Risk, Returns, and Risk Aversion
 Return and Risk Measures






Real versus Nominal Rates
EAR versus APR
Holding Period Returns
Excess Return and Risk Premium
Variance
Sharpe Ratio
 Risk Aversion and Capital Allocations



Risk Aversion and Utility Function
Capital allocation line
Optimal Allocations
Risk, Return and Risk Aversion
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Road map in this and the next lecture
 Risk and return
 Optimal allocation given risk and return tradeoff
 Two-asset allocation
 Efficient frontier
 Multiple asset allocation
 Capital asset pricing models (CAPM)
 Arbitrage pricing theory (APT)
 Fama-French three-factor model
Risk, Return and Risk Aversion
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Nominal and Real Rates
 Nominal rate
 Real rate
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Example 5.2 Annualized Rates of Return
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Formula for EARs and APRs
1
EAR  [1 r f (T ) T ] 1
T
(1 EAR) 1
APR 
T
See page 128-129
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Rates of Return: Single Period
P
1  P0  D1
HPR 
P0
HPR = Holding Period Return
P0 = Beginning price
P1 = Ending price
D1 = Dividend during period one
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Rates of Return: Single Period
Example
Ending Price =
Beginning Price = 40
Dividend =
48
2
HPR = (48 - 40 + 2 )/ (40) =
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Excess Return
 Risk free rate
 Excess return
 Also known as risk premium
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Scenario or Subjective Returns:
Example
State
.1
2
3
4
5
E(r) =
Prob. of State
-.05
.2
.4
.2
.1
Risk, Return and Risk Aversion
r in State
.10
.05
.15
.25
.35
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Variance or Dispersion of Returns
  
   p s  r s  E r 
2
s

2
Standard deviation = [variance]1/2
Using Our Example:
Var=
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Mean and Variance of Historical
Returns
Arithmetic average or rates of return
1 n
E (r )  s 1 p( s )r ( s )  s 1 r ( s )
n
n

n
2 1
s 1
n
r(s)r
2
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Geometric Average Returns
TV
n
 (1  r1)(1  r 2)  (1  r n)
TV = Terminal Value of the Investment
g  TV
1/ n
1
g= geometric average rate of return
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Sharpe Ratio
Sharpe Ratio for Portfolios=
Risk _ Pr emium
STD _ excess _ ret
Measure of risk-return tradeoff
Other concepts: Skewness and Kurtosis – page 142-143
Check out the statistics from page 147-151
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Page 187
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Figure 5.4 The Normal Distribution
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Figure 5.4 The Normal Distribution
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Normality and Risk Measures
 What if excess returns are not normally
distributed?



Standard deviation is no longer a complete measure
of risk
Sharpe ratio is not a complete measure of portfolio
performance
Need to consider skew and kurtosis
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Skew and Kurtosis
Skew
Equation 5.19
Kurtosis
 Equation 5.20
_ 3

_ 4



 R  R  
R

R
 

skew  average  ^   kurtosis  average  ^    3




3
4
 

 





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Figure 5.5A Normal and Skewed Distributions
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Figure 5.5B Normal and Fat-Tailed
Distributions (mean = .1, SD =.2)
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Value at Risk (VaR)
 A measure of loss most frequently associated with
extreme negative returns
 VaR is the quantile of a distribution below which
lies q % of the possible values of that distribution

The 5% VaR , commonly estimated in practice, is
the return at the 5th percentile when returns are
sorted from high to low.
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Expected Shortfall (ES)
 Also called conditional tail expectation (CTE)
 More conservative measure of downside risk
than VaR


VaR takes the highest return from the worst cases
ES takes an average return of the worst cases
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Lower Partial Standard Deviation (LPSD)
and the Sortino Ratio
 Issues:


Need to consider negative deviations separately
Need to consider deviations of returns from the
risk-free rate.
 LPSD: similar to usual standard deviation, but
uses only negative deviations from rf
 Sortino Ratio replaces Sharpe Ratio
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Historic Returns on Risky
Portfolios
 Returns appear normally distributed
 Returns are lower over the most recent half of the
period (1986-2009)
 SD for small stocks became smaller; SD for long-
term bonds got bigger
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Historic Returns on Risky
Portfolios
 Better diversified portfolios have higher Sharpe
Ratios
 Negative skew
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Figure 5.10 Annually Compounded,
25-Year HPRs
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Risk Aversion
 Risk Aversion
 Risk Love
 Risk Neutral
 Utility function
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Utility Function
Utility Function: U = E ( r ) – 1/2 A σ2
Where U = utility
E ( r ) = expected return on the asset or portfolio
A = coefficient of risk aversion
σ2 = variance of returns
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Computing Utility Scores
If A=2, then
See page 168
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Figure 6.2 The Indifference Curve
Page 166, Table 6.3
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Allocating Capital: Risky & Risk Free Assets
 It’s possible to split investment funds between
safe and risky assets.
 Risk free asset: proxy; T-bills
 Risky asset: stock (or a portfolio)
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Example Using Chapter 6.4
Numbers
The total market value of an initial portfolio is $300,000, of
which $90,000 is invested in the Ready Asset money
market fund, a risk-free asset. The remaining $210,000 is
invested in risky securities – $113,400 in equity and
$96,600 in long-term bonds. Find the distribution of this
portfolio.
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Expected Returns for
Combinations
E(rc) = yE(rp) + (1 - y)rf
rc = complete or combined portfolio
For example, y = .75
E(rc) =
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Combinations Without Leverage
If y = .75, then
c =
If y = 1

c
=
If y = 0
c =
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Capital Allocation Line with
Leverage
Borrow at the Risk-Free Rate and invest in stock.
Using 50% Leverage,
rc = (-.5) (.07) + (1.5) (.15) = .19
c = (1.5) (.22) = .33
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Risk, Return and Risk Aversion
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Table 6.5 Utility Levels
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Optimal Portfolio
 Maximize the mean-variance utility function


U=E(R)-1/2Aσ2
Based on the expressions for expected return and σ,
we have the expression for optimal allocation y*:
 Example 6.4 (page 175):
Risk, Return and Risk Aversion
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Figure 6.6 Utility as a Function of Allocation
to the Risky Asset, y
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Figure 6.7 Indifference Curves for U = .05
and U = .09 with A = 2 and A = 4
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Optimal Complete Portfolio on Indifference Curves
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