Chapter (5)Chapter (5)

Risk and ReturnRisk and Return

Chapter 5: Learning GoalsChapter 5: Learning Goals

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

2. Describe procedures for assessing and measuring the risk of a single asset.

3. Discuss the measurement of return & standard deviation for a portfolio and the concept of correlation.

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

Chapter 5: Learning GoalsChapter 5: Learning Goals

5. Review the two types of risk and thederivation and role of beta in measuringthe relevant risk of both a security and aportfolio.

6. Explain the capital asset pricing model(CAPM) and its relationship to thesecurity market line (SML), and themajor forces causing shifts in the SML.

Risk and Return Fundamentals If everyone knew ahead of time how much a stock

would sell for some time in the future, investing would be simple endeavour.

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 (collection/group of assets).

Risk Defined 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

ofloss are considered more risky than those with a lowerchance of loss.

Risk may be used interchangeably with the termuncertainty to refer to the variability of returnsassociated with a given asset (Gov. bonds vs. shares).

The more nearly certain the return from an asset, the less variability and therefore the less risk.

Other sources of risk are listed on the following slide (table 5.1 – page 227).

Return Defined We can assess risk on the basis of

variability of return. Return represents the total gain or loss on an

investment over a given period of time The most basic way to calculate return is

as follows: Kt = Ct + Pt – Pt-1

Where:Where: PPt-1t-1KKtt = actual, expected, or required rate of return during period = actual, expected, or required rate of return during period ttCCtt = cash (flow) received from the asset investment in the time period = cash (flow) received from the asset investment in the time period t-1t-1 to to ttPPtt = price (value) of asset at time t, the end-of-period value. = price (value) of asset at time t, the end-of-period value.PPt-1t-1 = price (value) of asset at time t-1, beginning-of-period value = price (value) of asset at time t-1, beginning-of-period valueNotes:Notes:1) Expected return and required return are used interchangeably, 1) Expected return and required return are used interchangeably, because in an efficient market they would be expected to be because in an efficient market they would be expected to be equal.equal.2) actual return is an ex post value, whereas expected/required 2) actual return is an ex post value, whereas expected/required returns are ex ante values, therefore, actual return maybe returns are ex ante values, therefore, actual return maybe greater/less/equal to expected required return.greater/less/equal to expected required return.

Return Defined (cont.) - Example

Robin’s Gameroom wishes to determine the returns on two of its video machines, Conqueror and Demolition. Conqueror was purchased 1 year ago for $20,000 and currently has a market value of $21,500. During the year, it generated $800 worth of after-tax receipts. Demolition was purchased 4 years ago; its value in the year just completed declined from $12,000 to $11,800. During the year, it generated $1,700 of after-tax receipts.

Return Defined (cont.)

Though the market value of Demolition decreased Though the market value of Demolition decreased during the year, its cash flow brought about a higher during the year, its cash flow brought about a higher rate of return than Conqueror during the same rate of return than Conqueror during the same period.period.

Therefore, the combination of cash flow and changes Therefore, the combination of cash flow and changes in value impacts is clearly very importantin value impacts is clearly very important

Historical Returns

We can eliminate the We can eliminate the impact of market and impact of market and other types of risk other types of risk thru’ averaging thru’ averaging historical returns over historical returns over a long period of time.a long period of time.

This makes the FDM to This makes the FDM to focus on differences in focus on differences in return that pertain to return that pertain to primarily to the types primarily to the types of investment.of investment.

Risk Preferences Agency problem: risk Agency problem: risk

preferences of FM vs. firmspreferences of FM vs. firms 3 basic risk preferences 3 basic risk preferences

behaviours:behaviours:1- risk-averse (shy away)1- risk-averse (shy away)2- risk-indifferent 2- risk-indifferent

(nonsensical)(nonsensical)3- risk-seeking (enjoying!)3- risk-seeking (enjoying!)

Most FM are risk-averse Most FM are risk-averse (like shareholders) for a (like shareholders) for a given increase in risk, they given increase in risk, they require an increase in require an increase in return on their investment return on their investment in that firmin that firmLook figure 5.1, page 230Look figure 5.1, page 230

Risk of a Single Asset Risk Assessment:Risk Assessment: Sensitivity/Scenario Analysis & Probability Sensitivity/Scenario Analysis & Probability

Distributions can be used to assess the general Distributions can be used to assess the general level of risk embodied in a given asset.level of risk embodied in a given asset.(1) Sensitivity Analysis:(1) Sensitivity Analysis: uses several possible- uses several possible-return estimates (pessimistic/worst, most return estimates (pessimistic/worst, most likely/expected, optimistic/best) to obtain the likely/expected, optimistic/best) to obtain the variability among outcomes.variability among outcomes.The assets can be measured by the ‘range of The assets can be measured by the ‘range of return’ which can be found by subtracting the return’ which can be found by subtracting the pessimistic outcome from the optimistic outcome. pessimistic outcome from the optimistic outcome. The greater the range, the more variability/risk The greater the range, the more variability/risk the asset is said.the asset is said.

Risk of a Single Asset (cont.) Norman Company, a custom golf equipment

manufacturer, wants to choose the better of two investments, A and B. Each requires an initial outlay of $10,000 and each has a most likely annual rate of return of 15%. Management has made pessimistic and optimistic estimates of the returns associated with each. The three estimates for each assets, along with its range, is given in Table 5.3. Asset A appears to be less risky than asset B. The risk averse decision maker would prefer asset A over asset B, because A offers the same most likely return with a lower range (risk).

Risk of a Single Asset (cont.)

Risk of a Single Asset:Discrete Probability Distributions (2) Probabilities Distributions (PD):(2) Probabilities Distributions (PD): provide provide

a more quantitative insight into an asset’s a more quantitative insight into an asset’s risk.risk.

Probability of outcomeProbability of outcome is the chance that a is the chance that a given outcome will occur.given outcome will occur.

A PD is a model that relates probabilities A PD is a model that relates probabilities to the associated outcomes.to the associated outcomes.

The simplest type of PD is the bar chart The simplest type of PD is the bar chart that shows only a limited No. of outcome-that shows only a limited No. of outcome-probability coordinates.probability coordinates.

Risk of a Single Asset:Discrete Probability Distributions Norman Company’s past estimates Norman Company’s past estimates

indicate that probabilities of the indicate that probabilities of the pessimistic, most likely and optimistic pessimistic, most likely and optimistic outcomes are 25%, 50%, and 25% outcomes are 25%, 50%, and 25% respectively.respectively.

The bar chart for Norman Company’s The bar chart for Norman Company’s assets A and B are shown in the assets A and B are shown in the following figure (5.2 – page 232) following figure (5.2 – page 232)

Note: the sum of these probabilities must equal Note: the sum of these probabilities must equal 100%100%

Risk of a Single Asset:Discrete Probability Distributions Though both assets have the same most likely return, Though both assets have the same most likely return,

the range of return is much greater (more dispersed) the range of return is much greater (more dispersed) for asset B than for asset A – 16% versus (vs.) 4%for asset B than for asset A – 16% versus (vs.) 4%

Risk of a Single Asset:Continuous Probability Distributions If all probabilities are available, a If all probabilities are available, a

‘continuous probability distribution’ can be ‘continuous probability distribution’ can be developed for a very large No. of outcomes. developed for a very large No. of outcomes.

The following figure (5.3 – page 233) The following figure (5.3 – page 233) presents CPD for the assets A & B.presents CPD for the assets A & B.

Though both assets have the same most Though both assets have the same most likely return (15%), the dist. of returns for likely return (15%), the dist. of returns for asset B has much greater dispersion than asset B has much greater dispersion than the dist. for asset A; thus asset B is more the dist. for asset A; thus asset B is more risky than asset A.risky than asset A.

Risk of a Single Asset:Continuous Probability Distributions

Return Measurement for a SingleAsset: Expected ReturnThe expected value

of a return, k-bar, is the most likely return of an asset.

* When all outcomes, Kjs, are known and their related probabilities are assumed to be equal then expected return = Summation of Kj /n

Return Measurement for a SingleAsset: Expected Return (cont.)

The expected values of returns for The expected values of returns for Norman Company’s assets A & B are Norman Company’s assets A & B are presented in Table 5.4, following slide presented in Table 5.4, following slide (page 234). (page 234).

Col. 1 gives the Col. 1 gives the PrPrj’sj’s (probabilities), (probabilities), Col. 2 gives Col. 2 gives the the KKj’sj’s (returns). (returns). In each case n equals 3.In each case n equals 3.

The expected value, applying the The expected value, applying the previous equation, for each asset’s return previous equation, for each asset’s return is 15%.is 15%.

Note: when the total probabilities equal 1, it means that 100% of Note: when the total probabilities equal 1, it means that 100% of outcomes, or all possible outcomes are considered.outcomes, or all possible outcomes are considered.

Return Measurement for a SingleAsset: Expected Return (cont.)

Risk Measurement for a Single Asset: Standard Deviation

The most common statistical indicator of an asset’s risk is the standard deviation, σk, which measures the dispersion around the expected value

The expression for the standard deviation of returns, σk, is given in Equation 5.3 below.

In general, the higher the standard deviation the greater z risk!

Risk Measurement for a Single Asset: Standard Deviation

The following slide/table 5.5 (page The following slide/table 5.5 (page

235) presents the 235) presents the σk for Norman

Company’s assets A & B. The σk for asset A is 1.41%, & for asset B is 5.66%.

The higher risk of asset B is clearly reflected in its higher σk.

Risk Measurement for a Single Asset: Standard Deviation (cont.)

Risk Measurement for a Single Asset: Standard Deviation (cont.)

Risk Measurement for a Single Asset: Coefficient of Variation

The coefficient of variation, CV, is ameasure of relative dispersion that is useful in comparing risks of assets with different expected returns.

Equation 5.4 below gives the expression of the coefficient of variation.

The higher the CV, the greater the risk and therefore the higher the expected return.

This can be seen in the following slide (table 5.6, P. 235) which shows historical 1926-2003 investment data.

The historical data confirms the existence of positive relationship between risk and return.

Risk Measurement for a Single Asset: Coefficient of Variation

Risk Measurement for a Single Asset: Coefficient of Variation When SDs (from table 5.5) and When SDs (from table 5.5) and KsKs (from (from

table 5.4) for assets A & B are substituted table 5.4) for assets A & B are substituted into CV’s equation, the CV for A is 0.094 into CV’s equation, the CV for A is 0.094 (1.41% / 15%) while the CV for B 0.377 (1.41% / 15%) while the CV for B 0.377 (5.66% / 15%).(5.66% / 15%).

Asset B has the higher CV and is therefore Asset B has the higher CV and is therefore more risky than asset A which we already more risky than asset A which we already know from the SD (know from the SD (where both assets have where both assets have the same expected returnthe same expected return))

The real utility of the coefficient of variation The real utility of the coefficient of variation come in comparing the risks of assets that come in comparing the risks of assets that have have different expected return.different expected return.

Risk Measurement for a Single Asset: Coefficient of Variation (cont.) A firm wants to select the less risky of two alternatives A firm wants to select the less risky of two alternatives

assets – C & D. The expected return, SD and CV for each of assets – C & D. The expected return, SD and CV for each of these assets’ returns are shown in the table below (p. 237). these assets’ returns are shown in the table below (p. 237).

The firm would prefer asset C with lower SD than asset D The firm would prefer asset C with lower SD than asset D (9% vs. 10%).(9% vs. 10%).

However, management would be mistaken in choosing C However, management would be mistaken in choosing C over D, because the dispersion/risk of the asset is lower for over D, because the dispersion/risk of the asset is lower for D (0.50) than for C (0.75)D (0.50) than for C (0.75)

Therefore, using CV to compare asset risk is effective Therefore, using CV to compare asset risk is effective because it also considers the relative size/expected return because it also considers the relative size/expected return of the asset.of the asset.

Portfolio Risk and Return 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 willdiversify 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 fullysuffer the consequences of poor performance.

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

Efficient portfolio is a portfolio that maximizes return for a given level of risk or minimize risk for a given level of return.

Portfolio Return

The return of a portfolio is a weightedaverage of the returns on the individualassets from which it is formed and can becalculated as shown in Equation 5.5.

Portfolio Risk and Return: Expected Return and Standard Deviation Assume that we wish to determine the

expected value and standard deviation of returns for portfolio XY, created by combining equal portions (50%) of assets X and Y. The expected returns of assets X and Y for each of the next 5 years are given in columns 1 and 2, respectively in part A of Table 5.7. In column 3, the weights of 50% for both assets Xand Y along with their respective returns from columns 1 and 2 are substituted into equation 5.5. Column 4 shows the results of the calculation – an expected portfolio return of 12%.

Portfolio Risk and Return: ExpectedReturn and Standard Deviation (cont.)

Portfolio Risk and Return: ExpectedReturn and Standard Deviation (cont.) As shown in part B of Table 5.7, the

expected value of these portfolio returns over the 5-year period is also 12%. In part C of Table 5.7, Portfolio XY’s standard deviation is calculated tobe 0%. This value should not be surprising because the expected return each year is the same at 12%. No variability is exhibited in the expected returns from year to year.

Portfolio Risk and Return: ExpectedReturn and Standard Deviation (cont.)

Risk of a Portfolio: Correlation Correlation is a statistical measure of the Correlation is a statistical measure of the

relationship between any two series of Nos.relationship between any two series of Nos. If two series move in the same direction, they are If two series move in the same direction, they are

positively correlated.positively correlated. If the series move in opposite directions, they are If the series move in opposite directions, they are

negatively correlated.negatively correlated. The degree of correlation is measured by the The degree of correlation is measured by the

correlation coefficient, which ranges from +1 for correlation coefficient, which ranges from +1 for perfectly correlated series to -1 for perfectly perfectly correlated series to -1 for perfectly negatively correlated series.negatively correlated series.

The concept of correlation is essential to developing The concept of correlation is essential to developing an efficient portfolio.an efficient portfolio.

In general, the lower the correlation between asset In general, the lower the correlation between asset returns, the greater the potential diversification of returns, the greater the potential diversification of riskrisk

Risk of a Portfolio: Correlation Combining un-correlated assets (no interaction between their Combining un-correlated assets (no interaction between their

returns) can reduce risk, not so effectively as combining returns) can reduce risk, not so effectively as combining negatively correlated assets, but more effectively than negatively correlated assets, but more effectively than combining positively correlated assets.combining positively correlated assets.

The correlation coefficient for uncorrelated assets is close to The correlation coefficient for uncorrelated assets is close to zero.zero.

The creation of a portfolio that combines two assets with The creation of a portfolio that combines two assets with perfectly positively correlated returns results in overall portfolio perfectly positively correlated returns results in overall portfolio risk that at minimum equals that of the least risky asset and at risk that at minimum equals that of the least risky asset and at maximum equals that of the most risky assetmaximum equals that of the most risky asset

However, a portfolio combining two assets with less than However, a portfolio combining two assets with less than perfectly positive correlation can reduce total risk to a level perfectly positive correlation can reduce total risk to a level below that o f either of the componentsbelow that o f either of the components

Example:Example: Embroidery sophisticated machines vs. sewing Embroidery sophisticated machines vs. sewing machine:machine:The advanced tool will have high sales when the economy is The advanced tool will have high sales when the economy is expanding and low sales during recession, however, visa versa expanding and low sales during recession, however, visa versa the sewing machine will have high sales during recession. the sewing machine will have high sales during recession.

Risk of a Portfolio (cont.) To reduce overall risk, it is best to diversify by

combining, or adding to the portfolio, assets that have a negative correlation (a low positive).

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 the figure below (5.5 – p. 240)

Risk of a Portfolio (cont.)

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 the figure 5.6 below. Combining negatively correlated assets can reduce the overall variability of returns.

Portfolio of assets FG has less risk (variability) then either of the individual assets, which have the same expected return.

Risk of a Portfolio (cont.) Example:Example: Table 5.8 (p. 242) presents the Table 5.8 (p. 242) presents the

forecasted returns from 3 different assets – X, Y & Z forecasted returns from 3 different assets – X, Y & Z – over the next 5 yrs, along with their expected – over the next 5 yrs, along with their expected values and standard deviation. Each of the assets values and standard deviation. Each of the assets has an expected value of return of 12% and SD of has an expected value of return of 12% and SD of 3.16% (having equal return and risk).3.16% (having equal return and risk).

The XY portfolio is perfectly negatively correlated. The XY portfolio is perfectly negatively correlated. The risk in this portfolio, as reflected by its SD is The risk in this portfolio, as reflected by its SD is reduced to 0% which leads to the complete reduced to 0% which leads to the complete elimination of risk (look at table 5.7, p. 239)elimination of risk (look at table 5.7, p. 239)

The XZ is perfectly positively correlated. The risk, The XZ is perfectly positively correlated. The risk, as reflected by its SD, is unaffected by this as reflected by its SD, is unaffected by this combination. Risk remains 3.16% because X & Z combination. Risk remains 3.16% because X & Z have the same SD.have the same SD.

Risk of a Portfolio (cont.)

Risk of a Portfolio (cont.)

Table 5.9 summarized the impact of Table 5.9 summarized the impact of correlation on the range of return and risk correlation on the range of return and risk for various two-asset portfolio for various two-asset portfolio combinations.combinations.

The table shows that as we move from The table shows that as we move from perfect positive correlation to uncorrelated perfect positive correlation to uncorrelated assets to perfect negative correlation, the assets to perfect negative correlation, the ability to reduce risk is improved.ability to reduce risk is improved.

Note that in NO CASE will a portfolio of Note that in NO CASE will a portfolio of assets be riskier than the riskiest asset assets be riskier than the riskiest asset included in the portfolio.included in the portfolio.

Risk of a Portfolio (cont.)

Risk of a Portfolio (cont.)

Example: a firm has calculated the expected Example: a firm has calculated the expected return and the risk fo reach of two-assets P & Q.return and the risk fo reach of two-assets P & Q.

AssetsAssets Expected ReturnExpected Return Risk/SDRisk/SD

PP 6%6% 3% 3%

QQ 8%8% 8% 8%

Risk and Return: The Capital AssetPricing Model (CAPM) Total Risk/Over All Risk = Div. + Non-div. This total risk affects investment opp./owners’

wealth. 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 throughdiversification because that variability is caused by events that affect most stocks similarly.

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

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.) In the early 1960s, finance 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 wholelot more.

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

The unsystematic (company-related) risk isirrelevant because it could easily be eliminated simplyby diversifying.

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.) 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 thereturns for an individual asset.

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

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

The beta coefficient of market, b, (relative measure of non-div. risk) is considered to equal 1 (look table 5.10, p. 249). The majority ob beta coefficient fall between .5 – 2.0 but positive betas are the norm

In general, cyclical companies like auto companies havehigh betas while relatively stable companies, like publicutilities, have low betas.

The calculation of beta is shown in figure 5.9 (p. 248) The higher the beta the higher respond to changing to market return the

more risky!

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.)

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.)

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.)

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.) 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 assetitself.

The risk-free rate (RF) is usually estimated from the return on US T-bills

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.)

The risk premium for a stock is composed of two parts:• The Market Risk Premium which is thereturn required for investing in any riskyasset rather than the risk-free rate• Beta, a risk coefficient which measuresthe sensitivity of the particular stock’sreturn to changes in market conditions.

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.)

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.) After estimating beta, which

measures a specific asset or portfolio’s systematic risk, estimates of the other variables in the model may be obtained to calculate an asset or portfolio’s required return.

Risk and Return: The Capital AssetPricing Model (CAPM) (cont.) Benjamin Corporation, a growing computer

software developer, wishes to determine the required return on asset Z, which has a beta of 1.5. The risk-free rate of return is 7%; the return on the market portfolio of assets is 11%.

Substituting bZ = 1.5, RF = 7%, and km = 11% into the CAPM yields a return of:kZ = 7% + 1. 5 [11% - 7%]kZ = 13%

Risk and Return:Some Comments on the CAPM The CAPM relies on historical data which means the

betas may or may not actually reflect the futurevariability of returns.

Therefore, the required returns specified by the modelshould be used only as rough approximations.

The CAPM also assumes markets are efficient. Although the perfect world of efficient markets appears

to be unrealistic, studies have provided support for theexistence of the expectational relationship described bythe CAPM in active markets such as the NYSE.