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Risk Aversion and Capital Allocation
Risk Tolerance
Asset Allocation
Capital Allocation Line
Investments 8 2
Risk Premium and Risk Aversion Risk Premium: E[r] - rf
It is compensation for risk
Risk Measure*: (Std. Dev.)
Risk Aversion coeff: A
221
frrEA
][
* just one of them
Investments 8 3
Risk Premium and Risk Aversion Example
Market portfolio E[r] = 12% Market portfolio = 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 4
Speculation vs. Gambling Speculation (i.e. Investing)
Taking risk for extra reward Higher investors’ risk aversion requires higher
expected returns
Risk premium: E[r] - rf > 0 Odds are in your favor
Gambling Risk is the reward Risk premium: E[r] - rf < 0
Odds are against you
Investments 8 5
Asset Allocation How to allocate your fund among the
following asset classes?
Investment Funds
Stock Bond T-Bills
Risky AssetsRisk-Free Asset
Investments 8 6
Asset Allocation Risky and Risk-Free Assets
Percentage to invest in risky asset Risky asset: a stock or a stock portfolio
Percentage in risk-free asset Risk-free asset: 30-day T-bill as proxy
Issues Examine risk/return tradeoff Demonstrate how different degrees of risk
aversion will affect allocations between risky and risk-free assets
Investments 8 7
Asset Allocation Moments of asset returns
Moments of portfolio C return
Example: w = 0.75
wr
wrE
ff
pp
107
2215
%%Free-Risk
%%][Risky
WeightDev StdMeanAssets
ww
wrwErwrE
pc
pfc
22.0 :Dev Std
08.007.0][)1(][ :Mean
%5.1616.022.0 :Dev Std
%1313.008.007.0][ :Mean
w
wrE
c
c
Investments 8 8
Capital Allocation Line How much in risky asset …
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0%
SD[r_c]
E[r
_c]
%][ 15prE
%7fr
%22p
Capital Allocation Line
Risky Portfolio
w = 0.75
Risk-Free Asset
Investments 8 9
Capital Allocation Line w > 1, what does that mean?
Find the E[rc] and SD[rc] with w = 1.2
Leverage 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
%4.26264.022.0 :Dev Std
%6.16166.008.007.0][ :Mean
w
wrE
c
c
Investments 8 10
Capital Allocation Line with Borrowing
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
20.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0%
%][ 15prE
%7fr
%22p
Capital Allocation Line: Borrowing at 10% Part
Risky Portfolio
w = 1.2
Risk-Free Asset
Investments 8 11
0.0%
2.0%
4.0%
6.0%
8.0%
10.0%
12.0%
14.0%
16.0%
18.0%
0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0%
SD[r_c]
E[r
_c]
Capital Allocation Line Sharpe (reward-to-variability) Ratio
%15][ prE
%7fr
%22p
%22p
%8%7%15][ fp rrE
36.0%22
%8][
:Ratio Sharpe
p
fp rrES
Investments 8 12
Capital Allocation Line Risk Tolerance and Allocation
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 13
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 14
Wrap-up 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?