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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 semiannually. We encourage candidates to follow the procedure 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

Summarily, 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. This effect is eliminated by the time-weighted rate of return. The money-weighted rate of return would only be superior to the TWRR if and only if the fund manager had complete control over cash flows and their timings.

QuestionA 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,

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

*Reading 7 LOS 7d*

*Calculate and compare the money-weighted and time-weighted rates of return of a portfolio and evaluate the performance of portfolios based on these measures. (Part two)*