2024 CFA Level I Exam: CFA Study Preparation

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Basic Question 2 of 6

A financial analyst is in the process of measuring the annualized return of an investment portfolio. Consider the following information:

t0: purchase an initial 1 share of Microscam for $65.40
t1: purchase an additional 1 share of Microscam for $68.12
t1: receive a dividend of $0.75
t2: purchase an additional 1 share of Microscam for $75.95
t2: receive a dividend of $0.77
t3: sell 3 shares for $82.76 per share

Assuming that there are no taxes or transaction costs, that dividends are not reinvested, and that each period represents one year, what is the time-weighted rate of return per year on this portfolio?

A. 8.27% per year
B. 10.73% per year
C. 8.92% per year

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Correct Answer: C

The time-weighted rate of return is the preferred method of return calculation in the investment management industry primarily because the return calculation it produces is not sensitive to significant additions and withdrawals of funds from portfolios under examination. The calculation of the time-weighted rate of return involves three steps:

Step 1: Price the portfolio immediately prior to any significant additions or withdrawals. Separate the portfolio into a series of sub-periods based on the dates of cash inflows and outflows.
Step 2: Calculate the holding period return for each sub-period.
Step 3: Determine the annualized holding period return by linking or compounding the holding period return of each sub-period. If the investment is for more than one year, use the geometric mean of the annual returns as the time-weighted rate of return. If the investment is for less than one year, compound the sub-period returns to obtain an annualized measurement.

To begin the process of determining the time-weighted rate of return, we would break the portfolio up into the following series of cash flows. However, in this example, the cash flows are already aggregated for us and we can move on to the next step: determining the holding period return for each sub-period. This process is detailed as follows:
t1: [($68.12 ending price + $0.75 dividend received - $65.40 beginning price) / $65.40 beginning price] = 5.306%
t2: [($75.95 ending price + $0.77 dividend received - $68.12 beginning price) / $68.12 ending price] = 12.625%
t3: [$82.76 ending price - $75.95 beginning price / $75.95 beginning price] = 8.966%

Now that the holding period returns for each sub-period have been calculated, the determination of the time-weighted rate of return can take place. Since the duration of these transactions exceeds one year, we must take the geometric mean of the annual returns to obtain the time-weighted rate of return. This is done by taking the cube root of [(1 + .05306) * (1 + .12625) * (1 + .08966], subtracting 1, and multiplying by 100%, which leads to a time-weighted rate of return of .0892 or 8.92%. If you chose 10.73%, remember that it is the geometric mean that is used in the time-weighted rate of return calculation, not the arithmetic mean.

User Contributed Comments 14

User	Comment
yly13	note that since dividends are not reinvested, it is only included in the ending price
0is4eva	Three steps, multiply, 3rd root: A. (68.12+0.75)/65.4 = 1.05306 B. (275.95+20.77)/(268.12)=1.12625 C. (382.76)/(375.95)=1.08966 D. 1.05306 1.12625 * 1.08966 = 1.29235 E. (1.29235)^(1/3) = 1.0892 ==> 8.92%
Cooltallgal	Very clear explaination, thanks Ois4eva!!
pierreE14	I know it is not the question but I got 9.35% for the money weighted rate of return.
julescruis	what you calculated is the IRR for this project not the time weighted rate of return
SriSri	I got lost thinking equation given in Theory was different than what is used in here, but took while to understand they are the same! :) ie. (div + endP)/begP - 1 is eqaul to (div + endp - begP )/begP
najm	Why take the qube root? I thought there is no qube root. please any one explain.
Rinoa86	qube root because it's three time periods.
hillrat	what is the key strokes for cube root on the ba 11 i haven't used this type of calc in so long, it's so much easier on ti 83 So, to calculate the 5th root of 100, we simply raise 100 to the 1/5th power. To do this: 100 yx 5 1/x =. In this example, the 5th root of 100 equals 2.51189.
Saxonomy	I prefer.. t1: -65.4 + 0.75 + 68.12 = 1.0531 t2: -136.24 + 1.54 + 151.9 = 1.1262 t3: -227.85 + 248.28 = 1.0897 (Note that 1.54 in t2 is the dividend for 2 shares of stock i.e. 0.77 * 2) You take the cube root of (1.05311.12621.0897) because you are trying to determine the combined effect/influence of the three separate period returns. Take the squareds and roots of 1 allows us to isolate the overall effect of the returns. Hope this helps.
Saxonomy	Oh crap, I forgot to divide the sums by each period's initial outlay (i.e. 65.4, 136.24 and 227.85 respectively). The answer is correct, just forgot to include that I divided the sum. Whoever has time, pls help me type it out. I have 100,000 more sections to study.
Ifi2703	To calculate cube root on TI BAII Plus, calculate the holding period returns and then multiply them all together. Next, using the "Y^x" button, enter the calculated value as "Y" and the cube root (1/3) as "x". This should give you the cube root and then you can subtract 1 and x 100 to get the answer.
bfeitosa	0is4Eva: Notice that you do not need to multiply each share price by the amount of shares you have in each different period. You are multiplying by 2 and 3 in both variables in your steps B. and C. That basically means that your return does not vary by taking into consideration the amount of shares you have (in a 1 stock portfolio). If it is a portfolio with multiple stocks than if you are overweight one stock that will make a difference in your total return.
Yrazzaq88	If you can't do this question properly the first time, keep practicing and take them step by step to learn what you are really trying to do 1) Divide into sub-periods (i.e: To, T1, T2, T3) 2) Calculate HPT for each period 3) Use geometric mean calculation 4) Cube Root >> Take the HPTs, multiply together, then use Y^x and input 3, then press 1/x, then press = sign. 5) Answer should be the same. Repeat, rinse, and enjoy.

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Learning Outcome Statements

compare the money-weighted and time-weighted rates of return and evaluate the performance of portfolios based on these measures

CFA® 2024 Level I Curriculum, Volume 1, Module 1.