## What is the money-weighted rate of return?

The money-weighted rate of return (MWRR) is a measure of the performance of an investment. The MWRR is calculated by finding the rate of return that will set the current values (PV) of all cash flows equal to the value of the initial investment.

MWRR is equivalent to internal rate of return (IRR). MWRR can be compared to time-weighted return (TWR), which removes the effects of cash inflows and outflows.

Key points to remember

- The money-weighted rate of return (MWRR) calculates the performance of an investment that takes into account the size and timing of deposits or withdrawals.
- MWRR is calculated by finding the rate of return that will set the present values of all cash flows equal to the value of the initial investment.
- The MWRR is equivalent to the internal rate of return (IRR).
- MWRR sets the initial value of an investment to be equal to future cash flows, such as added dividends, withdrawals, deposits and sale proceeds.

## Understanding the Money-Weighted Rate of Return

The MWRR formula is as follows:

Investpodia \, +\, \frac{CF_Instagram}{(1\, +\, IRR)}\, +\, \frac{CF_Pinterest}{(1\, +\, IRR)^Pinterest}\,\\ &\qquad\quad\, +\, \frac{CF_Google}{(1\, +\, IRR)^Google}\,\, +\,… \frac{CF_{n}}{(1\, +\, IRR)^{n}}\,\\ &\textbf{where:}\\ &PVO = \text{PV Outflows}\\ &PVI = \text{PV Inflows}\\ &CF_0 = \text{Initial cash outlay or investment}\\ &CF_1, CF_2, CF_3, … CF_n = \text{Cash flows}\\ &N = \text{Each period}\\ &IRR = \text{Initial rate of return}\\ \end{aligned}”>

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\begin{aligned} &PVO = PVI = CF_Investpodia \, +\, \frac{CF_Instagram}{(1\, +\, IRR)}\, +\, \frac{CF_Pinterest} {(1\, +\, IRR)^Pinterest}\,\\ &\qquad\quad\, +\, \frac{CF_Google}{(1\, +\, IRR)^{3 }}\,\, +\,… \frac{CF_{n}}{(1\, +\, IRR)^{n}}\,\\ &\textbf{where:}\\ &PVO = \text{Outgoing PV}\\ &PVI = \text{Incoming PV}\\ &CF_0 = \text{Initial expense or investment}\\ &CF_1, CF_2, CF_3, … CF_n = \text{Cash flow}\\ &N = \text{Each period}\\ &IRR = \text{Initial rate of return}\\ \end{aligned}

PVO=PVI=VSF0+(1+IRR)VSF1+(1+IRR)2VSF2+(1+IRR)3VSF3+...(1+IRR)notVSFnotwhere:PVO=PV outputsPVI=Contributions PVVSF0=Down payment or initial investmentVSF1,VSF2,VSF3,...VSFnot=Cash flowNOT=Each periodIRR=Initial rate of return

### How to Calculate the Currency Weighted Rate of Return

- To calculate the IRR using the formula, set the net present value (NPV) to zero and solve for the discount rate (r), which is the IRR.
- However, due to the nature of the formula, the IRR cannot be calculated analytically and must instead be calculated either by trial and error or by using software programmed to calculate the IRR.

## What does the money-weighted rate of return tell you?

There are many ways to measure asset returns, and it is important to know which method is used when examining return on assets. MWRR incorporates the size and timing of cash flows, so it is an effective measure of portfolio returns.

The MWRR fixes the initial value of an investment at the future value cash flowsuch as dividends added, withdrawals, deposits and sales proceeds. In other words, the MWRR helps determine the rate of return needed to start with the initial investment amount, taking into account all changes in cash flows over the investment period, including the sale proceeds.

## Cash flow and money-weighted rate of return

As stated above, the MWRR of an investment is identical in concept to the IRR. In other words, it’s the discount rate on which the actual net value (VAN) = 0, or the present value of the inputs = the present value of the outputs.

It is important to identify cash flows in and out of a portfolio, including the sale of the asset or investment. Some of the cash flows an investor may have in a portfolio include:

### Exits

- The cost of any investment purchased
- Dividends or reinvested interest
- Withdrawals

### Starters

- The proceeds of any investment sold
- Dividends or interest received
- Submissions

## Example of a dollar-weighted rate of return

Each input or output must be discounted to the present using a rate (r) which will make PV (inputs) = PV (outputs).

Let’s say an investor buys a stock of a stock for $50 which pays an annual dividend of $2 and sells it after two years for $65. Thus, you will discount the first dividend after the first year, and for the second year you will discount both the dividend and the sale price. The MWRR will be a rate that satisfies the following equation:

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\begin{aligned} PV \text{ Outflows} &= PV \text{ Inflows} \\ &= \frac{ \$2 }{ 1 + r } + \frac{ \$2 }{ 1 + r^2 } + \ frac{ \$65 }{ 1 + r^2} \\ &= \$50 \end{aligned}

PV Exits=PV Starters=1+r$2+1+r2$2+1+r2$65=$50

solve for *r* using a spreadsheet or financial calculator, we have an MWRR of 11.73%.

## The difference between the money-weighted rate of return and the time-weighted rate of return

MWRR is often compared to time-weighted rate of return (TWRR), but the two calculations have distinct differences. TWRR is a measure of the compound growth rate of a portfolio. The TWRR measure is often used to compare the returns of investment managers because it eliminates the distorting effects on growth rates created by cash inflows and outflows.

It can be difficult to determine how much money has been earned on a portfolio because deposits and withdrawals distort the value of portfolio performance. Investors cannot simply subtract the initial balance, after the initial deposit, from the ending balance since the ending balance reflects both the rate of return on investments and any deposits or withdrawals during the term invested in the fund.

The TWRR divides the return of an investment portfolio into separate intervals depending on whether money has been added to or taken out of the fund. The MWRR differs in that it takes into account the behavior of investors via the impact of cash inflows and outflows on performance, but does not separate the intervals where cash flows have occurred, as does the TWRR. Therefore, cash outflows or cash inflows can have an impact on the MWRR. If there is no cash flow, both methods should produce the same or similar results.

## Limitations on using the dollar-weighted rate of return

The MWRR takes into account all cash flows of the fund or contribution, including withdrawals. If an investment spans multiple quarters, for example, the MWRR gives more weight to the performance of the fund when it is at its peak, hence the description “currency-weighted.”

Weighting can penalize fund managers with cash flows over which they have no control. In other words, if an investor adds a large sum of money to a portfolio just before its performance increases, it is equivalent to a positive action. This is because the larger portfolio benefits more (in dollars) from the growth of the portfolio than if the contribution had not been made.

On the other hand, if an investor withdraws funds from a portfolio just before an increase in performance, this is equivalent to negative action. The now smaller fund derives less benefit (in dollars) from portfolio growth than if the withdrawal had not occurred.

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