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Investors purchase financial assets such as shares of stock because they desire to increase their wealth, i.e., earn a positive rate of return on their investments. The future, however, is uncertain; investors do not know what rate of return their investments will realize.
In finance, we assume that individuals base their decisions on what they expect to happen and their assessment of how likely it is that what actually occurs will be close to what they expected to happen. When evaluating potential investments in financial assets, these two dimensions of the decision making process are called expected return and risk.
The concepts presented in this paper include the development of measures of expected return and risk on an indivdual financial asset and on a portfolio of financial assets, the principle of diversification, and the Captial Asset Pricing Model (CAPM).
The future is uncertain. Investors do not know with certainty whether the economy will be growing rapidly or be in recession. As such, they do not know what rate of return their investments will yield. Therefore, they base their decisions on their expectations concerning the future.
The expected rate of return on a stock represents the mean of a probability distribution of possible future returns on the stock. The table below provides a probability distribution for the returns on stocks A and B.
|
State |
Probability |
Return on |
Return on |
|
1 |
20% |
5% |
50% |
|
2 |
30% |
10% |
30% |
|
3 |
30% |
15% |
10% |
|
3 |
20% |
20% |
-10% |
In this probability distribution, there are four possible states of the world one period into the future. For example, state 1 may correspond to a recession. A probability is assigned to each state. The probability reflects how likely it is that the state will occur. The sum of the probabilities must equal 100%, indicating that something must happen. The last two columns present the returns or outcomes for stocks A and B that will occur in the four states.
Given a probability distribution of returns, the expected return can be calculated using the following equation:
where
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Expected Return on Stocks A and B |
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Stock A
Stock B
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So we see that Stock B offers a higher expected return than Stock A. However, that is only part of the story; we haven't yet considered risk.
Risk reflects the chance that the actual return on an investment may be very different than the expected return. One way to measure risk is to calculate the variance and standard deviation of the distribution of returns.
Consider the probability distribution for the returns on stocks A and B provided below.
|
State |
Probability |
Return on |
Return on |
|
1 |
20% |
5% |
50% |
|
2 |
30% |
10% |
30% |
|
3 |
30% |
15% |
10% |
|
3 |
20% |
20% |
-10% |
The expected returns on stocks A and B were calculated on the Expected Return page. The expected return on Stock A was found to be 12.5% and the expected return on Stock B was found to be 20%.
Given an asset's expected return, its variance can be calculated using the following equation:
where
The standard deviation is calculated as the positive square root of the variance.
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Variance and Standard Deviation on Stocks A and B |
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Note: E[RA] = 12.5% and E[RB] = 20% Stock A
Stock B
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Although Stock B offers a higher expected return than Stock A, it also is riskier since its variance and standard deviation are greater than Stock A's. This, however, is only part of the picture because most investors choose to hold securities as part of a diversified portfolio.
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