True time-weighted rate of return - AbsoluteAstronomy.com

True Time-Weighted Rate of Return(TWROR) is a way to measure the performance of an investing portfolio in the presence of external cash flows. It determines the return for an investor who has not invested any additional cash flows during the investment period:

Where

is the "True Time-Weighted Return" of the portfolio,

is the initial portfolio value,

is the portfolio value at the end of sub-period

, immediately after cash-flow

is the cash flow which occurs at the end of sub-period

, and

is the number of cash-flows (also the number of sub-periods). Cash flows into the portfolio are positive and cash flows out of the portfolio are negative.

Discussion/Derivation

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

Where

is the portfolio's final value,

is the portfolio's initial value, and

is the portfolio's return over the period.

2) The presence of external cash-flows during the period being analyzed complicates the performance calculation. If external cash-flows are not taken into account, a cash-flow into the portfolio would wrongly overstate the true performance, while cash-flows out of the portfolio would wrongly understate the true performance.

3) The best way of taking into account external cash flows is the Internal Rate of Return

Internal rate of return

The internal rate of return is a rate of return used in capital budgeting to measure and compare the profitability of investments. It is also called the discounted cash flow rate of return or the rate of return . In the context of savings and loans the IRR is also called the effective interest rate...

. The Internal Rate of Return

Internal rate of return

not only takes into account the timing of cash flows but also the value invested at any time in the portfolio. Internal rates of return are however computationally awkward.

Money-Weighted Rate Of Return(MWROR) - also measures the rate of return for an asset or portfolio of assets. It is calculated by finding the rate of return that will set the present values of all cash flows and terminal values equal to the value of the initial investment. The money-weighted rate of return is equivalent to the internal rate of return (IRR).

Linked Internal Rate of Return(LIROR) - is another measure that is sometimes used. It is a variant of the time-weighted rate of return which approximates the ﬂow of new money into and out of the fund. In this TWROR will be approximated with MWROR over reasonably frequent time intervals and then those returns will be chain-linked. eg: LIROR = (1.04) (1.09) (1.05) (1.11) – 1 = 32.12%; when MWROR are 4%, 9%, 5% and 11%

4) Simple Dietz Method

Simple Dietz Method

Simple Dietz Method is a means of calculating an approximation of investment portfolio performance during a period of external cash flows into/out of the portfolio...

is an approximation of the Internal Rate of Return, in that it takes into account timing of external cash flows, assuming that they are made in the middle of the time period in question. However, this assumption is still generally not accurate, leading to undesired errors.

5) Modified Dietz Method

Modified Dietz Method

The Modified Dietz Method is a calculation used to determine an approximation of the performance of an investment portfolio based on money-weighted cash flow...

is another approximation of return, weighting the external cash flow for where in the time period in question it actually occurred. While this is certainly an improvement over Simple Dietz, it is still an imprecise approximation.

6) Another method of calculating returns in the presence of external cash flows is to divide the period of analysis into contiguous sub-periods, with cash flows occurring at the end of sub-periods. A return would be calculated for each sub-period and then the returns of each sub-period would be "chained together" to give the "True Time-Weighted Return" of the portfolio for the period in question.

Let

be the initial portfolio value and

be the portfolio value immediately after the cash flow

at the end of the sub-period. Then the return

for that sub-period would be:

Rearranging,

All such sub-period returns must be chained together to give the True Time-Weighted Rate of Return for the overall period:

Where

is the True Time-Weighted Rate of Return for the portfolio,

is the initial portfolio value,

is the portfolio value at the end of sub-period

, immediately after cash-flow

is the cash flow which occurs at the end of sub-period

, and

is the number of cash-flows (also the number of sub-periods).

7) Unfortunately the "True Time-Weighted Rate of Return" depends on portfolio valuations during the investment period that are not necessarily realized by an investor. It can therefore substantially overstate or understate the true portfolio performance and should not be used as a way of measuring an investor's return. A simple example illustrates this problem: If an investor invests $500 at the beginning of year 1, $1000 at the beginning of year 2 and his portfolio has a total value of $1500 at the end of the year 2, the total return is clearly 0%. This is also the internal rate of return. If we assume that the initial investment has gone up by 100% over the first year, but the portfolio has declined by 25% over the second year we obtain the same cash flows but a "True Time-Weighted Rate of Return" of

, which massively overstates the return realized by the investor. Similary if the intermediate valuations of the portfolio are below the valuation at the end of the investment period we will understate the real portfolio return.

Discussion/Derivation

See also

Further reading