Complete Guide To Corporate Finance

AAA

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
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:

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 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:

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 = 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%.

Scenario
Probability
Return
Expected Return
Worst Case
10%
10%
0.01
Base Case
80%
14%
0.112
Best Case
10%
18%
0.018

Answer:
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.)

Portfolios


Related Articles
  1. Mutual Funds & ETFs

    High Dividend ETF: iShares Core High Dividend vs. Vanguard Dividend Appreciation (HDV, VIG)

    Get a comparison review of two high dividend yield ETFs: the iShares Core High Dividend ETF and the Vanguard Dividend Appreciation ETF.
  2. Retirement

    A Guide to Arizona's State Retirement System

    The Arizona State Retirement System offers a defined-benefit plan for former teachers, state workers and public employees.
  3. Mutual Funds & ETFs

    2 Reasons to Be Wary of New ETFs (IFLY, EQLT)

    Pay attention to the number of thematic and smart beta ETFs that may fail to survive as a result of poor performance and thin trading volume.
  4. Forex

    Global Utilities: Exploring Revenue Trends & Fundamentals

    Analyze global revenue exposure in the utilities sector to learn about the impact of currency, regulation and economic growth on geographic contributions.
  5. Home & Auto

    4 Alternatives to a Traditional Mortgage

    If you can't qualify for or don't want a traditional mortgage, one of these options might be right for you.
  6. Home & Auto

    Understanding Mortgage Impound Accounts

    Home buyers with low down payments may get stuck with higher mortgage payments. Find out what you get for the extra money.
  7. Investing

    Municipal Bonds Offer Something More for Everyone

    Are municipal bonds really for me? The popular perception is that tax-exempt income only benefits those investors in the highest tax brackets.
  8. Retirement

    5 Top Alternatives to a Reverse Mortgage

    If you have substantial home equity and don't want to do a reverse mortgage to tap it for retirement expenses, cost out these viable alternatives.
  9. Credit & Loans

    What Is an Alt-A Mortgage?

    Called "liar loans" for their low documentation requirements, Alt-A mortgages were hot until the subprime crisis. Now Wall Street wants to bring them back.
  10. Home & Auto

    Understanding Mortgage-Backed Securities

    Find out the meaning of this popular asset-backed security and its benefits for banks and investors.
RELATED TERMS
  1. Primary Mortgage Market

    The market where borrowers and mortgage originators come together ...
  2. 100% Mortgage

    A mortgage loan in which the borrower receives a loan amount ...
  3. Reverse Mortgage

    A type of mortgage in which a homeowner can borrow money against ...
  4. Mortgage Originator

    An institution or individual that works with a borrower to complete ...
  5. Secondary Mortgage Market

    The market where mortgage loans and servicing rights are bought ...
  6. Mortgage Pool

    A group of mortgages held in trust as collateral for the issuance ...
RELATED FAQS
  1. How safe are money market accounts?

    Learn the difference between a money market account and a money market fund. Both savings vehicles are relatively safe, but ... Read Answer >>
  2. Why is Belize considered a tax haven?

    Explore the factors that make Belize one of the most modern and corporate-friendly tax havens in the world, including its ... Read Answer >>
  3. What is an assumable mortgage?

    The purchase of a home is a very expensive undertaking and usually requires some form of financing to make the purchase possible. ... Read Answer >>
  4. Why would a homebuyer need to take out PMI (private mortgage insurance)?

    Learn why some home buyers are required to take out private mortgage insurance (PMI), and how it affects the total monthly ... Read Answer >>
  5. Why does the majority of my mortgage payment start out as interest and gradually ...

    When you make a mortgage payment, the amount paid is a combination of an interest charge and principal repayment. Over the ... Read Answer >>
  6. What are the disadvantages of a Roth IRA?

    Get informed about Roth IRAs, which have a few disadvantages, including limited access to funds and contribution limits based ... Read Answer >>
Hot Definitions
  1. Hawk

    A policymaker or advisor who is predominantly concerned with interest rates as they relate to fiscal policy. A hawk generally ...
  2. Physical Capital

    Physical capital is one of the three main factors of production in economic theory. It consists of manmade goods that assist ...
  3. Reverse Mortgage

    A type of mortgage in which a homeowner can borrow money against the value of his or her home. No repayment of the mortgage ...
  4. Labor Market

    The labor market refers to the supply and demand for labor, in which employees provide the supply and employers the demand. ...
  5. Demand Curve

    The demand curve is a graphical representation of the relationship between the price of a good or service and the quantity ...
  6. Goldilocks Economy

    An economy that is not so hot that it causes inflation, and not so cold that it causes a recession. This term is used to ...
Trading Center