###### Calculating Returns Using CAPM

Given as asset systematic risk, the expected return can be computed using the... **Read More**

The money-weighted rate of return (MWRR) refers to a portfolio’s internal rate of return. 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 value equation 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.
- Withdrawals made.

Cash outflows, on the other hand, refer to:

- The final value of the fund.
- Dividends or interest received.
- Contributions.

Assume you bought a stock at $100 and sold it a year later at $110. Further, assume that the stock pays an annual dividend of $1 per year. The money-weighted rate of return is *closest* to:

**Solution**

In this case, the dividends received are outflows, and 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.*

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

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:

- Establish the holding period return (HPR) for each sub-period.
- Add 1 to each HPR.
- Multiply all the (1 + HPR) terms.
- Subtract 1 from the final product to get the compounded TWRR.

In a summary, compounded TWRR = {(1 + HPR_{1})*(1 + HPR_{2})*(1 + HPR_{3})…*(1 + HPR_{n-1})*(1 + HPR_{n})} – 1

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

Where n is the number of years.

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.

Ending value = 220.

*Holding period 2:*

beginning value = 440 (2 shares * 220).

dividends paid = 6 (2 shares * 3).

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,

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

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. The time-weighted rate of return eliminates this effect. The money-weighted rate of return would only be superior to the TWRR if the fund manager had complete control over cash flows and their timings.

Question 1A stock was valued at $20 on January 1, 2015 and $22 on December 31, 2015, when 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%.

SolutionThe 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 2A chartered analyst buys a share of stock at time t = 0 for $50. 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 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%.

SolutionThe correct answer is

A.We have two 1-year holding periods:

HP_{1}:P

_{0}= 50.D= 0.5.

P

_{1}= 53.

HP_{2}:P

_{0}= 106.D = 1.2.

P

_{1}= 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,

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