# Introduction to Utility Theory

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```							               Risk aversion, Capital Allocation to Risky Assets
Chapter 6

Part (a): Utility Theory

 How do people react to risk? They demand compensation to take it on:

Ex: Imagine coin toss where you get \$50,000 if heads, pay
–20,000 if tails.

 Expected outcome = .50(50,000) + .50(-20,000) = 15,000
Suppose you can choose between taking the 15,000 with certainty or the gamble …
Most people will pick the 15,000 even if the expected outcome of the coin toss is
the same.

Why?? Most people dislike risk (and have to be compensated for bearing it).

Assumption: Investors are usually thought to be Risk Averse: Choose portfolios
that offer least amount of risk for a given level of return.

Definitions:

• Utility of wealth function – Quantifies amount of satisfaction derived from total
amount of wealth

dollar of wealth – always positive

Assumption: An additional dollar increases a risk averse investor’s total utility.
However, the marginal utility from this dollar is less than the marginal utility of
the previous dollar of wealth.

 Total utility always increases with more wealth, but marginal utility
decreases with more wealth:

1
Total
Utility

Total Utility of Wealth

ΔW     ΔW ΔW
Wealth

Decreasing MU  Risk Averse Behavior

Why??

 Idea that “losses hurt more than gains”

Implication: Given the choice of two uncertain outcomes (a gamble) and the mean
of those two outcomes (with certainty), a risk investor will always choose the
certain choice …

 Offers higher utility!

Why? Decrease in utility from losing gamble is greater than increase in utility
from winning gamble for risk averse investors (feel more of a loss) … utility drops
more going to left than it increases going to right

Certainty Equivalent:

Definition: Amount of wealth (obtained with certainty) that gives you the same
utility as the utility from taking a gamble

2
       Ex: Suppose there is a gamble with outcomes of 0% or 10% (50/50 odds):

Alternatively, if you are risk averse, you may decide that a 3% certain return
gives you the same utility as the 5% average return from the gamble

 For risk averse investors, this certain return (the “certainty equivalent”) is
always less than the average outcome from the gamble (means you require
extra compensation to take the risk of the gamble).

Definition: The difference between the mean outcome and the CE: This additional
amount makes the investor indifferent between the certain and risky choices

       For risk averse investors, the RP will always be positive

Indifference curves:

Main idea: Risk averse investors only accept more risk if they are compensated by
higher return. An indifference curve illustrates investments that offer the same
utility to the investor:

I1
E(r)
C
A

B



Compared to B, other investments (like C) offer the same utility – they give higher
return but at the expense of higher risk.

3
          If risk averse, will accept more risk only if return is higher.

          Therefore, asset C (higher risk and return) would be about as good.

Investment A: Better than B (or C)        higher return, less risk
 Risk averse investors use the Mean Variance Criterion to choose
investments: A gives more utility than B if …

E (rA )  E (rB )
and                 - And one of these inequalities is strict
A B

 Utility can also be measured with a utility score function:

1
U  E (r ) A 2
2
Just plug in E(r),  , and extent of risk aversion (A).
2

 Higher levels of A mean more risk aversion (so react more negatively to
more risk)

 Investments on the same indifference curve have to give the same utility score.
Investments on higher curves have greater utility than those on lower curve

4
E(r)
I2
I1          Preferred Direction
I0


Characteristics of Indifference Curves:

    Upward sloping, higher utility going up to left
    Infinite number (Indifference curve map)
    Non-crossing
    Extent of risk aversion measured by steepness (slope)
    Convex

E(r)

I0


          You require more and more compensation as risk increases to maintain the same
level of utility – chance of losing wealth greater from increase from C to D than
from A to B

5
Part B: Capital Allocation with a Risky Asset and Risk-Free Asset (6.2)

•Risk-free assets … Are there such a thing? (i.e. should we consider ST or LT
maturities)?

•Types of risk for a fixed-income security:

–Default;
–Interest rate risk;
–Reinvestment rate risk;
–Inflation.

•Risk-free asset and a risky security:

–Expected Return is a weighted average

–Standard deviation is also a weighted average!!!

Objective: What criteria do investors use to evaluate investments?

 Optimal investments offer the highest possible expected return
for the least possible expected risk

Suppose you have the following information:

E(r)          SD

T-Bills       5.5 %         0

S&P500        6.36 %        8.9

Problem is to decide how much to allocate to the risky asset (P):

6
Return = rP (so expected return = E(rP) )

Allocation = y

And to the T-Bills:

Return = rF

Allocation = (1-y)

(1) Expected Return of the Combination Portfolio

The expected return of the combination portfolio is

E(rC) = y*E(rP) + (1-y)*rF = y*E(rP) + rF – y*rF

Factoring out the y and plugging in values results in:

E(rC) = rF + y*(E(rP) – rF)    = 5.5 + y*(6.36 – 5.5)

 Conclusion: How much you want above 5.5% depends on how much is
invested in the risky S&P500 (P) portfolio:

E(rC)

6.36
S&P500

5.5

0%                         100 %           y (% allocated to risky asset)

7
 As you allocate more money to the risky S&P portfolio, your combination
portfolio return E(rC) rises proportionately.

(2) Standard Deviation of the Combination Portfolio

 When a risky asset is combined with a risk-free asset, the portfolio’s
standard deviation depends on the % you put into the risky asset -

 C  y 2 P  1 / 2  y P
2

Therefore,    C  y P   = y(8.9)

C

8.9
S&P500

0

0%                       100 %               y (% allocated to risky asset)

As you allocate more money to the S&P500, your combination portfolio’s
risk  C rises proportionately.

8
(3) Now, combine the graphs from (1) and (2) together, to see how expected return
of the combination portfolio changes as the standard deviation changes. To
C
do this, substitute y       into E(rC) from step (1) :
P
C                       E (rP )  rF            
 E(rC) = rF +  ( E (r )  r )  rF + 
                          C

P
P

F
P                  

The resulting graph shows return vs. risk of the combination portfolio and is called
the Capital Allocation Line :

Plugging in numbers from above,
 E (rP )  rF 
E(rC) = rF + 
               C = 5.5 +
                       (0.097)   C
     P       
Intercept         Slope

 Interpretation:       Expected return depends on how much how much you put into
the risky asset.

E(rC)

CAL = Capital Allocation Line
P
6.36                                           borrow

lend
slope = 0.097 = “reward-to-variability ratio”
5.5

 P  8.9                        C

Note: The “reward to variability ratio” is also called the Sharpe ratio – more on this
later in the course.

9
Q: What if you want to increase your return beyond 6.36 % (i.e. 100% in the
S&P500)?

 Borrow money to increase the risky holding:
For example, suppose you borrow 50% of all of your funds (at the rF rate) and

invest this in the S&P500 along with the original 100%. This makes the weight

(y) invested in the S&P500 = 150% and the weight of the risk-free asset = -50%

 y  1.5       so (1  y)  .50

E(rC) =     5.5 + y*(6.36 – 5.5)       =        5.5 + 1.5*(6.36-5.5)

=        5.5 + 1.29 = 6.79

 C  y P = y * (8.9) = 1.5*(8.9) = 13.35

Q: Suppose you borrow at a higher rate than you lend at?

E(rC)

P                  CAL is kinked
6.36                                      borrow

rF(B)
lend
rF(L)

 P  8.9                       C

10
Next Question: How do you allocate your money?

 Depends on how risk averse you are!

 First, recall that you can measure utility from an asset with a utility function,
such as:

1
U  E (r )      A 2
2

 Second, we know that the idea is to maximize our utility. Therefore, just
need to find the value of y that gives you the highest possible value of U

 After a few substitutions and some algebra, this value of y is:

E (rP )  r f
y* 
A P
2

 Once you have the optimal weightings, can calculate the E (rC ) and  C of
the best portfolio.

11
The portfolio selection problem in pictures

 First, recall that higher indifference curves represent higher utility

 Next, recall that how risk averse you are determines how steep your
indifference curves are

Indifference Curves

E(rC)

CAL
P*

6.36

P*

5.5

 P  8.9           C

 Your Optimal Portfolio (P*) corresponds to point of tangency on the highest
possible indifference curve.

12

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