Difference between Money Weighted Return and Time Weighted Return?

Very often students of finance are confused between Time Weighted Return and Money Weighted Return. In this article we have tried to clarify both of them and also show how both these type of returns differ from each other.

This article highlights how the timing of the returns earned on the portfolio affects the overall financial position. In other words it is not just the size of the returns that has to be considered but also the timing of the returns while assessing the overall returns on the portfolio. We would explain this phenomenon with the help of examples covering two 2 concepts-Time Weighted Rate of Return and Money Weighted rate of Return.

Time Weighted Return (TWR)

It allows an investor to directly measure their portfolio’s true performance and compare the performances of different money managers over a given time frame.It computes the return for each period and takes the average of the results. It finds the holding period for each period and averages them. If the investment is for more than one year, the geometric mean of the annual returns is taken to find the time-weighted rate of return for the measurement period.

Let us explain this with an example:

Suppose we have 3 annual returns of -10%, +15% and +5% then the average (time weighted) return after the 3 years can be calculated as follows:

       Average time-weighted = [(1+ r1) ´ (1+r2)´ (1+r3)]^(1/3)-1

                                                        = [(1-10%) ´ (1+15%) ´ (1+5%)]^(1/3)-1

                                                        = [0.9´ 1.15 ´ 1.05]^(1/3)-1

                                                        = 2.81% per annum

It is interesting to note that when working with time weighted returns, the order in which the returns occur does not matter. Let us change r1 and r3 around. Then we get:

           Average time-weighted = [(1+5%) ´ (1+15%) ´ (1-10%)]^(1/3)-1

                                                           = [1.05 ´ 1.15´ 0.9]^(1/3)-1

                                                           = 2.81% per annum

We get the same answer because we have applied the same weighting to each return item, which in this case, was based on the amount of time it applies to i.e. one year. Hence the name “time-weighted” returns.

In general, the arithmetic and time-weighted average returns do not provide the same answers, because computation of the arithmetic average assumes the initial amount invested to be maintained (through additions or withdrawals) at its initial investment value. The time weighted return, on the other hand, is the return on a portfolio that varies in size because of the assumptions that all proceeds are reinvested.

Money-Weighted Return

When it comes to what one actually ends up with in his/herkitty, time-weighted returns is mostly irrelevant. This is wheremoney-weighted return is the correct measure of returns and this is where the timing of the returns affects the final result.It is defined as the internal rate of return on a portfolio taking into account all cash inflows and outflows. The IRR is the discount rate that equates the ending investment with the compounded value of the beginning market value as well as all net contributions made during the life of the investment.

The MWRR can be calculated by solving the expression below for r(T)...

MW(T) = MV(0)´ {1+r(T)}^T + sum [C(t)´ {1+r(T)}^{T-t}]

MW(T)            :       Ending market value portfolio

MW(0)           :       Beginning market value portfolio

T                      :       Ending time T

r(T)                  :       IRR at time T over time period {0,T}

C(t)                  :       Net contribution at time t

Note that the time period (0,T) is assumed to be divided in n equally spaced time periods. Other formulas with different time conventions exist.

Let us look at two different members of the same retirement fund.

Mr. A just started his first job and has started saving towards retirement. His starting assets are therefore Rs 0. Let us furtherassume that he contributes Rs10 000 at the end of each of the next 3 years. Mr. B is nearing retirement and thus far accumulated Rs750 000. He also contributes Rs10 000 at the end of each of the next 3years.Both Mr. A and Mr. B are invested in the same balanced fund.

Scenario 1

Now let us assume the annual returns on the balanced fund over the following 3 years are:

Year 1                     -10%

Year 2                     +15%

Year 3                     +5%

What is the average return earned on their assets? And what are the total assets after 3 years?

For Mr. A his total assets are Rs32575.00 which implies an average money-weighted return of 8.35% per annum.

For Mr. B his total assets are Rs847637.50 which implies an average money-weighted return of 2.88% per annum.

Scenario 2

Now let us change the annual returns of year 1 and year 3 around and perform the same calculations:

Year 1                     +5%

  Year 2                   +15%

 Year 3                    -10%

What is the average return earned on their assets? And what are the total assets after 3 years?

For Mr. A his total assets are Rs29350.00 which implies an average money-weighted return of -2.18% per annum.

For Mr. Bhis total assets are Rs844412.50 which implies an average money-weighted return of 2.75% per annum.

Summary of results

The reason for the big difference in money-weighted returns for Mr. A is the fact that he had Rs0 in the first year and was therefore not affected by the first years' results. When the negative return of -10% was earned in the 1st year, he was not affected at all and only benefited from the two positive years. However, when the negative year followed when his accumulated assets was at its highest level i.e. in the last year, the effect was devastating, bringing his total return into negative zone. This is because now the return is not based on time, but on the amount of money it is applied to. Hence the name "money-weighted" returns.

As we can see, Mr.B was also affected but not as much as Mr. A. The reason for this is that his assets in year 3 compared toher assets in year 1 was relatively similar which means that the effect of the returns was similar, even though it was worseunder Scenario 2 when the negative return was applied to a higher asset value.

Graphical Representation of Calculation of TWRR and MWRR

SNAPSHOT

ADVANTAGES AND DISADVANTAGES OF TIME AND MONEY WEIGHTED RETURNS

Advantages of using Money Weighted Rates of Return:

Investors can easily determine if they are making a consistent month on month return and place an equivalent interest rate value on the return. If you are not generating a consistent return, your internal rate of return will fall. There can be no doubt about the importance of making a consistent return over time, because as time goes by, the value of money depreciates due to the effects of inflation.

Ideal for comparing investment performance over time regardless of the size of the investment or when you deposit or withdraw money; for example the Internal Rate of Return is ideally suited to comparing the performance of stocks within a portfolio, or comparing your portfolio with a given market index such as the NIFTY or SENSEX.

Disadvantages of using Money Weighted Rates of Return:

They are not suited to determining the change in the portfolio value between two consecutive dates within a given date range.

Advantages of using Time Weighted Rates of Return

It enables investors to determine rates of return independent of when capital is added or withdrawn from the available investment fund. More commonly this relates to fund managers and not private investors, as fund managers have limited control over when they receive funds from investors, or when the investor choose to withdraw their funds.

Ideally suited to environments where you have shared ownership, as they enable ownership to be allocated based on the value of the assets and the amount invested or withdrawn at any point in time.

Relatively simple to understand and calculate; for example when using the Unit Valuation System the unit value is the sum of the assets divided by the number of units in circulation. If you want to invest more money, you simply buy more units at the current unit value. This is primarily used in all mutual fund schemes.

Disadvantages of using Time Weighted Rates of Return

They do not factor in how long money has been invested and therefore when it was invested. As an investor, the Money Weighted Return measures, such as the Internal Rate of Return enable you to track your performance over time. For example the Unit Value might reflect that you have made a return of 100% but if your unit value does not consistently increase, you can find yourself in a position where your Internal Rate of Return is falling month after month, which results in your investment capital devaluing with time.

Time Weighted metrics are not suited to comparing investment performance for different investment portfolio.