###### Measuring and Modifying Risks

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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
- withdrawals made.

Cash outflows, on the other hand, refer to:

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

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 $1 per year. Determine the money-weighted rate of return.

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.*

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.

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; and
- 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; 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,

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

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.

Question 1A 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%

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 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%

SolutionThe correct answer is A.

We have two 1-year holding periods:

HP_{1}:P

_{0}= 50D= 0.5

P

_{1}= 53

HP_{2}:P

_{0}= 106D = 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,

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