Problem_Set__4_Solutions

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Jan 9, 2024

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Fin 401 Problem Set #4 Financial Instruments Solutions 1. Do Higgins, Chapter 5, #5. Answer provided in the back of the book. 2. A firm has debt with a face (book) value of 80. The future market value of the firm’s assets is uncertain, but the distribution of possible outcomes is shown in the table below. a. What is the market value of the firm’s assets, debt, and equity? Throughout this question, ignore any need to discount to present value. Payoffs to debt and equity are shown below. Assets = 0.1*50 + 0.2*100 + 0.4*200 + 0.2*300 + 0.1*350 = 200 Debt = 0.1*50 + 0.9*80 = 77 Equity = 0.2*20 + 0.4*120 + 0.2*220 + 0.1*270 = 123 (Or, Equity = Assets – Debt) b. Suppose that there is bad news about the firm. The bad news decreases the asset value in each case by 5%, as shown in the new table. What is the new market value of the firm’s assets, debt, and equity? What was the proportional change in each of these? What affect would this news have on the accounting balance sheet? Prob. 0.1 0.2 0.4 0.2 0.1 Asset Value 50 100 200 300 350 Debt payoffs 50 80 80 80 80 Equity payoffs 0 20 120 220 270 Prob. 0.1 0.2 0.4 0.2 0.1
Repeat the process from part a. Assets = 0.1*47.5 + 0.2*95 + 0.4*190 + 0.2*285 + 0.1*332.5 = 190 Debt = 0.1*47.5 + 0.9*80 = 76.75 Equity = Assets – Debt = 113.25 Percentage change: Assets: -5% Debt: -0.4% Equity: -9.2% Equity is the residual claimant, so its value is most sensitive to new information. Nothing happens to the accounting balance sheet as a consequence of this news. c. Repeat part a under the assumption that the face value of the debt was initially 150 instead of 80. Payoffs to debt and equity are shown below. Assets = 0.1*50 + 0.2*100 + 0.4*200 + 0.2*300 + 0.1*350 = 200 Debt = 0.1*50 + 0.2*100 + 0.7*150 = 130 Equity = Assets – Debt = 70 d. Why is the difference between the book and market values of debt higher in part c than in part a? In part a, the firm had a 10% chance of default; in part c, it had a 30% chance. The credit risk is higher because of the higher leverage. Asset Value 47.5 95 190 285 332.5 Debt payoffs 47.5 80 80 80 80 Equity payoffs 0 15 110 205 252.5 Prob. 0.1 0.2 0.4 0.2 0.1 Asset Value 50 100 200 300 350 Debt payoffs 50 100 150 150 150 Equity payoffs 0 0 50 150 200 2
e. Repeat part b under the assumption that the face value of the debt was initially 150 instead of 80. Assets = 0.1*47.5 + 0.2*95 + 0.4*190 + 0.2*285 + 0.1*332.5 = 190 Debt = 0.1*47.5 + 0.2*95 + 0.7*150 = 128.75 Equity = Assets – Debt = 61.25 Percentage change: Assets: -5% Debt: -1.0% Equity: -12.5% Nothing happens to the accounting balance sheet as a consequence of this news. f. Why is the proportional change in the values of debt and equity in part e different from part b? With more leverage, both debt and equity are more sensitive to new information. 3. a. A $1,000 face value zero-coupon bond that has 20 years to maturity has a yield to maturity of 6.2%. What is the price of the bond? Please solve without a financial calculator. Price = 1000 / (1.062)^20 = 300.27 b. What is the price of a $1,000 face value bond that has 4 years to maturity and that pays annual coupon payments at a 4.4% rate, if the current yield to maturity is 7%? What would be the new price if yields fell to 6%? Please solve without a financial calculator. Price = 44/(1.07) + 44/(1.07)^2 + 44/(1.07)^3 + 1044/(1.07)^4 = 41.12 + 38.43 + 35.92 + 796.46 = 911.93 Prob. 0.1 0.2 0.4 0.2 0.1 Asset Value 47.5 95 190 285 332.5 Debt payoffs 47.5 95 150 150 150 Equity payoffs 0 0 40 135 182.5 3
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For the new price, replace “1.07” above with “1.06.” Price = 944.56 4. a. What is the price of a $1,000 face value 6% coupon bond with annual interest payments that has 3 years to maturity if the interest rate is 5.6%? Please solve without a financial calculator. Price = 60/(1.056) + 60/(1.056)^2 + 1060/(1.056)^3 = 56.82 + 53.81 + 900.15 = 1010.78 b. Now find the present value assuming that coupon payments are made semiannually. (This is the more typical case in the U.S.) In order to do this, you must remember some conventions. In the U.S. if a bond with semiannual coupon payments is said to have a coupon rate of 6%, this usually means that 3% ($30 in this case) will be paid every six months. Likewise, if the bond’s interest rate is stated as 5.6%, this usually means that the 6-month rate is 5.6/2=2.8%. That is, you can think of the stated rate as the APR, not the effective annual rate. Price = 30/(1.028) + 30/(1.028)^2 + 30/(1.028)^3 + 30/(1.028)^4 + 30/(1.082)^5 + 1030/(1.082)^6 = 29.18 + 28.39 + 27.61 + 26.86 + 26.13 + 872.73 = 1010.90 c. Now replicate your answers from (a) and (b) using Excel. To do so, enter the data from (a) in a worksheet that looks like this: Bond Valuation Settlement Date 1/25/2017 Maturity Date 1/25/2020 Rate (Coupon) 6.00% Yield 5.60% Redemption 100 Frequency 1 Basis 1 Value 4
Explanation of terms Settlement: Date when bond would be sold Maturity: Date when bond matures Rate: Annual coupon rate Yield: Yield to maturity Redemption: Amount received at maturity as a % of par Frequency: Number of payments per year Basis: Code for date convention (Just set this to 1). Note that here the dates have been set somewhat arbitrarily just to be 3 years apart. (Any dates 3 years apart would give the same answer.) One of the nice things about bond valuation in Excel is the flexibility allowed in setting these dates. Now solve for the value using the PRICE function and the inputs as listed on the worksheet. (Note: if you don’t have PRICE available you need Analysis ToolPak add-in. Go to Tools | Add-ins.) You should get the same answer as in (a) above, but it will be as a % of par. Multiply the answer by 10 to get the price for a standard $1,000 face value bond. Now switch the frequency to 2 and you should get the same answer as in (b) above. d. Use the YIELD function in Excel, the inputs above, and the price found in (c) to verify that the yield to maturity is 5.6% when the price is as found in (c). e. What is the value of a $1,000 face value bond to be sold on Feb. 22, 2016, that matures on November 15, 2028, that has semiannual coupon payments and an annual coupon rate of 7.25%, if the yield to maturity is 8.43%? Plugging these values into Excel, and multiplying the answer by 10, gives $908.77. 5