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Summer 2012 Test 2 solution sketches

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1(a) You are paid $500 per year for three years, starting today. If the stated annual discount rate is 5%, compounded continuously, what is the total present value of the three payments? PV = $500 + $500/exp(0.05) + $500/exp(0.05 * 2) = $1,428.03

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1(b) If Jacob takes out a mortgage of $300,000 and only pays off the interest incurred each month, how much will his monthly payment be if the effective annual interest rate is 17%? The 12 th root of 1.17 is 1.0131696 Monthly rate is 1.31696% Monthly payment is $300,000(0.0131696) $3,950.88

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1(c) Stark’s Sizzling Steaks is considering opening a new location in Walla Walla, WA. In order to open, $2 million must be invested today. If the restaurant opens, the yearly operating profits are $300,000 per year forever, starting 1 year from today. (Profits are achieved only once a year in this example.) For what discount rates would the net present value be positive if the new restaurant is opened? All numbers are in millions of dollars –2 + 0.3/r > 0 r < 0.15

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2 Belly Batteries, Inc. Belly Batteries, Inc. has just paid out its annual dividend of $5 earlier today. The annual dividend will go up by 10% each of the next 5 years, followed by no growth after that. What will the price of this stock be 3 years from today if the effective annual discount rate is 9%? (Note: Provide the price AFTER the dividend has been paid.) Note that we need to calculate the future value, 3 years from today Also note that payments made between now and Year 3 are NOT counted in the value of the stock

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2 Belly Batteries, Inc. Dividend in Year 4 $5(1.1) 4 = $7.3205 FV in Year 3 dollars: $7.3205/1.09 = $6.7161 Dividend in Years 5 onward $5(1.1) 5 = $8.0525 FV in Year 3 dollars for all dividends paid in Years 5 onward (note that we have to discount the perpetuity formula used by 1 year) ($8.05255/0.09)(1/1.09) = $82.0851 Total of all dividends paid in Years 4 onward: Add up bolded numbers $88.80 (rounded to the nearest cent)

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3: A sample of a stock’s returns over the past five years was 200%, 20%, -40%, 40%, -100%. (a) What is the geometric average return over the five-year period? The fifth root of (1+2)(1+.2)(1-.4)(1+.4)(1-1), minus 1 0 – 1 = -1 = -100%

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3: A sample of a stock’s returns over the past five years was 200%, 20%, -40%, 40%, -100%. (b) What is the standard deviation of this sample? Arithmetic mean is (2+.2-.4+.4-1)/5 0.24 = 24% Variance (note that we lose a degree of freedom since we have a sample here) (1/4)[(2-.24) 2 + (.2-.24) 2 + (-.4-.24) 2 + (.4-.24) 2 + (-1-.24) 2 ] 1.268 Standard deviation is the square root of the variance 1.126, or 112.6%

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3: A sample of a stock’s returns over the past five years was 200%, 20%, -40%, 40%, -100%. (c) If someone invested $100 in this stock five years ago, how much would this stock be worth today? Two ways to find the answer $100(1+2)(1+.2)(1-.4)(1+.4)(1-1) = 0 You have a negative 100% return in your final year, which means you lose your entire amount You are left with 0

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4: Nominal return/inflation/real return In 1946, the nominal return for large- company stocks was –8.18%, and the Consumer Price Index (CPI) went up by 18.13%. Assuming that the CPI represents the inflation rate, what was the real rate of return of large-company stocks in 1946? 1 + nominal = (1 + real)(1 + inflation) 1 – 0.0818 = (1 + real)(1 + 0.1813) Solve for real to be -0.22272 or -22.272%

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5 Lotta Love/Soren Lotta Love will receive many payments from Soren. She will receive $1,000 today, $2,000 two years from today, $X four years from today and $4,000 every year for 8 years, starting six years from today. The present value of all payments is $40,000, and her effective annual discount rate is 24%. Find X. PV of all future payments (including today’s) must be $40,000

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5 Lotta Love/Soren 40,000 = 1,000 + 2,000/1.24 2 + X/1.24 4 + (4000/.24)[1 – 1/1.24 8 ]/1.24 5 Solve for X $78,092.93 Annuity formula 5 years of discounting

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6 Zero-coupon bond A zero-coupon bond is purchased for $900 at 10 am today, with a face value of $1,100 to be paid two years from today. Later today, at 1 pm, the yield to maturity (calculated on a yearly basis) changes to 12%. How much does the value of the bond change between 10 am and 1 pm? (Make sure to clearly state if the value goes up or down.) The new value is $1,100/1.12 2 = $876.91 The old value was $900 The change in value $876.91 - $900 = -$23.09 Down by $23.09

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