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Risk premium based on:1 Arithmetic average returns Geometric average returns

1 nr'.U.S. rnarkel reliirn over -> -year U.S. Irea^nry bond. Source; IbbOtSOil itsStKiillti [1999).

Exhibit 10.5 Five-Year Rolling Average of Market Risk Premia

Exhibit 10.5 Five-Year Rolling Average of Market Risk Premia

Geometric Versus Arithmetic Average

Let's turn to the question of geometric versus arithmetic average rates of return. An arithmetic average of rates of return is the simple average of the single period rates of return. Suppose you buy a share of a non-dividend-paying stock for $50. After one year the stock is worth $100. After two years the stock falls to $50 once again. The first period return is 100 percent; the second period return is -50 percent. The arithmetic average return is 25 percent—100 percent -50 percent divided by 2. The geometric average is the compound rate of return that equates the beginning and ending value, zero in our example.

What can we infer from these data? If we are willing to make the strong assumption that each return is an independent observation from a stationary underlying probability distribution, then we can infer that four equally likely return paths actually exist: 100 percent followed by 100 percent, 100 percent followed by -50 percent, -50 percent followed by 100 percent, and -50 percent followed by -50 percent. These possibilities are illustrated in Exhibit 10.6. The shaded area represents what we have actually observed, and the remainder of the binomial tree is what we have inferred by assuming independence.

The difference between the arithmetic and geometric averages is that the former infers expected returns by assuming independence, and the latter treats the observed historical path as the single best estimate of the

future. If you believe that it is proper to apply equal weighting to all branches in the binomial tree, and if your starting position is $50, then your expected wealth is as follows:

Exactly the same value can be obtained by computing the arithmetic average return and applying it to the starting wealth as follows: $50(1.25) (1.25) = 178.125

The arithmetic average is the best estimate of future expected returns because all possible paths are given equal weighting. The single geometric average return is 0 percent, but this is the historical return along a single path that was realized by chance. Although the geometric return is the correct measure of historical performance, it is not forward looking.

The arithmetic return is always higher than the geometric return. The difference between them becomes greater as the variance of returns increases. Also, the arithmetic average depends on the interval chosen. For example, an average of monthly returns will be higher than an average of annual returns. The geometric average, being a single estimate for the entire time interval, is the same regardless of the interval chosen.

Exhibit 10.7 shows illustrative returns during 10 periods, and their arithmetic and geometric average during various intervals. The geometric average is independent of the time interval that is chosen for averaging, but the arithmetic average declines as a function of the time interval.

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