Consider the investment in the following table:
Start of Year 1 | One share purchased at $100 |
End of Year 1 | $5.00 dividend/share paid and one additional share purchased at $125 |
End of Year 2 | $5.00 dividend/share paid and both shares sold for $140 per share |
Assuming dividends are not reinvested, compared with the time-weighted return, the money-weighted return is:
- the same.
Solution
A is correct. The following table represents cash flows of the investment:
Year | Contribution | Start-of-Year Value after Contribution | End-of-Year Dividend | End-of-Year Value after Dividend |
1 | 1 × $100 | 1 × $100 = $100 | 1 × $5 = $5 | $125 |
2 | 1 × $125 | 2 × $125 = $250 | 2 × $5 = $10 | (2 × 140) + 10 = $290 |
The time-weighted rate of return (TWR) on this investment is found by taking the geometric mean of the two holding period returns (HPRs):TWR = [(1 + HPRYear 1) × (1 + HPRYear 2)]1/2 − 1where
HPRYear 1 = ($125 − $100 + $5)/$100 = 30.0%
HPRYear 2 = ($280 − $250 + $10)/$250 = 16.0%
TWR = [(1 + 0.30) × (1 + 0.16)]1/2 − 1 = 22.80%
The money-weighted rate of return (MWR) is the internal rate of return (IRR) of the cash flows associated with the investment:
0=−100+(−125+5)(1+r1)+(280+10)(1+r2)0=−100+−125+51+�1+280+101+�2, where r = MWR.
Using the cash flow (CF) function of a financial calculator:
CF0 = −100, CF1 = (−125 + 5), CF2 = (280 + 10), and solving for IRR: MWR or IRR = 20.55%.
The difference between the TWR and MWR of this investment = 22.80% − 20.55% = 2.25%, or 225 bps, with MWR being lower than TWR.
B is incorrect. The difference between MWR and TWR is either higher or lower, and could hardly be equal.
C is incorrect, as per the calculation above.
Why in this question they used period calculation (i.e. power to the returns) generally we are using it for geometric mean only.
IT HAS BEEN SPECIFIED IN THE CORE THAT MWROR IS THE IRR AND THE TWROR IS GEOMETRIC MEAN.