Equilibrium Pricing
• Roadmap…
• Review of different approaches to valuing securities
• The role of utility theory and it’s important results
• Generalizing utility to the marketplace and
development of Capital Asset Pricing Model
Approaches to Valuing Assets
No-Arbitrage Approach
Pro
Based on replicating payoff of an asset with payoffs from
related assets
Since payoffs of the asset and the replicating portfolio are the
same, they should have equal values
Widely used for valuation of derivatives
Con
Must be able to replicate the payoff with existing securities
Not very useful when a new security is introduced and cannot
be replicated
Approaches to Valuing Assets
Equilibrium Approach
More general framework, applicable to wider array of assets
Prices related to economic concepts
More of “here’s where the price comes from” than “here’s
how to derive the price”
May require more structure, however, since not just taking
underlying asset prices as given
Equilibrium Approach
General Assumptions in Equilibrium Model:
A set of individuals (often called “agents”) trade a set of securities with
fixed characteristics
Each agent has an initial amount of resources (often called an
“endowment”)
A market exists to trade the securities
Agents look to maximize their own utility
The Result:
Equilibrium is obtained when no agent has an incentive to trade
Equilibrium prices are reached when each agent’s expected utility is
maximized
Equilibrium Approach
What if something changes?
If any of the conditions in the market change, the equilibrium
prices will change as well.
So if conditions change to temporarily allow arbitrage, agents
will change their activities to again maximize their utility.
Prices will react quickly to remove the arbitrage. This
leads to consistent results with the no-arbitrage approach.
Equilibrium Approach
The underlying issue in the Equilibrium Approach is to find
the best way of modeling how the agents will act and react.
In order to do this, we will look into the concept of Expected
Utility
The Expected Utility Hypothesis gives some sort of
framework of decision-making when outcomes are
uncertain
Expected Utility Hypothesis
Basic assumptions:
Agent preferences can be measured using a defined utility
function that produces utility amounts given each amount
of wealth
Agents act to maximize the expected value of the utility
function – they create their own probabilities about future
states of wealth and will work to maximize the expected
value
Utility Functions
Basic assumptions about Utility Functions, u(x):
We assume that agents prefer more wealth to less wealth –
that utility functions are increasing
This implies that u’(x) > 0
We assume that agents are risk averse – that the difference in
utility for a given increase in wealth in much smaller for
larger amount sof wealth than for smaller amount of wealth
This implies that u’’(x) < 0 --- concave utility function
Utility Functions
General results for insurance and investment using these basic
assumptions about Utility Functions:
Risk aversion implies that an investor will invest in safer
assets – there is no enticement to put wealth at risk to
generate greater wealth because there is only small gains in
utility
Risk aversion implies that an insurance purchaser will be
willing to pay more than the expected value of losses for
insurance protection – there is a desire to protect the
current wealth position since lower wealth positions have
lower utility
Utility Functions
Example: Power Utility
u(x) = (x α – 1) / α, for 0 < α < 1
u´(x) = x α-1
u´´(x) = α-1 (x α-2)
We can see the severe risk aversion as α moves from being
close to 1 to being close to 0
Utility Functions
We can use the utility function to generate measures of risk
aversion:
Absolute Risk Aversion
RA(x) = - u´´(x) / u´(x)
Relative Risk Aversion
RR(x) = - (x u´´(x)) / u´(x)
Utility Functions
For Power Utility…
Absolute Risk Aversion
RA(x) = - u´´(x) / u´(x) = (1 – α) / x
RA(x) decreases as x increases. Potentially not a bad model
since people tend allocate more wealth towards more risky
assets as their wealth increases.
Relative Risk Aversion
RR(x) = - (x u´´(x)) / u´(x) = (1 – α)
Constant
Utility Functions
Some other potential Utility Functions (not an exhaustive
list):
Quadratic: u(x) = x – (x2 / 2b) for x < b
Increasing risk aversion properties make less desirable, but
often used since we can show that decision makers only
care about mean and variance of return
Exponential: u(x) = 1 – e-ax, a > 0
Constant absolute risk aversion
Utility Functions
Examples of utility in making decisions:
We can talk about the concept of utility in a few scenarios
If we want to increase our wealth, we can consider how utility will help
us make investment decisions – how will our utility function help us
decide whether to invest our wealth in risk-free versus risky assets?
If we want to protect our wealth, we can consider how utility will help us
make insurance decisions – how will our utility function help us decide
how much we are willing to spend on premiums to protect our wealth?
Utility Functions
The investment decision…
Assume you have w in wealth. You can invest in a risk free
security for a return of rf, or a risky security with two
possible outcomes... ru with probability p and rd with
probability 1 – p. We can compare final utilities to help
make the decision.
Expected Utilities:
Risk free: u(w(1 + rf)) – no probability since it’s a certain
outcome
Risky: p u(w(1 + ru)) + (1 – p) u(w(1 + rd))
Utility Functions
Let w = 100, rf = .05, ru = .10, rd = -.05, p = .30
u(x) = 2x1/2 – 2
Note that ending expected wealth is 100(1.05) = 105 when investing in
risk-free asset, and .30(110)+.70(95) = 99.5
Expected Utility
Risk free: u(w(1 + rf)) = u(105) = 18.494
Risky: p u(w(1 + ru)) + (1 – p) u(w(1 + rd))
= .30 u(110) + .70 u(95)
= .30 (18.976) + .70 (17.494)
= 17.938
Utility Functions
Let w = 100, rf = .05, ru = .10, rd = -.05, p = .70
u(x) = 2x1/2 - 2
Expected Utility
Risk free: u(w(1 + rf)) = u(105) = 18.494
Risky: p u(w(1 + ru)) + (1 – p) u(w(1 + rd))
= .70 u(110) + .30 u(95)
= .70 (18.976) + .30 (17.494)
= 18.531
So answer is dependent on utility function and probability of potential
outcomes
Utility Functions
The insurance decision…
Assume you have w in wealth. You are exposed to a random
loss X. You can buy insurance for a premium B to fully
cover the loss, or you can be uninsured. We can compare
final utilities to help make the decision.
Expected Utilities:
Buy Insurance: u(w – B) – no probabilities since a certain
outcome
Don’t Buy Insurance: E[u(w – X)]
Utility Functions
Assume that both the frequency of loss follow Bernoulli
distributions
Frequency:
With probability 0.10, a loss occurs
With probability 0.90, no loss occurs
Severity:
With probability 0.40, the loss is .10w
With probability 0.60, the loss is .25w
Utility Functions
Aggregate Loss Distribution
Loss Amount
0
.10w
.25w
Probability
.90
.04
.06
Expected Loss = .90(0) + .04(.10w) + .06(.25w) = .019w
Let u(x) = 2x1/2 – 2
E[u(w – X)] = .90(u(w)) + .04(u(.90w)) + .06(u(.75w))
Utility Functions
If w = 100, B = 2… The insurer will charge a premium of 2
to cover the loss fully
Expected Loss = .019w = 1.90
E[u(100 – X)] = .90(u(100)) + .04(u(90)) + .06(u(75))
= .90(18) + .04(16.974) + .06(15.321)
= 17.798
u(w – B) = u(100 – 2) = u(98) = 17.799
Utility Functions
So in this example, u(w – B) > E[u(w – X)] and the decision
would be to buy the insurance
The expected utility from buying the insurance is greater than
the expected utility for incurring the loss…
Even though what is being paid for the insurance (B = 2) is
larger than the expected loss (1.90)
This is due to the risk aversion. Just as in the investment
decision, the final decision made is dependent on the utility
function and the probabilities of the potential outcomes
Utility Functions
Could we also find out how much we could raise the premium
and still have this particular client still purchase the
coverage?
Where does u(w – B) = E[u(w – X)]?
u(100 – B) = 17.798
Using utility function, we find that u(97.99) = 17.798
So B = 2.01
Jensen’s Inequality
Given some general risk aversion properties, an individual
will always be willing to pay a premium that is higher than
the expected value of the loss.
Jensen’s Inequality:
If u´´(x) < 0, then u(E[X]) > E[u(X)]
Can look at graphic representation….
Jensen’s Inequality
u(x)
u(E[X]) > E[u(X)]
slope is u´(x)
Equation of line is
u(m) + u´(m) (x – m)
m = E[X]
x
Jensen’s Inequality
We can tell from the graph that:
u(x) < u(m) + u´(m) (x – m)
E[u(x)] < E[u(m) + u´(m) (x – m)]
E[u(x)] < E[u(m)] + E[u´(m) (x)] – E[u´(m) (m)]
E[u(x)] < u(m) + u´(m) E[x] – u´(m) (m)
E[u(x)] < u(m) + u´(m) (m) – u´(m) (m)
E[u(x)] < u(E[x])
If u´´(x) < 0, then u(E[X]) > E[u(X)]
Jensen’s Inequality
Expanding on the Jensen Inequality in our insurance purchase terms, we
have that for risk averse individuals:
Compare… The insurance purchase decision
u(w – B) > E[u(w – X)]
And… The facts from Jensen’s Inequality
u(w – E[X]) > E[u(w – X)]
In words: The expected ending utility state by paying an insurance
premium equal to the expected loss (which is a “certain” state since it
avoids the random loss) will always be at least as large as the expected
utility from exposing current wealth to the random loss
We also saw that in many cases, the individual will also pay a premium
greater than the expected loss
Jensen’s Inequality
Commentary:
Risk averse individuals purchase insurance because they know there is much
happiness / satisfaction / utility to lose when their wealth is depleted.
Some common risks that are insured:
Property
Auto Collision
Liability
Life
Disability
Long Term Care
Longevity
What are the large financial losses that can occur from these risks?
Maximum Insurance Premiums
We saw through Jensen’s Inequality that clients will always buy insurance
if it is offered for a premium equal to the expected loss.
We also saw an example where clients often buy insurance if it offered for
a premium greater than the expected loss.
So, what is the maximum premium clients would be willing to spend?
Clearly, the answer should be a function of:
The riskiness of the loss
The risk tolerance / utility function of the client
Maximum Insurance Premiums
We can show that the maximum premium a risk-averse client is willing to
pay to protect wealth w from a loss of X is:
π = μ + (σ2 / 2) RA(w – μ)
Where
μ = mean of X
σ2 = variance of X
RA = Absolute risk aversion function
This assumes that higher moments of the loss distribution are negligible
Maximum Insurance Premiums
Remarks: If we think about π-μ as the “risk premium”, the
amount greater than the expected loss we would pay to be
insured, then we have:
Risk Premium = π – μ = (σ2 / 2) RA(w – μ)
• Risk Premium gets larger as the variance of the loss
increases
• Risk Premium gets larger as the absolute risk aversion
increases
Maximum Insurance Premiums
Example:
Let u(x) = 2x1/2, w = 100, μ = 10, σ2 = 10
What does this mean graphically on utility and loss distribution?
First, what is RA(x)?
RA(x) = - u´´(x) / u´(x) = 1 / 2x
So RA(w – μ) = RA(90) = 1 / 180
Maximum insurance premium:
π = μ + (σ2 / 2) RA(w – μ)
π = 10 + (10 / 2) (1 / 180)
= 10.0278
Maximum Insurance Premiums
Example:
Let u(x) = 1000x1/1000, w = 100, μ = 10, σ2 = 10
RA(x) = - u´´(x) / u´(x) = 999 / 1000x
So RA(w – μ) = RA(90) = 999 / 90000
Maximum insurance premium:
π = μ + (σ2 / 2) RA(w – μ)
π = 10 + (10 / 2) (999 / 90000)
= 10.0555
Maximum Insurance Premiums
Example:
Let u(x) = 2x1/2, w = 100, μ = 10, σ2 = 100
RA(x) = - u´´(x) / u´(x) = 1 / 2 x
So RA(w – μ) = RA(90) = 1 / 180
Maximum insurance premium:
π = μ + (σ2 / 2) RA(w – μ)
π = 10 + (100 / 2) (1 / 180)
= 10.2778
Allocation between Risk-Free and
Risky Assets
We can use Expected Utility theory to determine how we should distribute
an investment between risk-free and risky assets
Consider a risky asset that has two possible rates of return, u and d, with a
probability of rate of return u being p and a probability of rate of return
d being (1 – p)
You get an expected return m:
m
m–d
p
= (p) u + (1 - p) d
= (p) u + d – (p) d
= p (u – d)
= (m – d) / (u – d)
Allocation between Risk-Free and
Risky Assets
Let x be the percent of wealth put in the risky asset. This means we put xw in the
risky asset and (1-x)w in the risk free asset.
Our ending wealth in up state will be the sum of earning a risk free rate of return
on (1-x)w and the up state rate of return on xw:
(1-x)w(1+r) + xw(1+u)
= (1-x)(w+wr) + xw + xwu
= w + wr – xw – xwr + xw + xwu
= w + wr + wxu – wxr
= w (1+r) + w(xu – xr)
= w [(1+r) + x(u-r)]
You can think of this as “earning the entire risk free rate on all the wealth, plus an
additional risk premium on x percent of the wealth”
Similarly, ending wealth in down state = w[(1+r) + x(d-r)]
Allocation between Risk-Free and
Risky Assets
Assume utility function is u(x) = (x α) / α, for 0 < α < 1
We always want to maximize our expected utility
The goal then would be to create the expected utility by plugging in the
two possible ending wealth states into the utility function and
weighting by the probabilities p and 1-p. This would give us the
expected utility.
You would then differentiate and solve for x to maximize the expected
utility.
We can show under these assumptions that….
Allocation between Risk-Free and
Risky Assets
x = (1 + r) (Z – 1) / [u – r + Z(r – d)]
Where Z =
p (u – r)
(1-p) (r – d)
1
(1-α)
Allocation between Risk-Free and
Risky Assets
x = (1 + r) (Z – 1) / [u – r + Z(r – d)]
Z = [p (u – r)]1/(1-α) / [(1-p) (r – d)]1/(1-α)
Remarks on x and Z:
• x > 0 when Z > 1…. And Z > 1 when u > r
“Positive weighting to the risky asset when it provides an
attraction over the risk free rate”
•
•
x = 0 when Z = 1… And Z = 1 when u gets close to r
x < 1 when Z < (1+u) / (1+d) --- Note that r not a factor
Allocation between Risk-Free and
Risky Assets
x = (1 + r) (Z – 1) / [u – r + Z(r – d)]
Z = [p (u – r)]1/(1-α) / [(1-p) (r – d)]1/(1-α)
Remarks on x and Z:
x < 1 (less than full weighting to the risky asset) when
(1 + r) (Z-1) < u – r + Z(r – d)
Z + Zr – r – 1 < u – r + Zr – Zd
Z – 1 < u – Zd
Z (1 + d) < 1 + u
Z < (1+u) / (1+d) --- Note that r not a factor
Allocation between Risk-Free and
Risky Assets
A numerical example:
Assume utility function is u(x) = 2x1/2, w = 100
u = 6.41%
d = 3.60%
r = 5.00%
p = .50
This implies m = 5.005% - just larger than the risk free rate
Allocation between Risk-Free and
Risky Assets
Ending wealth in up state
= w[(1+r) + x(u-r)]
= 100 [(1.05) + x(.0141)]
= 105 + 1.41x
Ending wealth in down state
= w[(1+r) + x(d-r)]
= 100 [(1.05) + x(-.0140)]
= 105 – 1.40x
Allocation between Risk-Free and
Risky Assets
Expected Utility
= 1/2 [2(105 + 1.41x)½] + 1/2 [2(105 – 1.40x)½]
End up with…
Z = [p (u – r)]1/(1-α) / [(1-p) (r – d)]1/(1-α)
Z = [.5 (.0641 – .05)]2 / [.5 (.05 – .036)]2
Z = [.0141]2 / [.014]2
Z = 1.0143
x=
x=
x=
x=
(1 + r) (Z – 1) / [u – r + Z(r – d)]
(1.05) (.0143) / [.0141 + 1.0143(.014)]
.0150 / .0283
.5319
x = .5319 implies 53.19% of wealth goes into risky asset and remaining 46.81%
goes to risk free asset
Allocation between Risk-Free and
Risky Assets
Alternatively…
Expected Utility = 1/2 [2(105 + 1.41x)½] + 1/2 [2(105 – 1.40x)½]
Which is a fairly easily differentiable function
Take first derivative, set = 0 and again find that x = .5319 to maximize the
expected utility
x = .5319 implies 53.19% of wealth goes into risky asset and remaining 46.81%
goes to risk free asset
Allocation between Risk-Free and
Risky Assets
Note that this exercise was a “one period” example – there
were only two possible outcomes for the risky asset
following a Bernoulli distribution
If we expand this to multiple periods of small lengths and
take some limits on our results, we’ll find that:
x* = (μ – r) / [σ2 (1 – α)]
This is called the Merton ratio
Allocation between Risk-Free and
Risky Assets
x* = (μ – r) / [σ2 (1 – α)]
Remarks:
• The higher the risk premium, the more above the risk free
rate you expect on the risky asset, and x* goes up
• The higher the variability of the return on the risky asset,
x* goes down
• The higher the investor’s risk aversion, x* goes down
Deriving the Capital Asset Pricing Model
Some initial definitions:
xj = Current price of a risky security j
Xj = Future price of risky security j – unknown – random
variable
Rj = Xj / xj - 1 = Return on risky security j
Xj / xj = 1+ Rj
Deriving the Capital Asset Pricing Model
Let’s assume that xj = E[Z Xj]
xj = Current price of a risky security j
= Expected value of the product of
1) The future payoff of the security, and
2) A random variable Z derived from the general
utility function of the marketplace
This expected value brings into play the subjective
probabilities associated with the potential payoffs of the
security
Deriving the Capital Asset Pricing Model
If xj = E[Z Xj], then dividing both sides by xj gives
1 = E[Z (1 + Rj)]
Similarly, for a risk-free asset we would have
1 = E[Z (1 + r)]
Both right hand sides of the above equations are equal to 1
Deriving the Capital Asset Pricing Model
E[Z (1 + r)] = E[Z (1 + Rj)]
E[Z + Z r] = E[Z + Z Rj]
E[Z] + E[Zr] = E[Z] + E[Z Rj]
r E[Z] = E[Z Rj]
r E[Z] = E[Z] E[Rj] + Cov[Rj , Z]
- Cov[Rj , Z] = E[Z] E[Rj] - r E[Z]
E[Z] [ E[Rj] – r] = -Cov[Rj , Z]
[E[Rj] – r] = [- 1 / E[Z] ] • Cov[Rj , Z]
Note that from the risk free equation 1 = E[Z (1 + r)], we
would get that [1 / E[Z] ] = 1 + r
Deriving the Capital Asset Pricing Model
[E[Rj] – r] = [- 1 / E[Z] ] • Cov[Rj , Z]
[E[Rj] – r] = -(1+r) • Cov[Rj , Z]
This gives as a way to look at the Expected Risk Premium of
a risky security j
Deriving the Capital Asset Pricing Model
If instead of talking about a single risky security j, we were to
talk about investing in the entire market m – perhaps
analogous to investing in an index fund – we would get
[E[Rm] – r] = -(1+r) • Cov[Rm , Z]
This gives as a way to look at the Expected Risk Premium of
a investing in the entire market of risky securities
Deriving the Capital Asset Pricing Model
Since
[E[Rj] – r] = -(1+r) • Cov[Rj , Z]
implies
-(1+r) = [E[Rj] – r] / Cov[Rj , Z]
AND
[E[Rm] – r] = -(1+r) • Cov[Rm , Z]
implies
-(1+r) = [E[Rm] – r] / Cov[Rm , Z]
Then
[E[Rj] – r] / Cov[Rj , Z] = [E[Rm] – r] / Cov[Rm , Z]
Deriving the Capital Asset Pricing Model
[E[Rj] – r] / Cov[Rj , Z] = [E[Rm] – r] / Cov[Rm , Z]
E[Rj] – r = (Cov[Rj , Z] / Cov[Rm , Z]) [E[Rm] – r]
In words:
We have found a way to relate the excess return for a risky
security j over the risk free rate to the excess return for the
market over the risk free rate
The excess return on security j may be higher or lower than
the excess return on the market depending on the value of
the covariance factors
Deriving the Capital Asset Pricing Model
One more nice simplifying assumption:
Assume that the utility function inherent in the random
variable Z is a quadratic utility function
If we do this, we can show that
Cov[Rj , Z] = k Cov[Rj , Rm]
Similarly,
Cov[Rm , Z] = k Cov[Rm , Rm] = k Var[Rm]
Deriving the Capital Asset Pricing Model
This now gives:
E[Rj] – r = (Cov[Rj , Z] / Cov[Rm , Z]) [E[Rm] – r]
E[Rj] – r = (k Cov[Rj , Rm] / k Var[Rm]) [E[Rm] – r]
E[Rj] – r = (Cov[Rj , Rm] / Var[Rm]) [E[Rm] – r]
So we can calculate expected excess returns on a security j by
looking at the expected return and variance of the market
combined with how security j moves when the market
moves
The term “Cov[Rj , Rm] / Var[Rm]” is often called βj. This is a
measure of the relative movement of the security compared
to the baseline measure of the market
Deriving the Capital Asset Pricing Model
Example:
Market Expected Return = 10%
Market Variance of Return = 12%
Covariance of Security Returns and Market Returns = 24%
Risk Free Rate = 5%
E[Rj] – r = (Cov[Rj , Rm] / Var[Rm]) [E[Rm] – r]
E[Rj] – .05 = (.24 / .12) [.10 – .05]
E[Rj] – .05 = (.24 / .12) [.10 – .05]
E[Rj] =.15
Expected Return on Security = 15%