# Money-Weighted and Time-Weighted Rate of Return

I have a question on Time-Weighted Rate of Return (TWRR) and then a question on the links between MWRR and TWRR,

An investor invested £100 in a fund on Jan 1st 1998 and another £100 on Jan 1st 1999. The following gives the price of a unit in the fund on Jan 1st:

Year   Unit Price
1998   100
1998   125
2000   130


Calculate TWRR and MWRR for the period 1998-2000 (1st Jan).

I am aware of the definition, but when I looked at the solution to this question, it was rather bizarre.

For MWRR they got:

100 + 100(1+i)^-1 = 234(1+i)^-2 so i=10.93%


For TWRR they got:

125/100 x 234/225 = (1+i)^2 so i=14.02%


The ‘234’ comes from a ratio I think. They had these calculations:

Year   Unit Price  Invest
1998   100            100          1                   100
1998   125            100          1+ 100/125 = 1.8    225
2000   130                         1.8                 234


Can you please explain this to me?

Also, the last part was a general proof. It asked ‘when is MWRR > TWRR’. Therefore, I am guessing we need to find a general interest rate (It) to show this is true. The questions hint is: Assume you invest 1 unit at time 0 and 1 unit at time 1. What’s the accumulation at time 2? This is guess but is the accumulation 1(1+I1)(1+I2) + 1(1+I2). But what’s this got to do with MWRR/TWRR? And how does it help? I am not sure how to do this and hope someone can help me! :)

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## migrated from stats.stackexchange.comMay 22 '11 at 20:54

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Migrating to math.SE because this is a purely a question of math. – whuber May 22 '11 at 20:53

The derivation of 234 is explained in your grey box: At the start of the first year you invest 100. This rises to 125 at the end of the first year. You then add 100, so at the start of the second year you have 225 invested. This rises to 234 at the end of the second year because $225 \times \frac{130}{125} = 234.$

The TWRR is simply the average annual (compound) rate of return over the period: it takes no account of how much you have invested. The index goes from 100 to 130 in two years, which would also have happened if the compund rate of return had been around 14.02% each year. $100 \times 1.1402^2 \approx 130$.

The MWRR is simply the constant annual (compound) interest rate which would have provided the same overall return and here is around 10.93%. You have one investment of 100 over two years and another over one. $100 \times 1.1093^2 + 100 \times 1.1093 \approx 234$.

Broadly speaking the MWRR will be higher than the TWRR when higher returns coincide with higher levels of investment.

Added: You have $$(1+i_1)(1+i_2) = (1+i_{twrr})^2$$ and $$(1+i_1)(1+i_2) + (1+i_2) = (1+i_{mwrr})^2+(1+i_{mwrr})$$ so
$$i_2>i_1$$ $$\iff i_2>i_{twrr}$$ $$\iff (1+i_1)(1+i_2) + (1+i_2) >(1+i_{twrr})^2 + (1+i_{twrr})$$ $$\iff (1+i_{mwrr})^2+(1+i_{mwrr}) >(1+i_{twrr})^2 + (1+i_{twrr})$$ $$\iff i_{mwrr} > i_{twrr}.$$

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Thanks!!! I understand the calculation above. Thanks for the example. The last par is perhaps the most important. I need to show MWRR>TWRR for some interest rate i. According to the book it's only when i2>i1 but i don't know how to show that... Can you help me out please? – user4645 May 26 '11 at 0:26
@user4645: I added something – Henry May 27 '11 at 17:40