 # Money-weighted vs. Time-weighted Rates of Return

## Money-weighted Rate of Return

The money-weighted rate of return (MWRR) refers to the internal rate of return on a portfolio. It is the rate of discount, r, at which:

$$\text{PV of cash outflows} = \text{PV of cash inflows}$$

The money-weighted rate of return on a fund satisfies the equation of value by taking the initial and final fund values, as well as the intermediate cash flows, into account . When dealing with an investment portfolio, cash inflows comprise:

• the beginning value;
• dividends /interest reinvested; and

Cash outflows, on the other hand, refer to:

• the final value of the fund;
• dividends or interest received; and
• contributions.

#### Example: Money-weighted Rate of Return

Suppose you buy a stock at $100 and sell it a year later at$110. Let’s assume that the stock pays an annual dividend of per year. geembi.com The money-weighted rate of return is closest to:

Solution

In this case, the dividends received are outflows, so is the final value of the stock. The cost of the stock is the only inflow. Therefore,

If we let our MWRR = r,

$$\text{PV of outflow}=\text {PV of inflow}$$

$$1(1 + r)^{-1} + 110(1 + r)^{-1} = 100$$

Now, if we let (1 + r) to be ‘x’, then:

\begin{align*} & \frac {1}{x} + \frac {110}{x} = 100 \\ & \frac {111}{x} = 100 \\ \end{align*}

Therefore,

$$x = 1.11$$

But x = 1 + r

\begin{align*} 1 + r & = 1.11 \\ r & = 0.11 \text{ or } 11\% \\ \end{align*}

Exam tip: The exam usually tests the candidate’s understanding of the concept of money-weighted rate of return. Any calculations are unlikely to require the use of a calculator.

### Shortcoming of the Money-weighted Rate of Return

As we stated earlier, the money-weighted rate of return takes all the cash flows, including any withdrawal from the fund or contribution, into account. Assuming an investment extends to several periods, the MWRR puts more weight on the performance of the fund during periods when the size of the account is biggest. This is a disadvantage to fund managers as they may be unfairly penalized because of cash flows that are beyond their control.

## Time-weighted Rate of Return

The time-weighted rate of return (TWRR) measures the compound growth rate of an investment portfolio.  Unlike the money-weighted rate of return, TWRR is not sensitive to withdrawals or contributions. Essentially, the time-weighted rate of return is the geometric mean of the holding period returns of the respective sub-periods involved.

When working out time-weighted measurements, we break down the total investment period into many sub-periods. Each sub-period ends at the point where we have a significant withdrawal or contribution. It could also end after a month, quarterly, or even semi-annually. We encourage candidates to follow the steps below when computing TWRR:

1. establish the holding period return (HPR) for each sub-period;
2. add 1 to each HPR;
3. multiply all the (1 + HPR) terms; and
4. subtract 1 from the final product to get the compounded TWRR.

In a summary, compounded TWRR = {(1 + HPR1)*(1 + HPR2)*(1 + HPR3)…*(1 + HPRn-1)*(1 + HPRn)} – 1

Finally, annual time-weighted rate of return = (1 + compounded TWRR) 1/n – 1

Where n is the number of years.

#### Example: Time-weighted Rate of Return

An investor purchases a share of stock at t = 0 for $200. At the end of the year (at t = 1) the investor purchases an additional share of the same stock, this time for$220. She then sells both shares at the end of the second year for $230 each. She also received annual dividends of$3 per share at the end of each year. Calculate the annual time-weighted rate of return on her investment.

Solution

First, we break down the 2-year period into two 1-year periods.

Holding period 1:

beginning value  = 200;

dividends paid = 3; and

ending value = 220.

Holding period 2:

beginning value = 440 (2 shares * 220);

dividends paid = 6 (2 shares * 3); and

ending value  = 460 (2 shares * 230).

Secondly, we calculate the HPR for each period:

$$\text{HPR}_1 =\cfrac {(220 – 200 + 3)}{200} = 11.5\%$$

$$\text{HPR}_2 =\cfrac {(460 – 440 + 6)}{440} = 5.9\%$$

Lastly,

$$(1 + \text{annual TWRR})^2 = 1.115 * 1.059$$

Therefore,

$$\text {annual TWRR} = (1.115 * 1.059)^{0.5} – 1 = 8.7\%$$

## Money-weighted Rate of Return vs. Time-weighted Rates of Return

The money-weighted rate of return is sensitive to the amount and timing of cash flows and could lead to an unfair rating of the fund manager – they have no control over the amount or timing of cash flows. This effect is eliminated by the time-weighted rate of return. The money-weighted rate of return would only be superior to the TWRR only if the fund manager had complete control over cash flows and their timings.

A stock was valued at $20 on 1 January 2015 and$22 on 31 December 2015, at which time the holder sold his stake. During the year, a dividend of 0.4 per share was paid out to shareholders. Determine the money-weighted rate of return. A. 1.12 B. 12% C. 200% Solution The correct answer is B. $$\text{PV of outgo} =\text {PV of income}$$ $$0.4(1 + r)^{-1} + 22(1 + r)^{-1} = 20$$ If we let (1 + r) to be ‘x’, \begin{align*} \frac {0.4}{x} + \frac {22}{x} & = 20 \\ \frac {22.4}{x} & = 20 \\ x & =\frac {22.4}{20} = 1.12 \\ r & = 1.12 – 1 = 0.12 \text{ or } 12\% \end{align*} ## Question 2 A chartered analyst buys a share of stock at time t = 0 for50. At t = 1, he purchases an extra share of the same stock for $53. The share gives a dividend of$0.50 per share for the first year and $0.60 per share for the second year. He sells the shares at the end of the second year for$55 per share. Calculate the annual time-weighted rate of return.

A. 5.9%

B. 12.24%

C. 7%

Solution

We have two 1-year holding periods:

HP1:

P0 = 50

D= 0.5

P1 = 53

HP2:

P0= 106

D = 1.2

P1 = 110

We now calculate the holding period returns:

\begin{align*} \text{HPR}_1 & =\cfrac {(53 – 50 + 0.5)}{50} = 7\% \\ \text{HPR}_2 & =\cfrac {(110 – 106 + 1.2)}{106} = 4.9\% \\ \text{Compounded TWRR} & = 1.07 * 1.049 = 12.24\% \end{align*}

Therefore,

$$\text {Annual TWRR} = (1 + 0.1224)^{0.5} – 1 = 5.9\%$$

Shop CFA® Exam Prep

Offered by AnalystPrep Level I
Level II
Level III
All Three Levels
Featured Shop FRM® Exam Prep FRM Part I
FRM Part II
FRM Part I & Part II
Learn with Us

Subscribe to our newsletter and keep up with the latest and greatest tips for success
Shop Actuarial Exams Prep Exam P (Probability)
Exam FM (Financial Mathematics)
Exams P & FM
Shop GMAT® Exam Prep Complete Course Sergio Torrico
2021-07-23
Excelente para el FRM 2 Escribo esta revisión en español para los hispanohablantes, soy de Bolivia, y utilicé AnalystPrep para dudas y consultas sobre mi preparación para el FRM nivel 2 (lo tomé una sola vez y aprobé muy bien), siempre tuve un soporte claro, directo y rápido, el material sale rápido cuando hay cambios en el temario de GARP, y los ejercicios y exámenes son muy útiles para practicar. diana
2021-07-17
So helpful. I have been using the videos to prepare for the CFA Level II exam. The videos signpost the reading contents, explain the concepts and provide additional context for specific concepts. The fun light-hearted analogies are also a welcome break to some very dry content. I usually watch the videos before going into more in-depth reading and they are a good way to avoid being overwhelmed by the sheer volume of content when you look at the readings. Kriti Dhawan
2021-07-16
A great curriculum provider. James sir explains the concept so well that rather than memorising it, you tend to intuitively understand and absorb them. Thank you ! Grateful I saw this at the right time for my CFA prep. nikhil kumar
2021-06-28
Very well explained and gives a great insight about topics in a very short time. Glad to have found Professor Forjan's lectures. Marwan
2021-06-22
Great support throughout the course by the team, did not feel neglected Benjamin anonymous
2021-05-10
I loved using AnalystPrep for FRM. QBank is huge, videos are great. Would recommend to a friend Daniel Glyn
2021-03-24
I have finished my FRM1 thanks to AnalystPrep. And now using AnalystPrep for my FRM2 preparation. Professor Forjan is brilliant. He gives such good explanations and analogies. And more than anything makes learning fun. A big thank you to Analystprep and Professor Forjan. 5 stars all the way! michael walshe
2021-03-18
Professor James' videos are excellent for understanding the underlying theories behind financial engineering / financial analysis. The AnalystPrep videos were better than any of the others that I searched through on YouTube for providing a clear explanation of some concepts, such as Portfolio theory, CAPM, and Arbitrage Pricing theory. Watching these cleared up many of the unclarities I had in my head. Highly recommended.