Download Risk & Rates of Return

Risk and Return
5/25/2017
Richard MacMinn
1
Objectives:
 Define Expected Return
 Define Risk
Systematic versus Unsystematic
 Examine the relationship between Asset Risk and Return
 Understand the effect of diversification on the Risk and
Return of a portfolio
 Determine an investor’s required rate of return on a
security as a function of its risk
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Expected Returns
 The expected benefits or returns that an investment
generates come in the form of cash flows.
 Cash flows, not accounting profits, are the relevant
variable used to measure returns.
 Conventionally, we measure the expected cash flow as
n
follows:
EX   pi X i  p1 X1  p2 X 2    pn X n
EX = X1 P1i +1 X2 P2 + . . . + X n Pn
where Xi is the cash flow in the ith state of the economy
and Pi is the probability of the ith state of the economy
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Expected Returns
 Similarly, the Expected Rate of Return is given by:
 n
ER =  R i Pi
n
ER   pi Ri i=1 p1 R1  p2 R2    pn Rn
i 1
 where Ri is the rate of return in the ith state of the
economy and pi is the probability of ith state of the
economy.
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Risk
 Risk can be defined as the possible variation in cash
flow about an expected cash flow.
 Statistically, risk may be measured by the standard
deviation of the random cash flow.
 The standard deviation of an asset a is denoted by a
and is calculated as follows:
n

n
2
2


p
R

ER
=
 R – ERa P  i ai a
i =1
i
i
i 1
a
f
where n is the number of states of the economy, Rai is
the return in the ith state and pi is the probability of the
ith state of the economy
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Risk
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Summary Statistics Example
State
Probability
Recession
20%
Moderate Growth
30%
Strong Growth
50%
Cash Flow
$ 1,000
$ 1,200
$ 1,400
EX
ER
Var X
Var R
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$
1,260
13%
Richard MacMinn
Return
10%
12%
14%
24400
0.000244
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Risk
 The attractiveness of a security cannot be determined
by standard deviation alone.
 The risk and return of a security has to be compared
with the alternatives available for investment.
 If two securities have the same risk, the one with the
higher return is preferable.
 Alternatively, if two securities have the same return,
then the one with lower risk is preferable.
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Risk & Diversification
 Total Risk or variability of returns can be divided into:
The variability of returns unique to the security
 Commonly referred to as Firm Specific Risk or Unique Risk or Diversifiable Risk or
Unsystematic Risk
The risk related to market movements
 Also referred to as Market Risk or Non-diversifiable Risk or Systematic Risk
 By diversifying, the investor can eliminate the “unique”
security risk. The systematic risk, however, cannot be
diversified.
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Effects of Diversification
Standard
Deviation
Total Risk = Unique Risk
+ Systematic Risk
Unique Risk
Systematic Risk
Number of Stocks in Portfolio
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Benefits of Diversification
State
Rainy Season
Sunny Season
ERU
ERR
Var RU
Cov(RR, RU)
Correlation
Portfolio
ERP
Var RP
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Probability
Umbrella
Manufacturer
50%
50.0%
50%
-25.0%
Resort
Portfolio
Owner
-25.0% 12.5%
50.0% 12.5%
12.5%
Cov(RU, RR)
Var RR
14.06%
-14.06%
-1
50%
12.5%
-14.06%
14.06%
-1
50%
12.5%
0.0%
Richard MacMinn
0.0%
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How do investors diversify?




The Portfolio Problem
Markowitz (Portfolio Selection)
Tobin (Portfolio Separation)
Sharpe (CAPM)
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What risk is priced?
 Risk averse investors demand higher returns, or
equivalently, a risk premium for undertaking risk.
 Investors cannot expect the market to compensate them
for risk that they can eliminate through diversification.
 Because stocks can be combined in portfolios to
eliminate specific risk, only diversifiable or systematic
risk matters, i.e. commands a risk premium.
 Only systematic risk contributes to the riskiness of a
portfolio and cannot be eliminated through
diversification.
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Measuring Systematic Risk
 Systematic Risk affects all securities.
 To measure systematic risk, we measure the tendency
of a stock to move relative to the market.
 The plot of firm excess returns versus market excess
returns is called the characteristic line, i.e., ERa - Rf
=  (ERm - Rf)
 The measure of a stock’s systematic risk or market risk
is commonly called beta
 Beta is also the slope of the characteristic line.
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Characteristic Line
Stock/Portfolio Returns (%)
Characteristic
line
Market Returns (%)
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Measuring Beta
 The beta of stock A is calculated as follows:
 Cov(R , R )
a
m ,R
Cov
R
a =
aVar
 Rm a m
Var Ra
a
f
where Ra is the return on stock A and Rm is the
return on the market portfolio and where
a
f
n
a
fa
Cov Ra ,Rm   pi Rai  ERa Rmi  ERm
i 1
f
 The beta of a portfolio is the weighted average
of the individual securities’ betas
 The beta of the market is 1.
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Required Rate of Return
 The required rate of return equals the risk free rate plus
a return to compensate for the additional risk.
 The required rate of return can be expressed as
R = Rf + RP,
where R is the investor’s required rate of return, Rf is
the risk-free rate, and RP is the risk premium
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The Required Rate of Return
 Capital Asset Pricing Model (CAPM)
 According to the CAPM
ERa = Rf + a [ERm - Rf ]
where ERa is the expected rate of return on stock “a”
and ERm is the expected rate of return on the market
portfolio.
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The Security Market Line
ERa (%)
ERm - Rf
Rf
a
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Criticisms of CAPM
 Can Beta capture all dimensions of risk? Is beta the
appropriate measure of risk?
 Some empirical research has shown that the CAPM does
not hold. Various anomalies such as the size effect and
the friday the 13th effect have been known and
investigated for some time.
 How do we determine the market portfolio?
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