equilibrium pricing roadmap… review of different approaches to valuing securities the role of...

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 AssetsNo-Arbitrage Approach

ProBased on replicating payoff of an asset with payoffs from

related assetsSince payoffs of the asset and the replicating portfolio are the

same, they should have equal valuesWidely used for valuation of derivatives

ConMust be able to replicate the payoff with existing securitiesNot 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 ApproachGeneral 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 FunctionsGeneral 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 FunctionsSome 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 FunctionsExamples 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 FunctionsThe 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 FunctionsLet 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 FunctionsLet 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 FunctionsAggregate Loss Distribution

Loss Amount Probability0 .90.10w .04.25w .06

Expected Loss = .90(0) + .04(.10w) + .06(.25w) = .019w

Let u(x) = 2x1/2 – 2E[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 FunctionsSo 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

x

u(x)

= E[X]

u(E[X]) > E[u(X)] slope is u´(x)

Equation of line is u() + u´() (x – )

Jensen’s InequalityWe can tell from the graph that:

u(x) < u() + u´() (x – )

E[u(x)] < E[u() + u´() (x – )]

E[u(x)] < E[u()] + E[u´ (x)] – E[u´ ()]

E[u(x)] < u() + u´() E[x] – u´() ()

E[u(x)] < u() + u´() () – u´() ()

E[u(x)] < u(E[x])

If u´´(x) < 0, then u(E[X]) > E[u(X)]

Jensen’s InequalityExpanding on the Jensen Inequality in our insurance purchase terms, we have

that for risk averse individuals:

Compare… The insurance purchase decisionu(w – B) > E[u(w – X)]

And… The facts from Jensen’s Inequalityu(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 InequalityCommentary:

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:PropertyAuto CollisionLiabilityLifeDisabilityLong Term CareLongevity

What are the large financial losses that can occur from these risks?

Maximum Insurance PremiumsWe 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 PremiumsWe 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 PremiumsExample:

Let u(x) = 2x1/2, w = 100, μ = 10, σ2 = 10What 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 PremiumsExample:

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 PremiumsExample:

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 = (p) u + (1 - p) d= (p) u + d – (p) d

m – d = p (u – d)p = (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 =

1(1-α)

p (u – r)

(1-p) (r – d)

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 – ZdZ – 1 < u – ZdZ (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 = (1 + r) (Z – 1) / [u – r + Z(r – d)] x = (1.05) (.0143) / [.0141 + 1.0143(.014)] x = .0150 / .0283 x = .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%