Constant and Decreasing Risk-Aversion

Constant Risk-Aversion


The risk premium for a gamble does not depend on the initial wealth held,
Can be represented using an exponential utility function
You have $x in your pocket, and you are facing a bet: 1) win $15 with
probability 0.5, or 2) lose $15 with probability 0.5. Suppose your
utility function can be modeled as an exponential function with risk
tolerance R=35.
RISK ATTITUDE
$ in Pocket
Utility Functions: Examples
(0.5)
(0.5)
$x+15
Probability Tree of the Bet
$x –15
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Suppose you face the following gamble: 1) win $2000 with probability
0.4; 2) win $1000 with probability 0.4, or 3) win $500 with probability
0.2, and your utility can be modeled as an exponential function with
R=900. What is your CE of this gamble?
The expected utility of the gamble is:
EU = 0.4∙U($2000)+0.4 ∙U($1000)+ 0.2∙U($500)
= 0.4∙(1-e-2000/900) +0.4 ∙(1-e-1000/900)+ 0.2∙(1-e1-500/900) = 0.710
Solve
0.710=1-e-CE/900
3
When x=$25
EU=0.5∙U($10)+0.5∙U($40)=0.5∙(1-e-10/35)+0.5∙(1-e-40/35)=0.4648
EU=U(CE) 0.4648=1-e-CE/35CE=$21.88
EMV= 0.5∙10+0.5∙40=$25
Risk Premium=EMV-CE=25-21.88=$3.12
$x
for CE, you can get CE=$1114.71
2
EV ($) CE ($)
Risk Premium ($)
25
25
21.88
3.12
35
35
31.88
3.12
45
45
41.88
3.12
55
55
51.88
3.12
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Constant and Decreasing Risk-Aversion
(Cont.)
Some Caveats


Decreasing Risk-Aversion



Utilities DO NOT Add Up

Typically, people’s attitude towards risk changes with their initial wealth
The risk premium for a gamble decreases along with the increase of wealth held
Can be represented with the logarithmic utility function

Utility Difference Does Not Express Strength of Preferences


U ( x)  ln( x)

U(A+B)≠U(A)+U(B) (why?)
U(A1 )-U(A2) > U(B1)-U(B2 ) does not mean we would rather go from A1 to A2
instead of from B1 to B2
Utility only provides a numerical scale for ordering preferences, not a measure
of their strengths
Utilities are Not Comparable from Person to Person

A utility function is a subjective personal statement of an individual’s
preference
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Exercise
In the previous example of betting, suppose your utility function
can be modeled as a logarithmic function
An investor with assets of $10,000 has an opportunity to invest $5,000
in a venture that is equally likely to pay either $15,000 or nothing. The
investor’s utility function can be described by the log utility function
U(x) =ln(x), where x is the total wealth.
When x=$25
EU=0.5∙U($10)+0.5∙U($40)=0.5∙ln(10)+0.5∙ln(40)=2.9957
EU=U(CE) 2.9957=ln(CE)CE=$20
EMV= 0.5∙10+0.5∙40=$25
Risk Premium=EMV-CE=25-20=$5
a.What should the investor do?
$x
EV ($) CE ($) Risk Premium ($)
25
25
20
5.00
35
35
31.62
3.38
45
45
42.43
2.57
55
55
52.92
2.08
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b.
a.
Total Wealth
success (0.5)
10,000-5,000+15,000 =$20,000
$15,000
Failure (0.5) 10,000-5,000= $5,000
Invest
- $5,000
Don’t
Invest
$10,000
EU(invest) = 0.5∙U($20,000)+0.5∙U($5,000)=0.5∙ln($20,000)+0.5∙ln($5000) = 9.21
EU(Don’t invest) = U($10,000) = ln($10,000) = 9.21
Therefore, the investor is indifferent between the two alternatives
EU(Invest|Win) = 9.326
Total Wealth
Success (0.5)
10,000+1,000-5,000+15,000 =$21,000
$15,000
Invest
Failure (0.5) 10,000+1,000-5,000= $6,000
Win (0.5)
-$5,000
$1,000
Don’t EU(Don’t Invest|Win) = 9.306
10,000+1,000 = $11,000
Invest
Success
(0.5)
EU(Bet) = 9.216
EU(Invest|Lose) = 9.073 $15,000 10,000-1,000-5,000+15,000 =$19,000
Invest
Failure (0.5) 10,000-1,000-5,000= $4,000
Bet
Lose (0.5)
-$5,000
-$1,000
Don’t EU(Don’t Invest|Lose) = 9.105
10,000-1,000 = $9,000
Invest
Don’t
Success (0.5)
10,000-5,000+15,000 =$20,000
Bet
$15,000
Invest
EU(Don’t Bet) = 9.21
Failure (0.5)
10,000-5,000= $5,000
-$5,000
Don’t
$10,000
Invest
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b. Suppose the investor places a bet with a friend before making the
investment decision. The bet is for $1,000; if a fair coin lands heads
up, the investor wins $1,000, but if it lands tails up, the investor
pays $1,000 to his friend. Only after the bet has been resolved will
the investor decide whether or not to invest in the venture. If he
wins the bet, should he invest? What if he loses the bet? Should he
toss the coin in the first place?
If he wins the bet:
EU(Invest) = 0.5∙ln($21,000) + 0.5∙ln($6,000) = 9.326
EU(Don’t Invest) = ln($11,000) = 9.306
Therefore, if he wins the bet, he should invest the venture
If he loses the bet:
EU(Invest) = 0.5∙ln($19,000) + 0.5∙ln($4,000) = 9.073
EU(Don’t Invest) = ln($9,000) = 9.105
Therefore, if he losses the bet, he should not invest the venture
EU(Bet) = 0.5∙EU(Invest|win) + 0.5∙EU(Don’t Invest |lose)
= 0.5(9.326)+0.5(9.105) = 9.216
EU(Don’t Bet) = 9.21 (from part a)
Therefore, he should bet
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