Calculating Money-Weighted Returns with Dated Cash Flows
Summary
The document shows how to calculate a money-weighted rate of return for a pension fund with dated inflows and outflows. The method sets the future value equation to zero at the fund’s closing date, compounds the starting balance over twelve months, adjusts the May benefit payment for the eight months it would otherwise have remained invested, and counts the year-end contribution for zero months. The closing fund value is treated as an amount transferred out.
Solving for the monthly rate gives approximately 0.0124671, which annualizes to about 16.03%, consistent with the stated 16% result. The initial attempt used inconsistent timing exponents and cash-flow signs, producing a lower annual figure. This is a single worked example; it relies on the stated monthly compounding convention and does not discuss alternative day-count or annualization conventions.
Key ideas
- Money-weighted return is found by setting the dated cash flows and ending value to a common valuation date.
- The opening balance is compounded for the full investment period.
- A benefit paid before year-end is adjusted for the months it would otherwise have remained invested.
- A contribution received on the valuation date has no time to earn a return within the period.
- The example converts the solved monthly rate to an annual return of about 16.03%.
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Full text
# Problems with Money Weighted Rate of Return
# Problems with Money Weighted Rate of Return
The market value of a small pension fund’s assets was 2.7m on 1 January 2000 and 3.1 m on 31 December 2000. During 2000 the only cash flows were:
- Bank interest and dividends totalling 125,000 received on 30 June
- A lump sum retirement benefit of 75,000 paid on 1 May
- A contribution of 50,000 paid to the fund on 31 December
Show that the MWRR is 16%
What I have tried:
$3.1*10^{6}*(1+i)^{-12}-75,000*(1+i)^{-5}+50,000*(1+i)^{-12}-2.7*10^{6}=0$
solving this in wolfram alpha we get $i=0.0107371$, converting this from a monthly interest rate we find that MWRR is 13.7% which is $ \neq 16$%.
Can someone please explain/show me where I went wrong in my attempted solution of the problem?
## Answer by Value at Risk (score 1, accepted)
https://quant.stackexchange.com/a/37540
In Wolfram Alpha language ...
2.7∗10^6*(1+x)^12 -75000*(1+x)^8 +50,000 -3.1∗10^6 = 0
... gives x≈0.0124671 per month, which is 16.03% per annum.
I.e. All incomings and outgoings must add up to zero, after adjusting for the monthly interest rate "x" over the number of months invested.
- 2.7m remains in the fund for the full 13-1 = 12 months, and would normally be positive;
- 75k benefit is "missing" for 13-5 = 8 months, and is therefore negative;
- 50k incoming is positive, but does not remain in for any length of time, so the exponent is 0
- 3.1m is the final value, and is negative because it is transferred out at the end of the period.
Thanks for showing that Wolfram Alpha handles polynomials so well. Seeing that, my New Year's resolution is use it more !Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.