Chapter 6: Risky Securities and Utility Theory

Topics

1. Principle of Expected Return

2. St. Petersburg Paradox

3. Utility Theory

4. Principle of Expected Utility

5. The Certainty Equivalent

6. Utility and Risk Attitudes

7. Measures of Risk Aversity

Background 1

1. A financial security is a document that records some

entitlement or obligation and which can be traded for money.

2. E.g. stock, bonds, options, treasury bills etc.

3. The return and fair price or value of a riskless security are

known with certainty

4. If the return of a (financial) security is risky, i.e. it is a random

variable, then we ask how to value or price the (risky) security.

Background 2

1. To this end, consider a security which pays a single amount A.

Since the security is risk-less its value is clearly S0 = A.

2. If, however, A is a random variable then things change

somewhat. To make things clear suppose that a (risky)

security has value A1 if some event E1 occurs and A2

otherwise. In this scenario the securities value is a discrete

binomial random variable. Define probabilities

P(E1) = p; P(E2) = 1− p.

Background 3

A sensible approach to pricing might be to take the value of the

security as

S0 = pA1 + (1− p)A2,

i.e. weighted sum of the possible payoff where the weights are the

same as the probability of that event occuring. This is just pricing

by expectation!

Principle of Expected Return

1. We have a risky security whose payoff is some random

variable, say R. We assume that the distribution of R is

known.

2. We then price the security by

S0 = E(R).

3. This way of pricing is called the Principle of Expected Return.

4. This method of pricing has some problems which we turn to

now.

St. Petersburg Paradox 1

1. There is a problem with pricing with the Principle of Expected

Return:

2. QUESTION: What is the fair price of a coin tossing game

which pays an amount

Ct = 2t−1A‘

if the first head appears on toss number t?

3. This is an example of a single pay-off at an unknown time

(toss) in the future, with zero interest rates.

4. Payoff is time(toss)-dependent

St. Petersburg Paradox 2

1. The first t tosses must be TTT . . .TH which occurs with

probability pt = 1/2t .

2. The payoff for this example is Ct = 2t−1A

3. By the Principle of Expected Return

S0 = E (CT ) =∞∑t=1

ptCt =∞∑t=1

2t−1A

2t=∞∑t=1

A

2

=A

2+

A

2+

A

2+ . . .

=∞

4. This appears nonsensical.

Utility Theory

1. Utility is a hypothetical quantity that is meant to measure

how much satisfaction a sum of money is meant to give,

2. There is obviously more than one utility function.

3. There are a number of properties that utility functions are

generally supposed to possess.

3.1 The law of non-satiety: “you can never have too much

money”, i.e. U ′(x) > 0

3.2 The law of diminishing returns: an increment of wealth on a

large fortune has less utility than an equal increment on a

small fortune, i.e. U ′′(x) < 0

Utility Theory 2

1. The law of diminishing returns may be interpreted as follows:

If we thnk of U ′(x) as a measure of the satisfaction associated

with a small increment in cash returns, U ′′(x) < 0 means that

the prospect of an extra dollar or two is less good if you are

getting a lot of cash anyway.

2. The two conditions we supposed implies that U(x) is a

monotonc increasing, concave function of wealth x .

Utility Theory 3

1. Since U is an increasing function, an infinitesimal increment in

x leads to a positive increment dU in utility.

2. Some people have argued that a good model for risk-averse

investors is one for which the increment in utility dU is

proportional to the increment dx and inversely proportional to

the resulting wealth x + W wher W denotes the individual’s

current wealth.

3. This ensures that an individual’s satisfaction is determined by

the proportion by which the investor’s total wealth increases,

rather than the actual value of the increase.

Utility Theory 4

This model implies that there exists a positive constant b such

that dU = bdx/(x + W ). The resulting ODE for U has solution

U(x) = b ln(x + W

W) + a = b ln(1 +

x

W) + a,

where a is a constant of integration and U(0) = 0 means that

a = 0

Principle of Expected Utility 1

1. Principle of Expected Utility is that an individual will aim to

maximise expected utility rather than expected return.

2. The Principle of Expected Utility can be used to price

securities.

Principle of Expected Utility 2

1. Suppose current purchase price of a security is S and that

after its purchase the security will produce cash-flows with a

combined value X

2. The net return from the security is X − S and so the expected

utility is

E (U(X − S)) .

3. Is the investment better than doing nothing? If we do nothing

then

E (U(0)) = U(0).

4. According to the Principle of Expected Utility we must

compare the expected utilities.

Principle of Expected Utility 3

1. Clearly, the purchase of the security is worthwhile if and only

if E (U(X − S)) > U(0).

2. it is not worthwhile if E (U(X − S)) < U(0)

3. if E (U(X − S)) = U(0) then neither alternative is preferrable.

4. Hence the root S0 of the equation

E (U(X − S0)) = U(0) (1)

is the price at which buying the security is neither

advantageuos or disadvantageous.

5. The root S0 of Equation (1) is clearly the maximum price at

which the purchase of the security might be considered.

Principle of Expected Utility 4

Leave end of calculations as home work. [Do equations on

black-board]

1. The price from the Principle Expected Utility is less than the

price based on the Principle of Expected Return.

2. The discrepancy between the two prices is called the risk

premium because it is a monetary measure of the extent to

which the investors valuation of of the security is diminished

due to the uncertainty in the reu=turn. In most cases, the

risk premium is a decreasing function of the investor’s wealth.

Principle of Expected Utility 5

Theorem

Investment decisions based on the Principle of Expected Utility are

unchanged by positive linear transformation.

1. Say that U(x) = a + bU(x) is a positive linear transformation

of the function U(x) if a and b are constants with b > 0.

2. Let X1 and X2 be two random variables that represent net

returns from two different investments. Also suppose that

E (U(X1)) > E (U(X2)) ,

i.e. choose investment 1 over investment 2 by the Principle of

Expected Utility.

Principle of Expected Utility 6

1. Then

E(U(X1)

)− E

(U(X2)

)= E (a + bU(X1))− E (a + bU(X2))

= b(E (U(X1))− E (U(X2)))

> 0

so

E(U(X1)

)> E

(U(X2)

)and the investor will make the same decision. NOTE: utility

functions which are related by positive linear transformations

are said to be equivalent

Principle of Expected Utility 8

Corollary Prices calculated using the principle of expected utility

are unaffected by positive linear transformations.

Principle of Expected Utility 7

St. Petersburg Paradox To determine the maximum price S0

that an investor should consider paying to play the game we need

to solve the equation

U(0) = E (U(CT − S0)) =∞∑t=1

ptU(CT − S0)

St Petersburg Paradox

If we price via expected utility the fair price of the coin tossing

game is finite.

The Certainty Equivalent 1

1. A security received as a gift may be valued differently from

one which is purchased.

2. Consider the scenario:

2.1 A fixed amount of cash C

2.2 a risky security with net return X

If the investor takes the cash the utility will be U(C ) if she takes

the risky security the expected utility is E(U(X )) According to the

Principle of Expected Utility chose the security if E(U(X )) > U(C )

and the cash if

E(U(X )) < U(C )

The Certainty Equivalent 2

But if E(U(X )) = U(C ) the cash and the security are equally

attractive. THEREFORE the particular value C may be thought as

the cash value of the security when received by the investor as a

gift. The valuie of C which solves this equation is referred to as

the “certainty equivalent” value of the security to the investor

The Certainty Equivalent 3

NOTE: the certainty equivalent may differ slightly from the

previously introduced quantity S0 which represented the maximum

price at which the investor would consider purchasing the security.

(In general, C is slightly higher than S0 because the investor will

be less concerned about the risks associated with a gift than a

purchase.)

The certainty equivalent is generally a very good approximation to

the max. price S0 the investor might consider paying. As it is

usually easier to calculate than S0, it is much more widely used.

Utility and Risk Attitudes 1

1. Since we have concave utility functions we always have

U(E(X )) > E(U(X ))

by Jensen’s Inequality

2. This inequality is generally true for any random variable and

any concave funcion, for e.g. U

3. The economic interpretation is that an investor with utility

function U will prefer the certainty of receiving an amount of

cash, here EX , to a risky security with return X .

Utility and Risk Attitudes 2

1. The risk premium is given by

ρ = E(X )− U−1(E(U(X )))

2. It represents the amount an investor with utility function U

must be compensated to accept the risky investment instead

of a riskless investment with the same expected net return.

3. It’s also the difference between the expected return price and

certainty equivalent of the security.

4. Note that we have used that U is monotonic implies that U

has a well-defined inverse

5. For a risk-averse investor the risk premium ρ is always positive

Utility and Risk Attitudes 3

Type U ′′(x) ρ Risk Attitudes

Risk averse < 0 (concave) ρ > 0 Prefers certainty to risk

for the same expected return

Risk neutral 0 (linear) ρ = 0 Indifferent between certainty

and risk for the same

expected return

Risk loving > 0 (convex) ρ < 0 Prefers risk to certainty for

the same expected return.

(Black-board U plots)

Utility and Risk Attitudes 4

Note here that risk-neutral investors have linear utility functions

U(x) = a + bx (equivalently U(x) = x . and the Principle of

Expected Utility reduces to the Principle of Expected Return.

Blakcboard: Example

Measures of Risk Aversity 1

1. Risk premium

ρ = E(X )− U−1(E(U(X )))

2. Absolute risk aversion

ρabs = −U ′′(x)

U ′(x)

3. Relative risk aversion

ρrel = −xU ′′(x)

U ′(x)

Measures of Risk Aversity 2

In all cases we have the classification

1. ρ > 0 (risk aversion)

2. ρ = 0 (risk neutrality)

3. ρ < 0 (risk loving)

Example: p84