# Money-Weighted and Time-Weighted Rate of Return

I have an inquiry promptly - Weighted Rate of Return (TWRR) and afterwards an inquiry on the web links in between MWRR and also TWRR,

A capitalist spent ₤ 100 in a fund on Jan 1st 1998 and also an additional ₤ 100 on Jan 1st 1999. The adhering to offers the rate of a device in the fund on Jan 1st:

```
Year Unit Price
1998 100
1998 125
2000 130
```

Calculate TWRR and also MWRR through 1998 - 2000 (1st Jan).

I recognize the definition, yet when I considered the remedy to this inquiry, it was instead peculiar.

For MWRR they obtained:

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

For TWRR they obtained:

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

The '234 originates from a proportion I assume. They had these estimations:

```
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 clarify this to me?

Additionally, the tail end was a basic evidence. It asked 'when is MWRR > TWRR'. Consequently, I am presuming we require to locate a basic rates of interest (It) to show this holds true. The inquiries hint is: Assume you spend 1 device sometimes 0 and also 1 device sometimes 1. What' s the buildup sometimes 2? This is hunch yet is the buildup 1 (1+I1) (1+I2)+1 (1+I2). Yet what' s this reached perform with MWRR/TWRR? And also just how does it aid? I am not exactly sure just how to do this and also wish a person can aid me!

The derivation of 234 is clarified in your grey box: At the start of the first year you spend 100. This climbs to 125 at the end of the first year. You after that add 100, so at the beginning of the 2nd year you have actually 225 spent. This climbs to 234 at the end of the 2nd year due to the fact that $225 \times \frac{130}{125} = 234.$

The TWRR is merely the ordinary yearly (substance) price of return over the duration: it takes no account of just how much you have actually spent. The index goes from 100 to 130 in 2 years, which would certainly additionally have actually taken place if the compund price of return had actually been around 14.02% yearly, $100 \times 1.1402^2 \approx 130$

The MWRR is merely the constant yearly (substance) rates of interest which would certainly have given the very same total return and also below is around 10.93%. You have one financial investment of 100 over 2 years and also an additional over one. $100 \times 1.1093^2 + 100 \times 1.1093 \approx 234$.

Generally talking the MWRR will certainly be greater than the TWRR when greater returns accompany greater degrees of financial investment.

*Included: * You have $$(1+i_1)(1+i_2) = (1+i_{twrr})^2 $$ and also $$(1+i_1)(1+i_2) + (1+i_2) = (1+i_{mwrr})^2+(1+i_{mwrr})$$ 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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