Risk and Returns: part 1 Economics 71a: Spring 2007 Mayo chapter 8 Malkiel, Chap 9-10 Lecture notes...
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Transcript of Risk and Returns: part 1 Economics 71a: Spring 2007 Mayo chapter 8 Malkiel, Chap 9-10 Lecture notes...
Risk and Returns: part 1
Economics 71a: Spring 2007
Mayo chapter 8Malkiel, Chap 9-10Lecture notes 3.2a
Returns
Total percentage gain on investment Includes dividends for stocksExample
Price : 100 - 105 Pays: $2 dividend Return = (105+2-100)/100 = 7%
What is Risk?
Hard questionNo good answerStatistical Features
Variance/Standard deviation Histograms (probability distributions)
Histogram
Frequency with which random variable takes value
Gives a pictorial indication of riskMany possible shapesMost famous is the Normal Distribution
Expected Return, E(R)
N states of the world, 1, 2, ..,N
pi = probability of state i Ri = return in state i Also called “Expected value”
€
E(R) = piRi
i=1
N
∑
Example
3 States Good: return = 20%, prob 1/4 Normal: return = 10%, prob = 1/2 Bad: return = -15%, prob = 1/4
Expected return (1/4)(20)+(1/2)(10)+(1/4)(-15)
6.25
Variance
Expected return = E(R) Variance
Standard Deviation = Square Root (Variance)
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Var (R) = pi (Ri − E(R))2
i=1
N
∑
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Std(R) = pi (Ri − E(R))2
i=1
N
∑
Example
3 States Good: return = 20%, prob 1/4 Normal: return = 10%, prob = 1/2 Bad: return = -15%, prob = 1/4
Variance (1/4)(20-6.25)^2+
(1/2)(10-6.25)^2+(1/4)(-15-6.25)^2
= 167.19 Standard deviation = 12.93
Sharpe Ratio
Reward/Risk Ratio
RF = Risk free returnRelation between expected return and
standard deviation or risk
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S =E(R) − RF
std(R)
Covariance and Correlation
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x = mean(x)
y = mean(y)
cov(x,y) = (1
N −1) (xi − x )(yi − y )
i=1
N
∑
corr(x,y) =(
1N −1
) (xi − x )(yi − y )i=1
N
∑
std(x)std(y)
−1 < corr(x,y) < 1
Expectations of Linear Functions
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x = random variable
y = random variable
E(ax) = aE(x)
E(ax + c) = aE(x) + c
E(ax + by + c) = aE(x) + bE(y) + c
Variances
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x = random variable
y = random variable
var(ax) = a2 var(x)
std(ax) = a std(x)
var(ax + c) = a2 var(x)
var(x + y) = var(x) + var(y) + 2 cov(x, y)
Where do we stand?
Expected return Best probabilistic guess of what the future return will be
Variance One measure of how good that guess may turn out, or in
other words, sort of a RISK measure
Histogram Complete measure of what the return could be, and RISK
Risk Versus Return
Efficient market world Return is fair payout for bearing risk Higher Risk -> Higher Return “No free lunch”
Not so efficient market world Opportunities might exist for low risk and high
return Need to search these out
Returns
Realized returns Actually returns on your investments from
the pastExpected returns
Mathematical forecast of future returnsRequired returns
Theoretically predicted return for a security given its risk level
Risk versus Return Not so Efficient Market View: Investment Opportunities
Risk
Return*
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