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On January 1, Jonathan Wood invests $50,000. At the end of March, his investment is worth $51,000. On April 1, Wood deposits $10,000 into his account, and by the end of June, his account is worth $60,000. Wood withdraws $30,000 on July 1 and makes no additional deposits or withdrawals the rest of the year. By the end of the year, his account is worth $33,000. The time-weighted return for the year is closest to:

Which of the following is most accurate with respect to the relationship of the money-weighted return to the time-weighted return? If funds are contributed to a portfolio just prior to a period of favorable performance, the:

A)	money-weighted rate of return will tend to be elevated.

B)	time-weighted rate of return will tend to be elevated.

C)	money-weighted rate of return will tend to be depressed.

The time-weighted returns are what they are and will not be affected by cash inflows or outflows. The money-weighted return is susceptible to distortions resulting from cash inflows and outflows. The money-weighted return will be biased upward if the funds are invested just prior to a period of favorable performance and will be biased downward if funds are invested just prior to a period of relatively unfavorable performance. The opposite will be true for cash outflows.

A)	return on invested capital.

B)	internal rate of return (IRR) of a portfolio.

C)	measure of the compound rate of growth of $1 over a stated measurement period.

Which of the following statements regarding the money-weighted and time-weighted rates of return is least accurate?

A)	The money-weighted rate of return removes the effects of the timing of additions and withdrawals to a portfolio.

B)	The time-weighted rate of return reflects the compound rate of growth of one unit of currency over a stated measurement period.

C)	The time-weighted rate of return is the standard in the investment management industry.

The money-weighted return is actually highly sensitive to the timing and amount of withdrawals and additions to a portfolio. The time-weighted return removes the effects of timing and amount of withdrawals to a portfolio and reflects the compound rate of growth of $1 over a stated measurement period. Because the time-weighted rate of return removes the effects of timing, it is the standard in the investment management industry.

An analyst managed a portfolio for many years and then liquidated it. Computing the internal rate of return of the inflows and outflows of a portfolio would give the:

Time-weighted returns are used by the investment management industry because they:

A)	result in higher returns versus the money-weighted return calculation.

B)	are not affected by the timing of cash flows.

C)	take all cash inflows and outflows into account using the internal rate of return.

Time-weighted returns are not affected by the timing of cash flows. Money-weighted returns, by contrast, will be higher when funds are added at a favorable investment period or will be lower when funds are added during an unfavorable period. Thus, time-weighted returns offer a better performance measure because they are not affected by the timing of flows into and out of the account.

Why is the time-weighted rate of return the preferred method of performance measurement?

A)	Time weighted allows for inter-period measurement and therefore is more flexible in determining exactly how a portfolio performed during a specific interval of time.

B)	Time-weighted returns are not influenced by the timing of cash flows.

C)	There is no preference for time-weighted versus money-weighted.

Which of the following statements about money-weighted and time-weighted returns is least accurate?

A)	The money-weighted return applies the concept of internal rate of return to investment portfolios.

B)	If the investment period is greater than one year, an analyst must use the geometric mean to calculate the annual time-weighted return.

C)	If a client adds funds to an investment prior to an unfavorable market, the time-weighted return will be depressed.

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?

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?

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%.

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%.

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.

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?

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

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?

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?

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.

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

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%

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