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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