Risk Aversion and
Capital Allocation
Risk Tolerance
Asset Allocation
Capital Allocation Line
Risk Premium and Risk Aversion
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E[r] - rf
Risk Premium:
It is compensation for risk
s (Std. Dev.)
Risk Measure*:
Risk Aversion coeff:
A
A
E[ r ]  r f
1
2
s
2
* just one of them
Investments 8
2
Risk Premium and Risk Aversion
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Example
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Market portfolio E[r] = 12%
Market portfolio s = 20%
Risk-free rate (T-bill) = 4%
Risk premium: E[r] - rf = 8%
Risk aversion coefficient:
A = 0.08/(0.5*0.20^2) = 4
Investments 8
3
Speculation vs. Gambling
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Speculation (i.e. Investing)
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Taking risk for extra reward
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Risk premium: E[r] - rf > 0
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Higher investors’ risk aversion requires higher
expected returns
Odds are in your favor
Gambling
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Risk is the reward
Risk premium: E[r] - rf < 0

Investments 8
Odds are against you
4
Asset Allocation
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How to allocate your fund among the
following asset classes?
Investment Funds
Stock
Risky Assets
Investments 8
Bond
T-Bills
Risk-Free Asset
5
Asset Allocation
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Risky and Risk-Free Assets
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Percentage to invest in risky asset
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Percentage in risk-free asset
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Risky asset: a stock or a stock portfolio
Risk-free asset: 30-day T-bill as proxy
Issues
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
Investments 8
Examine risk/return tradeoff
Demonstrate how different degrees of risk
aversion will affect allocations between risky
and risk-free assets
6
Asset Allocation
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Moments of asset returns
Assets
Risky
Risk - Free
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Mean
Std Dev
Weight
E[rp ]  15% s p  22%
r f  7%
s f  0%
w
1 w
Moments of portfolio C return
Mean : E[rc ]  (1  w)rf  wE[rp ]  0.07  0.08w
Std Dev : s c  ws p  0.22w

Example: w = 0.75
Mean : E[rc ]  0.07  0.08w  0.13  13%
Std Dev : s c  0.22w  0.16  16.5%
Investments 8
7
Capital Allocation Line
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How much in risky asset …
18.0%
Capital Allocation Line
16.0%
14.0%
E[rp ]  15%
Risky Portfolio
E[r_c]
12.0%
w = 0.75
10.0%
8.0%
6.0%
4.0%
r f  7%
Risk-Free Asset
2.0%
s p  22%
0.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
SD[r_c]
Investments 8
8
Capital Allocation Line
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w > 1, what does that mean?
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Find the E[rc] and SD[rc] with w = 1.2
Mean : E[rc ]  0.07  0.08w  0.166  16.6%
Std Dev : s c  0.22w  0.264  26.4%
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Leverage
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Investments 8
Investing 120% of wealth in risky asset
Using margin borrowing
Higher expected return than the risky asset
Higher volatility to go with the higher return
9
Capital Allocation Line with Borrowing
Capital Allocation Line:
Borrowing at 10% Part
20.0%
18.0%
16.0%
14.0%
w = 1.2
E[rp ]  15%
Risky Portfolio
12.0%
10.0%
8.0%
6.0%
4.0%
r f  7%
Risk-Free Asset
2.0%
0.0%
0.0%
Investments 8
s p  22%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
10
Capital Allocation Line
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Sharpe (reward-to-variability) Ratio
18.0%
16.0%
14.0%
E[rp ]  15%
E[r_c]
12.0%
E[rp ]  rf  15%  7%  8%
10.0%
8.0%
Sharpe Ratio :
E[rp ]  rf
8%
S

 0.36
sp
22%
r f  7%
6.0%
s p  22%
4.0%
2.0%
0.0%
0.0%
s p  22%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
SD[r_c]
Investments 8
11
Capital Allocation Line
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Risk Tolerance and Allocation
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Greater risk aversion leads to higher
allocation to risk-free asset
Lower risk aversion leads to greater
allocation to risky asset
Willingness to accept extremely higher risk
for higher return may lead to leveraged
position
Investments 8
12
How to find your portfolio allocation?
Example 1
 You desire 12% return for your portfolio:
12% = (1-w)*7%+w*15% or w = 62.5%
Std. Dev. = 62.5%*22% = 13.75%
Example 2
 You desire risk no more than 10% for your
portfolio:
w*Std. Dev. = 10% or w = 45.45%
Return = (1-45.45%)*7%+45.45*15% = 10.64%
Investments 8
13
Wrap-up
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How does risk aversion affect expected
returns?
Is investment a form of gambling???
What is the Capital Allocation Line?
How risk tolerance affects asset
allocation?
Investments 8
14