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Introduction to Bond Valuation The value of a bond is equal to the present value of its future cash flows. To calculate present value, an investor must forecast the bond’s future cash flows and discount them at a discount rate. The basic formula of a coupon bond is detailed below. The formula assumes that the bond is purchased and held until maturity.
The discount rate represents the investor’s required rate of return. It is calculated as the sum of the market’s risk-free rate, the expected rate of inflation, and a risk premium derived to compensate the investor for the risks associated with holding the investment. Note that the yield of a comparable U.S. Treasury security can be used as the risk-free rate. The following is the equation used to derive the discount rate:
Consider the following example: Calculate the discount rate of a bond with a risk premium of 2.5% and an expected inflation rate of 2.7%. Assume the market’s risk-free rate is 3.0%. Discount Rate (r) = 0.03 + 0.027 + 0.025 = 0.082 or 8.2% The discount rate, or required rate of return, equates to 8.2%. Now consider a basic valuation problem in which a discount rate of 8.2% is applied to derive the market value of a bond: Calculate the market value (present value) of an annual coupon bond with a $1,000 face value, a 10% coupon rate, and four years remaining until maturity. Use the discount rate derived above as the appropriate required rate of return.
This problem can be solved by hand tabulation. Each cash flow must be discounted by the discount rate of 8.2%. The discounted cash flows can then be summed to derive the market value of the bond. Hand tabulation requires a lengthy effort, particularly for bonds with many cash flows. Alternatively, the Time Value of Money functions of a financial calculator can be used to solve the problem easily and efficiently. The following are the basic entries required to solve the problem above:
The market value of the bond is $1,059.35. The bond will be quoted as 105-30 in the market (Reference the tutorial, Quotation Conventions). Note that the calculator should display the value as negative. This is because the bond’s present value represents the cash outflow required to purchase the bond. The bond’s coupon payments and principal repayment (cash inflows) should be entered into your calculator as positive figures.
Consider a second example: An investor wishes to purchase a ten-year annual coupon bond with a face value of $1,000 and a coupon rate of 8%. The market’s risk free rate is 2.0%. Inflation is expected to be 2.5%. The investor considers a 1.5% risk premium to be appropriate considering the bond’s various risks. Calculate the investor’s required rate of return and the maximum price he should pay for the bond.
The investor’s required rate of return is 6.0%. The maximum price he should pay for the bond is $1,147.20 or 114-23. Questions: 1. Calculate the discount rate of a bond with a risk premium of 4.2% and an expected inflation rate of 2.0%. Assume the market’s risk-free rate is 3.0%. 2. Calculate the risk premium of a bond with a required rate of return of 7% and an inflation premium of 1.5%. Assume the market’s risk-free rate is 4%. 3. Calculate the market value of an annual coupon bond with a $1,000 face value, a 4% required rate of return, a 7% coupon rate, and six years remaining until maturity. 4. Calculate the present value of an annual coupon bond with a $10,000 face value, a 5% discount rate, a coupon rate of 10%, and four years remaining until maturity. 5. Calculate the present value of an annual coupon bond with a $10,000 face value, a 9% discount rate, a coupon rate of 5%, and four years remaining until maturity. Answers: 1. 9.2% r = 3.0% + 2.0% + 4.2% 2. 1.5% rPremium = 7% - 4.0% - 1.5% 3. $1,157.26 N = 6; I = 4; PMT = $70; FV = $1,000; PV = Solve 4. $11,772.98 N = 4; I = 5; PMT = $1,000; FV = $10,000; PV = Solve 5. $8,704.11 N = 4; I = 9; PMT = $500; FV = $10,000; PV = Solve All
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