# 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 + 1^{st } period return) × (1 + 2^{nd} 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 January 2009 — 31 December 2009
- 1 January 2010 — 31 December 2010
- 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.