# Time-Weighted Return

## Formula

The Time-Weighted Return (also called the Geometric Average Return) is a way of calculating the rate of return for an investment when there are deposits and withdrawals (cash flows) during the period. You often want to exclude these cash flows so that we can find out how well the underlying investment has performed.

To calculate the time weighted return for a particular period, the period in question has to be divided into sub-periods, at each point that a cash flow occurred. You also need the value of the investment at the end of each sub-period.

### Calculating the Arithmetic Return for Each Sub-Period

First you need to calculate the Arithmetic Return for each sub-period, excluding the effect of any cash flows, which you can do with the following formula:

R_(arith) = (V_(f) - D + W - V_i)/V_i

where V_f is the value of the investment at the end of the sub-period, V_i is the value of the investment at the start of the sub-period, D is the total of any deposits made into the investment at the end of this sub-period, and W is the total of any withdrawals.

### Calculating the Time-Weighted Return

To calculate the overall return for the whole of the period, you multiply together the growth factors (1 + R_(arith)) for each sub-period, then subtract 1.

R_(tw) = prod_(i=1)^n(1 + R_(arith,i)) - 1

In other words:

R_(tw) = ((1 + R_(arith,1)) xx (1 + R_(arith,2)) xx (1 + R_(arith,3)) xx ... xx (1 + R_(arith,n))) - 1

This is the time-weighted return. Note that this is the return per dollar (or whatever unit of currency you are using). To get an annual rate, you need to do a further step.

### Annualization

If you want to know the time-weighted return expressed as an annual rate, then you need to annualize using the following formula:

R_{a\n\n\ual} = (1 + R_(tw))^(1/y) - 1

where y is the number of years for the period.

## Example

Sally has a mutual fund investment. Her statement shows the following transactions during 2010 and 2011:

DateTransactions
31st Dec 2009Initial deposit of 1000
30th June 2010Deposit of 100
Investment valuation 1300
31st December 2010Deposit of 100
Annual fee deducted 50
Investment valuation 1220
30th June 2011Deposit of 100
Investment valuation 1503
31st December 2011Deposit of 100
Annual fee deducted 50
Investment valuation 1703.30

Sally wants to calculate the time-weighted return, so she divides the period into 4 sub-periods and calculates the arithmetic return for each one.

### Period 1: 1st Jan to 30th June 2010

The deposit is a positive cash flow into the investment, so we subtract it from the valuation in this calculation.

R_1 = (1300 - 100 - 1000)/1000

R_1 = 0.2

### Period 2: 1st July to 31st December 2010

This period includes the annual fee, which is a withdrawal from the investment, so we add it in this calculation.

R_2 = (1220 - 100 + 50 - 1300)/1300

R_2 = -0.1

### Period 3: 1st Jan to 30th June 2011

R_3 = (1503 - 100 - 1220)/1220

R_3 = 0.15

### Period 4: 1st July to 31st December 2011

R_4 = (1703.30 - 100 + 50 - 1503)/1503

R_4 = 0.1

### Time-Weighted Return

Sally can now calculate the time-weighted return:

R_(tw) = ((1 + 0.2) xx (1 - 0.1) xx (1 + 0.15) xx (1 + 0.1)) - 1

R_(tw) = (1.2 xx 0.9 xx 1.15 xx 1.1) - 1

R_(tw) = 0.3662

Or as a percentage:

R_(tw) = 36.62%

### Annualization

The period covered is 2 years. Sally now calculates the annual rate:

R_{a\n\n\ual} = (1 + 0.3662)^(1/2) - 1

R_{a\n\n\ual} = 0.1688

Or as a percentage:

R_{a\n\n\ual} = 16.88%