# LO 70.1: Differentiate between time-weighted and dollar-weighted returns of a

LO 70.1: Differentiate between time-weighted and dollar-weighted returns of a portfolio and describe their appropriate uses.
The dollar-weighted rate of return is defined as the internal rate of return (IRR) on a portfolio, taking into account all cash inflows and outflows. The beginning value of the account is an inflow as are all deposits into the account. All withdrawals from the account are outflows, as is the ending value.
Example: Dollar-weighted rate of return
Assume an investor buys a share of stock for $100 at t = 0, and at the end of the next year (t = 1), she buys an additional share for$120. At the end of year 2, the investor sells both shares for $130 each. At the end of each year in the holding period, the stock paid a$2.00 per share dividend. What is the investors dollar-weighted rate of return?
2018 Kaplan, Inc.
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Topic 70 Cross Reference to GARP Assigned Reading – Bodie, Kane, and Marcus, Chapter 24
Answer:
Step 1: Determine the timing of each cash flow and whether the cash flow is an inflow
(+) or an outflow (). t = 0:
purchase of first share
= $100 = +$2 = $120$118
dividend from first share purchase of second share subtotal, t = 1
t = 1:
t = 2:
dividend from two shares proceeds from selling shares = +$260 +$264 subtotal, t = 2
= +$4 Step 2: Net the cash flows for each time period, and set the PV of cash inflows equal to the present value of cash outflows. PV:inflows = PVoutflow s$100 + $120 U + r)$2
(1 + r)
$264 (1 + r)2 Step 3: Solve for r to find the dollar-weighted rate of return. This can be done using trial and error or by using the IRR function on a financial calculator or spreadsheet. The intuition here is that we deposited$100 into the account at t = 0, then added $118 to the account at t = 1 (which, with the$2 dividend, funded the purchase of one more share at $120), and ended with a total value of$264.
To compute this value with a financial calculator, use these net cash flows and follow the procedure described in Figure 1 to calculate the IRR.
Net cash flows: CFQ = 100; CFj = 120 + 2 = 118; CF2 = 260 + 4 = 264
Figure 1: Calculating Dollar-Weighted Return With the TI Business Analyst II Plus
Calculator
Key Strokes
Explanation
[CF] [2nd] [CLR WORK] Clear cash flow registers
100 [+/-] [ENTER] [4-] 118 [+/-] [ENTER]   264 [ENTER] [IRR] [CPT] Initial cash outlay Period 1 cash flow Period 2 cash flow
Calculate IRR
Display
CFO = 0.00000
CFO = -100.00000 C01 =-118.00000 C02 = 264.00000 IRR = 13.86122
The dollar-weighted rate of return for this problem is 13.86%.
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2018 Kaplan, Inc.
Topic 70 Cross Reference to GARP Assigned Reading – Bodie, Kane, and Marcus, Chapter 24
Time-weighted rate of return measures compound growth. It is the rate at which $1.00 compounds over a specified time horizon. Time-weighting is the process of averaging a set of values over time. The annual time-weighted return for an investment may be computed by performing the following steps: Step 1: Value the portfolio immediately preceding significant addition or withdrawals. Form subperiods over the evaluation period that correspond to the dates of deposits and withdrawals. Step 2: Compute the holding period return (HPR) of the portfolio for each subperiod. Step 3: Compute the product of (1 + HPRt) for each subperiod t to obtain a total return for the entire measurement period [i.e., (1 + HPRt) x (1 + HPR2) … (1 + HPRn)]. If the total investment period is greater than one year, you must take the geometric mean of the measurement period return to find the annual time-weighted rate of return. Example: Time-weighted rate of return A share of stock is purchased at t = 0 for$100. At the end of the next year, t = 1, another share is purchased for $120. At the end of year 2, both shares are sold for$130 each. At the end of years 1 and 2, the stock paid a $2.00 per share dividend. What is the time- weighted rate of return for this investment? (This is the same data as presented in the dollar-weighted rate-of-return example.) Answer: Step 1: Break the evaluation period into two subperiods based on timing of cash flows. Holding period 1: Holding period 2: beginning price =$100.00 dividends paid = $2.00 ending price =$120.00
beginning price = $240.00 (2 shares) dividends paid =$4.00 ($2 per share) ending price =$260.00 (2 shares)
Step 2: Calculate the HPR for each holding period.
HPRj = [($120 + 2) /$100] – 1 = 22%
HPR2 = [($260 + 4) /$240] – 1 = 10%
Step 3: Take the geometric mean of the annual returns to find the annualized time-
weighted rate of return over the measurement period.
(1 + time-weighted rate of return) = (1.22) (1.10)
time-weighted rate of return = y/( 1.22)(1.10) – 1 = 15.84%
2018 Kaplan, Inc.
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Topic 70 Cross Reference to GARP Assigned Reading – Bodie, Kane, and Marcus, Chapter 24
In the investment management industry, the time-weighted rate of return is the preferred method of performance measurement for a portfolio manager because it is not affected by the timing of cash inflows and outflows, which may be beyond the managers control.
In the preceding examples, the time-weighted rate of return for the portfolio was 15.84%, while the dollar-weighted rate of return for the same portfolio was 13.86%. The difference in the results is attributable to the fact that the procedure for determining the dollar- weighted rate of return gave a larger weight to the year 2 HPR, which was 10% versus the 22% HPR for year 1.
If funds are contributed to an investment portfolio just before a period of relatively poor portfolio performance, the dollar-weighted rate of return will tend to be depressed. Conversely, if funds are contributed to a portfolio at a favorable time, the dollar-weighted rate of return will increase. The use of the time-weighted return removes these distortions, providing a better measure of a managers ability to select investments over the period. If a private investor has complete control over money flows into and out of an account, the dollar-weighted rate of return may be the more appropriate performance measure.
Therefore, the dollar-weighted return will exceed the time-weighted return for a manager who has superior market timing ability.
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