chapter 6 return and risk lawrence j. gitman jeff madura introduction to finance

Chapter

6

Return and Risk

Lawrence J. GitmanJeff Madura

Introduction to Finance

6-2Copyright © 2001 Addison-Wesley

Understand the meaning and fundamentals of risk, return, and risk aversion.

Describe procedures for measuring the risk of a single asset.

Discuss the risk measurement for a single asset using the standard deviation and coefficient of variation.

Understand the risk and return characteristics of a portfolio in terms of correlation and diversification, and the impact of international assets on a portfolio.

Learning Goals


Learning Goals

Review the two types of risk and the derivation and role of beta in measuring the relevant risk of both an individual security and a portfolio.

Explain the capital asset pricing model (CAPM) and its relationship to the security market line (SML).


Introduction

If everyone knew ahead of time how much a stock would sell for some time in the future, investing would be a simple endeavor.

Unfortunately, it is difficult—if not impossible—to make such predictions with any degree of certainty.

As a result, investors often use history as a basis for predicting the future.

We will begin this chapter by evaluating the risk and return characteristics of individual assets, and end by looking at portfolios of assets.


Fundamental Concepts about Return and Risk

In the context of business and finance, risk is defined as the chance of suffering a financial loss.

Assets (real or financial) which have a greater chance of loss are considered more risky than those with a lower chance of loss.

Risk may be used interchangeably with the term uncertainty to refer to the variability of returns associated with a given asset.


Return represents the total gain or loss on an investment.

The most basic way to calculate return is as follows:


kt =

Pt - Pt-1 + Ct

Pt-1

Where:

kt = actual required or expected return during period t

Pt = current price

Pt-1 = price during the previous time period

Ct = any cash flow accruing from the investment


For example, compute the holding period return if you purchased a stock for $100, received a $5 dividend, and sold the stock for $110.


kt =

Pt - Pt-1 + Ct

Pt-1

kt =

110 - 100 + 5

100=

15

100= 15%


Chapter Example


Single Financial Assets

Historical Return Arithmetic average

• The historical average (also called arithmetic average or mean) return is simple to calculate.

• The accompanying text outlines how to calculate this and other measures of risk and return.

• All of these calculations were discussed and taught in your introductory statistics course.

• This slideshow will demonstrate the calculation of these statistics using Microsoft® Excel.


Historical Return

Arithmetic Average

What you type What you see



Historical Risk Variance

• Historical risk can be measured by the variability of an asset’s returns in relation to its average.

• Variance is computed by summing squared deviations and dividing by the number of observations minus one (n - 1).

• Squaring the differences ensures that both positive and negative deviations are given equal consideration.

• The sum of the squared differences is then divided by the number of observations minus one (n - 1).


Historical Risk

Variance



Historical Risk Standard deviation

• Squaring the deviations makes the variance difficult to interpret.

• In other words, by squaring percentages, the resulting deviations are in percent squared terms.

• The standard deviation simplifies interpretation by taking the square root of the squared percentages.

• In other words, standard deviation is in the same units as the computed average.

• If the average is 10%, the standard deviation might be 20%, whereas the variance would be 20% squared.


Historical Risk

Standard deviation



Historical Risk

Normal Distribution



Expected Return and Risk Investors and analysts often look at historical returns

as a starting point for predicting the future.

However, they are much more interested in what the returns on their investments will be in the future.

For this reason, we need a method for estimating future or “ex-ante” returns.

One way of doing this is to assign probabilities for future states of nature and the returns that would be realized if a particular state of nature does occur.


Expected Return and Risk

Expected Return

E(R) = piRi

Where:

pi = probability of the ith scenario

Ri = forecasted return in the ith scenario

Also, the variance of E(R) may be computed as:

And the standard deviation as:



Coefficient of Variation (CV) One problem with using standard deviation as a

measure of risk is that we cannot easily make risk comparisons between two assets.

The coefficient of variation overcomes this problem by measuring the amount of risk per unit of return.

The higher the coefficient of variation, the more risk per return.

Therefore, if given a choice, an investor would select the asset with the lower coefficient of variation.


Coefficient of Variation


Portfolio of Assets

An investment portfolio is any collection or combination of financial assets.

If we assume all investors are rational and therefore risk averse, that investor will ALWAYS choose to invest in portfolios rather than in single assets.

Investors will hold portfolios because he or she will diversify away a portion of the risk that is inherent in “putting all your eggs in one basket.”

If an investor holds a single asset, he or she will fully sufferthe consequences of poor performance.

This is not the case for an investor who owns a diversified portfolio of assets.


Portfolio of Assets

Diversification is enhanced depending upon the extent to which the returns on assets “move” together.

This movement is typically measured by a statistic known as correlation as shown in Figure 6.3 below.

Figure 6.3


Portfolio of Assets

Even if two assets are not perfectly negatively correlated, an investor can still realize diversification benefits from combining them in a portfolio as shown in Figure 6.4 below.

Figure 6.4


Portfolio of Assets

Recall Stocks A and B


Portfolio AB

50% in A, 50% in B


Portfolio AB

50% in A, 50% in B

Here are cellsB12 and B13

Where the contentsof cell B12 and B13 = 50% in this case.


Portfolio AB

50% in A, 50% in B


Portfolio AB

40% in A, 60% in B

Changing the weights


Portfolio AB

20% in A, 80% in B

And again


Portfolio Return and Risk

Summarizing Changes in Return and Risk as the Composition of the Portfolio Changes



Perfect Negative Correlation



Perfect Negative Correlation

Notice that if we weight the portfoliojust right (50/50 in this case), we cancompletely eliminate risk.


Adding Assets to a Portfolio


Portfolios of Assets

Capital Asset Pricing Model (CAPM) If you notice in the last slide, a good part of a portfolio’s risk

(the standard deviation of returns) can be eliminated simply by holding a lot of stocks.

The risk you can’t get rid of by adding stocks (systematic) cannot be eliminated through diversification because that variability is caused by events that affect most stocks similarly.

Examples would include changes in macroeconomic factors such interest rates, inflation, and the business cycle.



Capital Asset Pricing Model (CAPM) In the early 1960s, researchers (Sharpe, Treynor, and Lintner)

developed an asset pricing model that measures only the amount of systematic risk a particular asset has.

In other words, they noticed that most stocks go down when interest rates go up, but some go down a whole lot more.

They reasoned that if they could measure this variability—the systematic risk—then they could develop a model to price assets using only this risk.

The unsystematic (company-related) risk is irrelevant because it could easily be eliminated simply by diversifying.



Capital Asset Pricing Model (CAPM) To measure the amount of systematic risk an asset has,

they simply regressed the returns for the “market portfolio”—the portfolio of ALL assets—against the returns for an individual asset.

The slope of the regression line—beta—measures an asset’s systematic (non-diversifiable) risk.

In general, cyclical companies like auto companies have high betas while relatively stable companies, like public utilities, have low betas.

Let’s look at an example to see how this works.



Capital Asset Pricing Model (CAPM)

We will demonstrate the calculation using the regression analysis feature in Microsoft® Excel.



This slide is the result of a regression using Microsoft® Excel. The slope of the regression (beta) in this case is 1.92. Apparently, this stock has a considerable amount of systematic risk.



Table 6.6



Table 6.7



Capital Asset Pricing Model (CAPM) The required return for all assets is composed

of two parts: the risk-free rate and a risk premium.

The risk premium is a function of both market conditions and the asset itself.

The risk-free rate (rf) is usually estimated from the return on U.S. T-Bills.



Capital Asset Pricing Model (CAPM) The risk premium for a stock is composed of two

parts:

• The market risk premium, which is the return required for investing in any risky asset rather than the risk-free rate.

• Beta, a risk coefficient which measures the sensitivity of the particular stock’s return to changes in market conditions.



Capital Asset Pricing Model (CAPM) After estimating beta, which measures a specific asset’s

systematic risk, relatively easy to estimate variables may be obtained to calculate an asset’s required return.

E(Ri) = RFR + b[E(Rm) - RFR]

Where:

E(Ri) = asset’s expected or required return

RFR = risk-free rate of return

b = an asset or portfolio’s beta

E(Rm) = expected return on the market portfolio


E(Ri) = 5.07 + 1.25 [15% - 5.07%]

E(Ri) = 17.48%


Capital Asset Pricing Model (CAPM) Example

• Calculate the required return for Federal Express assuming it has a beta of 1.25, the rate on U.S. T-Bills is 5.07%, and the expected return for the S&P 500 is 15%.


Some Comments on the CAPM The CAPM is based on an assumed efficient

market with the following characteristics:

• Many small investors each having the same information and expectations with respect to securities

• Rational investors who are risk averse

• No restrictions on investment

• No taxes

• No transactions costs




Some Comments on the CAPM Because these assumptions do not strictly hold,

the CAPM should only be viewed as a useful conceptual framework for understanding the risk and return on an investment.

In addition, the data used to approximate the beta and other components of the CAPM are historical.

Because the CAPM is an expectational model, these variables may or may not reflect the future variability of returns.


Using Microsoft® Excel

The Microsoft® Excel Spreadsheets used in the this presentation can be downloaded from the Introduction to Finance companion web site: http://www.awl.com/gitman_madura

Chapter

6

End of Chapter

Lawrence J. GitmanJeff Madura

Introduction to Finance

chapter 6 return and risk lawrence j. gitman jeff madura introduction to finance

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