1215

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?

A) 6.35%.

B) 0.06%.

C) 11.24%.

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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%.

1215

Assume an investor makes the following investments:

• Today, she purchases a share of stock in Redwood Alternatives for $50.00.
• After one year, she purchases an additional share for $75.00.
• After one more year, she sells both shares for $100.00 each.

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:

A) 51.4%.

B) 23.2%.

C) 51.7%.

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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%.

1215

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?

A) 7.7%; 7.7%.

B) 9.0%; 15.0%.

C) 10.8%; 9.4%.

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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^?

• Enter CF₀: 200, +/-, Enter, down arrow
• Enter CF₁: 220, +/-, Enter, down arrow, down arrow
• Enter CF₂: 480, Enter, down arrow, down arrow, 
• Compute IRR: IRR, CPT
• Result:  9.39

HP 12C^?

• Enter CF₀: 200, CHS, g, CF₀
• Enter CF₁: 220, CHS, g, CF_j
• Enter CF₂: 480, g, CF_j 
• Compute IRR: f, IRR
• Result:  9.39

1215

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?

Average cost

Time-weighted return

A) Incorrect Correct

B) Correct Correct

C) Correct Incorrect

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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%.

1215

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?

Average cost

Money-weighted return

A) Correct Correct

B) Incorrect Correct

C) Correct Incorrect

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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.

1215

 

An investor makes the following investments:

• She purchases a share of stock for $50.00.
• After one year, she purchases an additional share for $75.00.
• After one more year, she sells both shares for $100.00 each.
• There are no transaction costs or taxes.

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:

A) 48.9%.

B) -7.5%.

C) 16.1%.

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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
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

Enter the cash flows and compute IRR:
CF0 = -50; CF1 = -70; CF2 = +215; CPT IRR = 48.8607