Time Series

Expected Risk

Expected Returns

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Exposure to Loss

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Factor Analysis

Cash Flow Analysis

Estimating Future Value: Arithmetic or Geometric

Why do we use the arithmetic average to estimate expected return (assuming we believe in the historical sample as a good estimator)

We should compound at the arithmetic average if we wish to estimate an investmentâ€™s expected value (return). We should compound at the geometric average when we wish to estimate the likelihood that an investment will exceed or fall below a target value.

Consider an investment that has an equal chance of increasing by 25% and decreasing by 5%. There is an equal chance that a dollar invested will grow to $1.25 or decline to $0.94 after one period. The expected value after one period is $1.10 which is equal to (1+arithmetic average of the two possible returns). Subsequently, there are four equally likely outcomes at the end of two periods. See the following diagram for the possible paths of $1.00 investment.

Binomial Tree of Investment Paths

The expected value at the end of the second period is equal to $1.21 which is the probability-weighted outcome.This also corresponds precisely to the quantity (1+arithmetic average of 10%)Â². If we calculate the geometric average for this example and compound it forward for two periods, we arrive at a terminal value of $1.1875, which does not equal to the expected value.

The expected value assumes that there is an equal chance of experiencing any of the possible paths. A path of high returns raises the expected value over multiple periods more than a path of equal-magnitude low returns lowers it. This disproportionate effect is the result of compounding.

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