For instance, you may have a need for cash between years or maybe an unexpected windfall which changes how your cash flows from investing affect your current position as well as your investment portfolios overall performance. Understanding this, we should look more closely at time-weighted versus dollar weighted returns.

Time-weighted return is the easiest and is the fairest measure for understanding a portfolio manager's performance. Looking at any 3 given years, assume the following annual returns:

Year 1: 8%

Year 2: 10%

Year 3: -4%

The annual time weighted return is the cube root of the combined returns or:

(1.08 x 1.10 x 0.96)^(1/3) = 1.14^(1/3) = 1.0448 or 4.47% gain

However, if assessing performance becomes more difficult for dollar-weighted return if there is cash flows happening between years not directly related to investment gains or losses (adding or taking capital from the investment portfolio). For example, assuming the same percentage gains, assume the following cash flows:

Year 1: $10M initial investment inflow

Year 2: $3M withdrawal outflow

Year 3: $5M investment inflow

(assume cash flows happen at the beginning of the year)

What happens? What model can we use do understand the dollar-weighted gains? By using a cash flow model (which incorporates gains and losses from portfolio management) looking at what happens between each year:

- Year 1 Starting Balance:

$10M - Year 1 Ending Balance:

($10M x 1.08) = $10.8M - Year 2 Starting Balance:

$10.8M - $3M = $7.08M - Year 2 Ending Balance:

($7.08 x 1.10) = $7.788M - Year 3 Starting Balance:

$7.788M + $5M = $12.788M - Year 3 Ending Balance:

($12.788 x .96) = $12.276M

Recall that:

NPV = CF0 + [CF1 / (1 + IRR)] + [CF2 / (1 + IRR)^2] + + [CF3 / (1 + IRR)^3] + ...

0 = $10M + [-$3M / (1 + IRR)] + [$5M / (1 + IRR)^2] + [-$12.276 / (1 + IRR)^3]

Also recall that for accurate reporting, the terminal year is assume to be withdrawn in full. Using Excel or a Financial calculator to solve for IRR, the dollar weighted return. In this example, the result is calculated as 0.943%. Why is this value so low compared to a time weighted return? As it turns out, the investor in this scenario made a few unlucky mistakes:

A few assumptions / extensions:

0 = $10M + [-$3M / (1 + IRR)] + [$5M / (1 + IRR)^2] + [-$12.276 / (1 + IRR)^3]

Also recall that for accurate reporting, the terminal year is assume to be withdrawn in full. Using Excel or a Financial calculator to solve for IRR, the dollar weighted return. In this example, the result is calculated as 0.943%. Why is this value so low compared to a time weighted return? As it turns out, the investor in this scenario made a few unlucky mistakes:

- Took money out after an ok year ($3M withdrawal after a year gaining 8%) and didn't capitalize on the upcoming year with 10% gains. The investor didn't gain $3M x 10%. In otherwords, they missed out on $300k in gains.
- Put more money ($5M) in after a good year (10% gains) just in time for a bad year (4% loss or 96% retention). This action lost $5M x 4% or $200k.

A few assumptions / extensions:

- The portfolio doesn't have to be "managed". The time-weighted returns can simply be based on market indicies such as S&P 500 or a portfolio constructed against the DJIA.
- Good timing can reward as much as bad timing punishes.
- There is a liquidty premium in the form of economic opportunity cost (in the case of the $3M being withdrawn for "investor needs", that was unavailble to participate in the 10% increase that year).

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