7-1 copyright 2009 mcgraw-hill australia pty ltd ppts t/a business finance 10e by peirson slides...

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Copyright 2009 McGraw-Hill Australia Pty Ltd PPTs t/a Business Finance 10e by PeirsonSlides prepared by Farida Akhtar and Barry Oliver, Australian National University

Chapter 7

Risk and Return

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Learning Objectives

• Understand how risk and return are defined and measured.

• Understand the concept of risk aversion by investors.

• Explain how diversification reduces risk.

• Understand the importance of covariance between returns on assets to determine the risk of a portfolio.

• Explain the concept of efficient portfolios.

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Learning Objectives (cont.)

• Understand distinction between systematic and unsystematic risk and significance of systematic risk.

• Explain the relationship between returns and risk proposed by the capital asset pricing model (CAPM).

• Understand the relationship between CAPM and the Fama-French three-factor model.

• Explain the development of the Fama-French three-factor model.

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Return • There is uncertainty associated with returns on shares.

Assume we can assign probabilities to the possible returns — given the following set of circumstances, the expected return is:

Example Solution

%11

)1.013.0(

)2.012.0()4.011.0(

)242110.0()1.009.0(

1

RE

x

xx

xx

PRREn

iii

Percentage Return, Ri

Probability, Pi

9 0.1

10 0.2

11 0.4

12 0.2

13 0.1

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Risk• Risk is present whenever investors are not certain about the

outcome an investment will produce.• Risk measured by variance — how much a particular return

deviates from an expected return. Use standard deviation to measure risk, which is simply the square root of the variance:

• Using previous example, risk is given by:

n

iii PRER

1

22

2 22

2 2

2

Variance:

= 0.09-0.11 0.1 0.10-0.11 0.2

0.11-0.11 0.4 0.12-0.11 0.2

0.13-0.11 0.1

= 0.000 12

Standard Deviation: = 0.000 12 0.01095 1.095%

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Risk Attitudes• Risk-neutral investor: (p. 175, Figure 7.3)

– One whose utility is unaffected by risk; when chooses to invest, investor focuses only on expected return.

• Risk-averse investor: (p. 175, Figure 7.3)

– One who demands compensation in the form of higher expected returns in order to be induced into taking on more risk.

• Risk-seeking investor: (p. 175, Figure 7.3)

– One who derives utility from being exposed to risk, and hence, may be willing to give up some expected return in order to be exposed to additional risk.

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Risk Attitudes (cont.)• The standard assumption in finance theory is all

investors are risk averse.

– This does not mean an investor will refuse to bear any risk at all.

– Rather, investors regards risk as something undesirable,

but may take up on board if compensated with sufficient return; trade-off between risk and return.

• Investors’ risk preferences

– Indifference curve — which represents those combinations of expected return and risk that result in a fixed level of expected utility for an investor. (p. 177, Figure 7.5 — Risk averse investor)

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Risk of Assets and Portfolios

• We now know that the risk of an individual asset is summarised by standard deviation (or variance) of returns.

• Investors usually invest in a number of assets (a portfolio) and will be concerned about the risk of their overall portfolio.

• Now concerned about how these individual risks will interact to provide us with overall portfolio risk.

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Portfolio Theory• Assumptions:

– Investors perceive investment opportunities in terms of a probability distribution defined by expected return and risk.

– Investors’ expected utility is an increasing function of return and a decreasing function of risk (risk aversion).

• Measuring return of portfolio:

– Portfolio return (Rp) is a weighted average of all the expected returns of the assets held in the portfolio:

n

jjjp REwRE

1

where:

= the proportion of the portfolio

invested in asset j

= the number of securities in the portfolio

jw

n

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Portfolio Risk• Portfolio (comprising two assets) risk depends on:

– The proportion of funds invested in each asset (w).– The riskiness of the individual assets (2).– The relationship between each asset in the portfolio

with respect to risk, correlation (– For a two-asset portfolio the variance is:

212,12122

22

21

21

2 2 wwwwp where:

= the proportion of the portfolio

invested in asset

= the standard deviation of asset

correlation between asset and returns

i

i

ij

w

i

i

i j

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Portfolio Risk and Return Measurement• Assume 60% of the portfolio is invested in

security 1 and 40% in security 2. If variances of security 1 and security 2 are 0.0016 and 0.0036, respectively, and the correlation (p1,2) is –0.5: Find expected return and risk of portfolio.

• The expected returns of the securities are 0.08 and 0.12 respectively.

• The standard deviation is 0.024.

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Relationship Measures• Covariance:

– Statistic describing the relationship between two variables.

– If positive, when one of the variables takes on a value above its expected value, the other has a propensity to do the same.

– If the covariance is negative, the deviations tend to be of an opposite sign.

• Correlation coefficient: – Is another measure of the strength of a relationship between

two variables? The correlation is equal to the covariance divided by the product of the asset’s standard deviations.

• It is simply a standardisation of the covariance, and for this reason is bounded by the range +1 to –1.

yx

xyyx

,cov

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Gains from Diversification• Diversification gain is related to correlation coefficient

value.

• The degree of risk reduction increases as the correlation between the rates of return on two securities decreases.

• r = +1, Risk reduction does not occur by combining securities whose returns are perfectly positively correlated.

• r = 1, Risk reduction occurs by combining securities whose returns are less than perfectly positively correlated.

• 0 < r < 1, If the correlation coefficient is less than 1, the third term in the portfolio variance equation is reduced, reducing portfolio risk.

• r = –1 If the correlation coefficient is negative, risk is reduced even more, but this is not a necessary prerequisite for diversification gains.

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Diversification with Multiple Assets• The more assets we incorporate into the portfolio,

the greater the diversification benefits are.

• The key is the correlation between each pair of assets in the portfolio.

• With n assets, there will be an n × n covariance matrix.

• The properties of the variance-covariance matrix are:– It will contain n2 terms.

– The two covariance terms for each pair of assets are identical.

– It is symmetrical about the main diagonal that contains n variance terms.

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Diversification with Multiple Assets (cont.)

• For a diversified portfolio, the variance of the individual assets contributes little to the risk of the portfolio.

– For example, in a 50-asset portfolio there are 50 (n) variance terms and 2450 (n2 − n) covariance terms.

• The risk depends largely on the covariances between the returns on the assets.

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Systematic and Unsystematic Risk• Intuitively, we should think of risk as comprising:

• Systematic risk: Component of total risk that is due to economy-wide factors. (non-diversifiable risk)

• Unsystematic risk: Component of total risk that is unique to firm and is removed by holding a well-diversified portfolio.

• The returns on a well-diversified portfolio will vary due to the effects of market-wide or economy-wide factors.

• Systematic risk of a security or portfolio will depend on its sensitivity to the effects of these market-wide factors.

Total Risk = Systematic Risk + Unsystematic Risk

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Risk of an Individual Asset• The risk contribution of an asset to a portfolio is largely

determined by the covariance between the return on that asset and the return on the holder’s existing portfolio:

• Well-diversified portfolios will be representative of the market as a whole. Thus, the relevant measure of risk is the covariance between the return on the asset and the return on the market:

pipipi RRCov , ,

Mi RRCov ,

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Beta Beta is a measure of a security’s systematic risk,

describing the amount of risk contributed by the security to the market portfolio.

Cov(Ri , RM) can be scaled by dividing it by the variance of the return on the market. This is the asset’s beta (i):

2

, i Mi

M

Cov R R

where:

= return on the market portfolio

= return on the particular assetM

i

R

R

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Construction of a Portfolio• The opportunity set:

– The set of all feasible portfolios that can be constructed from a given set of risky assets.

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Construction of a Portfolio (cont.)• The efficient frontier:

– Investor will try to secure a portfolio on the efficient frontier.

– The efficient frontier is determined on the basis of dominance.

• A portfolio is efficient if:

– No other portfolio has a higher return for the same risk, or

– No other portfolio has a lower risk for the same return.

• Investors are a diverse group and, therefore, each investor may prefer a different point along the efficient frontier.

• Investor risk preferences will determine the preferred portfolio on the efficient frontier.

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Value at Risk (VaR)• A relatively new measure of the riskiness of an

asset or portfolio.

• Defined as ‘the worst loss that is possible under normal market conditions during a given time period’.

• Requires standard deviation of the return on the asset or portfolio.

• Typically assumes returns are normally distributed.

• Using the normal distribution and the standard deviation, can calculate a worst-case scenario.

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Value at Risk (cont.)

• Investment of $10m in Curzon has an estimated return of zero and a standard deviation of 20% ($2m).

• Assume returns are normally distributed and bad market conditions expected 5% of the time.

• Worst outcome under normal conditions is a loss of 1.645 (from normal tables) multiplied by standard deviation of $2m.

• Worst outcome is loss of $3.29m or an investment value of $6.71m.

• VaR was not used effectively by NAB in the foreign exchange scandal — poor implementation and execution.

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The Pricing of Risky Assets• What determines the expected rate of return

on an individual asset?

• Risky assets will be priced such that there is a relationship between returns and systematic risk.

• Investors need to be sufficiently compensated for taking on the risks associated with the investment.

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The Capital Market Line• Combining the efficient frontier with preferences,

investors choose an optimal portfolio.

• This can be enhanced by introducing a risk-free asset:

– The opportunity set for investors is expanded and results in a new efficient frontier — capital market line (CML).

• The CML represents the efficient set of all portfolios that provides the investor with the best possible investment opportunities when a risk-free asset is available.

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The Capital Market Line (cont.)

• The CML links the risk-free asset with the optimal risky portfolio (M): p. 191, Figure 7.11.

• Investors can then vary the riskiness of their portfolio investment by changing weights in the risk-free asset and portfolio M.

• This changes their return according to the CML:

M f

p f pM

E R RE R R

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The CAPM and the Security Market Line (SML)• In equilibrium, the expected return on a risky asset i

(or an inefficient portfolio), is given by the security market line:

MiM

fMfi RR

RRERRE ,cov

where:

( ) = the expected return on the th risky asset

( , ) = the covariance between returns

on th risky asset and the market portfolio

i

i M

E R i

Cov R R

i

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• The covariance term is the only explanatory factor in the equation that is specific to asset i.

• As Cov(Ri,RM) is the risk of an asset held as part of the market portfolio, and

M is the risk of the market portfolio, beta measures the risk of i relative to the risk of the market as a whole.

• We can thus write the Security Market Line (SML) as the Capital Assets Pricing Model (CAPM) equation:

• See p.193, Figure 7.12 for graphical depiction of the CAPM and the SML.

i f i M fE R R E R R

The CAPM and the SML (cont.)

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Portfolio Beta• The systematic risk (beta) of a portfolio is

calculated as the weighted average of the betas of the individual assets in the portfolio:

n

iiip w

1

where:

number of assets in the portfolio

= proportion of the current market value

of portfolio p constituted by the i th asseti

n

w

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Implementation of the CAPM• The three components of the CAMP: Rf, eand E(Rm)

– Rf: The government securities current yield whose term to maturity matches the life of the proposed project.

– e: Use market model to estimate beta by obtaining time series data on the rates of return on shares and market portfolio. Hence, number of years and the length of period is significant over which returns are calculated.

– E(Rm): Two ways to calculate:

(1) Use average return in share market index over a long period of time.

(2) Estimate market risk premium directly over a long period of time.

• However, test and empirical studies have found problems with CAPM implementation, and concerns have led to introducing new model introduction.

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Risk, Return and the CAPM• The capital market will only reward investors for

bearing risk that cannot be eliminated by diversification.

• Unsystematic risk can be diversified away, so capital market will not reward investors for taking this type of firm specific risk.

• However, CAPM states the reward for bearing systematic risk is a higher expected return, consistent with the idea of higher risk requires higher return.

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Tests of the CAPM• Early empirical evidence was supportive of CAPM

in explaining asset pricing.

• Roll’s critique (1977) criticised methodology of testing CAPM empirically.

• Most tests of the CAPM can only determine whether the market portfolio used is efficient.

• In response, researchers implemented methodological refinements — CAPM seems untestable, given Roll’s critique.

• However, CAPM is a useful tool when thinking about asset returns.

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Fama-French Three-Factor Model• Fama and French (1992) provide evidence on

factors that explain asset returns — no support for CAPM, support for firm size, leverage, P/E, BV/MV, though not definitive.

• Fama and French (1995) leads to the most common three-factor model:

• Includes the CAPM, market factor, a small minus large portfolio factor (SML) and a high minus low market to book portfolio (HML).

+ it ft iM Mt ft iS ihE R R E R R E SMB E HML

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Fama-French Three-Factor Model (cont.)• This model is supported by Australian data relative

to CAPM: Gaunt (2004).

• While the three-factor model is empirically robust, it suffers from difficult economic interpretation — Why do company size and BV/MV explain asset returns?

• The fact that Fama-French includes market factor, along with ambiguity of role of other factors is supportive of CAPM.

• The three-factor model is now very common in empirical research.

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Summary• Portfolio theory: diversification reduces risk.

– Diversification works best with negative or low positive correlations between assets and asset classes.

• Risk can be divided into two categories:

– Systematic risk — cannot be diversified away.

– Unsystematic risk — can be diversified away.

• Systematic risk of an asset is measured by the asset’s beta. Risk of asset is relative to market.

• CAPM provides the relationship between risk and expected return for risky assets.

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Summary (cont.)• CAPM uses asset’s beta and assumes linear

relationship between expected return and risk relative to market, measured by beta.

• Fama–French three-factor model is a contemporary version of the multi-factor (APT) model.

– Key factors are the market excess return, return on a small minus large portfolio, return on a high minus low market to book portfolio.

• Although CAPM has its shortfall, it does provides some important insights into the link between risk and return. Hence, there is no perfect model.

7-1 copyright 2009 mcgraw-hill australia pty ltd ppts t/a business finance 10e by peirson slides...

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