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# 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.

## Time-weighted Rate of Return Formula

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:

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

Summarily, 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

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,

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

## Money-weighted Rate of Return Vs Time-weighted Rate 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 if and only if the fund manager had complete control over cash flows and their timings.

## Question

A 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%

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,

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

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)

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