Money-Weighted Returns
Internal Rate of Return (IRR)
The Money-Weighted Return is the same as the Internal Rate of Return (IRR). 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 IRR can be calculated by solving the expression below for r(T)...
MV(T) = MV(0)*{1+r(T)}^T + sum[C(t)*{1+r(T)}^{T-t}]
MV(T)... Ending market value portfolio
MV(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.
Unfortunately, there exists no "formula" for IRR. The above expression has to be solved numerically (typically with the Newton-Raphson method). Fortunately, there is plenty of software available for free that can do that task for you. In Microsoft Excel, you can use the function 'XIRR(values; dates; guess)'; be aware that Excel uses 365 day count, not effective days. Alternatively, one can calculate money-weighted returns with Excel's function 'YIELD(settlement,maturity,rate,pr,redemption,frequency,basis)', the only difference being that YIELD is assuming 30 days per month.
MWR measures the return on your money invested. So periods in which more funds are invested contribute more to the overall return - returns are 'money-weighted'. MWR is also said to reflect the 'timing' of the money invested.
If investment management is delegated (as it is the case when giving away a mandate to a portfolio manager or buying a mutual fund), then you can expect that the investment manager will only report TWR. Reason for this is that the timing of contributions is outside his control and responsibility for the timing is with the client (or his advisor). This can lead to the rather absurd situation in which the investment manager presents a very good track record, while the return actually earned on his client money is negative. One can interpret TWR as the 'manager's return', while MWR is the 'client's return'. This important distinction is very much neglected in the financial industry.
The properties of the IRR are well known and can be found in the literature related to financial decision making, investment project evaluation. Further, note that IRR is equivalent to the 'yield' concept used in bond analysis and the fixed income literature in general.
Since calculating IRR can be a bit tricky, a lot of effort has been put into developing approximations.
All Contributions at the End: eopIRR
Assuming that all net contributions take place at the same time and at the end of the investment period, the IRR expression becomes...
MV(T) = MV(0)*{1+r(T)} + C(T)*{1+r(T)}^{T-T}
This can now be solved analytically for r(T)...
r(T) = {MV(T) - MV(0) - C(T)} / MV(0)
The above expression is used in the 'BAI Method' for TWR approximations; we call it the 'eopIRR'. See the next section...
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