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 + 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

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

The correct answer is A.

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,

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

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)