Complete Guide To Corporate Finance
Return, Risk And The Security Market Line  Expected Return, Variance And Standard Deviation Of A Portfolio
Expected return is calculated as the weighted average of the likely profits of the assets in the portfolio, weighted by the likely profits of each asset class.
Expected return is calculated by using the following formula:
Written another way, the same formula is as follows: E(R) = w_{1}R_{1} + w_{2}R_{q} + ...+ w_{n}R_{n}
Example: Expected Return
For a simple portfolio of two mutual funds, one investing in stocks and the other in bonds, if we expect the stock fund to return 10% and the bond fund to return 6% and our allocation is 50% to each asset class, we have the following:
Expected return (portfolio) = (0.1)*(0.5) + (0.06)*(0.5) = 0.08, or 8%
Expected return is by no means a guaranteed rate of return. However, it can be used to forecast the future value of a portfolio, and it also provides a guide from which to measure actual returns.
Let's look at another example. Assume an investment manager has created a portfolio with Stock A and Stock B. Stock A has an expected return of 20% and a weight of 30% in the portfolio. Stock B has an expected return of 15% and a weight of 70%. What is the expected return of the portfolio?
E(R) = (0.30)(0.20) + (0.70)(0.15)
= 6% + 10.5% = 16.5%
The expected return of the portfolio is 16.5%.
Now, let's build on our knowledge of expected returns with the concept of variance.
Variance
Variance (σ^{2}) is a measure of the dispersion of a set of data points around their mean value. In other words, variance is a mathematical expectation of the average squared deviations from the mean. It is computed by finding the probabilityweighted average of squared deviations from the expected value. Variance measures the variability from an average (volatility). Volatility is a measure of risk, so this statistic can help determine the risk an investor might take on when purchasing a specific security.
Example: Variance
Assume that an analyst writes a report on a company and, based on the research, assigns the following probabilities to next year's sales:
The analyst's expected value for next year's sales is (0.1)*(16.0) + (0.3)*(15.0) + (0.3)*(14.0) + (0.3)*(13.0) = $14.2 million.
Calculating variance starts by computing the difference in each potential sales outcome from $14.2 million, then squaring:
Variance then weights each squared deviation by its probability, giving us the following calculation:
(0.1)*(3.24) + (0.3)*(0.64) + (0.3)*(0.04) + (0.3)*(1.44) = 0.96
Portfolio Variance
Now that we've gone over a simple example of how to calculate variance, let's look at portfolio variance.
The variance of a portfolio's return is a function of the variance of the component assets as well as the covariance between each of them. Covariance is a measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns move inversely. Covariance is closely related to "correlation," wherein the difference between the two is that the latter factors in the standard deviation.
Modern portfolio theory says that portfolio variance can be reduced by choosing asset classes with a low or negative covariance, such as stocks and bonds. This type of diversification is used to reduce risk.
Portfolio variance looks at the covariance or correlation coefficient for the securities in the portfolio. Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding two times the weighted average weight multiplied by the covariance of all individual security pairs. Thus, we get the following formula to calculate portfolio variance in a simple twoasset portfolio:
(weight(1)^2*variance(1) + weight(2)^2*variance(2) + 2*weight(1)*weight(2)*covariance(1,2)
Here is the formula stated another way:
Example: Portfolio Variance
Data on both variance and covariance may be displayed in a covariance matrix. Assume the following covariance matrix for our twoasset case:
From this matrix, we know that the variance on stocks is 350 (the covariance of any asset to itself equals its variance), the variance on bonds is 150 and the covariance between stocks and bonds is 80. Given our portfolio weights of 0.5 for both stocks and bonds, we have all the terms needed to solve for portfolio variance.
Portfolio variance = w^{2}_{A}*σ^{2}(R_{A}) + w^{2}_{B}*σ^{2}(R_{B}) + 2*(w_{A})*(w_{B})*Cov(R_{A}, R_{B}) =(0.5)^{2}*(350) + (0.5)^{2}*(150) + 2*(0.5)*(0.5)*(80) = 87.5 + 37.5 + 40 = 165.
Standard Deviation
Standard deviation can be defined in two ways:
1. A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard deviation is calculated as the square root of variance.
2. In finance, standard deviation is applied to the annual rate of return of an investment to measure the investment's volatility. Standard deviation is also known as historical volatility and is used by investors as a gauge for the amount of expected volatility.
Standard deviation is a statistical measurement that sheds light on historical volatility. For example, a volatile stock will have a high standard deviation while a stable blue chip stock will have a lower standard deviation. A large dispersion tells us how much the fund's return is deviating from the expected normal returns.
Example: Standard Deviation
Standard deviation (σ) is found by taking the square root of variance:
(165)^{1/2 }= 12.85%.
We used a twoasset portfolio to illustrate this principle, but most portfolios contain far more than two assets. The formula for variance becomes more complicated for multiasset portfolios. All terms in a covariance matrix need to be added to the calculation.
Let's look at a second example that puts the concepts of variance and standard deviation together.
Example: Variance and Standard Deviation of an Investment
Given the following data for Newco's stock, calculate the stock's variance and standard deviation. The expected return based on the data is 14%.
Answer:
σ^{2 }= (0.10)(0.10  0.14)^{2} + (0.80)(0.14  0.14)^{2 }+ (0.10)(0.18  0.14)^{2 }= 0.0003
The variance for Newco's stock is 0.0003.
Given that the standard deviation of Newco's stock is simply the square root of the variance, the standard deviation is 0.0179, or 1.79%.
(For further reading, see Investing Myths That Need To Go and 5 Reasons You Should Love Stocks.)
Written another way, the same formula is as follows: E(R) = w_{1}R_{1} + w_{2}R_{q} + ...+ w_{n}R_{n}
Example: Expected Return
For a simple portfolio of two mutual funds, one investing in stocks and the other in bonds, if we expect the stock fund to return 10% and the bond fund to return 6% and our allocation is 50% to each asset class, we have the following:
Expected return (portfolio) = (0.1)*(0.5) + (0.06)*(0.5) = 0.08, or 8%
Expected return is by no means a guaranteed rate of return. However, it can be used to forecast the future value of a portfolio, and it also provides a guide from which to measure actual returns.
Let's look at another example. Assume an investment manager has created a portfolio with Stock A and Stock B. Stock A has an expected return of 20% and a weight of 30% in the portfolio. Stock B has an expected return of 15% and a weight of 70%. What is the expected return of the portfolio?
E(R) = (0.30)(0.20) + (0.70)(0.15)
= 6% + 10.5% = 16.5%
The expected return of the portfolio is 16.5%.
Now, let's build on our knowledge of expected returns with the concept of variance.
Variance
Variance (σ^{2}) is a measure of the dispersion of a set of data points around their mean value. In other words, variance is a mathematical expectation of the average squared deviations from the mean. It is computed by finding the probabilityweighted average of squared deviations from the expected value. Variance measures the variability from an average (volatility). Volatility is a measure of risk, so this statistic can help determine the risk an investor might take on when purchasing a specific security.
Example: Variance
Assume that an analyst writes a report on a company and, based on the research, assigns the following probabilities to next year's sales:
Scenario 
Probability 
Sales ($ Millions) 
1 
0.10 
$16 
2 
0.30 
$15 
3 
0.30 
$14 
3 
0.30 
$13 
The analyst's expected value for next year's sales is (0.1)*(16.0) + (0.3)*(15.0) + (0.3)*(14.0) + (0.3)*(13.0) = $14.2 million.
Calculating variance starts by computing the difference in each potential sales outcome from $14.2 million, then squaring:
Scenario 
Probability 
Deviation from Expected Value 
Squared 
1 
0.1 
(16.0  14.2) = 1.8 
3.24 
2 
0.30 
(15.0  14.2) = 0.8 
0.64 
3 
0.30 
(14.0  14.2) =  0.2 
0.04 
4 
0.30 
(13.0  14.2) =  1.2 
1.44 
Variance then weights each squared deviation by its probability, giving us the following calculation:
(0.1)*(3.24) + (0.3)*(0.64) + (0.3)*(0.04) + (0.3)*(1.44) = 0.96
Portfolio Variance
Now that we've gone over a simple example of how to calculate variance, let's look at portfolio variance.
The variance of a portfolio's return is a function of the variance of the component assets as well as the covariance between each of them. Covariance is a measure of the degree to which returns on two risky assets move in tandem. A positive covariance means that asset returns move together. A negative covariance means returns move inversely. Covariance is closely related to "correlation," wherein the difference between the two is that the latter factors in the standard deviation.
Modern portfolio theory says that portfolio variance can be reduced by choosing asset classes with a low or negative covariance, such as stocks and bonds. This type of diversification is used to reduce risk.
Portfolio variance looks at the covariance or correlation coefficient for the securities in the portfolio. Portfolio variance is calculated by multiplying the squared weight of each security by its corresponding variance and adding two times the weighted average weight multiplied by the covariance of all individual security pairs. Thus, we get the following formula to calculate portfolio variance in a simple twoasset portfolio:
(weight(1)^2*variance(1) + weight(2)^2*variance(2) + 2*weight(1)*weight(2)*covariance(1,2)
Here is the formula stated another way:
Portfolio Variance = w^{2}_{A}*σ^{2}(R_{A}) + w^{2}_{B}*σ^{2}(R_{B}) + 2*(w_{A})*(w_{B})*Cov(R_{A}, R_{B}) Where: w_{A} and w_{B }are portfolio weights, σ^{2}(R_{A}) and σ^{2}(R_{B}) are variances and Cov(R_{A}, R_{B}) is the covariance 
Example: Portfolio Variance
Data on both variance and covariance may be displayed in a covariance matrix. Assume the following covariance matrix for our twoasset case:
Stock 
Bond 

Stock 
350 
80 
Bond 
150 

From this matrix, we know that the variance on stocks is 350 (the covariance of any asset to itself equals its variance), the variance on bonds is 150 and the covariance between stocks and bonds is 80. Given our portfolio weights of 0.5 for both stocks and bonds, we have all the terms needed to solve for portfolio variance.
Portfolio variance = w^{2}_{A}*σ^{2}(R_{A}) + w^{2}_{B}*σ^{2}(R_{B}) + 2*(w_{A})*(w_{B})*Cov(R_{A}, R_{B}) =(0.5)^{2}*(350) + (0.5)^{2}*(150) + 2*(0.5)*(0.5)*(80) = 87.5 + 37.5 + 40 = 165.
Standard Deviation
Standard deviation can be defined in two ways:
1. A measure of the dispersion of a set of data from its mean. The more spread apart the data, the higher the deviation. Standard deviation is calculated as the square root of variance.
2. In finance, standard deviation is applied to the annual rate of return of an investment to measure the investment's volatility. Standard deviation is also known as historical volatility and is used by investors as a gauge for the amount of expected volatility.
Standard deviation is a statistical measurement that sheds light on historical volatility. For example, a volatile stock will have a high standard deviation while a stable blue chip stock will have a lower standard deviation. A large dispersion tells us how much the fund's return is deviating from the expected normal returns.
Example: Standard Deviation
Standard deviation (σ) is found by taking the square root of variance:
(165)^{1/2 }= 12.85%.
We used a twoasset portfolio to illustrate this principle, but most portfolios contain far more than two assets. The formula for variance becomes more complicated for multiasset portfolios. All terms in a covariance matrix need to be added to the calculation.
Let's look at a second example that puts the concepts of variance and standard deviation together.
Example: Variance and Standard Deviation of an Investment
Given the following data for Newco's stock, calculate the stock's variance and standard deviation. The expected return based on the data is 14%.
Scenario 
Probability 
Return 
Expected Return 
Worst Case 
10% 
10% 
0.01 
Base Case 
80% 
14% 
0.112 
Best Case 
10% 
18% 
0.018 
Answer:
σ^{2 }= (0.10)(0.10  0.14)^{2} + (0.80)(0.14  0.14)^{2 }+ (0.10)(0.18  0.14)^{2 }= 0.0003
The variance for Newco's stock is 0.0003.
Given that the standard deviation of Newco's stock is simply the square root of the variance, the standard deviation is 0.0179, or 1.79%.
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