# Money-weighted rate of return

Money-weighted rate of return |
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See also |

**Money-weighted rate of return MWWR** measures the performance of assets. Money-weighted rate of return shows the rate at which present value of cash flows and terminal values is equal to initial investment. This rate of return is calculated on the basis of the initial and final value and cash flows, if they occur in a given period. It is a rate at which investor earns and loses exactly zero. Therefore, it is equal to **internal rate of return (IRR)** over the assessed period (P. Moles, N. Terry 2002, s.359; S. Illmery, W. Marty 2003 s.42).

## How to calculate MWRR

It is calculated as follows\[MV0=\frac{CF_1}{(1 - r_m)}+\frac{CF_2}{(1 - r_m)^2}+…+\frac{CF_n + MV_n}{(1 + r_m)^n}\]

Where:

MV0 - the initial value of the fund,

CFt - the cash flow at period t,

rm - the internal rate of return that equates the value of the cash flows 1 to m to the current value.

MVn - the final value of the fund,

To calculate money-weighted rate of return investor should take into account outflows (e.g. costs of investment, reinvested dividend, withdrawals) and inflows (investments sold, dividends received, contributions). The net present value (NPV) should be calculated at the rate which results in NPV equal to zero. It can be calculated using IRR equation, however that equation has some limitations due to non-linear characteristics of NPV. The most often money-weighted rate of return is using for venture capital and private equity asset categories. The reason is that the initial investment appraisal for non-quoted investments often uses a MWRR approach and the assets are illiquid and difficult to value accurately (P. Moles, N. Terry 2002, s.359).

## Example

The example of using money-weighted rate of return is from *CFA Program Curriculum 2018 Level I*

„To illustrate the money-weighted rate of return, consider an investment that cover two-year horizon. At time t=0, an inwestor buys one share at $200. At time t=1, he purchases an additional share at $225. At the end of year 2, t=2, for $235 each. During both years, the stock pays a per share dividend of $5. The t=1 dividend is not reinvested(…)

\(200 + \frac{225}{(1 + r)} = \frac{5}{(1 + r)} + \frac{480}{(1 + r)^2}\)

The left-hand side of this equation details the outflows: $200 at time t=0 and $225 at time t=1. The $225 outflow is discounted back one period because it occurs at t=1. The right-hand side of the equation shows the present value of thr inflows: $5 at time t=1 (discounted back one period) and $480 (the $10 dividend plus the $470 sale proceeds) at time t=2 (discounted back two periods).”(CFA Institute 2017, s.370-371) There are two alternatives methods for calculating portfolio returns in a most period setting when the portfolio is an object of additions and withdrawal – time-weighted rate of return and money-weighted rate of return. Time-weighted rate of return is calculated normally in the case in the investment management industry. Money-weighted rate of return can be appropriate if the investor exercises control over additions and withdrawals to the portfolio and have control over the timing because these have greatly influence the mentioned rate of return ( C. Bacon 2013 ).

## Advantages

Advantages of using money-weighted rate of return:

- Easy to interpret (mainly due to the percentage expression of profitability,
- The assessment takes into account the net benefit from the entire life cycle of the investment,
- It is possible to specify the limit cost of capital used to finance the project,
- The level of the meter does not depend directly on the discount rate, so it can be used in the efficiency assessment also when the discount rate is not known (S. Garrett 2013, s.109).

## Disadvantages

There are also some disadvantages of using the money-weighted rate of return:

- Equation may not have a unique solution, o rany solution at all,
- In fund appraisal this method is sensitive to both amount and timing of cash flows,
- Calculations require information about all the cash flows of the fund during the period of concern,

The MWR is the true return from a client's point of view if it is the client's decision to invest money into or to withdraw money from the portfolio (S. Garrett 2013, s.111).

Introduction: Apart from Money-weighted rate of return (MWWR), there are several other approaches used to assess the performance of assets.

- The time-weighted rate of return (TWRR) measures the amount of excess return generated over the period of interest, regardless the contributions and withdrawals made by the investor. It is calculated as the portfolio’s compound annual rate of return over the period.
- The modified Dietz method gives the return of the portfolio by taking into account the contributions and withdrawals, but not the timing of the cash flows.
- The geometric mean return is a measure of average return per period, that does not take into account the cash flows.
- The simple rate of return is the arithmetic mean of the returns over the period, also not taking cash flows into account.

In summary, Money-weighted rate of return is one of the approaches used to measure the performance of assets. Other approaches include time-weighted rate of return, modified Dietz method, geometric mean return and simple rate of return.

## References

- Bacon C., (2013).
*Practical Portfolio Performance Measurement and Attribution*. - CFA Institute, (2017).
*CFA Program Curriculum 2018 Level I*. Willey, 370-371. - Garrett S., (2013).
*An Introduction to the Mathematics of Finance: A Deterministic Approach*, Butterworth - Heinemann, 108-112. - Illmer S. & Marty W., (2003).
*Decomposing the money-weighted rate of return*. Journal of Performance Measurement, 7(4), 42-50. - Moles P., Terry N., (2002).
*The Handbook of International Financial Terms*, Oxford, 359-360.

**Author:** Marta Mieszczak