Risk and Return
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).
1 Expected Return
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 Stock A |
Return on Stock B |
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
- E[R] = the expected return on the stock,
- N = the number of states,
- p_{i} = the probability of state i, and
- R_{i} = the return on the stock in state i.
Expected Return on Stocks A and B |
Stock AStock B |
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.
2 Measures of Risk – Variance and Standard Deviation
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 Stock A |
Return on Stock B |
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
- N = the number of states,
- p_{i} = the probability of state i,
- R_{i} = the return on the stock in state i, and
- E[R] = the expected return on the stock.
The standard deviation is calculated as the positive square root of the variance.
Variance and Standard Deviation on Stocks A and B |
Note: E[R_{A}] = 12.5% and E[R_{B}] = 20%Stock A
Stock B |
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.
Portfolio Risk and Return
Most investors do not hold stocks in isolation. Instead, they choose to hold a portfolio of several stocks. When this is the case, a portion of an individual stock’s risk can be eliminated, i.e., diversified away. This principle is presented on the Diversification page. First, the computation of the expected return, variance, and standard deviation of a portfolio must be illustrated.
Once again, we will be using the probability distribution for the returns on stocks A and B.
State | Probability | Return on Stock A |
Return on Stock B |
1 | 20% | 5% | 50% |
2 | 30% | 10% | 30% |
3 | 30% | 15% | 10% |
3 | 20% | 20% | -10% |
From the Expected Return and Measures of Risk pages we know that the expected return on Stock A is 12.5%, the expected return on Stock B is 20%, the variance on Stock A is .00263, the variance on Stock B is .04200, the standard deviation on Stock S is 5.12%, and the standard deviation on Stock B is 20.49%.
1 Portfolio Expected Return
The Expected Return on a Portfolio is computed as the weighted average of the expected returns on the stocks which comprise the portfolio. The weights reflect the proportion of the portfolio invested in the stocks. This can be expressed as follows:
where
- E[R_{p}] = the expected return on the portfolio,
- N = the number of stocks in the portfolio,
- w_{i} = the proportion of the portfolio invested in stock i, and
- E[R_{i}] = the expected return on stock i.
For a portfolio consisting of two assets, the above equation can be expressed as
Expected Return on a Portfolio of Stocks A and B |
Note: E[R_{A}] = 12.5% and E[R_{B}] = 20%Portfolio consisting of 50% Stock A and 50% Stock B
Portfolio consisting of 75% Stock A and 25% Stock B |
2 Portfolio Variance and Standard Deviation
The variance/standard deviation of a portfolio reflects not only the variance/standard deviation of the stocks that make up the portfolio but also how the returns on the stocks which comprise the portfolio vary together. Two measures of how the returns on a pair of stocks vary together are the covariance and the correlation coefficient.
The Covariance between the returns on two stocks can be calculated using the following equation:
where
- s_{12} = the covariance between the returns on stocks 1 and 2,
- N = the number of states,
- p_{i} = the probability of state i,
- R_{1i} = the return on stock 1 in state i,
- E[R_{1}] = the expected return on stock 1,
- R_{2i} = the return on stock 2 in state i, and
- E[R_{2}] = the expected return on stock 2.
The Correlation Coefficient between the returns on two stocks can be calculated using the following equation:
where
- r_{12} = the correlation coefficient between the returns on stocks 1 and 2,
- s_{12} = the covariance between the returns on stocks 1 and 2,
- s_{1} = the standard deviation on stock 1, and
- s_{2} = the standard deviation on stock 2.
Covariance and Correlation Coefficent between the Returns on Stocks A and B |
Note: E[R_{A}] = 12.5%, E[R_{B}] = 20%, s_{A} = 5.12%, and s_{B} = 20.49%. |
Using either the correlation coefficient or the covariance, the Variance on a Two-Asset Portfolio can be calculated as follows:
The standard deviation on the porfolio equals the positive square root of the the variance.
Variance and Standard Deviation on a Portfolio of Stocks A and B |
Note: E[R_{A}] = 12.5%, E[R_{B}] = 20%, s_{A} = 5.12%, s_{B} = 20.49%, and r_{AB} = -1.Portfolio consisting of 50% Stock A and 50% Stock B
Portfolio consisting of 75% Stock A and 25% Stock B |
Notice that the portfolio formed by investing 75% in Stock A and 25% in Stock B has a lower variance and standard deviation than either Stocks A or B and the portfolio has a higher expected return than Stock A. This is the essence of Diversification, by forming portfolios some of the risk inherent in the individual stocks can be eliminated.
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