Risk Preferences
and Utility CIS 418
Source: S. Bodily, 2007
A dream of chances
Those dreams are built from losing lottery tickets, by Brooklyn-based artists Adam Eckstrom and Lauren Was and it’s entitled Ghost of a Dream. The tickets were discarded by unlucky patrons. “Chance city” was built by the artist Jean Shin.
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Are you risk averse?
Suppose you owned a lottery ticket that was equally likely to result in $100 loss and $125 gain.
How much would you accept for this ticket?
Expected Monetary Value (EMV)
50%x$125+50%x(-$100) = $12.5
Certainty Equivalent (CE)
the price at which you’d be willing to sell the ticket.
Risk premium (RP)
how much of the EMV you’d be willing to give up to avoid the risk of losing money.
RP = EMV-CE
If RP>0 you are risk averse. If RP<0 you are risk prone. If RP=0 you are risk neutral.
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A monetary utility function translates wealth into utility
Does this utility function represents the utility of a risk-averse, risk-neutral or risk-prone decision maker?
Explain by showing an example.
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Expected Utility vs. Certainty Equivalent
Risk Premium
Under the expected utility model, decision makers make choices that maximize their expected utility. The same choices also maximize the certainty equivalent.
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Various functions can be used to model risk
• Risk-averse function: Concave.
• Risk-neutral function: Linear.
• Risk-prone function: Convex.
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Are you risk averse?
Here is an explanation by Veritasium:
https://www.youtube.com/watch?v=vBX-KulgJ1o
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Example 1: A simple portfolio problem
There is an investment that for every 1$ invested returns $4.3 or $0 with equal probability.
My current wealth is $14,000. How much of $14,000 should I invest?
-- The more I invest, the higher my expected net wealth, but my risk goes up as well.
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My optimal investment would depend on my risk preferences
• A decision maker shows constant risk aversion if she has the same positive risk premium for any two risky opportunities that have respective outcomes that differ only by a constant amount.
Therefore, her expected utility would be modeled by a negative exponential function:
• A decision maker shows decreasing risk aversion if she has decreasing risk premium for any two risky opportunities that have respective outcomes that differ only by a constant amount.
Therefore, her expected utility would be modeled by a logarithmic function:
EU 1eCE R
EU ln CE A
• R, A = Risk tolerance.
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Risk premium as a function of initial wealth
Lower risk premiums = higher risk
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Risk as a function of initial wealth
With the logarithmic utility the percentage of total wealth invested in the risky investment stays the same as the amount of wealth changes.
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Optimize by maximizing EU
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Optimize by maximizing CE
Certainty equivalent and expected utility graphs (previous slide)
have different shapes. But are maximized by the same
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investment amount.
Calculate the Certainty Equivalent
The expected utility of the uncertain investment is equal to the utility of certainty
equivalent.
Negative Exponential Utility
Logarithmic utility
Logarithmic Utility
EU lnCEA gamble
expEUgamble CE A
EU
gamble
1eCE R
utility of certainty equivalent
eCE R 1EUgamble
CE expEU
CE Rln 1EUgamble
CERln 1EUgamble
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A Ricky Roet-Green
CIS-418
gamble
Calculate the Expected Utility
Investment: for every 1$ invested, returns $4.3 or $0 with equal probability. My current wealth is $14,000.
Calculate EMV, CE and RP for $2000 investment.
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CIS-418
Ricky Roet-Green
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Initial Wealth Investment
R (risk tolerance)
$ 14,000.00 $ 2,000.00 $ 4,166.00
Probability
Expected
Lose 0.5 $ 12,000.00 0.943891 Win 0.5 $ 20,600.00 0.99288
CE $ 14,389.93 RP $ 1,910.07
Net Wealth
$ 16,300.00
Utility
0.968385
EU 1exp(12,000/R)
CE Rln(10.968)
Example 2: Keep the investment?
The decision maker owns an investment that will result in personal wealth of either $21,000 or $11,000 in today’s dollars with equal probability
Q1. The decision maker can choose to
o Keep the investment
o Sell this investment now for $14,000
o Sell half of this investment now for $7,000 and keep the other half
Which option will be preferred?
Q2. What is the minimum price that the decision maker will accept now for the entire investment?
Assume the decision maker has constant risk aversion with risk tolerance parameter R = $4166 .
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Option 1: Write VBA Functions
• To increase efficiently, we can write a VBA function to calculate the Negative Exponential Utility and the Certainty Equivalent:
Function NEXPEU(CE, R) NEXPEU = 1 - Exp(-CE / R)
End Function
Function CE_NEXPEU(EU, R) CE_NEXPEU = -R * Log(1 - EU)
End Function
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Option 2: Compare the expected utilities (or CEs)
keep
Sell for 14K
Decision: Sell 1⁄2 for 7K
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Option 3: Use a decision tree
50%
Win
21000
Keep
21000
21000
0.993531471
0
13526.0248
50%
0.961100171
Lose
11000
11000
11000
0.928668872
3
Sell
14291
14000
0.967626
14000
14000
0.965283413
50%
Win
17500
Sell Half
10500
17500
0.985014498
7000
14291.00156
50%
0.967625663
Lose
12500
5500
12500
0.950236828
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For this investor, the second 50% of this investment are worth less than 7K in CE, but not the first 50%
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Q2: The minimum this investor will accept right now for the investment is $13,256
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Example 3: Bet on what horse?
I would like to bet $5000 on a horse. The odds, and my beliefs about the probability of winning are given below.
So if I place $100 bet on Tea Biscuit, and Tea Biscuit comes in first I would get $1800. That will happen with 15% probability. With 85% probability Tea Biscuit will not come in first, and I will lose my $100.
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Sample spreadsheet to compare utilities of betting on different horses
EU 1eCE R
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What if you could split the bet among the horses?
Built a spreadsheet that includes the above data.
• What would be your objective?
• What are the decision variables?
• What are the constraints?
Optimize using two different objectives: risk-averse vs. risk-neutral. Compare the results.
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