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) = w1R1 + w2Rq + ...+ wnRn

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 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 probability-weighted 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:

Sales ($ Millions)

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:

Deviation from Expected Value
(16.0 - 14.2) = 1.8
(15.0 - 14.2) = 0.8
(14.0 - 14.2) = - 0.2
(13.0 - 14.2) = - 1.2

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 two-asset 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 = w2A2(RA) + w2B2(RB) + 2*(wA)*(wB)*Cov(RA, RB)
Where: wA and wB are portfolio weights, σ2(RA) and σ2(RB) are variances and
Cov(RA, RB) 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 two-asset 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 = w2A2(RA) + w2B2(RB) + 2*(wA)*(wB)*Cov(RA, RB) =(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 two-asset portfolio to illustrate this principle, but most portfolios contain far more than two assets. The formula for variance becomes more complicated for multi-asset 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%.

Expected Return
Worst Case
Base Case
Best Case

σ2 = (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.)

Related Articles
  1. Fundamental Analysis

    Calculating Return on Net Assets

    Return on net assets measures a company’s financial performance.
  2. Economics

    Understanding Cost of Revenue

    The cost of revenue is the total costs a business incurs to manufacture and deliver a product or service.
  3. Economics

    Explaining Carrying Cost of Inventory

    The carrying cost of inventory is the cost a business pays for holding goods in stock.
  4. Fundamental Analysis

    Is India the Next Emerging Markets Superstar?

    With a shift towards manufacturing and services, India could be the next emerging market superstar. Here, we provide a detailed breakdown of its GDP.
  5. Investing

    How To Calculate Minority Interest

    Minority interest calculations require the use of minority shareholders’ percentage ownership of a subsidiary, after controlling interest is acquired.
  6. Term

    Estimating with Subjective Probability

    Subjective probability is someone’s estimation that an event will occur.
  7. Economics

    Explaining Replacement Cost

    The replacement cost is the cost you’d have to pay to replace an asset with a similar asset at the present time and value.
  8. Economics

    How Does National Income Accounting Work?

    National income accounting is an economic term describing the system used by a country to gather data and determine aggregate economic activity.
  9. Investing Basics

    Understanding the Modigliani-Miller Theorem

    The Modigliani-Miller (M&M) theorem is used in financial and economic studies to analyze the value of a firm, such as a business or a corporation.
  10. Economics

    Explaining Kurtosis

    Kurtosis describes the distribution of data around an average.
  1. Zero-Sum Game

    A situation in which one person’s gain is equivalent to another’s ...
  2. Supply

    A fundamental economic concept that describes the total amount ...
  3. Receivables Turnover Ratio

    An accounting measure used to quantify a firm's effectiveness ...
  4. International Financial Reporting ...

    A set of international accounting standards stating how particular ...
  5. Days Sales Outstanding - DSO

    A measure of the average number of days that a company takes ...
  6. Principal-Agent Problem

    The principal-agent problem develops when a principal creates ...
  1. What should I study in school to prepare for a career in corporate finance?

    Depending on which area you want to specialize in, corporate finance can be one of the most competitive fields in business. ... Read Full Answer >>
  2. Why would a company issue preference shares instead of common shares?

    Preference shares, or preferred stock, act as a hybrid between common shares and bond issues. As with any produced good or ... Read Full Answer >>
  3. What is the difference between cost of debt capital and cost of equity?

    In corporate finance, capital – the money a business uses to fund operations – comes from two sources: debt and equity. While ... Read Full Answer >>
  4. What is the difference between gross profit, operating profit and net income?

    The terms profit and income are often used interchangeably in day-to-day life. In corporate finance, however, these terms ... Read Full Answer >>
  5. Can I use my IRA to pay for my college loans?

    If you are older than 59.5 and have been contributing to your IRA for more than five years, you may withdraw funds to pay ... Read Full Answer >>
  6. Can I use my 401(k) to pay for my college loans?

    If you are over 59.5, or separate from your plan-sponsoring employer after age 55, you are free to use your 401(k) to pay ... Read Full Answer >>

You May Also Like

Trading Center

You are using adblocking software

Want access to all of Investopedia? Add us to your “whitelist”
so you'll never miss a feature!