# Time-Weighted Rate of Return

Time-weighted rate of return is the compound rate of growth over a period on one unit of currency invested at the start of the period. It is called time-weighted because it gives equal weightage to each of the sub-period returns.

It is one of the two methods for calculating rate of return over multiple periods: the other being the money-weighted rate of return which calculates the compound rate such that the rate of return in the periods of high amount of invested money is given higher weightage and vice versa.

## Formulas

Time-weighted return for a single period is calculated using the following formula:

$$Time\text{-}weighted\ Rate\ of\ Return \\= \frac{Ending\ Value − Beginning\ Value}{Beginning\ Value}$$

In case of any external cash flows: new investments, withdrawals, dividends, etc., the time-weighted return is calculated for each sub-period and then chain-linked using the following formula:

Time-weighted return for multiple periods
= (1 + 1st period return) × (1 + 2nd period return) – 1

## Example

Guddu Gupta is a client of RAK Asset Management. He started his portfolio with an investment of $10,000 on 1 January 2009. The value of his portfolio on 31 December 2009 was$13,050. Impressed with the performance, he invested a further amount of $10,000 on 1 January 2010 with the firm. On 31 December 2010, he withdrew$5,000. The value of his portfolio before withdrawal was $25,000. The value at the end of 31 December 2011 was$22,500. Calculate the time-weighted rate of return for his portfolio and tell how it differs from money-weighted return.

Solution

We have to calculate the compound rate of return from 1 January 2009 to 31 December 2011. The period has three sub-periods:

1. 1 January 2009 — 31 December 2009
2. 1 January 2010 — 31 December 2010
3. 1 January 2011 — 31 December 2011

Sub-period rates of return are calculated as follows:

$$Return\ in\ Period\ 1 \\ = \frac{13,050 − 10,000}{10,000} \\ = 30.50\%$$

In Period 2, $10,000 is added because it is a new investment that increased the beginning portfolio value. $$Return\ in\ Period\ 2 \\ = \frac{25,000 − \left(13,050 + 10,000\right)}{13,050 + 10,000} \\ = 8.46\%$$ At the start of Period 3, there is a deduction of$5,000 so beginning portfolio value for Period 3 should be $20,000 [=$25,000 minus \$5,000].

$$Return\ in\ Period\ 3 \\ = \frac{22,500 − 20,000}{20,000} \\ = 12.5\%$$

The sub-period returns are then chain-linked as follows:

Time-weighted return for the whole period
= (1 + 30.50%) × (1 + 8.46%) × (1 + 12.5%) – 1
= 59.23%

Please note that this is not an annual rate. However, it can be converted to an annual rate.

The point to note here is that time-weighted rate of return assigns equal weight to all the periods. The money-weighted rate of return on the other hand will weight Period 2 higher because the highest amount of money that remained invested during that period.