Measuring investment returns over multiple periods

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The holding-period return is a simple and unambiguous measure of investment return over a single period. But often you will be interested in average returns over longer periods of time. For example, you might want to measure how well a mutual fund has performed over the preceding five-year period. In this case, return measurement is more ambiguous. Consider a fund that starts with $1 million under management. It receives additional funds from new and existing shareholders and also redeems shares of existing shareholders so that net cash inflow can be positive or negative. The fund’s quarterly results are as given in Table 5.1 , with negative numbers in parentheses. The numbers indicate that when the firm does well (i.e., achieves a high HPR), it attracts new funds; otherwise it may suffer a net outflow. For example, the 10% return in the first quarter by itself increased assets under management by .10 3 $1 million 5 $100,000; it also elicited new investments of $100,000, thus bringing assets under management to $1.2 million

by the end of the quarter. An even better HPR in the second quarter elicited a larger net inflow, and the second quarter ended with $2 million under management. However, HPR in the third quarter was negative, and net inflows were negative. How would we characterize fund performance over the year, given that the fund experienced both cash inflows and outflows? There are several candidate measures of performance, each with its own advantages and shortcomings. These are the arithmetic average, the geometric average, and the dollar-weighted return. These measures may vary considerably, so it is important to  understand their differences.

Arithmetic average The arithmetic average of the quarterly returns is just the sum of the quarterly returns divided by the number of quarters; in the above example: (10 + 25 - 20 + 20)/4 5 8.75%. Since this statistic ignores compounding, it does not represent an equivalent, single quarterly rate for the year. However, without information beyond the historical sample, the arithmetic average is the best forecast of performance for the next quarter.

Geometric average The geometric average of the quarterly returns is equal to the single per-period return that would give the same cumulative performance as the sequence of actual returns. We calculate the geometric average by compounding the actual period-byperiod returns and then finding the per-period rate that will compound to the same final  value. In our example, the geometric average quarterly return, r G , is defined by

The left-hand side of this equation is the compounded year-end value of a $1 investment earning the four quarterly returns. The right-hand side is the compounded value of a $1 investment earning r G each quarter. We solve for r G :

The geometric return is also called a time-weighted average return because it ignores the quarter-to-quarter variation in funds under management. In fact, an investor will obtain a larger cumulative return when high returns are earned in periods when larger sums have been invested and low returns are earned when less money is at risk. In Table 5.1 , the higher returns (25% and 20%) were achieved in quarters 2 and 4, when the fund managed $1,200,000 and $800,000, respectively. The lower returns ( 2 20% and 10%) occurred when the fund managed $2,000,000 and $1,000,000, respectively. In this case, better returns were earned when less money was under management an unfavorable combination. Published data on past returns earned by mutual funds actually are required to be timeweighted returns. The rationale for this practice is that since the fund manager does not have full control over the amount of assets under management, we should not weight returns in one  period more heavily than those in other periods when assessing “typical” past performance.

Dollar-weighted return To account for varying amounts under management, we treat the fund cash flows as we would a capital budgeting problem in corporate finance and compute the portfolio manager’s internal rate of return (IRR). The initial value of $1 million and the net cash inflows are treated as the cash flows associated with an investment “project.” The year-end “liquidation value” of the portfolio is the final cash flow of the project. In our example, the investor’s net cash flows are as follows:

 

The entry for time 0 reflects the starting contribution of $1 million; the negative entries for times 1 and 2 are additional net inflows in those quarters, while the positive value for quarter 3 signifies a withdrawal of funds. Finally, the entry for time 4 represents the sum of the final (negative) cash inflow plus the value of the portfolio at the end of the fourth quarter. The latter is the value for which the portfolio could have been liquidated at year-end. The dollar-weighted average return is the internal rate of return of the project, which is 3.38%. The IRR is the interest rate that sets the present value of the cash flows realized on the portfolio (including the $1.56 million for which the portfolio can be liquidated at the end of the year) equal to the initial cost of establishing the portfolio. It therefore is the interest rate that satisfies the following equation

The dollar-weighted return in this example is less than the time-weighted return of 7.19% because, as we noted, the portfolio returns were higher when less money was under management. The difference between the dollar- and time-weighted average return in this case is quite large.

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