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Overview of Risk and ReturnOverview of Risk and Return

Timothy R. Mayes, Ph.D.FIN 3300: Chapter 8

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What is Risk?What is Risk?

A risky situation is one which has some probability of loss

The higher the probability of loss, the greater the risk

The riskiness of an investment can be judged by describing the probability distribution of its possible returns

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Probability DistributionsProbability Distributions

A probability distribution is simply a listing of the probabilities and their associated outcomes

Probability distributions are often presented graphically as in these examples Potential Outcomes

Potential Outcomes

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The Normal DistributionThe Normal Distribution

For many reasons, we usually assume that the underlying distribution of returns is normal

The normal distribution is a bell-shaped curve with finite variance and mean

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The Expected ValueThe Expected Value

The expected value of a distribution is the most likely outcome

For the normal dist., the expected value is the same as the arithmetic mean

All other things being equal, we assume that people prefer higher expected returns

E(R)

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The Expected Return: An The Expected Return: An ExampleExample

Suppose that a particular investment has the following probability distribution:• 25% chance of -5% return• 50% chance of 5% return• 25% chance of 15% return

This investment has an expected return of 5%

0%

20%

40%

60%

-5% 5% 15%Rate of Return

Pro

ba

bili

ty

05.0)15.0(25.0)05.0(50.0)05.0(25.0)( iRE

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The Variance & Standard The Variance & Standard DeviationDeviation

The variance and standard deviation describe the dispersion (spread) of the potential outcomes around the expected value

Greater dispersion generally means greater uncertainty and therefore higher risk

Riskier

Less Risky

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Calculating Calculating 22 and and : An : An ExampleExample

Using the same example as for the expected return, we can calculate the variance and standard deviation:

Note: In this example, we know the probabilities. However, often we have only historical data to work with and don’t know the probabilities. In these cases, we assume that each outcome is equally likely so the probabilities for each possible outcome are 1/N or (more commonly) 1/(N-1).

071.0)05.015.0(25.0)05.005.0(50.0)05.005.0(25.0

005.)05.015.0(25.0)05.005.0(50.0)05.005.0(25.0

22i

222i

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The Scale Problem The Scale Problem

The variance and standard deviation suffer from a couple of problems

The most tractable of these is the scale problem:• Scale problem - The magnitude of the

returns used to calculate the variance impacts the size of the variance possibly giving an incorrect impression of the riskiness of an investment

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The Scale Problem: an The Scale Problem: an ExampleExample

Is XYZ really twice as risky as ABC?

No!

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The Coefficient of VariationThe Coefficient of Variation

The coefficient of variation (CV)provides a scale-free measure of the riskiness of a security

It removes the scaling by dividing the standard deviation my the expected return (risk per unit of return):

In the previous example, the CV for XYZ and ABC are identical, indicating that they have exactly the same degree of riskiness

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Determining the Required Determining the Required ReturnReturn

The required rate of return for a particular investment depends on several factors, each of which depends on several other factors (i.e., it is pretty complex!):

The two main factors for any investment are:• The perceived riskiness of the investment• The required returns on alternative investments

An alternative way to look at this is that the required return is the sum of the RFR and a risk premium:

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The Risk-free Rate of ReturnThe Risk-free Rate of Return

The risk-free rate is the rate of interest that is earned for simply delaying consumption

It is also referred to as the pure time value of money The risk-free rate is determined by:

• The time preferences of individuals for consumption Relative ease or tightness in money market (supply &

demand) Expected inflation

• The long-run growth rate of the economy Long-run growth of labor force Long-run growth of hours worked Long-run growth of productivity

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The Risk PremiumThe Risk Premium

The risk premium is the return required in excess of the risk-free rate

Theoretically, a risk premium could be assigned to every risk factor, but in practice this is impossible

Therefore, we can say that the risk premium is a function of several major sources of risk:• Business risk• Financial leverage• Liquidity risk• Exchange rate risk

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The MPT View of Required The MPT View of Required ReturnsReturns

Modern portfolio theory assumes that the required return is a function of the RFR, the market risk premium, and an index of systematic risk:

This model is known as the Capital Asset Pricing Model (CAPM).It is also the equation for the Security Market Line (SML)

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Risk and Return GraphicallyRisk and Return GraphicallyR

ate

of

Retu

rn

RFR

Risk

The Market Line

or

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Portfolio Risk and ReturnPortfolio Risk and Return

A portfolio is a collection of assets (stocks, bonds, cars, houses, diamonds, etc)

It is often convenient to think of a person owning several “portfolios,” but in reality you have only one portfolio (the one that comprises everything you own)

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Expected Return of a Expected Return of a PortfolioPortfolio

The expected return of a portfolio is a weighted average of the expected returns of its components:

E R w RP i ii

N

1

Note: wi is the proportion of the portfolio that is invested in security I, and Ri is the expected return for security I.

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Portfolio RiskPortfolio Risk

The standard deviation of a portfolio is not a weighted average of the standard deviations of the individual securities.

The riskiness of a portfolio depends on both the riskiness of the securities, and the way that they move together over time (correlation)

This is because the riskiness of one asset may tend to be canceled by that of another asset

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The Correlation CoefficientThe Correlation Coefficient

The correlation coefficient can range from -1.00 to +1.00 and describes how the returns move together through time.

Stock 2 Stock 4

Stock 1 Stock 3

Time Time

Retu

rns (

%)

Retu

rns (

%)

Perfect Negative CorrelationPerfect Positive Correlation(r = 1) (r = -1)

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The Portfolio Standard Deviation

The portfolio standard deviation can be thought of as a weighted average of the individual standard deviations plus terms that account for the co-movement of returns

For a two-security portfolio:

P w w r w w 12

12

22

22

1 2 1 2 1 22 ,

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An Example: Perfect Pos. Correlation

Potential ReturnsState of Economy Probability ABC XYZ 50/50 Portfolio

Recession 25% 2% 2% 2%Moderate Growth 50% 8% 8% 8%Boom 25% 14% 14% 14%Expected Return 8% 8% 8%Standard Deviation 4.24% 4.24% 4.24%Correlation 1.00

P . . . . . . . . . .5 0 0424 5 0 0424 2 100 0 0424 0 0424 0 5 0 5 0 04242 2 2 2

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An Example: Perfect Neg. Correlation

Potential ReturnsState of Economy Probability ABC XYZ 50/50 Portfolio

Recession 25% 2% 14% 8%Moderate Growth 50% 8% 8% 8%Boom 25% 14% 2% 8%Expected Return 8% 8% 8%Standard Deviation 4.24% 4.24% 0.00%Correlation -1.00

P . . . . . . . . . .5 0 0424 5 0 0424 2 100 0 0424 0 0424 0 5 0 5 0 002 2 2 2

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An Example: Zero Correlation

Potential ReturnsState of Economy Probability ABC XYZ 50/50 Portfolio

Recession 25% 2% 2% 2%Moderate Growth 50% 8% 2% 5%Boom 25% 14% 2% 8%Expected Return 8% 2% 5%Standard Deviation 4.24% 0.00% 2.12%Correlation 0.00

P . . . . . . . . .5 0 0424 5 0 0424 2 0 0 0424 0 0424 0 5 0 5 0 02122 2 2 2

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Interpreting the ExamplesInterpreting the Examples

In the three previous examples, we calculated the portfolio standard deviation under three alternative correlations.

Here’s the moral: The lower the correlation, the more risk reduction (diversification) you will achieve.