# Money-Weighted Rate of Return: Definition, Formula, and Example

## 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 present values (PV) of all cash flows equal to the value of the initial investment.

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

### Key Takeaways

• The money-weighted rate of return (MWRR) calculates the performance of an investment that accounts for the size and timing of deposits or withdrawals.
• The 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).
• The MWRR sets the initial value of an investment to equal future cash flows, such as dividends added, withdrawals, deposits, and sale proceeds.

## Understanding the Money-Weighted Rate of Return

The formula for the MWRR is as follows:

﻿ \begin{aligned} &PVO = PVI = CF_{0} \, +\, \frac{CF_{1}}{(1\, +\, IRR)}\, +\, \frac{CF_{2}}{(1\, +\, IRR)^{2}}\,\\ &\qquad\quad\, +\, \frac{CF_{3}}{(1\, +\, IRR)^{3}}\,\, +\,... \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}﻿

### How to Calculate the Money-Weighted Rate of Return

1. To calculate the IRR using the formula, set the net present value (NPV) equal to zero and solve for the discount rate (r), which is the IRR.
2. However, because of the nature of the formula, the IRR cannot be calculated analytically and instead must be calculated either through 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 being used when reviewing asset performance. The MWRR incorporates the size and timing of cash flows, so it is an effective measure of portfolio returns

The MWRR sets the initial value of an investment to equal future cash flows, such as dividends added, withdrawals, deposits, and sale proceeds. In other words, the MWRR helps to determine the rate of return needed to start with the initial investment amount, factoring all of the changes to cash flows during the investment period, including the sale proceeds.

## Cash Flows and the Money-Weighted Rate of Return

As stated above, the MWRR for an investment is identical in concept to the IRR. In other words, it is the discount rate on which the net present value (NPV) = 0, or the present value of inflows = the present value of outflows.

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

### Outflows

• The cost of any investment purchased
• Reinvested dividends or interest
• Withdrawals

### Inflows

• The proceeds from any investment sold
• Contributions

## Example of the Money-Weighted Rate of Return

Each inflow or outflow must be discounted back to the present by using a rate (r) that will make PV (inflows) = PV (outflows).

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

\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}

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

## The Difference Between Money-Weighted Rate of Return and Time-Weighted Rate of Return

The MWRR is often compared to the time-weighted rate of return (TWRR), but the two calculations have distinct differences. The TWRR is a measure of the compound rate of growth in 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 inflows and outflows of money.

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

The TWRR breaks up the return on an investment portfolio into separate intervals based on whether money was added to or withdrawn from the fund. The MWRR differs in that it takes into account investor behavior via the impact of fund inflows and outflows on performance but doesn’t separate the intervals where cash flows occurred, as the TWRR does. Therefore, cash outflows or inflows can impact the MWRR. If there are no cash flows, then both methods should deliver the same or similar results.

## Limitations of Using Money-Weighted Rate of Return

The MWRR considers all the cash flows from the fund or contribution, including withdrawals. Should an investment extend over several quarters, for example, the MWRR lends more weight to the performance of the fund when it is at its largest—hence, the description “money-weighted.”

The weighting can penalize fund managers because of 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 rises, then it equates to positive action. This is because the larger portfolio benefits more (in dollar terms) 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 a surge in performance, then it equates to a negative action. The now-smaller fund sees less benefit (in dollar terms) from the growth of the portfolio than if the withdrawal had not occurred.

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