Risk and Rates of Return

1
Risk and Rates of Return

• What does it mean to take risk when investing?
• How are risk and return of an investment
  measured?
• For what type of risk is an average investor
  rewarded?
• How can investors reduce risk?
• What actions do investors take when the return
  they require to purchase an investment is
  different from the return the investment is
  expected to produce?

2
RISK AND RATES OF RETURN

• Definitions and General Concepts
• Probability Distributions
• Expected return
• Standard deviation, ?
• Risk Attitudes
• Coefficient of Variation
• Portfolio Risk and Return
• Diversification
• Relevant risk
• Beta coefficients
• Determining ReturnCapital Asset Pricing Model
• Real (Physical) Assets Versus Financial Assets

3
What is Risk?

• Dictionary definitionchance of loss
• In finance we define risk as the chance that
  something other than what is expected occursthat
  is, variability of returns
• Risk can be considered badthat is, when the
  results are worse than expected
  (lower-than-expected returns)or goodthat is,
  when the results are better than expected
  (higher-than-expected returns)

• Dictionary definitionchance of loss
• In finance we define risk as the chance that
  something other than what is expected occursthat
  is, variability of returns

• Dictionary definitionchance of loss
• In finance we define risk as the chance that
  something other than what is expected occursthat
  is, variability of returns

• Dictionary definitionchance of loss

4
Risk

• Stand-alone riskrisk of an investment if it was
  held by itself, or alone
• Portfolio riskrisk of an investment when it is
  combined in a portfolio with other investments

5
Risk

• We know that an investment is risky if more than
  one future outcome is possiblethat is, there are
  two or more possible payoffs associated with the
  investment
• A probability distribution summarizes each
  possible outcome along with the chance, or
  probability, that the outcome will occur

6
Probability DistributionsExample
Economy Probability Payoff Booming 0.2 18.0 No
rmal 0.5 8.0 Recession 0.3 -2.0
Economy Probability Payoff Booming 0.2 18.0 No
rmal 0.5 8.0 Recession 0.3 -2.0 1.0
Risky Risk-Free Economy Probability Asset
Asset Booming 0.2 18.0 5.0 Normal 0.5 8.0
5.0 Recession 0.3 -2.0 5.0 1.0

7
Probability Distributions
Discrete Distributions
Continuous Distributions

8
Expected Return

• Weighted average of the various possible outcomes
  based on the probability that each outcome will
  occur
• Average of the outcomes if the actionfor
  example, an investmentwas continued over and
  over again and the probability for each outcome
  remained the samethat is, the probability
  distribution does not change

9
Expected Return
ri the result of outcome i Pri the
probability that outcome i will occur
10
Expected Return
Boom 0.2 18.0 3.6 Normal 0.5 8.0 4.0 Recession 0
.3 -2.0 -0.6

11
Measuring Stand-Alone RiskStandard Deviation, ?

• Measures the tightness, or variability, of a set
  of outcomes, or a probability distribution
• The tighter the distribution, the less the
  variability of the outcomes and the less risk
  associated with the event
• Measures risk for a single investmentthat is an
  investment held by itself (standing alone)

12
Measuring Stand-Alone RiskStandard Deviation, ?
Variance, ?2measures the variability of outcomes
Standard deviation, ?

13
Standard Deviation, ?
18.0
18.0 7.0
18.0 7.0 11.0
18.0 7.0 11.0 121.0
18.0 7.0 11.0 121.0 x 0.2
18.0 7.0 11.0 121.0 x 0.2 24.2
8.0 7.0 1.0 1.0 x 0.5 0.5 -2.0 7.0
-9.0 81.0 x 0.3 24.3
8.0 7.0 1.0 1.0 x 0.5 0.5 -2.0 7.0
-9.0 81.0 x 0.3 24.3 ?2 49.0

14
Risk Attitudes

• Risk Aversionall else equal, risk averse
  investors prefer higher returns to lower returns
  as well as less risk to more risk
• Risk averse investors demand higher returns for
  investments with higher risk

15
Risk Aversion
Risk Premium RP
r rRF RP
rRF
Risk-Free Return rRF

16
Coefficient of Variation

• Measures the relationship between risk and return
• Allows for comparisons among various investments
  that have different risks and different returns

17
Coefficient of Variation
Economy Probability A Boom 0.2 18.0
Normal 0.5 8.0 Recession 0.3 -2.0
Payoffs Economy Probability A
B C Boom 0.2 18.0 -5.0 55.0 Normal
0.5 8.0 7.0 14.0 Recession 0.3 -2.0 15.0
-10.0
18
Portfolio Risk

• By combining investments to form a portfolio, or
  collection of investments, diversification can be
  achieved
• When evaluated in a portfolio, the performance of
  a single investment is not very important,
  because some investments will perform better than
  expected while others will perform worse than
  expected
• The performance of the portfolio as a whole is
  important

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Portfolio Return

• Expected return of a portfolio weighted average
  of the expected returns of the individual
  investments in the portfolio

wj proportion of funds invested in Asset j

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Portfolio Return
Payoffs Portfolio Probability A
B wA0.6 wB0.4 0.2 18.0 -5.0 0.
5 8.0 7.0 0.3 -2.0 15.0 7.0 7.0 ? 7
.0 6.9 CV 1.00 0.99
Payoffs Portfolio Probability
A B wA0.6 wB0.4 0.2 18.0 -5.0
18(0.6) (-5)(0.4) 8.8 0.5 8.0 7.0 0.3
-2.0 15.0 7.0 7.0 ? 7.0 6.9 CV 1.
00 0.99
Payoffs Portfolio Probability
A B wA0.6 wB0.4 0.2 18.0 -5.0
18(0.6) (-5)(0.4) 8.8 0.5 8.0 7.0 8(0.6
) 7(0.4) 7.6 0.3 -2.0 15.0 7.0 7.
0 ? 7.0 6.9 CV 1.00 0.99
Payoffs Portfolio Probability
A B wA0.6 wB0.4 0.2 18.0 -5.0
18(0.6) (-5)(0.4) 8.8 0.5 8.0 7.0 8(0.6
) 7(0.4) 7.6 0.3 -2.0 15.0
(-2)(0.6) 15(0.4) 4.8 7.0 7.0 ? 7.0
6.9 CV 1.00 0.99
Payoffs Portfolio Probability
A B wA0.6 wB0.4 0.2 18.0 -5.0
18(0.6) (-5)(0.4) 8.8 0.5 8.0 7.0 8(0.6
) 7(0.4) 7.6 0.3 -2.0 15.0
(-2)(0.6) 15(0.4) 4.8 7.0 7.0 7(0.6)
7(0.4) 7.0 ? 7.0 6.9 CV 1.00 0
.99
Payoffs Portfolio Probability
A B wA0.6 wB0.4 0.2 18.0 -5.0
18(0.6) (-5)(0.4) 8.8 0.5 8.0 7.0 8(0.6
) 7(0.4) 7.6 0.3 -2.0 15.0
(-2)(0.6) 15(0.4) 4.8 7.0 7.0 7(0.6)
7(0.4) 7.0 ? 7.0 6.9 1.5 CV 1.00
0.99 0.22
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Portfolio RiskDiversification

• When investments that are not perfectly
  correlatedthat is, do not mirror each others
  movements on a relative basisare combined to
  form a portfolio, the risk of the portfolio can
  be reduced (diversification)
• The amount of the risk reduction depends on how
  the investments in a portfolio are related
• The smaller (greater) the positive (negative)
  relationship among the various investments
  included in a portfolio, the greater the
  diversification
• Diversificationinvesting in a combination of
  stocks generally reduces risk overall

22
Risk

• Stand-alone risk
• ?

Stand-alone risk ? total risk
firm-specific risk
Stand-alone risk ? total risk
Stand-alone risk ? total risk
firm-specific risk market risk
Stand-alone risk ? total risk
firm-specific risk market risk diversifiable

Stand-alone risk ? total risk
firm-specific risk market risk diversifiable
nondiversifiable
23
Firm-Specific Risk

• Caused by actions that are specific to the
  firmmanagement decisions, labor characteristics,
  etc.
• The impact of this type of risk on the expected
  return associated with an investment is generally
  fairly random
• This risk component is often called unsystematic
  risk
• This risk is also called diversifiable risk,
  because this portion of total risk can be reduced
  in a portfolio of investments

24
Market Risk

• Results from movements in factors that affect the
  economy as a wholeinterest rates, employment,
  etc.
• This risk affects all companies, thus all
  investments it is a system wide risk that cannot
  be diversified away
• This risk is called systematic, or
  nondiversifiable, risk
• Even though all investments are affected by
  systematic risk, they are not all affected to the
  same degree

25
Relevant Risk

• Risk that cannot be reduced or diversified away
• Relevant risk systematic, or market risk
• Irrelevant risk firm-specific, or
  unsystematic risk, because this portion of total
  risk can be reduced through diversification
• Investors should not be rewarded for taking
  irrelevant riskthat is, for not diversifying
• Risk premiums are based on the amount of
  systematic risk associated with an investment

26
Relevant Risk
r rRF RP

27
Concept of Beta

• Market, or systematic, risk is measured by
  comparing the return on an investment with the
  return on the market in general, or an average
  stock
• The market is very well diversified so that any
  movements should be the result on systematic risk
  only
• Beta coefficient, ßmeasures the relationship
  between an individual investments returns and
  the markets returns, thus the systematic risk of
  the investment

28
Concept of Beta
29
Portfolio Beta Coefficients

• A portfolios beta, ßp is a function of the betas
  of the individual investments in the portfolio
• A portfolio beta is the weighted average of the
  betas associated with the individual investments
  contained in the portfolio

wj of total funds invested in asset j ßj
asset js beta coefficient
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Portfolio Beta CoefficientsExample
Stock A 30,000 Stock B 20,000 Stock
C 10,000 Stock D 40,000 100,000
Stock A 30,000 2.0 Stock
B 20,000 Stock C 10,000 Stock D
40,000 100,000
Stock A 30,000 2.0 Stock
B 20,000 1.5 Stock C 10,000 Stock D
40,000 100,000
Stock A 30,000 2.0 Stock
B 20,000 1.5 Stock C 10,000 1.0 Stock D
40,000 100,000
Stock A 30,000 2.0 Stock
B 20,000 1.5 Stock C 10,000 1.0 Stock D
40,000 0.5 100,000
Stock A 30,000 2.0 0.3 Stock
B 20,000 1.5 Stock C 10,000 1.0 Stoc
k D 40,000 0.5 100,000
Stock A 30,000 2.0 0.3 Stock
B 20,000 1.5 0.2 Stock C 10,000 1.0 0.1
Stock D 40,000 0.5 0.4
100,000
Stock A 30,000 2.0 0.3 Stock
B 20,000 1.5 0.2 Stock C 10,000 1.0 0.1
Stock D 40,000 0.5 0.4
100,000 1.0
Stock A 30,000 2.0 0.3 0.6 Stock
B 20,000 1.5 0.2 Stock C 10,000 1.0 0.1
Stock D 40,000 0.5 0.4
100,000 1.0
Stock A 30,000 2.0 0.3 0.6 Stock
B 20,000 1.5 0.2 0.3 Stock
C 10,000 1.0 0.1 0.1 Stock D
40,000 0.5 0.4 0.2 100,000 1.0
Stock A 30,000 2.0 0.3 0.6 Stock
B 20,000 1.5 0.2 0.3 Stock
C 10,000 1.0 0.1 0.1 Stock D
40,000 0.5 0.4 0.2 100,000 1.0 ?p 1.2


31
Relationship between Risk and Rates of Return

• Market risk premium RPM rM - rRF
• RPM return associated with the riskiness of a
  portfolio that contains all the investments in
  the market
• RPM is based on how risk averse investors are on
  average
• Because an investments beta coefficient
  indicates volatility relative to the market, we
  can use ß to determine the risk premium for an
  investment
• Investment risk premium RPInvest RPM x
  ßInvest
• A more volatile investmentthat is, an investment
  with a higher ßwill earn a higher return than a
  less volatile investment

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Relationship between Risk and Rates of Return
Return Risk-free rate Risk
Premium rInvest rRF RPInvest
rRF ( RPM ) ßInvest rRF (
rM rRF ) ßInvest Capital Asset Pricing
Model (CAPM)
5.0
6.0
11.0
5.0
1.5
4.0
5.0
9.0
5.0
1.5
?j 1.5
rRF 5.0
rM 9

33
CAPM GraphSML
Security Market Line, SML
RPM
rj rRF RPM?j

34
CAPMInflation Effects
r1 6 4(1.5) 12 r2 8 4(1.5) 14
rM2 12
rRF2 8

35
CAPMChanges in Risk Aversion
r1 6 4(1.5) 12.0 r2 6 5(1.5) 13.5
rM2 11
36
CAPMChanges in Beta
rB2 11
rB1 12
1.5
1.25

37
Changes in Equilibrium Stock Prices

• Stock prices are not constant due to changes in
  rRF, RPM, bx, and so forth.

38
Risk and Rates of Return

• What does it mean to take risk when investing?
• More than one outcome is possible
• How are risk and return of an investment
  measured?
• Variability of its possible outcomes greater
  variability greater risk
• How can investors reduce risk?
• Risk can be reduced through diversification

39
Risk and Rates of Return

• For what type of risk is an average investor
  rewarded?
• Investors should only be rewarded for risks they
  must take
• What actions do investors take when the return
  they require to purchase an investment is
  different from the return the investment is
  expected to produce?
• Investors will purchase a security only when its
  expected return is greater than or equal to its
  required return