FIN 350 – Business Finance Homework 1 Fall 2015 Solutions 1. You currently have $100,000 saved and the annual interest rate is 8%. You plan to make a down payment on a home in 10 years. Assume that you do not make any additional contributions to your savings. How large We also discuss several other topics like When does Rome controlled Italian peninsula (both Etruscans and Greeks) award citizenship to these people to make allies?
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of a down payment will you be able to afford? Using the future value formula, F Vt = P V ∗ (1 + r)t = $100, 000 ∗ (1 + 0.08)10 = $215, 892.50 2. McDonald’s Filet-o-Fish sandwich has increased in price by 5.45% annually since 1965. As of 2017, it costs $3.79. If this is true, what was the price of a Filet-o-Fish sandwich in 1965? P V =F Vt (1 + r)t =$3.79 (1 + 0.0545)2017−1965 = $0.24 3. You’re saving money to buy a new $276,000 Ferrari. (Assume the car price will not increase.) You have $95,000 today that can be invested at your bank. The bank pays 2.4% annual interest on its accounts. How long will it be before you have enough money to buy the car? By rearranging and taking the natural logarithm of the present value formula, we find that t =ln F Vt P V ln(1 + r) $276,000 ln $95,000 = ln(1 + 0.024) = 44.97 years ≈ 45 years 4. A friend is willing to lend $2,000 to you. She demands repayment of $3,500 in three years. What annual interest rate is she charging you?Rearranging the present value formula, r = F Vt P V 1t− 1 = $3500 $2000 13− 1 = 0.2051 or 20.51% 5. Let’s say that social security promises you $39,000 per year starting when you retire 45 years from today (the first $39,000 will come 45 years from now). If your discount rate is 8%, compounded annually, and you plan to live for 15 years after retiring (so that you will get a total of 16 payments including the first one), what is the value today of social security’s promise? This is a deferred annuity, since the payments begin in year 45. The value of the annuity in year 44 is: P V44 =Cr 1 −1 (1 + r)T =$39, 000 0.08 1 −1 (1 + 0.08)16 = $345, 203.40 Since this is the value of the annuity in year 44, we need to discount this back 44 years to the present: P V0 =P V44 (1 + r)44 =$345, 203.40 (1 + 0.08)44 = $11, 679.65 6. You won a lottery and are offered to receive $50,000 today or $100,000 in 10 years. If the interest rate is 12% per year, which option is preferable? Rearranging the present value formula, the present value of $100,000 in 10 years is: P V =F Vt (1 + r)t =$100, 000 (1 + 0.12)10 = $32, 197.32 The option for $50,000 today is preferable since it is greater than the present value of $100,000 in 10 years. (Alternatively, you could have calculated the FV of $50,000 in 10 years at 12% and compared to $100,000. Another option would be to calculate the rate the lottery is offering and comparing it to 12%) 27. Your uncle developed a new video game franchise and left the rights to you. Under current copyright law, your intellectual property is protected for 95 years. You expect that the profits from the franchise will be $1 million in its first year and that this amount will grow at a rate of 5% per year during the life of the copyright. Assume each year’s profits are received at the end of the period. Further assume that once the copyright expires, imitations and knockoffs will drive the profits from the intellectual property to zero. What is the present value of the video game franchise if the interest rate is 9% per year? The profits from the franchise are a growing annuity. Using the formula for a growing annuity, we find that the present value is: P V =C 1 − 1 + g N! r − g 1 + r 95! =$1 million 0.09 − 0.051 − 1.05 1.09= $24.283 million 8. When Alfred Nobel died, he left the majority of his estate to fund 5 annual prizes starting one year after he died (the 6th one, in economics, was added later). The winners of the 2013 economics award were three finance professors (Eugene Fama, Lars Hansen and Robert Shiller). (a) If he wanted the cash award of each of the 5 prizes to be $45,000 and his estate could earn 7% per year, how much would he need to fund his prizes (assuming that they are awarded forever)? The total award is 5 ∗ $45, 000 = $225, 000. To endow a perpetuity of $225,000 per year that earns 7% per year, Alfred Nobel would need: P V =Cr=$225, 000 0.07 = $3, 214, 285.71 (b) How much would he need to fund his prizes today if the first cash award started in year 10? If the first award was in year 10, rather than year 1, he would need: P V9 =C10 r =$225, 000 0.07 = $3, 214, 285.71 Discounting this amount to today, P V0 =P V9 (1 + r)9 =$3, 214, 285.71 (1.07)9 = $1, 748, 358.46 3(c) If he wanted the value of each prize to keep up with inflation by growing 4% per year, how much would he need to leave? Assume that the award of $45,000 for each prize starts next year, as in part (a). Using the perpetuity formula when the cash flow grows, he would need: P V =C r − g =$225, 000 0.07 − 0.04 = $7, 500, 000 (d) If the awards ended after 50 years (rather than forever) and he wanted the value to keep up with inflation by growing at 4% per year, how much would he need to leave now? Using the growing annuity formula, he would need: P V =C 1 − 1 + g N! r − g 1 + r 50! =$225, 000 0.07 − 0.041 − 1 + 0.04 1 + 0.07= $5, 690, 580.18 (e) His heirs were surprised by his will and fought it. If they had been able to keep the amount of money you calculated in part (c) until today, 121 years after he died, and had invested it at 7% per year, how much would they have now? The amount from part (c) is $7,500,000. Then, today they would have P V =F V (1 + r)121 ⇐⇒ F V = P V ∗ (1 + r)121 = $7, 500, 000 ∗ (1.07)121 = $26, 946, 251, 772.69 4