Additional+practice+exercises+topics+2+and+3

Additional+practice+exercises+topics+2+and+3 - Additional...

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Unformatted text preview: Additional practice exercises topics 2 (time value of money) and 3 (bond valuation) Solutions are provided at the end. Question 1: (TVM) You can buy property today for $3 million and sell it in 5 years for $4 million. You will earn no rental income on the property. a. If the interest rate is 8%, should you buy the property? b. Suppose the property earns $200,000 rent, at the beginning of each year. Should you buy it? c. Suppose you think that there is only a 75% chance the property will sell for $4,000,000 and a 25% chance that it will sell for $3,000,000 in 5 years. Should you buy the property? Question 2: (TVM) Jennifer JackPot has just won lottery. She will get $50,000 every 6 months for 10 years (20 payments). The first payment will be coming 4.5 years from today; subsequent payments will be every 6 months thereafter. If the discount rate is 13.61% compounded monthly, a. What is the value of this annuity three years from now? b. What is today’s value of the annuity? Question 3: (TVM) Tim Crony is the Knight errant of the Peachland in the Court of Luxemstein. He is 60 years old, with an average life expectancy of 87, and is paid $60,000 annually. King Vincent Nepotism wants him to resign so that a distant cousin of King Vincent can have the job. Tim refuses, since the appointment is for life. He suggests instead that he retires in 5 years with a pension of $50,000 per annum, with his wife (who is expected to outlive him by 20 years) getting half on his death. In addition he wants to stay on staff for 5 years after his retirement as an adviser to the court at a stipend of $5,000 per year. Tim is the traditional director of Luxemstein’s once a century anniversary celebrations, coming up in 10 years. He would receive an additional honorarium of $40,000 per year for 2 years before the celebrations (years 9 and 10), and he wants to receive those payments as well. Finally, he wants to receive a lump sum payment of $100,000 at the retirement date. King Vincent makes a counter offer, “We will pay you an extra $130,000 per year for the next 5 years and a lump sum of $40,000 at age 65 if you retire now. All payments come at year end. The interest rate is 7%. Which alternative gives Tim the highest present value, using expected lives? Question 4: (Bonds) Donald has just purchased a 10 year bond of WorldCheese Corporation. The bond has a coupon rate of 8% payable annually. The required rate of return is 9%. Suppose one year later required rate of return declined to 7% and stayed at that level for the remaining years. If Donald sells the bonds one year after purchase then what would his current yield, capital gain yield and the total yield be? 1 Solutions Answer 1: a. If you invest $3,000,000 at 8% in 5 years you will have $3,000,000(1.08)5 = $4,407,984. After 5 years property can only be sold for $4,000,000, therefore, you should not buy it. OR, alternatively: PV of $4,000,000 property selling price @ 8% is $4,000,000/(1.08)5 = $2,722,333. Since the price of property today is $3,000,000, therefore, you should not buy it. b. The rental payments are an annuity due. Calculate the present value: 1 ⎛ ⎜1− (1.08)5 PV(rental payments) = $200,000⎜ ⎜ 0.08 ⎜ ⎝ ⎞ ⎟ ⎟(1.08) = $862,425 ⎟ ⎟ ⎠ PVof total benefits = PV of building selling price + PV of rental payments = $2,722,333 + 862,425 = $3,584,758 Since PV of benefits is more than the purchase price of the building, you should buy the building. c. Expected selling price = = 0.75( 4,000,000) + 0.25(3,000,000) = $3,750,000 PV of expected selling price = $3,750,000/(1.08)5 = $2,552,187 PV of total benefits = $2,552,187 + 862,425 = $3,415,312 Since PV of benefits is more than the purchase price of the building, therefore, you should buy the building. Answer 2: a. Monthly rate = 0.1361/12 = 0.01134 Semi annual rate = (1.01134)6 ‐1= 7% Consider everything in 6‐month periods: Cash Flow 50K 50K 50K 50K 50K … 50K Time 1 2 3 4 5 6 7 8 9 10 11 12 13 … 28 First, calculate the value at t=8 (i.e. one period before the first cash flow): 1 ⎛ ⎜1− (1.07) 20 PV(time 8) = $50,000⎜ ⎜ 0.07 ⎜ ⎝ ⎞ ⎟ ⎟ = $529,701 ⎟ ⎟ ⎠ Next, calculate the value at t=6, i.e. three years from now: PV(time 6) = $529,701/(1.07)2 = $462,661 b. PV(t=0) = $462,661/(1.07)6 = $308,291 2 Answer 3: First, draw a timeline. retire 60 65 69 70 87 107 |‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐|‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐| 40,000 40,000 | $5000 p.a. for 5 years| |$50,000 p.a. for 22 years | $25,000 p.a. for 20 yrs | $100,000 PV of Tim’s idea At age 65 1 ⎛ ⎜1− (1.07 )22 $100,000 + 50,000⎜ ⎜ 0.07 ⎜ ⎝ 1 ⎛ ⎜1− (1.07 )20 ⎜ ⎜ 0.07 ⎞ ⎟ ⎜ ⎟ + 25,000 ⎝ ⎟ (1.07 )22 ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ 1 ⎛ ⎟ ⎜1− (1.07 )5 ⎠ + 5,000⎜ ⎜ 0.07 ⎜ ⎝ ⎞ ⎟ ⎟ + 40,000 + 40,000 ⎟ (1.07 )4 (1.07 )5 ⎟ ⎠ = $100,000 + 553,062 + 59,780 + 20,501 + 30,516 + 28,519 = $792,378 At age 60 $792,378/ (1.07)5 = $564,955 PV of King’s offer At age 65 ⎛ (1.07 ) − 1 ⎞ ⎟ $130,000⎜ ⎜ 0.07 ⎟ + 40,000 = $745,596 + 40,000 = $787,596 ⎝ ⎠ At age 60 $787,596/ (1.07)5 = $561,545 Whether we compare at age 60 or at 65, Tim is getting a higher value with his idea. We have ignored $60,000 p.a. salary as we are assuming that in both cases he will keep getting the salary till age 65. Answer 4: 1 ⎛ ⎞ ⎜1− 10 ⎟ (1.09) ⎟ $1000 P0 = $80⎜ + = $935.82 ⎜ ⎟ 1.0910 0.09 ⎜ ⎟ ⎝ ⎠ 1⎞ ⎛ ⎜1− ⎟ (1.07) 9 ⎟ $1000 ⎜ P1 = $80 + = $1065.12 ⎜ 0.07 ⎟ 1.07 9 ⎜ ⎟ ⎝ ⎠ Current Yield = Capital Gain Yield= Total Yield = $80/935.82 (1065.12‐935.82)/935.82 0.0855+0.1382 = = = 3 0.0855 0.1382 0.2237 ...
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