# Time-weighted return

(Redirected from True time-weighted rate of return)

The time-weighted return (TWR)[1][2] (or true time-weighted rate of return (TWRR)) is a method of calculating investment return. To apply the time-weighted return method, combine the return over sub-periods, by compounding them together, resulting in the overall period return. The rate of return over each different sub-period is weighted according to the duration of the sub-period.

The time-weighted method differs from other methods of calculating investment return only in the particular way it compensates for external flows - see below.

## Explanation

### Why It's Called "Time Weighted"

#### Example 1

For an understanding of why this method is called "time weighted", consider an example where we are tasked with calculating the annualized rate of return over a five-year period of an investment which returns 10% p.a. for two of the five years, and -3% p.a. for the other three. The time-weighted return over the five-year period is:

${\displaystyle (1+0.10)(1+0.10)(1-0.03)(1-0.03)(1-0.03)-1}$
${\displaystyle =1.1^{2}\times 0.97^{3}-1}$
${\displaystyle =0.104334...}$
${\displaystyle =10.4334...\%}$

and after annualization, the rate of return is:

${\displaystyle (1.1^{2}\times 0.97^{3})^{1/(2+3)}-1}$
${\displaystyle =1.0200...-1}$
${\displaystyle =2.00...\%}$

The reason why this is called "time-weighted" can be partly understood by observing that the length of time over which the rate of return was 10% was two years, and the two-year "weight" appears in the power of two on the 1.1 factor:

${\displaystyle 1.1^{2}}$

Likewise, the rate of return was -3% for three years, and the three-year "weight" appears in the power of three on the 0.97 factor. The result is then annualized over the overall five-year period.

### General Formula for Ordinary Time-Weighted Return

More generally, if an annualized rate of return ${\displaystyle r_{1}}$ applies over a period of ${\displaystyle t_{1}}$ years, etc. then the time-weighted return over the overall period ${\displaystyle T=t_{1}+t_{2}+t_{3}+...+t_{n}}$ is:

${\displaystyle (1+r_{1})^{t_{1}}(1+r_{2})^{t_{2}}(1+r_{3})^{t_{3}}...(1+r_{n})^{t_{n}}-1}$

and the annualized time-weighted rate of return is:

${\displaystyle (1+r_{1})^{\frac {t_{1}}{T}}(1+r_{2})^{\frac {t_{2}}{T}}(1+r_{3})^{\frac {t_{3}}{T}}...(1+r_{n})^{\frac {t_{n}}{T}}-1}$

The powers ${\displaystyle {\frac {t_{1}}{T}},{\frac {t_{2}}{T}},{\frac {t_{3}}{T}}...{\frac {t_{n}}{T}}}$ can be thought of as weights.

#### Example 2

The overall period ${\displaystyle T}$ and sub-periods ${\displaystyle t_{i}}$ where ${\displaystyle i=1,2,3,...n}$ are not necessarily whole years.

For example, suppose the overall period ${\displaystyle T}$ is the 674 days from the year-end of 2014, to the end of the day on 4 November 2016. Over the sub-period of ${\displaystyle t_{1}=}$ 337 days between the end of 2014 and the end of the day on 3 December 2015, the rate of return on a portfolio was 10.25% p.a., and over the remaining ${\displaystyle t_{2}=}$ 337 days of the period, it fell to -19% p.a.

The rate of return over the whole 674 day period was:

${\displaystyle 1.1025^{337/674}\times 0.81^{(674-337)/674}-1}$
${\displaystyle =1.05\times 0.9-1}$
${\displaystyle =0.945-1}$
${\displaystyle =-5.5\%p.a.}$

### Continuous Time-Weighted Rate of Return

In terms of continuous (logarithmic) returns, it is apparent why it is called the time-weighted rate of return. The general formula is:

${\displaystyle \sum _{i=1}^{n}{{\frac {t_{i}}{T}}\times r_{i}}={\frac {t_{1}}{T}}\times r_{1}+{\frac {t_{2}}{T}}\times r_{2}+{\frac {t_{3}}{T}}\times r_{3}+...+{\frac {t_{n}}{T}}\times r_{n}}$

The continuous time-weighted rate of return is the weighted average of the sub-period returns. The weight ${\displaystyle {\frac {t_{i}}{T}}}$ assigned to the return ${\displaystyle r_{i}}$ in sub-period ${\displaystyle i}$ is the duration of the respective sub-periods, as a proportion of the overall period ${\displaystyle T}$.

#### Example 3

Consider the following example of calculating the continuous (logarithmic) rate of return using the time-weighted method: over a period of a decade, a portfolio returns 5% p.a. (continuous) over three of those years, and 10% p.a. over the other seven years. The continuous time-weighted rate of return over the ten-year period is:

${\displaystyle {\frac {3}{10}}\times 5\%+{\frac {7}{10}}\times 10\%={\frac {3\times 5\%+7\times 10\%}{10}}=8.5\%}$

## External Flows

The time-weighted return is a measure of the historical performance of an investment portfolio which compensates for external flows. External flows are net movements of value which result from transfers of cash, securities or other instruments, into or out of the portfolio, with no simultaneous equal and opposite movement of value in the opposite direction, as in the case of a purchase or sale, and which are not income from the investments in the portfolio, such as interest, coupons or dividends.

To compensate for external flows, the overall time interval under analysis is divided into contiguous sub-periods at each point in time within the overall time period whenever there is an external flow. In general, these sub-periods will be of unequal lengths. The returns over the sub-periods between external flows are linked geometrically (compounded) together, i.e. by multiplying together the growth factors in all the sub-periods. (The growth factor in each sub-period is equal to 1 plus the return over the sub-period.)

Investment managers are judged on investment activity which is under their control. If they have no control over the timing of flows, then compensating for the timing of flows using the true time-weighted return method is a superior measure of the performance of the investment manager.

### The Problem of External Flows

To illustrate the problem of external flows, consider the following example.

#### Example 4

Suppose an investor transfers $500 into a portfolio at the beginning of Year 1, and another$1,000 at the beginning of Year 2, and the portfolio has a total value of $1,500 at the end of the Year 2. The net gain over the two-year period is zero, so intuitively, we might expect that the return over the whole 2-year period to be 0%. If the cash flow of$1,000 at the beginning of Year 2 is ignored, then the simple method of calculating the return without compensating for the flow will be 100% ($1,000 divided by$500). Intuitively, a 100% rate of return is incorrect.

If we add further information however, a different picture emerges. If the initial investment gained 100% in value over the first year, but the portfolio then declined by 25% during the second year, we would expect the overall return over the two-year period to be the result of compounding a 100% gain ($500) with a 25% loss (also$500). The time-weighted return is found by multiplying together the growth factors for each year, i.e. the growth factors before and after the second transfer into the portfolio, then subtracting one, and expressing the result as a percentage:

${\displaystyle (1+1.0)(1-0.25)-1=2.0\times 0.75-1=1.5-1=0.5=50\%}$.

We can see from the time-weighted return that the absence of any net gain over the two-year period was due to bad timing of the cash inflow at the beginning of the second year.

The time-weighted return appears in this example to overstate the return to the investor, because he sees no net gain. However, by reflecting the performance each year compounded together on an equalized basis, the time-weighted return recognizes the performance of the investment activity independently of the poor timing of the cash flow at the beginning of Year 2. If all the money had been invested at the beginning of Year 1, the return by any measure would most likely have been 50%. $1,500 would have grown by 100% to$3,000 at the end of Year 1, and then declined by 25% to $2,250 at the end of Year 2, resulting in an overall gain of$750, i.e. 50% of \$1,500.

Measuring the performance of a portfolio in the absence of flows is trivial:

${\displaystyle r={\frac {M_{2}-M_{1}}{M_{1}}}}$

where ${\displaystyle M_{2}}$ is the portfolio's final value, ${\displaystyle M_{1}}$ is the portfolio's initial value, and ${\displaystyle r}$ is the portfolio's return over the period.

The growth factor is:

${\displaystyle 1+r={\frac {M_{2}}{M_{1}}}}$

External flows during the period being analyzed complicate the performance calculation. If external flows are not taken into account, the performance measurement is distorted: a flow into the portfolio would cause this method to overstate the true performance, while flows out of the portfolio would cause it to understate the true performance.

To compensate for an external flow ${\displaystyle C_{1}}$ into the portfolio at the beginning of the period, adjust the portfolio's initial value ${\displaystyle M_{1}}$ by adding ${\displaystyle C_{1}}$. The return is:

${\displaystyle r={\frac {M_{2}-(M_{1}+C_{1})}{M_{1}+C_{1}}}}$

and the corresponding growth factor is:

${\displaystyle 1+r={\frac {M_{2}}{M_{1}+C_{1}}}}$

To compensate for an external flow ${\displaystyle C_{2}}$ into the portfolio just before the valuation ${\displaystyle M_{2}}$ at the end of the period, adjust the portfolio's final value ${\displaystyle M_{2}}$ by subtracting ${\displaystyle C_{2}}$. The return is:

${\displaystyle r={\frac {(M_{2}-C_{2})-M_{1}}{M_{1}}}}$

and the corresponding growth factor is:

${\displaystyle 1+r={\frac {M_{2}-C_{2}}{M_{1}}}}$

### Time-Weighted Return Compensating for External Flows

Suppose that the portfolio is valued immediately after each external flow. The value of the portfolio at the end of each sub-period is adjusted for the external flow which takes place immediately before. External flows into the portfolio are considered positive and flows out of the portfolio are negative.

${\displaystyle 1+r={\frac {M_{1}-C_{1}}{M_{0}}}\times {\frac {M_{2}-C_{2}}{M_{1}}}\times {\frac {M_{3}-C_{3}}{M_{2}}}\times \cdots \times {\frac {M_{n-1}-C_{n-1}}{M_{n-2}}}\times {\frac {M_{n}-C_{n}}{M_{n-1}}}}$

where:

${\displaystyle r}$ is the "true time-weighted return" of the portfolio,
${\displaystyle M_{0}}$ is the initial portfolio value,
${\displaystyle M_{t}}$ is the portfolio value at the end of sub-period ${\displaystyle t}$, immediately after external flow ${\displaystyle C_{t}}$,
${\displaystyle M_{n}}$ is the final portfolio value,
${\displaystyle C_{t}}$ is the net external flow into the portfolio which occurs just before the end of sub-period ${\displaystyle t}$,

and

${\displaystyle n}$ is the number of sub-periods.

Note that if there is an external flow occurring at the end of the overall period, then the number of sub-periods ${\displaystyle n}$ matches the number of flows. However, if there is no flow at the end of the overall period, then ${\displaystyle C_{n}}$ is zero, and the number of sub-periods ${\displaystyle n}$ is one greater than the number of flows.

Note also that if the portfolio is valued immediately before each flow instead of immediately after, then each flow should be used to adjust the starting value within each sub-period, instead of the ending value, resulting in a different formula:

${\displaystyle 1+r={\frac {M_{1}}{M_{0}+C_{0}}}\times {\frac {M_{2}}{M_{1}+C_{1}}}\times {\frac {M_{3}}{M_{2}+C_{2}}}\times ...\times {\frac {M_{n-1}}{M_{n-2}+C_{n-2}}}\times {\frac {M_{n}}{M_{n-1}+C_{n-1}}}}$

where:

${\displaystyle r}$ is the "true time-weighted return" of the portfolio,
${\displaystyle M_{0}}$ is the initial portfolio value,
${\displaystyle M_{t}}$ is the portfolio value at the end of sub-period ${\displaystyle t}$, immediately before external flow ${\displaystyle C_{t}}$,
${\displaystyle M_{n}}$ is the final portfolio value,
${\displaystyle C_{t}}$ is the net external flow into the portfolio which occurs at the beginning of sub-period ${\displaystyle {t+1}}$,

and

${\displaystyle n}$ is the number of sub-periods.

## Comparison With Other Returns Methods

Other methods exist to compensate for external flows when calculating investment returns. Such methods are known as "money-weighted" or "dollar-weighted" methods. The time-weighted return is higher than the result of other methods of calculating the investment return when external flows are badly timed - refer to Example 4 above.

### Internal Rate of Return

One of these methods is the internal rate of return. Like the true time-weighted return method, the internal rate of return is also based on a compounding principle. It is the discount rate that will set the net present value of all external flows and the terminal value equal to the value of the initial investment. However, solving the equation to find an estimate of the internal rate of return generally requires an iterative numerical method.

The internal rate of return is commonly used for measuring the performance of private equity investments, because the principal partner (the investment manager) has greater control over the timing of cash flows, rather than the limited partner (the end investor).

### Simple Dietz Method

The Simple Dietz method[3] applies a simple rate of interest principle, as opposed to the compounding principle underlying the internal rate of return method, and further assumes that flows occur at the midpoint within the time interval (or equivalently that they are distributed evenly throughout the time interval). However, the Simple Dietz method is unsuitable when such assumptions are invalid, and will produce different results to other methods in such a case.

The simple Dietz returns of two or more different constituent assets in a portfolio over the same period can be combined together to derive the simple Dietz portfolio return, by taking the weighted average. The weights are the start value plus half the net inflow.

### Modified Dietz Method

The Modified Dietz method is another method which, like the Simple Dietz method, applies a simple rate of interest principle. Instead of comparing the gain in value (net of flows) with the initial value of the portfolio, it compares the net gain in value with average capital over the time interval. Average capital allows for the timing of each external flow. As the difference between the Modified Dietz method and the internal rate of return method is that the Modified Dietz method is based on a simple rate of interest principle, whereas the internal rate of return method applies a compounding principle, the two methods produce similar results over short time intervals, if the rates of return are low. Over longer time periods, with significant flows relative to the size of the portfolio, and where the returns are not low, then the differences are more significant.

Like the simple Dietz method, the Modified Dietz returns of two or more different constituent assets in a portfolio over the same period can be combined together to derive the Modified Dietz portfolio return, by taking the weighted average. The weight to be applied to the return on each asset in this case is the average capital of the asset.

Calculating the "true time-weighted return" depends on the availability of portfolio valuations during the investment period. If valuations are not available when each flow occurs, the time-weighted return can only be estimated by linking returns for contiguous sub-periods together geometrically, using sub-periods at the end of which valuations are available. Such an approximate time-weighted return method is prone to overstate or understate the true time-weighted return.

Linked Internal Rate of Return (LIROR) is another such method which is sometimes used to approximate the true time-weighted return. It combines the true time-weighted rate of return method with the internal rate of return (IRR) method. The internal rate of return is estimated over regular time intervals, and then the results are linked geometrically. For example, if the internal rate of return over successive years is 4%, 9%, 5% and 11%, then the LIROR equals 1.04 x 1.09 x 1.05 x 1.11 – 1 = 32.12%. If the regular time periods are not years, then either calculate the un-annualized holding period version of the IRR for each time interval, or calculate the IRR for each time interval firstly, and then convert each one to a holding period return over the time interval, then link together these holding period returns to obtain the LIROR.

### Returns Methods in the Absence of Flows

If there are no external flows, then all these methods (time-weighted return, internal rate of return, Modified Dietz Method etc.) give identical results - it is only the various ways they handle flows which makes them different from each other.

## Logarithmic Returns

The continuous or logarithmic return method is not a competing method of compensating for flows. It is simply the natural logarithm ${\displaystyle ln({\frac {M_{2}}{M_{1}}})}$ of the growth factor.

## Fees

To measure returns net of fees, allow the value of the portfolio to be reduced by the amount of the fees. To calculate returns gross of fees, compensate for them by treating them as an external flow, and exclude accrued fees from valuations.

## Annual Rate of Return

Any confusion over the meaning of the term return or rate of return should be avoided. The return calculated by these methods is the return per dollar (or per some other unit of currency), not per year (or other unit of time). Annualization, which means conversion to an annual rate of return, is a separate process. Refer to the article rate of return.