Homework Assignment #2: Answer Key


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1 Homework Assignment #2: Answer Key Chapter 4: #3 Assuming that the current interest rate is 3 percent, compute the value of a fiveyear, 5 percent coupon bond with a face value of $,000. What happens if the interest rate goes up to 4 percent? Answer: The value of a coupon bond comes from two components: the value of the stream of coupon payments and the value of the face value repaid at maturity. Note: the coupon rate need not be the same as the interest rate. The coupon rate is given at the beginning of the bond. So, the coupon rate gives a $50 interest payments every period. The present value calculation for 3 percent interest would be Present Value = Present Value Coupons + Present Value Principal [ 50 = (.03) + 50 (.03) (.03) (.03) ] (.03) (.03) 5 = = 9.60 Since the interest rate is less than the coupon rate, the value of the bond exceeds the face value of 00. If the interest where to rise to $ percent the present value of the bond would be Present Value = Present Value Coupons + Present Value Principal [ 50 = (.04) + 50 (.04) (.04) (.04) ] (.04) (.04) 5 = = The increase in the interest rate has reduced the value of the bond. #6: You decide you would like to retire at age 65 and expect to live until you are 85. You figure that you can live nicely on $50,000 per year.. (a) Describe the calculation you need to make to determine how much you must save to purchase an annuity paying $50,000 per year for the rest of your life. Assume the interest rate is 7 percent. Answer: You find this by simply getting the present value of the stream of $50,000 payments for the 20 years of your remaining life. This can be found by using the following formula. Present Value = ( ) ( (.07) 20 ) = = $529, 700 Thus, you are indifferent (or more importantly an annuity firm) is indifferent between the stream of 20, $50,000 payments and $529,700 in the bank. So you should be able to purchase the annuity for $529,700.
2 (b) How would your calculation change if you expected inflation to average 2 percent for the rest of your life? Answer: Assuming that you still believe you need $50,000 per year, the inflation rate would increase the nominal interest rate in this environment from 7 to 9 percent. Thus the present value calculation would be Present Value = ( ) ( ) (.09) 20 = = $456, 428 The price of the annuity has fallen the future $50,000 payments are discounted at a quicker rate. In reality how would this affect the price of the annuity? Typically, retirees don t think about keeping the same $50,000 per year in payments, they want the buying power of their payments to stay the same. So, if inflation were 2 percent, we would typically want our payments to also grow by 2 percent which would cancel out the increase in the nominal interest rate from 497 to 9 percent. Thus, if this is how households value annuities the price would be unchanged. #8: Some friends of yours have just had a child. Realizing the power of compound interest, they are considering investing for their child s college education, which will begin in 8 years. Assume that the cost of a college education today is $25,000; there is no inflation; and there are no taxes on interest income that is used to pay college tuition and expenses. a. If the interest rate is 5 percent, how much money will your friends need to put into their savings account today to have $25,000 in 8 years? Answer: This is an application of future value, in this case you know the future value and need to find the present amount which generates that value. You need to solve = x(.05) = x x = If we place $5, in the bank today, this would generate $25,000 in 8 years. b. What if the interest rate is 0 percent? Answer: Simply alter the previous question for a 0 percent interest rate. Thus, = x (.) = x x =
3 If we get a return of 0 percent we would only need to put $22, in the bank today to get 25,000 in 8 years c. The chance that a college education will cost the same in 8 years from now seems remote. Assuming that the price will rise 3 percent per year and that today s interest rate is 8 percent, what will your friend s investment need to be? Answer: This is really a two part question. First we need to figure out what the cost of a college education will be in 8 years. This is an application of future value. So, x = 25000(.03) 8 x = So, given the path of prices we should expect a college education to cost $22800 in 28 years. So now we need to find how much we should invest today to generate $22800 in 8 years given 8 percent interest. This is just like parts a and b = x(.08) = 3.996x x = We would need an initial investment of $53, d. Return to the case with 5 percent interest and no inflation. Assume that your friend doesn t have enough to make the initial investment. Instead, they think they will be able to split their investment into two equal parts, one invest immediately, and the second in five years. How would you compute the required size of the two equal investments made five years apart? Answer: This is not as difficult as you may think. The goal is simple two investment need to generate a combined future value of $25,000. The setup for this problem goes as follows = Future Value of Initial Investment + Future Value of Second Investment The key difference in the investments is that the first investment will compound for 8 years and the second will only compound for 3 years. So we have = x(.05) 8 + x(.05) = x x = x x = Your friend will need to make 2 investments of $29,22.59 to achieve $25,000 in 8 years. 3
4 #5 Recently, some lucky person won the lottery. The lottery winnings were reported to be $85.5 million. In reality, the winner got a choice of $2.85 million per year for 30 years or $46 million today.. (a) Explain briefly why winning $2.85 million per year for 30 years in not equivalent to winning $85.5 million. Answer: Sure the total dollars received are the same, but a dollar today is does not have the same value as a dollar tomorrow. Thus, because people discount the value of future money, the stream of 30 $2.85 million payments has a lower value than $85.5 million. (b) The evening news interviews a group of people the day after the winner was announced. When asked, most of them responded that, if they were the lucky winner, they would take the $46 million upfront payment. Suppose that you were the lucky winner. How would you decide between the annual installments or the upfront payment? Answer: This really comes down to how fast you discount the future. If you discount the future quickly, then you would be likely to take the up front payments. If you are more patient the likely choice is that you would take the stream of payments over the lump sum of $46 million. Actually, given the numbers in the problem, we can make an educated guess on how you discount the future. The key is to find the break even rate of discounting. That would occur at the point where the present value of the stream of 30 payments just equals $46 million.thus, we need to find ( $2, 850, 000 $46, 000, 000 = r ) ( ) ( + r) 40 Using this formulation we find r is approximately 5.62%. People with discount rates above 5.62% would choose the lump sum and those below would choose to take the stream of 30 payments. Chapter 5: # Consider a game in which a coin will be flipped three times. For each heads you will be paid 0. Assume that the coin has a twothirds probability of coming up heads.. (a) Construct a table of the possibilities and probabilities of this game.answer: The following table describes the possible outcomes of this game, in this version of the game, the ordering of heads and tails does not matter, but I will write out the order which will become important in part d. With ordering there 8 4
5 outcomes to this game. H stands for Heads and T for Tails Possibility Event Probability Payout H,H,H = 8 27 $300 2 H,H,T = H,T,H = H,T,T = T,H,H = T,H,T = T,T,H = T,T,T = 27 If you ignored the ordering in this case the table would simplify to only 4 cases Possibility Event Probability Payout 8 ( from above) 3 heads 27 $ (2+3+5 from above) 2 heads (4+6+7 from above) head (8 from above) 0 heads 27 (b) Compute the expected value of the game. Answer: The expected value is calculated as EV = = =.00 (c) How much would you be willing to pay to play the game? Answer: If you are a risk neutral person you would be willing to pay to play this game, if you are risk averse you would be willing to pay some amount less than to play the game. (d) Consider the effect of a change in the game so that if tails comes up two times in a row, you get nothing. How would your answers to the first three parts of this question change? Answer: In this case, the ordering of Heads and Tails does matter and when ever two tails come up together the payout goes to zero. This changes the payout of case #4 and #8 in this first table from 0 to.so the probability distribution changes to 5
6 Possibility Event Probability Payout H,H,H = 8 27 $300 2 H,H,T = H,T,H = H,T,T = T,H,H = T,H,T = T,T,H = T,T,T = 27 Given this change, the expected value of this game is going to drop. We can see this by recalculating the expected value. We find EV = = = $85.9 The price you would be willing to pay to play this game is going to fall. #4 Assume that the economy can experience high growth, normal growth, or recession. You expect the following stockmarket returns for the coming year under these conditions State of the Economy Probability Return High Growth % Normal Growth % Recession 0. 5% a. Compute the expected value of a 00 investment both in dollars and as a percentage over the coming year. Answer: Given the above information, we can construct a frequency distribution of the payoff profile of this investment. State of the Economy Probability Payoff High Growth 0.2 $300 Normal Growth 0.7 $20 Recession 0. $850 Given this we can construct the expected payoff from this investment EV = (300.2) + (20.7) + (850.) EV = $29 We find a expected value of $29 or and expected profit of $29 which as a percentage of the initial investment is = 2.9%. 6
7 b. Compute the standard deviation of the return as a percentage over the coming year. Answer: Here is the computation of the standard deviation for this investment SD = ((300 29) 2.2) + ((20 29) 2.7) + ((850 29) 2.) = 7.0 The standard deviation of the payoffs for this investment is $7.0 or as a percentage of the initial investment is.7%. c. If the riskfree rate of return is 7 percent, what is the risk premium for a stockmarket investment? Answer: The risk premium is an return over the risk free rate that is brought on by the risk in the investment. In this example, all of the extra return is generated by risk so the risk premium is Risk Premium =2.9%7%=5.9% #8 Mortgages increase the risk faced by homeowners.. (a) Explain how. Answer: Mortgages are collateralized loans, meaning that if the owner defaults on the loan, the bank gets the house. In this situation the bank only faces the risks associated with the value of the home and their ability to sell the home. The homeowner faces a potentially large loss if they default on the loan. The lose the house. (b) What happens to the homeowner s risk as the down payment on the house rises from 0 percent to 50 percent? Answer: In some ways this actually shifts more risk onto the homeowner and away from the bank. As more of the house is owned by the homeowner, and defualting would involving losing the entire 50% put down. At the same time the risk can also be thought of as decrease because the size of the mortgage payments should fall as the down payment goes up. This reduces the probability of a default taking place. # Which of hte investments in the table below would be most attractive to a risk averse investor? How would your answer differ if the investor was described as risk neutral? Investment Expected Value Standard Deviation A 75 0 B 00 0 C
8 Answer: For the risk averse invetor the clear choice is investment B, it offeres the highest value for the lowest risk. For the risk neutral investor, the answer only changes in one way. This investor would actually be indifferent between investments B and C. The risk neutral investor looks only at the expected value of an investment. The risk of the investment is ignored by the risk neutral investor. 8
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