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Summer 2012 Test 2 solution sketches

Summer 2012 Test 2 solution sketches. 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. 1(b).

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Summer 2012 Test 2 solution sketches

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  1. Summer 2012Test 2 solution sketches

  2. 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

  3. 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 12th root of 1.17 is 1.0131696 • Monthly rate is 1.31696% • Monthly payment is $300,000(0.0131696) • $3,950.88

  4. 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

  5. 2Belly 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

  6. 2Belly 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)

  7. 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%

  8. 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%

  9. 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

  10. 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%

  11. 5Lotta 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 

  12. 5Lotta Love/Soren • 40,000 = 1,000 + 2,000/1.242 + X/1.244 + (4000/.24)[1 – 1/1.248]/1.245 • Solve for X • $78,092.93 5 years of discounting Annuity formula

  13. 6Zero-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 $1,100/1.122 = $876.91 • The old value was $900 • The change in value • $876.91 - $900 = -$23.09 • Down by $23.09

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