ams-511-sol-05-m.nb
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AMS 511 - Foundations
Utility Theory and Portfolio Choice
Robert J Frey, Research Professor
Stony Brook University, Applied Mathematics and Statistics
frey@ams.sunysb.edu
Solutions to exercises for Class 5. Chapters refer to Luenberger’s Investment Science.
Chapter 9, Problem 2
Let f(x) be the pdf of the payoff x. Note that the terminal wealth given x is W – w + x. The expected utility of terminal
wealth is, therefore,
-‡ ‰
¶
-a HW -w+xL
f HxL „ x = - ‡ ‰
-¶
¶
-a W
-a Hx-wL
‰
f HxL „ x = -‰
-¶
-a W
-a Hx-wL
f HxL „ x
‡ ‰
¶
-¶
Thus, if we are comparing two investments with price and payoff {w1 , x1 } and {w2 , x2 }, then their relative ranking will
be independent of of W.
Chapter 9, Problem 6
We'll start out with the HARA expression itself.
g
1-g
ax
y ;
j
hara = ÅÅÅÅÅÅÅÅÅÅÅÅ i
j ÅÅÅÅÅÅÅÅÅÅÅÅ + bz
z
g k 1-g
{
‡ Linear
If we set a = 1 and then take the limit as g Ø 1, then
Limit@hara ê. a Ø 1, g Ø 1D
x
‡ Quadratic
If we look at HARAs with g = 2, then we get the quadratic polynomial
ams-511-sol-05-m.nb
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g
1-g
ax
y ê. g Ø 2E
j
ExpandA ÅÅÅÅÅÅÅÅÅÅÅÅ i
j ÅÅÅÅÅÅÅÅÅÅÅÅ + bz
z
g k 1-g
{
b2
a2 x2
- ÅÅÅÅÅÅÅ + a b x - ÅÅÅÅÅÅÅÅÅÅÅÅÅ
2
2
An affine transformation puts U(x) into the form we want with c = a / 2b.
1
b
SimplifyA% ÅÅÅÅÅÅÅÅ + ÅÅÅÅÅÅÅÅ E
ab
2a
a x2
x - ÅÅÅÅÅÅÅÅÅÅ
2b
‡ Exponential
If we set b =1 and then take the limit as g Ø -¶,then
Limit@hara ê. b Ø 1, g Ø -¶D
-‰-a x
‡ Power
For simplicity we set a =1 and then take the limit as b Ø 0.
Limit@hara ê. a Ø 1, b Ø 0D
-1+g
x
x H ÅÅÅÅÅÅÅ
1-g L
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅ
g
We can further simplify this to
H1 - gL1-g
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ xg
g
H1-gL
and then make the substitution that c = ÅÅÅÅÅÅÅÅ
ÅgÅÅÅÅÅÅÅÅÅÅÅ to bring it into its final form.
1-g
‡ Logarithmic
The text suggests looking at the function below. We'll take its limit as g Ø 0
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H1 - gL1-g
LimitA ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ Hxg - 1L, g Ø 0E
g
Log@xD
Note that for g ≠ 0 this function is equivalent to a power law utility (with an affine transformation of its standard form).
Thus, the log utility function is the dividing line between power laws with positive and negative exponents.
‡ Arrow-Pratt
The Arrow-Pratt risk aversion coefficient is shown below. Note that this can easily be restated in form desired.
D@#, 8x, 2<D
aHara = SimplifyA- ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ E &@haraD
D@#, xD
a-ag
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
b+ax-bg
a
b-bg
We now make the substituton c = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ and d = ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
a-ag
a-ag
1
a
b-bg
SimplifyAaHara ã ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ ê. 9c Ø ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ , d Ø ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ =E
cx +d
a-ag
a-ag
True
Chapter 9, Problem 14
Let p = 0.25 be the probability of winning the race. For a $1 bet we have w = 5 as the payoff on a win and l = 0 as the
è!!!
payoff on a loss. The individual's utility function is U(x) = x . The initial wealth is W. Our objective is to determine the
optimal bet b.
The expected utility is
E@UHxLD = H1 - pL
è!!!!!!!!!!!!!
è!!!!!!!!!!!!!!!!!!!!!!!!!!
W - b + p W + bHw - 1L
We differentiate with respect to b and solve for the root.
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sol = SolveA∂b IH1 - pL
è!!!!!!!!!!
è!!!!!!!!!!!!!!!!!!!!!!!!!!!
W - b + p W + b Hw - 1L M ã 0, bEP1, 1T
H-1 + 2 p - 2 p2 w + p2 w2 L W
b Ø ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
H-1 + wL H1 - 2 p + p2 wL
This is the general solution. We can now substitute the actual parameters into the rule and then recover the actual value
of b.
b ê. sol ê. 8p Ø 0.25, w Ø 5<
0.134615 W
Thus, Gavin ought to bet roughly 13.5% of his wealth on the bet. A plot of expect utility illustrates the result.
PlotA
IH1 - pL
è!!!!!!!!!!
è!!!!!!!!!!!!!!!!!!!!!!!!!!!
W - b + p W + b Hw - 1L M ê. 8W Ø 1, p Ø 0.25, w Ø 5<, 8b, 0, 0.3<E;
1.008
1.006
1.004
1.002
0.05
0.1
0.15
0.2
0.25
0.3
As an additional illustration, we'll repeat the analysis with log utility. Note that this gives a more conservative result.
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sol = Solve@∂b HH1 - pL Log@W - bD + p Log@W + b Hw - 1LDL ã 0, bDP1, 1T
b ê. sol ê. 8p Ø 0.25, w Ø 5<
Plot@HH1 - pL Log@W - bD + p Log@W + b Hw - 1LDL ê.
8W Ø 1, p Ø 0.25, w Ø 5<, 8b, 0, 0.15<D;
-W + p w W
b Ø ÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅÅ
-1 + w
0.0625 W
0.006
0.004
0.002
0.02 0.04 0.06 0.08
-0.002
-0.004
0.25 H5 - 1L + 0.75 H-1L
0.25
0.1
0.12 0.14