Utility
1
An economic term referring to the total satisfaction received from
consuming a good or service.
A consumer's utility is hard to measure. However,
we can determine it indirectly with consumer
behavior theories, which assume that consumers
will strive to maximize their utility.
Utility is a concept that was introduced by
Daniel Bernoulli. He believed that for the
usual person, utility increased with wealth
but at a decreasing rate.
Investopedia
Exposition of a New
Theory on the
Measurement of
Risk - 1738
Utility and Risk Aversion

An individual may value expected outcome differently based on their risk
aversion which may be based on wealth or preferences

The utility of a financial gain or loss to an individual is likely dependent on
current wealth
8
7
U(w)
6
5
4
3
U(w)=ln(1+w)
2
1
0
$0
$250
$500
$750
w
$1,000
$1,250
$1,500
Utility and Risk Aversion

An individual has wealth of 1000 and has the opportunity to
participate in a fair ‘financial game.’ 50% chance to gain 100 or lose
100. Assume her utility function is the natural log of her wealth
U(w)  ln(1000  1)  6.909
U(w)  .5  ln(900  1)  .5  ln(1100  1)  6.904
What probability of winning 100, p, would motivate her to play the financial game?
U(w)  (1  p)  ln(900  1)  p  ln(1100  1)  6.909
She needs a 52.5% probabilit y of winning
U(w)  .475  ln(900  1)  .525  ln(1100  1)  6.909
Her expected wealth after game is $1005.00
w  .475  900  .525  1100  $1005.00
Risk – Return Utility Curve
Expected Return & Utility of Expected Return [%]
11%
A=3
10%
9%
( )
3×s2
U r,s =r2
Note the same utility
for these assets
8%
7%
6%
u = 10% s = 20%
5%
u = 7% s = 14%
4%
u = 4% s = 0%
3%
2%
1%
0%
0%
2%
4%
6%
8%
10%
12% 14% 16% 18% 20%
Expected Risk [Std Dev %]
Attitude Towards Risk

A>0




A=0



Risk decreases utility of return
Individual is risk averse and is thus an ‘investor’
Investor will not participate in a ‘fair financial game’
Risk does not effect the utility of return
Individual is risk neutral and will participate in a ‘fair financial
game’
A<0


Risk increases utility of return
Individual will participate in an “unfair financial game”
 Las Vegas
Indifference Curves
Expected Return %
Risk – Return Indifference Curve
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
3  s2
u  uCE 
2
Note the investor’s
indifference between
these assets
uCE = 11% s = 0%
0
2
4
6
8
10
12
Expected Risk [Std Dev %]
14
16
18
20
u = 12%
s = 8%
u = 14%
s = 14%
Optimal Portfolio



50%
CAL line contains all
possible portfolios
45%
What’s your allocation
of funds between
assets
Depends on your “A”
and say its 5
 Set the shape and
orientation of
indifference curve
Your optimal portfolio
is at the tangent point
 Equal slopes
Indifference curve
with A=5 tangent
to the CAL
40%
35%
Expected Return

30%
l
25%
20%
Asset
A
CAL
15%
uCE
10%
5%
Asset
P
Asset
F
0%
0%
5%
10%
15%
20%
25%
Expected Std Dev
30%
35%
40%
Portfolios With Two Risky Assets
9
2.4%
2.2%
2.0%
rAB=-1
Expected Return Rate
1.8%
B
1.6%
1.4%
rAB=-.5
rAB=0
rAB=1
1.2%
1.0%
0.8%
A
0.6%
0.4%
0.2%
0.0%
0%
2%
4%
6%
8%
10%
12%
14%
16%
Expected Std Dev
sp2  w2A  s2A  wB2  sB2  2  wA  wB  sA  sB  ρAB
18%
Now Determine Your Optimal Portfolio
10
3.0%
2.5%
Indifference
curves
A=2 , 4, 7
Expected Return Rate
2.0%
B
1.5%
T: Optimal Risky Portfolio
V
1.0%
A
0.5%
P: Your optimal portfolio
F
0.0%
0%
2%
4%
6%
8%
10%
12%
14%
Expected Std Dev
16%
18%
20%
Portfolio with 2 Risky Assets
11
3.0%
2.5%
Indifference
curves
A=4
Return
2.0%
B
1.5%
T: Optimal Risky Portfolio
V
1.0%
A
0.5%
P: Your optimal portfolio
F
0.0%
0%
2%
4%
6%
8%
10%
Std Dev
12%
14%
16%
18%
20%