Which of the following statements about money-weighted and time-weighted returns is least accurate?
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The time-weighted method is not affected by the timing of cash flows. The other statements are true.
An investor buys four shares of stock for $50 per share. At the end of year one she sells two shares for $50 per share. At the end of year two she sells the two remaining shares for $80 each. The stock paid no dividend at the end of year one and a dividend of $5.00 per share at the end of year two. What is the difference between the time-weighted rate of return and the money-weighted rate of return?
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T = 0: Purchase of four shares = -$200.00 T = 1: Dividend from four shares = +$0.00 Sale of two shares = +$100.00 T = 2: Dividend from two shares = +$10.00 Proceeds from selling shares = +$160.00 The money-weighted return is the rate that solves the equation: $200.00 = $100.00 / (1 + r) + $170.00 / (1 + r)2. Cfo = -200, CF1 = 100, Cf2 = 170, CPT → IRR = 20.52%. The holding period return in year one is ($50.00 ? $50.00 + $0.00) / $50.00 = 0.00%. The holding period return in year two is ($80.00 ? $50.00 + $5.00) / $50 = 70.00%. The time-weighted return is [(1 + 0.00)(1 + 0.70)]1/2 ? 1 = 30.38%. The difference between the two is 30.38% ? 20.52% = 9.86%.
An investor buys one share of stock for $100. At the end of year one she buys three more shares at $89 per share. At the end of year two she sells all four shares for $98 each. The stock paid a dividend of $1.00 per share at the end of year one and year two. What is the investor’s money-weighted rate of return?
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T = 0: Purchase of first share = -$100.00 T = 1: Dividend from first share = +$1.00 Purchase of 3 more shares = -$267.00 T = 2: Dividend from four shares = +4.00 Proceeds from selling shares = +$392.00 The money-weighted return is the rate that solves the equation: $100.00 = -$266.00 / (1 + r) + 396.00 / (1 + r)2. CFO = -100; CF1 = -266; CF2 = 396; CPT → IRR = 6.35%.
An investor buys one share of stock for $100. At the end of year one she buys three more shares at $89 per share. At the end of year two she sells all four shares for $98 each. The stock paid a dividend of $1.00 per share at the end of year one and year two. What is the investor’s time-weighted rate of return?
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The holding period return in year one is ($89.00 ? $100.00 + $1.00) / $100.00 = -10.00%. The holding period return in year two is ($98.00 ? $89.00 + $1.00) / $89 = 11.24%. The time-weighted return is [{1 + (-0.1000)}{1 + 0.1124}]1/2 – 1 = 0.06%.
Assume an investor makes the following investments:
There are no transaction costs or taxes. The investor’s required return is 35.0%.
During year one, the stock paid a $5.00 per share dividend. In year two, the stock paid a $7.50 per share dividend.
The time-weighted return is:
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To calculate the time-weighted return: Step 1: Separate the time periods into holding periods and calculate the return over that period: Holding period 1: P0 = $50.00 D1 = $5.00 P1 = $75.00 (from information on second stock purchase) HPR1 = (75 ? 50 + 5) / 50 = 0.60, or 60% Holding period 2: P1 = $75.00 D2 = $7.50 P2 = $100.00 HPR2 = (100 ? 75 + 7.50) / 75 = 0.433, or 43.3%. Step 2: Use the geometric mean to calculate the return over both periods Return = [(1 + HPR1) × (1 + HPR2)]1/2 ? 1 = [(1.60) × (1.433)]1/2 ? 1 = 0.5142, or 51.4%.
An investor buys a share of stock for $200.00 at time t = 0. At time t = 1, the investor buys an additional share for $225.00. At time t = 2 the investor sells both shares for $235.00. During both years, the stock paid a per share dividend of $5.00. What are the approximate time-weighted and money-weighted returns respectively?
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Time-weighted return = (225 + 5 ? 200) / 200 = 15%; (470 + 10 ? 450) / 450 = 6.67%; [(1.15)(1.0667)]1/2 ? 1 = 10.8% Money-weighted return: 200 + [225 / (1 + return)] = [5 / (1 + return)] + [480 / (1 + return)2]; money return = approximately 9.4% Note that the easiest way to solve for the money-weighted return is to set up the equation and plug in the answer choices to find the discount rate that makes outflows equal to inflows. Using the financial calculators to calculate the money-weighted return: (The following keystrokes assume that the financial memory registers are cleared of prior work.) TI Business Analyst II Plus? HP 12C?
Miranda Cromwell, CFA, buys ?2,000 worth of Smith & Jones PLC shares at the beginning of each year for four years at prices of ?100, ?120, ?150 and ?130 respectively. At the end of the fourth year the price of Smith & Jones PLC is ?140. The shares do not pay a dividend. Cromwell calculates her average cost per share as [(?100 + ?120 + ?150 + ?130) / 4] = ?125. Cromwell then uses the geometric mean of annual holding period returns to conclude that her time-weighted annual rate of return is 8.8%. Has Cromwell correctly determined her average cost per share and time-weighted rate of return?
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Average cost |
Time-weighted return |
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Because Cromwell purchases shares each year for the same amount of money, she should calculate the average cost per share using the harmonic mean. Cromwell is correct to use the geometric mean to calculate the time-weighted rate of return. The calculation is as follows:
Year Beginning price Ending price Annual rate of return 1 ?100 ?120 20% 2 ?120 ?150 25% 3 ?150 ?130 ?13.33% 4 ?130 ?140 7.69%
TWR = [(1.20)(1.25)(0.8667)(1.0769)]1/4 ? 1 = 8.78%. Or, more simply, (140/100)1/4 ? 1 = 8.78%.
Robert Mackenzie, CFA, buys 100 shares of GWN Breweries each year for four years at prices of C$10, C$12, C$15 and C$13 respectively. GWN pays a dividend of C$1.00 at the end of each year. One year after his last purchase he sells all his GWN shares at C$14. Mackenzie calculates his average cost per share as [(C$10 + C$12 + C$15 + C$13) / 4] = C$12.50. Mackenzie then uses the internal rate of return technique to calculate that his money-weighted annual rate of return is 12.9%. Has Mackenzie correctly determined his average cost per share and money-weighted rate of return?
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Average cost |
Money-weighted return |
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Because Mackenzie purchased the same number of shares each year, the arithmetic mean is appropriate for calculating the average cost per share. If he had purchased shares for the same amount of money each year, the harmonic mean would be appropriate. Mackenzie is also correct in using the internal rate of return technique to calculate the money-weighted rate of return. The calculation is as follows:
Time Purchase/Sale Dividend Net cash flow 0 -1,000 0 -1,000 1 -1,200 +100 -1,100 2 -1,500 +200 -1,300 3 -1,300 +300 -1,000 4 400 × 14 = +5,600 +400 +6,000
CF0 = ?1,000; CF1 = ?1,100; CF2 = ?1,300; CF3 = ?1,000; CF4 = 6,000; CPT → IRR = 12.9452.
An investor makes the following investments:
During year one, the stock paid a $5.00 per share dividend. In year 2, the stock paid a $7.50 per share dividend. The investor’s required return is 35%. Her money-weighted return is closest to:
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To determine the money weighted rate of return, use your calculator's cash flow and IRR functions. The cash flows are as follows:
CF0: initial cash outflow for purchase = $50 Enter the cash flows and compute IRR:
CF1: dividend inflow of $5 - cash outflow for additional purchase of $75 = net cash outflow of -$70
CF2: dividend inflow (2 × $7.50 = $15) + cash inflow from sale (2 × $100 = $200) = net cash inflow of $215
CF0 = -50; CF1 = -70; CF2 = +215; CPT IRR = 48.8607
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