# Concepts Return, Risk, and Risk Aversion

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"Concepts Return, Risk, and Risk Aversion"

```					   Concepts: Return, Risk, and Risk
Aversion
       Introduce key concepts that are central to asset
allocation
A. Holding-period return, and probability
distributions
B. The historical record
C. Risk and risk aversion (intro to ch.6)
D. Portfolio mathematics (intro to ch.7)

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Slide 5-1
A. Holding Period Return (HPR)

HPR  P P D
1    0     1

P    0

HPR = Holding Period Return
P0 = Price at the beginning of the period
P1 = Price at the end of the period
D1 = Dividend or interest received during the period

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Slide 5-2
Rates of Return:
Single Period Example

Ending Price =          48
Beginning Price =       40
Dividend =               2

48  40  2
HPR               25%
40

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Slide 5-3
Characteristics of Probability Distributions
Moments of a probability distribution
e.g., distribution of stock returns

1) Mean: most likely value
2) Variance/standard deviation: dispersion from the
mean
3) Skewness e.g., centered?
4) Kurtosis e.g., fat tails?
* If a distribution is “normal”, it can be completely described by its
first two moments

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Slide 5-4
Normal + Skewness
 = 0.06,  = 0.17

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Slide 5-5
Normal + Kurtosis
 = 0.1,  = 0.2

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Slide 5-6
Measuring the Mean
Expected / mean return
s
E(r)     p
i 1
i    ri
‘s’               = number of possible outcomes or
“states of nature”, i
pi                = probability that outcome ‘i’ will occur
ri                = return if outcome ‘i’ occurs
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Slide 5-7
Numerical example:
Discrete Distribution
Outcome          Probability          Return
1                0.1                -5%
2                0.2                5%
3                0.4                15%
4                0.2               25%
5                0.1               35%
E(r) = (.1)(-.05)+(.2)(.05)...+(.1)(.35)
E(r) = .15 = 15%
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Slide 5-8
Measuring Variance or Dispersion of
Returns
s               2
2
Variance        p  [r  E(r)]
i 1
i   i

Standard deviation = variance 1/2
Using our example:
2=[(.1)(-.05-.15)2+(.2)(.05- .15)2+…+
=.01199
 = [ .01199]1/2 = .1095 = 10.95%
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Slide 5-9
Sample Statistics
 What we just looked at were the moments of a
probability distribution
 Before any realizations (i.e., before the fact, or ex-
ante)
 Calculating moments from actual results, e.g.,
historical records of financial asset returns – we use
sample statistics
 Weights are no longer the probabilities, but related
to sample size, N

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Slide 5-10
B. Historical records: Annual HPRs

Series   Mean (%)   St. Deviation (%)

Stocks                 11.13          16.12
LT Bonds                8.99          10.08
T-bills                 6.74           3.75
Inflation               4.21           3.22

Slide 5-11
Annual HP Risk Premiums
and Real Returns, Canada

Series   Risk Premium    Real Return
(%)           (%)
Stocks               4.48           7.01
LT Bonds             2.24           4.77

T-bills               -             2.53
Inflation             -               -

Slide 5-12
Annual HPRs
U.S., 1926-2002
G Mean   A Mean   Std Dev
(%)      (%)      (%)
Sm Stocks         11.64    17.74    39.30
Lg Stocks        10.01    12.04    20.55
LT Bonds (Gov)    5.38     5.68     8.24
T-bills           3.78     3.82     3.18
Inflation         3.05     3.14     4.37

Slide 5-13
Geometric vs. Arithmetic/Simple mean
 Geometric mean
 Example: 10% return in first year, 8% in second
year
 If arithmetic/simple: mean = ?
 If geometric: mean = √*(1+0.1)(1+0.8)+ – 1
= 0.08995
 Relationship if distribution is normal
 Geometric = arithmetic – 0.52

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Slide 5-14
Annual HP Risk Premiums and
Real Returns, U.S.
Risk Premium   Real Return
(%)          (%)
Sm Stocks             7.86         8.59
Lg Stocks            6.23          6.96
LT Bonds (Gov)       1.60          2.33
T-bills               -            0.73
Inflation             -              -

Slide 5-15
C. Risk and Risk Aversion
Uncertain Outcomes

W1 = 150; Profit = 50
W = 100
1-p = .4          W2 = 80; Profit = -20

E(R) = pR1 + (1-p)R2 = 22% return

2 = p[R1 – E(R)]2 + (1-p) [R2 – E(R)]2
 = 34.29%
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Slide 5-16
Risky Investments
with Risk-Free Investment

Risky                   W1 = 150 Profit = 50
Investment
1-p = .4   W2 = 80 Profit = -20
100

Risk Free T-bills               Profit = 5

Risk Premium = 22% - 5% = 17%
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Slide 5-17
Risk Aversion & Utility
 Investor’s view of risk
 Risk Averse
 Risk Neutral
 Risk Seeking / loving
 Utility Function
 Private value, in our case, of risk and return
 Example: One popular utility function used (e.g.,
in the CFA curriculum) is
U = E ( r ) – 0.5 A  2
 “A” measures the degree of risk aversion
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Slide 5-18
Risk Aversion and Value:
The Sample Investment
U = E ( r ) - 0.5 A  2
= 0.22 - 0.5 A (0.34) 2
Risk Aversion        A  Utility
High               5  -0.0690
3   0.0466         Compare to
Low                1   0.1622
T-bill = .05

* Utility: ordinal ranking. Actual value unimportant
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Slide 5-19
Dominance Principle
Expected Return

4
2         3
1

Standard Deviation
• 2 dominates 1; has a higher return
• 2 dominates 3; has a lower risk
• 4 dominates 3; has a higher return
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Slide 5-20
Utility and Indifference Curves
 Represent an investor’s willingness to trade-off
return and risk
Example (for an investor with A=4):

Exp Return(%)        St Deviation(%)   U=E(r)-0.5A2

10           20.0             0.02
15           25.5             0.02
20           30.0             0.02
25           33.9             0.02
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Slide 5-21
Indifference Curves
Expected Return
Increasing Utility

Standard Deviation
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Slide 5-22
D. Portfolio Mathematics:
Portfolio Return
 The rate of return on a portfolio is a weighted
average of the return of each asset in the portfolio,
with the portfolio proportions as weights

Consider a two-asset case:
rp = w1r1 + w2r2

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Slide 5-23
Portfolio Mathematics:
Risk of a Portfolio
 Risk as measured by the variance
 When two assets with variances, 12 and 22 , are
combined into a portfolio with portfolio weights w1
and w2, respectively, the portfolio variance is:
    p
2
 w12 12  w22 22  2w1w2Cov(r1, r2)

 What if one of the assets is risk-free?

Slide 5-24

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