The time-weighted return (TWR)^[1][2] is a method of calculating investment return, where returns over sub-periods are compounded together, with each sub-period weighted according to its duration. The time-weighted method differs from other methods of calculating investment return, in the particular way it compensates for external flows.

External flows

The time-weighted return is a measure of the historical performance of an investment portfolio which compensates for external flows. External flows refer to the net movements of value into or out of a portfolio, stemming from transfers of cash, securities, or other financial instruments. These flows are characterized by the absence of a concurrent, equal, and opposite value transaction, unlike what occurs in purchases or sales. Furthermore, they do not originate from the income generated by the portfolio's investments, 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.

The problem of external flows

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

Example 1

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% (which is incidentally the result of applying one of the money-weighted methods). 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 200% ($1,000 divided by $500). Intuitively, 200% 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 ($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. The difference is a matter of perspective.

Adjustment for flows

The return of a portfolio in the absence of flows is:

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

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.

${\displaystyle 1+R=\frac{M_1-<!--\; -\; -->C_1}{M_0}\times <!--\; \times \; -->\frac{M_2-<!--\; -\; -->C_2}{M_1}\times <!--\; \times \; -->\frac{M_3-<!--\; -\; -->C_3}{M_2}}$Zdroj:https://en.wikipedia.org?pojem=True_time-weighted_rate_of_return
Text je dostupný za podmienok Creative Commons Attribution/Share-Alike License 3.0 Unported; prípadne za ďalších podmienok. Podrobnejšie informácie nájdete na stránke Podmienky použitia.

---