Suppose we invest $100 at the beginning of the year and end up with
$140 at the end of the year. We make cash flows of $10 each at the end of January and February. What is the appropriate return for this situation?
The MWR that we are looking for will be the value that solves this equation:
100× (1 + MWR) + 10× (1 + MWR)0.92 + 10× (1 + MWR)0.83 = 140 MWR no cash flows Gain or Loss Investment made ---
×100
=
Money-Weighted Return 39
The return that causes the beginning value and intermediate cash flows to grow to the ending value is the Internal Rate of ReturnorIRR.
The return is the value that solves for IRR in the equation:
(3.1) where CF is the amount of the cash flow in or out of the portfolio and N is the percentage of the period that the CF was available for investment, or period weight.
The IRR is an MWR. It is approximately equal to a weighted aver- age of the returns for each subperiod within the total period measured, where the weights are a product of the length of the subperiod and the amount of money at work during the subperiod, which is equal to the MVB plus the net cash flows for the subperiod. The IRR is the rate of return implied by the growth in the observed market values of the fund, as well as additional cash flows. It explains the growth in assets over the time period being measured. The IRR is a constant rate over the mea- surement period; this means that we assume that each dollar invested grows at the same rate, no matter when it was invested.
The inputs to the calculation are simply the beginning and ending market values, the cash flows into or out of the portfolio, and the date that these cash flows occurred. Notice that the problem of calculating an IRR is the reverse of that for calculating the future value of an invest- ment. Here, the ending value is known and the return is unknown.
Unlike the formula we used to create future value given a return, we can- not directly calculate the return given the beginning and ending values.
Because we cannot use algebra to rearrange the terms of the equation to derive the solution, the IRR is calculated using a trial and error process—
an initial guess is made and then we iteratively try successive values until the beginning market value equals the sum of the discounted cash flows plus the ending market value. Techniques have been developed to per- form the iteration efficiently and converge on a solution. Exhibit 3.2 shows the calculation of the IRR using the Microsoft Excel solver.
EXHIBIT 3.2 Internal Rate of Return
MVE = MVB×(1+IRR)+CF1×(1+IRR)1…CFN×(1+IRR)N
40 RETURN MEASUREMENT
The IRR that resolves the flows used in Exhibit 3.2 is 17.05%. The steps taken to set up the spreadsheet were:
1. Inserted rows 2–5, which represent the beginning market value of 100, the two cash flows of 10, and the ending market value of 140.
2. Added cell E5, which is the sum of the future value of the cash flows.
3. Added cell E7, which is set as the difference between the ending market value and the sum of the future values.
4. Executed the Excel solver utility, with parameters set to change the value in cell E8 until the difference value in cell E7 was 0.
In terms of our original example:
In this example, each cash flow is compounded at 17.05% for the whole portion of the year invested; this illustrates the assumption made by using the IRR that the rate of return is constant within the period.
We can calculate an IRR for single periods of less than a year. The period weight used for each of the cash flows is the percentage of the total period under consideration. For example, a cash flow on the 5th of a 30- day month would be weighted at [(30 − 5)/30] = 0.8333 of the month.
Exhibit 3.3 shows the calculation of the monthly IRR where MVB = 1000 on 12/31, MVE = 1200 on 1/31, and we have two cash flows, 400 into the portfolio on 1/10, and 100 out of the portfolio on 1/20.
If the cash flows are out of the portfolio, the cash flow adjustment is negative. A time of day assumption must also be taken into account in the IRR calculation; in this example we are using a beginning of day assumption. If the cash flow out of the portfolio took place at the begin- ning of day on the 20th, the cash was not available for investment for 12 full days in the month. Solving for the IRR in this situation we get a –8.02% return for the month.
EXHIBIT 3.3 IRR for Periods Less than a Year 100× (1 + 0.1705) + 10× (1 + 0.1705)0.92 + 10× (1 + 0.1705)0.83 = 140
Money-Weighted Return 41
IRR is an MWR: it takes into account both the timing and the size of cash flows into the portfolio. It is an appropriate measure of the per- formance of the investment as experienced by the investor. But there are some drawbacks to using the IRR formula. The main problem with the IRR formula is that it cannot be calculated directly and needs to be solved via iteration. This was a problem when computer CPU time was very expensive and needed to be conserved. The need to save computing time led to the development of various IRR estimation techniques that did not require the iterative algorithm. One of these return calculation methods, the Modified Dietz method, is still the most common way of calculating periodic investment returns.