupon payments and the current price of the bond and the final repayment. Thus, it is a measure of your return if you buy the bond today and hold it until maturity. But what if you wanted to hold this bond for only one year … now how would you calculate your return?

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copyright © 2003 McGraw Hill Ryerson Limited 4-27 Bond Prices and Yields Rate of Return Let’s continue with our previous example: You buy a 3 year bond with a 10% coupon for $1,136.16. One year later, you sell it for $1,130. For this example we need to calculate the bond’s rate of return: Rate of return = coupon income + price change investment = $100 + ($1,130 - $1,136.16) $1,136.16 = 0.083 = 8.3%

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copyright © 2003 McGraw Hill Ryerson Limited 4-28 Bond Prices and Yields Yield to Maturity vs Rate of Return Note that the rate of return over a particular holding period is not the same as the yield to maturity. But, is there a relationship between yield to maturity and rate of return? Yes: If the bond’s yield to maturity is unchanged over the holding period, then the yield to maturity will equal the rate of return.

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copyright © 2003 McGraw Hill Ryerson Limited 4-29 Bond Prices and Yields Yield to Maturity vs Rate of Return Continuing with our previous example, suppose the yield to maturity stays at 5% for one year, then at the end of the holding period, its price would be: PV = PV (coupons) + PV (face value) = $100 x [1/0.05–1/(0.05(1 + 0.05) 2 )]+1000x[1/(1+ 0.05) 2 ] = $185.94 + 907.03 = $1,092.97 It rate of return= coupon income + price change investment = $100 + ($1,092.97 - $1,136.16) $1,136.16 = 0.05 = 5%

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copyright © 2003 McGraw Hill Ryerson Limited 4-30 Bond Prices and Yields Yield to Maturity vs Rate of Return Note that: The yield to maturity equals 5%. The rate of return over the one year holding period also equals 5%. This occurs because: The price of the bond changes with time so that: Total Return on the Bond = Yield to Maturity

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copyright © 2003 McGraw Hill Ryerson Limited 4-31 Bond Prices and Yields Yield to Maturity vs Rate of Return The rules with respect to rate of return and yield to maturity are: If interest rates do not change, the rate of return on the bond is equal to the yield to maturity. If interest rates increase, then the rate of return will be less than the yield to maturity. If interest rates decrease, then the rate of return will be more than the yield to maturity.

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copyright © 2003 McGraw Hill Ryerson Limited 4-32 Bond Prices and Yields Multiple Period Rate of Return So far we have only considered 1 year investments. How do you calculate the rate of return if the investment lasts for longer than 1 year? For example: You buy a 3 year bond with a 6.5% coupon for $1,038.05. Two years later, you sell it for $1,015.25.

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copyright © 2003 McGraw Hill Ryerson Limited 4-33 Bond Prices and Yields Multiple Period Rate of Return You will receive two coupon payments on this bond: $65 at the end of the first year and $65 at the end of the second year. Assume that you reinvest the first coupon payment at 4% for one year. Your coupon income for this investment will be: Coupon Income = ($65 x 1.04) + $65 = $67.60 + $65 = $132.60

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copyright © 2003 McGraw Hill Ryerson Limited 4-34 Bond Prices and Yields Multiple Period Rate of Return You can now calculate the bond’s rate of return: Annual Rate of return = (1.1058) 1/2 – 1 = 0.052 = 5.2% However, this is a two year rate of return. You must annualize it: Rate of return = coupon income + price change investment = $32.60 + ($1,015.25 - $1,038.05) $1,038.05 = 0.1058 = 10.58%

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copyright © 2003 McGraw Hill Ryerson Limited 4-35 Bond Prices and Yields Interest Rate Risk You have seen that: bond prices fall when interest rates rise and bond prices rise when interest rates fall. You have also seen that the rate of return on your bond investments will vary as interest rates change. This is why we say that bonds are subject to interest rate risk.

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copyright © 2003 McGraw Hill Ryerson Limited 4-36 Bond Prices and Yields Interest Rate Risk Interest rate risk is the risk in bond prices due to fluctuations in interest rates. Bond investors worry that if interest rates go up, they will have losses on their bonds. But bonds are not equally affected by changes in interest rates: Some bonds have big price changes in response to interest rate changes. Some bond prices react very little to a change in interest rates.

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copyright © 2003 McGraw Hill Ryerson Limited 4-37 Bond Prices and Yields Interest Rate Risk Compare the two curves on the next slide. The blue curve shows how the price of a 30 year bond varies according to interest rates. Notice how steep the curve is. Notice that a change in rates results in a large change in the price. The red curve shows how the price of a 3 year bond varies according to interest rates. Notice how flat the curve is. Notice that a change in rates results in a small change in price.

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copyright © 2003 McGraw Hill Ryerson Limited 4-38 Price change in response to interest rate changes (30 year bond vs 3 year bond): $0 $500 $1,000 $1,500 $2,000 $2,500 $3,000 $3,500 $4,000 $4,500 0.00%1.00%2.00%3.00% 4.00%5.00%6.00%7.00%8.00%9.00% 10.00% 11.00% 12.00% 13.00%14.00% Bond Price Interest Rates 30 year bond 3 year bond Price Rates Price Rates

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copyright © 2003 McGraw Hill Ryerson Limited 4-39 Bond Prices and Yields Interest Rate Risk Which bond would you rather own, the 30 year or the 3 year, if you expected interest rates to go up? Why? Which bond would you rather own, the 30 year or the 3 year, if you expected interest rates to go down? Why?

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copyright © 2003 McGraw Hill Ryerson Limited 4-40 Bond Prices and Yields Default Risk Both corporations and the Government of Canada borrow money by issuing bonds. There is an important difference between corporate borrowers and government borrowers: Corporate borrowers can run out of cash and default on their borrowings. The Government of Canada cannot default – it just prints more money to cover its debts.

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copyright © 2003 McGraw Hill Ryerson Limited 4-41 Bond Prices and Yields Default Risk Default risk (or credit risk) is the risk that a bond issuer may default on its bonds. To compensate investors for this additional risk, corporate borrowers must promise them a higher rate of interest than the Canadian government would pay. The default premium or credit spread is the difference between the promised yield on a corporate bond and the yield on a Canada bond with the same coupon and maturity.

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copyright © 2003 McGraw Hill Ryerson Limited 4-42 Bond Prices and Yields Default Risk You are thinking of investing in one of these bonds: Government of Canada 10 years, 6% coupon, 6% yield. Safe Corp 10 years, 6% coupon, 6.5% yield. Risky Corp 10 years, 6% coupon, 10% yield. The corporate bonds both have a higher yield than the Canada because of default risk. Safe Corp has little risk of default and a small default premium (credit spread) of 0.5%. Risky Corp has high default risk and thus a large default premium of 4%.

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copyright © 2003 McGraw Hill Ryerson Limited 4-43 Bond Prices and Yields Default Risk The safety of a corporate bond can be judged from its bond rating. Bond ratings are provided by companies such as: Dominion Bond Rating Service (DBRS). Moody’s. Standard and Poor’s.

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copyright © 2003 McGraw Hill Ryerson Limited 4-44 Bond Prices and Yields Default Risk Bond ratings list bonds in order from the lowest risk of default to the highest risk of default. For example: AAA (triple A) bonds have the least risk of default. Then comes AA (double A) and A bonds. Then comes BBB bonds, BB bonds and B bonds. Bonds rated BBB and above are called investment grade bonds. Bonds rated BB and below are called speculative grade, high yield, or junk bonds.

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copyright © 2003 McGraw Hill Ryerson Limited 4-45 Bond Prices and Yields Default Risk See table 4.1 in your text for a list of bond ratings and what they mean. Note that the yield on a bond varies inversely with its rating: Bonds with a high bond rating have less risk and thus lower yields. Bonds with a low bond rating have a lot of risk and higher yields. See figure 4.9 in your text for a graph of the relationship between bond rating and yield.

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copyright © 2003 McGraw Hill Ryerson Limited 4-46 Summary of Chapter 4 A bond’s coupon rate, current yield and yield (or yield to maturity) are not the same. Coupon rate is the bond’s coupon divided by its face value. Current yield is the bond’s coupon divided by its current price. Yield to maturity measures the average return to an investor who purchases the bond and holds it until maturity. Yield to maturity accounts for both coupon income and changes in the bond’s value over time.

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copyright © 2003 McGraw Hill Ryerson Limited 4-47 Summary of Chapter 4 The price of a bond is determined by calculating its PV. PV of a bond is calculated by discounting its cash flows at the current interest rate for similar bonds. You can also start with the bond price and calculate what interest rate the bond offers. The interest rate which equates the PV of the bond payments to the bond price is called the yield to maturity. Because PV’s are lower when discount rates are higher, bond prices and yield to maturity vary inversely.

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copyright © 2003 McGraw Hill Ryerson Limited 4-48 Summary of Chapter 4 Bond prices fluctuate in response to changes in interest rates. This risk of price change is called interest rate risk. Long term bonds have greater interest rate risk than short term bonds. Investors use bond ratings to determine the risk of default on a bond. The higher the risk of default (credit risk), the greater the rate of return the investor should demand. The additional return that investors demand for bearing credit risk is called the default premium. Bond ratings measure a bond’s credit risk.

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