Chapter 8Risk, Return, and Portfolio Theory

Chapter 8 Outline8.1

Measuring Returns

• Ex post vs. ex ante returns

• Total return• Measuring

average returns

8.2 Measuring

Risk

• The standard deviation: ex post

• The standard deviation: ex ante

8.3 Expected

Return and Risk for

Portfolios • Calculating

a portfolio’s return

• Covariance• Correlation

coefficient

8.5 Diversificati

on

• Random diversification

• Unique risk• Market risk• Total risk

8.4 The Efficient Frontier

• Modern Portfolio Theory

• Harry Markowitz

• Efficient portfolio

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Definitions: Ex post return-past or historical returns Ex ante returns-expected returns Income yield-return earned by investors as a

periodic cash flow Capital gain-measures the appreciation (or

depreciation) in the price of the asset from some starting price

8.1 Measuring Returns on Investment

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Where CF1 is the expected cash flows to be received, P0 is the purchase price today, and P1 = selling price 1 year from today

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Total return Total return- the sum of the income yield and the

capital gain (or loss) yield Example: At the end of 2009, IBM had a stock

price of $130.90 and at the end of 2010 IBM had a stock price $146.76. During 2010, IBM paid four dividends, totaling $2.50. The return on IBM stock for 2010 is:

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Answer:

=1.191% + 12.12% = 14.03%

Total Return

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Arithmetic mean or average mean-sum of all observations divided by the total number of observations

Geometric mean-average or compound growth rate over multiple time periods

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Measuring Average Returns

The arithmetic mean simply averages the annual rates of return without taking into account that the amount invested varies across time.

The geometric mean is a better average return estimate when we are interested in the rate of return performance of an investment over time.

Why do the arithmetic mean return and the geometric mean return differ?

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The expected return is often estimated based on historical averages, but the problem is that there is no guarantee that the past will repeat itself.

Estimating Expected Returns on Investment

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Example:Suppose you have two possible returns on investment:

The expected return is the weighted average of the possible returns, where the weights are the probabilities:

Expected return, E(r) = 0.07

Estimating Expected Returns on Investment

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Suppose we have the following estimates for the return on an investment:

Estimating expected returns

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Probability

Return

Best case 30% 25%Most likely case

60% 10%

Worst case 10% -5%What is the expected value return, based on these estimates?

Probability ReturnProbability x Return

Best case 30% 25% 7.5%Most likely case 60% 10% 6.0%Worst case 10% -5% -0.5%Total 100% 13.0%

Estimating expected returns

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Expected return

Risk is the probability of incurring harm, and for financial managers, harm generally means losing money or earning an inadequate rate of return.

Risk is the probability that the actual return from an investment is less than the expected return. This means that the more variable the possible return, the greater the risk.

8.2 Measuring Risk

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Standard Deviation: Ex PostUses information that has occurred (ex post)

Ex post standard deviation (σ) =

Where σ is the standard deviation, is the average return, xi is the observation in year i, and N is the number of observations

Methods of Measuring Risk

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The Standard Deviation: Ex AnteFormulated based on expectations about the future cash flows or returns of an asset.

Ex ante standard deviation (σ) =

Where r is one of the possible outcomes, E(r) is the calculated expected value of the possible outcomes, and pi is the probability of the occurrence

Methods of Measuring Risk

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Probability Return

Expected value

Deviation from

expected value

Squared deviation

Weighted

squared deviatio

nBest case 30% 25% 7.5% 12% 0.0144 0.00432Most likely case 60% 10% 6.0% -3% 0.0009 0.00054Worst case 10% -5% -0.5% -18% 0.0324 0.00324

Expected return = 13.0% Variance = 0.00810

Ex-ante risk

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𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛= 2√0.00810=0.09𝑜𝑟 9%

Ex Post Standard Deviation

Ex Ante Standard Deviation

Perspective Looking back on what has occurred.

Looking forward to what may happen.

Observations Observations from the past.

Possible outcomes in the future.

Calculation of the variance

Average of the squared deviations of observations from the mean of the observations.

Weighted average of the squared deviations of possible outcomes from the expected outcome, where the weights are the probabilities.

Formula for the standard deviation (σ)

Methods of Measuring Risk

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A portfolio is a collection of assets, such as stocks and bonds, that are combined and considered a single asset.

Investors should diversify their investments so that they are not unnecessarily exposed to a single negative event

“Don’t put all your eggs in one basket”

8.3 Expected Return and Risk for Portfolios

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The expected return on a portfolio is the weighted average of the expected returns on the individual assets in the portfolio:

where E(rp) represents the expected return on the portfolio, E(ri) represents the expected return on asset i, and wi represents the portfolio weight of asset i.

Calculating a Portfolio’s Return

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Calculating a Portfolio’s Return

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ProbabilityReturn on

AReturn on

B

Probability x Return

on A

Probability x Return

on BGood 25% 15% 20% 3.75% 5.00%OK 60% 10% 12% 6.00% 7.20%Bad 15% 5% -10% 0.75% -1.50%

10.50% 10.70%

Expected return on A

Expected return on B

Estimating standard deviation:

where σpis the portfolio standard deviation and COVxy is the covariance of the returns on X and Y

Covariance-a statistical measure of the degree to which two or more series move together

Calculating a Portfolio’s Return

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Estimating a standard deviations of returns

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Probability

Return on A

Return on B

Probability x Return

on A

Probability x Return

on B

Weighted deviation squared

for Return on A

Weighted deviation squared

for Return on A

Good 25% 15% 20% 3.75% 5.00% 0.000506 0.002162OK 60% 10% 12% 6.00% 7.20% 0.000015 0.000101Bad 15% 5% -10% 0.75% -1.50% 0.000454 0.006427

10.50% 10.70% 0.000975 0.008691

Standard deviation of A = 3.1225%Standard deviation of B = 9.3226%

Estimating a Covariance

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Probability

Probability x

Return on A

Probability x

Return on B

Deviation for Return

on A

Deviation for

Return on B

Product of

deviations

Probability

weighted products

Good 25% 3.75% 5.00% 4.500% 9.300% 0.0041850 0.0010463

OK 60% 6.00% 7.20% -0.500% 1.300%-

0.0000650-

0.0000390Bad 15% 0.75% -1.50% -5.500% -20.700% 0.0113850 0.0017078

10.50% 10.70% 0.0027150Covarianc

e15%-10.5%

20%-10.7%

Correlation coefficient- a statistical measure that identifies how asset returns move in relation to one another; denoted by p

Ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation)

Related to covariance and individual standard deviations:

where ρxy is the correlation coefficient, COV is covariance, subxy are the variables, and σ is the standard deviation

Correlation Coefficient

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No correlation:Correlation Examples

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Perfect positive and perfect negative correlation:

Correlation Examples

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Positive and negative correlation:

Correlation Examples

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From our example,

Correlation

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Good OK Bad-20%-10%

0%10%20%30%

Return on A Return on B

Scenario

Poss

ible

ret

urn

The essential problem is that our models are still too simple to capture the full array of governing variables that drive global economic reality. A model, of necessity, is an abstraction from the full detail of the real world.

Word of Caution

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8.4 The Efficient FrontierThere is a mix of assets that minimizes a

portfolio’s standard deviation

Modern portfolio theory is a set of theories that explain how rational investors, who are risk averse, can select a set of investments that maximize the expected return for a given level of risk

Harry Markowitz is the “father” of modern portfolio theory, was awarded the 1990 Nobel Prize in Economics

Modern Portfolio Theory

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Harry Markowitz showed investors how to diversify their portfolios based on several assumptions: Investors are rational decision makers Investors are risk averse Investor preferences are based on a portfolio’s

expected return and risk (as measured by variance or standard deviation)

Introduced the notion of an efficient portfolio

Modern Portfolio Theory

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The efficient portfolio is a collection of investments that offers the highest expected return for a given level of risk, or, equivalently, offers the lowest risk for a given expected return

The Efficient Portfolio

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The Efficient Frontier

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Portfolios are either: Attainable (lie on the minimum variance

frontier), or Dominated (lower level of expected return for a

given level of risk than another portfolio)

The Efficient Frontier

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Rational, risk-averse investors are interested in holding only those portfolios, such as Portfolio II, that offer the highest expected return for their given level of risk.

A more aggressive (i.e., less risk averse) investor might choose Portfolio II, whereas a more conservative (i.e., more risk averse) investor might prefer Portfolio V (i.e., the MVP).

The Efficient Frontier

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The reduction of risk by investing funds across several assets

8.5 Diversification

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Random diversification or naïve diversification is the act of randomly buying assets without regard to relevant investment characteristics (e.g., “dartboard”)

Types of Risk

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Unique risk, nonsystematic risk, or diversifiable risk-a company-specific part of total risk that is eliminated by diversification

Market risk, systematic risk, beta risk, or nondiversifiable risk-a systematic part of total risk that cannot be eliminated by diversification

Types of Risk

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Total risk = Market risk + Unique risk

Types of Risk

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Risk that DISAPPEARS as you diversify

Risk that REMAINS even when you diversify

• Diversifiable risk• Nonsystematic risk• Unique risk

• Market risk• Systematic risk• Nondiversifiable risk• Beta risk

Summarizing Risks

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In financial decision making and analysis, we use both ex post and ex ante returns on investments: We use ex post returns when we look back at what has happened, and we use ex ante returns when we look forward, into the future.

One measure of risk is the standard deviation, which is a measure of the dispersion of possible outcomes.

Summary

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When we invest in more than one investment, there may be some form of diversification, which is the reduction in risk from combining investments whose returns are not perfectly correlated.

If we consider all possible investments and their respective expected return and risk, there are sets of investments that are better than others in terms of return and risk. These sets make up the efficient frontier.

Summary

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If we consider a company as a portfolio of investments, diversification plays a role in financial decision making. Financial managers need to consider not only what an investment looks like in terms of its return and risk as a stand-alone investment, but more important, how it fits into the company’s portfolio of investments.

Summary

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Problem #1Scenario Probability OutcomeGood 30% $40 Normal 50% $20 Bad 20% $10 What is the expected value and standard deviation for this probability distribution?

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Problem #2The Key Company is evaluating two projects:

Project 1 has a 40% chance of generating a return of 20% and a 60% change of generating a return of -10%.

Project 2 has a 20% chance of generating a return of 30% and an 80% chance of return of -5%.

Which project is riskier? Why?

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Problem #3Suppose the covariance between the returns on project A and B is -0.0045. And suppose the standard deviations of A and B are 0.1 and 0.3, respectively. What is the correlation between A and B’s returns?