Quantitative Finance - PART III - ASSET PRICING

QUANTITATIVE FINANCEPART III - ASSET PRICING

Dr. Luo, Dan

June 16, 2020

Utility Functions Portfolio Choice UM vs MVA Asset Pricing

Investment Decision over a Single Period

Time 0 Time 1Initial Investment Random Future Payoff

Your future wealth will depend on the random realization of the assetprices.

The Expected Utility framework provides the weapon fordecision-making under uncertainty, that is, ranking the random payoffs.

It trades off expected returns with volatility, as we will show later, andmore things like skewness and kurtosis.

QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 1 / 83


Utility Function

A utility function U(x) is defined on real numbers and gives a real utilityscore. You compare two random outcomes by comparing the expectedutility E [U(x)] of the two.

John von Neumann and Oskar Morgenstern pioneered the expectedutility framework in 1944, and they showed that preferences overrandom investment opportunities can be represented by a certainexpected utility function when preferences satisfy some axiomaticconditions.

One general restriction on the utility function is that it is increasing.That is, if x > y , then U(x) > U(y). Other than this, utility functioncould in theory take any form. And it could vary among individualinvestors.



Commonly Used Utility Functions

1 Exponential:U(x) = −e−ax ,

for some a > 0.2 Power:

U(x) =x1−γ

1− γ,

for some γ > 0. When γ → 1, we obtain the logarithmic utilityU(x) = ln(x).

3 Quadratic:U(x) = x − bx2,

for some b > 0. We need x < 1/(2b) for it to be increasing.4 When γ = 0 or b = 0, we obtain the risk-neutral utility

E [U(x)] = E [x ] wherein investors only care about expectedreturns.



Commonly Used Utility Functions

x

U(x

)

ExponentialPowerQuadraticLogarithmicRisk Neutral



Decision under Uncertainty: An Illustration

Suppose an investor has a logarithmic utility U(x) = ln(x). She wouldinvest either in a riskfree bond which generates a future payoff of $100,or in a risky stock which produces a payoff of $80 and $120 withcorresponding probabilities 50% and 50%.

Her expected utility of investing in the bond isE [U(x)] = ln(100) = 4.6052, while her expected utility of investing inthe stock is E [U(x)] = 0.5 ln(80) + 0.5 ln(120) = 4.5848. Therefore,she would take the bond.



Pros for Utility Functions

1 Merits of simplicity: just compare the expected values of the utilityscores to make decisions.

2 Flexibility of various functional forms: not many restrictions on thefunctional forms (we will show how to choose a specific utilityfunction soon).

3 Strong theoretic justifications: can be derived from a small set ofaxioms describing the preferences of rational investors.



Equivalent Utility Function

If U(x) and V (x) always give the same ranking of all random payoffs,they are said to be equivalent. It can be proved that, the necessaryand sufficient condition for equivalency is that:

V (x) = aU(x) + b, (1)

where a and b are arbitrary constants with a > 0. Hence expectedutility functions are equivalent up to linear monotonic transformations.They are “cardinal” instead of “ordinal”.

This enables us to write utility function in convenient forms. Forinstance, we could also write the power utility U(x) = x1−γ

1−γ asU(x) = xγ . But in this case, we would not easily see that thelogarithmic utility is a special case of the power.

However, we could solve the problem by writing it as U(x) = xγ/γ. Wewill talk about another reason for further rewriting in the following.



Concave Utility and Risk Aversion

� A utility function on [x , x ] is concave if for any admissible x1, x2and 0 ≤ α ≤ 1,

U(αx1 + (1− α)x2) ≥ αU(x1) + (1− α)U(x2). (2)

A utility function is risk averse if it is concave on [x , x ].� Recall our discussion of convexity in Part II. A utility function on

[x , x ] is convex if for any admissible x1, x2 and 0 ≤ α ≤ 1,

U(αx1 + (1− α)x2) ≤ αU(x1) + (1− α)U(x2). (3)

A utility function is risk seeking/loving if it is convex on [x , x ].� A utility function on [x , x ] is linear if for any admissible x1, x2 and

0 ≤ α ≤ 1,

U(αx1 + (1− α)x2) = αU(x1) + (1− α)U(x2). (4)

A utility function is risk neutral if it is linear on [x , x ].QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 8 / 83


Concave utility and Fair Games

Unwillingness to accept a fair game (asset) with an expected payoff ofzero means that the utility function is concave.To see this, suppose an individual has current wealth W and utilityfunction U. She is offered a fair game with payoff x = x1 withprobability p and payoff x = x2 with probability 1− p. By definition,E [x ] = px1 + (1− p)x2 = 0.The individual’s refusal to accept the fair game means

U(W ) > E [U(W + x)] = pU(W + x1) + (1− p)U(W + x2).

Note thatU(W ) = U(W + px1 + (1− p)x2) = U(p(W + x1) + (1− p)(W + x2)).Hence

U(p(W + x1) + (1− p)(W + x2)) > pU(W + x1) + (1− p)U(W + x2),

which is just the definition of the concavity of U.QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 9 / 83


Concave utility and Fair Games

Conversely, concavity of utility functions also means unwillingness toaccept fair games.Simply apply Jensen’s inequality from statistics. For a concavefunction U and a random payoff ω,

E [U(ω)] < U(E [ω]).

Here,E [U(W + x)] < U(E [W + x ]) = U(W ).



Risk Aversion

A utility function is increasing if U ′(x) > 0. It is strictly concave ifU ′′(x) < 0. The stronger the concavity, or the bend of the function, thehigher the risk aversion.

We can formally measure risk aversion using the Arrow-Prattabsolute risk aversion coefficient:

A(x) = −U ′′(x)

U ′(x). (5)

The negative sign makes the coefficient positive for a risk averse utility– the larger the bend, the higher the coefficient.The denominator normalizes the coefficient such that an equivalentlinear transformation does not change the coefficient.



For the exponential utility,

U(x) = −e−ax , =⇒ A(x) = a.

And for the power utility,

U(x) =x1−γ

1− γ, =⇒ A(x) = γ/x .1

The power utility exhibits lower risk aversion for higher wealth,capturing the fact that people become less risk averse and willing totake more risk as their wealth grows.Still for the quadratic utility,

U(x) = x − bx2, =⇒ A(x) = 2b/(1− 2bx).

For b > 0 and x < 1/(2b), risk aversion increases with wealth.1Clearly, writing U(x) in this form delivers a simple expression and an intuitive

interpretation of γ.QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 12 / 83


Certainty Equivalent

We have established the link between risk aversion and utilityfunctions. How to quantify this risk aversion? The answer is riskpremium, which is defined as the difference between the expectedvalue and certainty equivalent of a random payoff (John W. Pratt 1964).

The certainty equivalent of a random variable x is defined as thecertain (riskfree) payoff C that achieves the same utility level. That is,

U(C) = E [U(C)] = E [U(x)].

Note that C corresponding to x is the same for all equivalent utilityfunctions.Define the risk premium π = E [x ]− C. Then

U(E [x ]− π) = E [U(x)].

Hence π is the amount the individual is willing to pay to avoid risk.QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 13 / 83


Certainty Equivalent

U(x

)

x1

x2E[x]C

For a risk averse investor,

U(C) = E [U(x)] = αU(x1) + (1− α)U(x2)

< U(αx1 + (1− α)x2) = U(E [x ]),

=⇒ C < αx1 + (1− α)x2 = E [x ], as the utility function is increasing.



Decision through Certainty Equivalent

As a utility function is increasing, an investor would prefer a payoff witha higher certainty equivalent value. This provides an alternative tocomparing expected utility values.

Suppose an investor has a logarithmic utility U(x) = ln(x). She wouldinvest either in a riskfree bond which generates a future payoff of $100,or in a risky stock which produces a payoff of $80 and $120 withcorresponding probabilities 50% and 50%.

Her expected utility of investing in the stock isE [U(x)] = 0.5 ln(80) + 0.5 ln(120) = 4.5848, with an accordingcertainty equivalent of C = exp(E [U(x)]) = 97.98 < 100.

Therefore, she would take the bond.



Specification of Utility Function

There are various systematic procedures to assigning utility functionsto investors. Here we discuss two relatively more “quantitative”methods.

(1) Direct Measurement through Certainty Equivalent Value

Select two totala wealth levels A and B.Now I offer you a lottery that awards you A with probability p and B withprobability 1− p. Hence your expected payoff is ep = pA + (1− p)B.

How much certain wealth would you pay to participate in this lottery?aNot wealth increments from, e.g., entering into a bet. The prospect theory,

developed by Daniel Kahneman and Amos Tversky in 1979, builds on theProspect Utility which derives felicity from wealth increments instead of totallevels. It challenges the expected utility theory and earned Daniel Kahnemanthe Nobel Memorial Prize in Economics in 2002.





Say you would pay cp for an expected payoff of ep = pA + (1− p)B.If you are risk averse, then cp < ep,0 < p < 1.The figure for the function c(e):

A (p=1) B (p=0) ep

c p





For any utility function V (x), we know that we can represent it in anequivalent form as U(x) = aV (x) + b,a > 0. We can choose a and bsuch that:

U(A) = A, and U(B) = B.

Then we know that

E [U(x)] = pU(A) + (1− p)U(B) = pA + (1− p)B = ep.

That is ep is the expected utility for the lottery p. Further, ep is theutility of the certain wealth cp.





Take the inverse of c(e) to get e(c), which is just U(x). Flipping theaxes we get the figure:

A (p=1) B (p=0) cp or x

e p o

r U

(x)




(2) Parameterized FamiliesWe assign a parameterized family for the utility function, and what’s leftis to determine the parameter value.We use the following three most commonly used families forillustration.

Exponential: U(x) = −e−ax , a > 0,

Power: U(x) =x1−γ

1− γ, γ > 0,

Quadratic: U(x) = x − bx2, b > 0, x < 1/(2b).

Remember: If your risk tolerance does not change with wealth, chooseExponential; If it increases with wealth, choose Power; If it decreaseswith wealth, choose Quadratic.




(2) Parameterized Families: A Simple Example

We take the Exponential utility U(x) = −e−ax , a > 0. Since it only hasone parameter, one single lottery would be enough to determine a.Say A = 1, B = 3, and p = 0.5. You think the lottery is equivalent to acertain wealth of c = 1.75. Then

−e−1.75a = −0.5e−a − 0.5e−3a,

=⇒ a = 0.52.

You can use Excel Solver to find the value of a.See CurveFitting.xlsx.




(2) Parameterized Families: An Example of Curve Fitting

We take the Power utility U(x) = x1−γ

1−γ , γ > 0.Suppose A = 1, B = 3, and γ = 3. We can calculate the certaintyequivalent values for p = 0,0.2,0.4,0.6,0.8,1. We list the results inthe following table.

p 0 0.2 0.4 0.6 0.8 1cp 3 1.86 1.46 1.25 1.10 1ep 3 2.6 2.2 1.8 1.4 1

Now, we ask the opposite question.We observe the certainty equivalents in the above table and we knowthat the investor has a power utility. But we do not know γ.How to infer it?





First, we transform U(x) into an equivalent V (x) = aU(x) + b, suchthat

V (A) = A and V (B) = B,

=⇒ aA1−γ

1− γ+ b = A and a

B1−γ

1− γ+ b = B,

=⇒ a =(1− γ)(A− B)

A1−γ − B1−γ and b =BA1−γ − AB1−γ

A1−γ − B1−γ .

Now we need V (cp) = ep.





Generally, we find the γ that solves:

minγ

∑p

(V (cp)− ep)2. (6)

See CurveFitting.xlsx. We get γ = 3.0030.



Exercise

HARA Utility FunctionThe class of HARA utility function is defined as

U(x) =1− γγ

(ax

1− γ+ b

)γ, b ≥ 0, (7)

where x ∈{

x∣∣ ax

1−γ + b > 0}

. Pick a, b, and γ (and equivalenttransformations when needed) to obtain the following special cases:

1 Linear or risk neutral: U(x) = x2 Quadratic: U(x) = x − 0.5cx2

3 Exponential: U(x) = −e−ax

4 Power: U(x) = cxγ

5 Logarithmic: U(x) = ln(x)



Exercise - Answer

HARA Utility Function

U(x) =1− γγ

(ax

1− γ+ b

)γ, b ≥ 0, x ∈

{x∣∣∣∣ ax1− γ

+ b > 0}.

1 Linear or risk neutral: U(x) = x

Answer: a = 1, b = 0, and γ = 1.

For a = 1 and b = 0,

U(x) = (1− γ)1−γxγ/γ.

Then

limγ→1−

(1− γ)1−γ = elim

y→0+

ln yy−1 L’Hopital’s rule

========== elim

y→0+

y−1

−y−2= e

limy→0+

−y= 1.



Exercise - Answer


U(x) =1− γγ

(ax

1− γ+ b

)γ, b ≥ 0, x ∈

{x∣∣∣∣ ax1− γ

+ b > 0}.

2 Quadratic: U(x) = x − 0.5cx2

Answer: b = 1/a and γ = 2.

For γ = 2,

U(x) = −0.5(−ax + b)2 = abx − 0.5a2x2 − 0.5b2.

Then c = a2 and b = 1/a give the equivalentU(x) = x − 0.5cx2 − 0.5b2.



Exercise - Answer


U(x) =1− γγ

(ax

1− γ+ b

)γ, b ≥ 0, x ∈

{x∣∣∣∣ ax1− γ

+ b > 0}.

3 Exponential: U(x) = −e−ax

Answer: b = 1 and γ →∞.

For b = 1 and γ →∞,

U(x) = limγ→∞

−(

1 +ax

1− γ

)γ= −e

limγ→∞

ax γ1−γ = −e−ax .



Exercise - Answer


U(x) =1− γγ

(ax

1− γ+ b

)γ, b ≥ 0, x ∈

{x∣∣∣∣ ax1− γ

+ b > 0}.

4 Power: U(x) = cxγ

Answer: b = 0.

For b = 0,

U(x) =(1− γ)1−γaγ

γxγ .

Let c = (1−γ)1−γaγ

γ .



Exercise - Answer


U(x) =1− γγ

(ax

1− γ+ b

)γ, b ≥ 0, x ∈

{x∣∣∣∣ ax1− γ

+ b > 0}.

5 Logarithmic: U(x) = ln(x)

Answer: a = 1, b = 0, and γ → 0.

For a = 1 and b = 0,

U(x) =(1− γ)1−γ

γxγ .

Consider the equivalent form U(x) = (xγ − 1)/γ. The for γ → 0,

U(x) = limγ→0

xγ − 1γ

= ln x .



Discussion on HARA

Now we calculate the absolute risk aversion coefficientA(x) = −U ′′(x)/U ′(x).

1 Linear or risk neutral: U(x) = x → A(x) = 02 Quadratic: U(x) = x − 0.5cx2 → A(x) = c/(1− cx)

3 Exponential: U(x) = −e−ax → A(x) = a4 Power: U(x) = cxγ → A(x) = (1− γ)/x5 Logarithmic: U(x) = ln(x)→ A(x) = 1/x

6 HARA: U(x) = 1−γγ

(ax

1−γ + b)γ→ A(x) = a(1−γ)

ax+b(1−γ) (HyperbolicAbsolute Risk Aversion)

Note that all the above utility functions have one thing in common: TheRisk Tolerance

T (x) ≡ 1A(x)

= Bx + C, (8)

where B and C are constants (including∞). That is, the risk toleranceis a linear function of x .



Discussion on the Definition of Risk Aversion

Arrow-Pratt absolute risk aversion coefficient is defined as

A(x) = −U ′′(x)

U ′(x).

A justification through certainty equivalent is as follows.

Let x be a random payoff with mean E(x) and variance Var(x). Asecond-order expansion near x = E(x) gives

U(x) ≈ U(E(x)) + U ′(E(x))(x − E(x)) +12

U ′′(E(x))(x − E(x))2. (9)

Hence,

E [U(x)] ≈ U(E(x)) +12

U ′′(E(x))Var(x). (10)




Meanwhile, let c be the certainty equivalent of x . c may be regarded tobe close to E(x). Then we can apply the first-order expansion to get

U(c) ≈ U(E(x)) + U ′(E(x))(c − E(x)). (11)

Since U(c) = E [U(x)], we can solve for c using the twoapproximations:

c ≈ E(x) +12

U ′′(E(x))

U ′(E(x))Var(x) = E(x)− 1

2A(E(x))Var(x). (12)

Note the intuitive meaning of A(x) in the above formula: It measuresthe response of c to Var(x). The higher A(x) is, the higher discountshe will impose to E(x) in her personal valuation.




Rearranging the previous equation, the risk premium π is

π = E(x)− c ≈ −12

U ′′(E(x))

U ′(E(x))Var(x) =

12

A(E(x))Var(x). (13)

The higher A(x) is, the higher the risk premium she pays.

However, although concavity of the utility function, U ′′(·), determinesher risk aversion, it cannot sufficiently determine the risk premium sheis willing to pay. The risk premium also depends on her marginal utility,U ′(·). Even if she is very risk averse (−U ′′(·) is large), she may beunwilling to pay a large risk premium if she is poor and her marginalutility is high (U ′(·) is also large).



The Portfolio Problem

Time 0 Time 1Initial Investment Random Future Payoff

� Your total wealth levels at times 0 and 1 are W0 and W1,respectively.� There are two assets available: a riskless bond with a gross return

Rf and a risky stock with a random gross return Rs, which isnormally distributed with mean µ and standard deviation σ.� You decide the amount θb to be invested in the riskless bond and

the amount θs to be invested in the risky stock, such that yourexpected utility E [U(W1)] = E [U(θbRf + θsRs)] is maximized.



The Portfolio Problem

maxθb,θs

E[U(θbRf + θsRs)

], (14)

s.t . θb + θs = W0.

We form the Lagrangian

L = E[U(θbRf + θsRs)

]− λ (θb + θs −W0) , (15)

where λ is the Lagrangian Multiplier.



The Portfolio Problem: Solution

F.O.C.s:∂L

∂θb= E

[U ′(θ∗bRf + θ∗sRs)

]Rf − λ = 0, (16)

∂L

∂θs= E

[U ′(θ∗bRf + θ∗sRs)Rs

]− λ = 0, (17)

∂L

∂λ= θ∗b + θ∗s −W0 = 0. (18)

From (17)−(16), we get

E[U ′(W ∗

1 )(Rs − Rf )]

= 0. (19)

Note that it holds for any risky asset and gives a fundamental assetpricing equation. Let Rs = Xs/Ps, where Xs is the total random payoffat time 1 and Ps is the price of the risky asset at time 0. Then

Ps = E [U ′(W ∗1 )Xs]/(E [U ′(W ∗

1 )]Rf ).



Optimal Portfolio: Comparative Statics

The budget constraint in (18) imposes the following restrictions.

∂θ∗s∂W0

+∂θ∗b∂W0

= 1, (20)

∂θ∗s∂µ

+∂θ∗b∂µ

=∂θ∗s∂σ

+∂θ∗b∂σ

=∂θ∗s∂Rf

+∂θ∗b∂Rf

= 0, (21)

∂θ∗s∂ς

+∂θ∗b∂ς

= 0, (22)

where ς is any parameter of the utility function.Therefore, any parameter other than W0 affects the optimalinvestments in the risky and riskfree assets in exactly oppositedirections.



Optimal Portfolio under Exponential Utility

For the exponential utility U(x) = −e−ax , the F.O.C.s are

∂L

∂θb= aRf e−aRf θ

∗b E [e−aθ∗s Rs ]− λ = 0,

∂L

∂θs= ae−aRf θ

∗b E [Rse−aθ∗s Rs ]− λ = 0,

∂L

∂λ= θ∗b + θ∗s −W0 = 0.

Let f (x , µ, σ) be the density of x ∼ N(µ, σ). We have

E [Rse−aθ∗s Rs ] =

∫ +∞

−∞Rse−aθ∗s Rs f (Rs, µ, σ)dRs

= e−aµθ∗s +0.5a2σ2θ∗2s

∫ +∞

−∞Rsf (Rs, µ− aσ2θ∗s , σ)dRs

= (µ− aσ2θ∗s)e−aµθ∗s +0.5a2σ2θ∗2s .



Optimal Portfolio under Exponential Utility

aRf e−aRf θ∗b e−aµθ∗s +0.5a2σ2θ∗2s = λ, (23)

=⇒ a(µ− aσ2θ∗s)e−aRf θ∗b e−aµθ∗s +0.5a2σ2θ∗2s = λ, (24)

θ∗b + θ∗s = W0. (25)

(24)÷(23), we get

θ∗s =µ− Rf

aσ2 , θ∗b = W0 −µ− Rf

aσ2 . (26)

Very intuitively, an investor increases her position in the risky assetwhen� its risk premium is higher (i.e., a higher µ and/or a lower Rf );� its risk is lower (i.e., a smaller σ);� she has lower absolute risk aversion (i.e., a lower A(x) = a).



Exercise

Note that E [U(x)] = −E [e−ax ] = −e−aE [x ]+0.5a2Var(x). Hence,

max E [U(x)]⇐⇒ max aE [x ]− 0.5a2Var(x).

Using this result to re-derive the optimal portfolio weights.



A Numerical Example

Choose the following baseline parameters: a = 3, µ = 1.12, σ = 0.2,Rf = 1.04, and W0 = 1.

1 2 3 4 5-1

0

1

2

1.05 1.1 1.15 1.2

0

0.5

1

0.1 0.15 0.2 0.25 0.3

-1

0

1

2

1.02 1.04 1.06

0.2

0.4

0.6

0.8



The Effect of Initial Wealth

An interesting observation:

∂θ∗s∂W0

= 0,∂θ∗b∂W0

= 1.

An investor armed with the exponential utility function invests a fixedamount in the risky asset no matter of her initial wealth levels.

0.5 1 1.5

0

0.2

0.4

0.6

0.8

0.5 1 1.5

0

0.5

1



Optimal Portfolio under Quadratic Utility

For the quadratic utility U(x) = x − 0.5cx2, the F.O.C.s are

∂L

∂θb= Rf − cR2

f θ∗b − cµRf θ

∗s − λ = 0, (27)

∂L

∂θs= µ− cµRf θ

∗b − c(σ2 + µ2)θ∗s − λ = 0, (28)

∂L

∂λ= θ∗b + θ∗s −W0 = 0. (29)




θ∗s =µ− Rf

c(σ2 + (µ− Rf )2)− (µ− Rf )Rf

σ2 + (µ− Rf )2 W0, (30)

θ∗b = − µ− Rf

c(σ2 + (µ− Rf )2)+σ2 + µ2 − µRf

σ2 + (µ− Rf )2 W0. (31)

Recall that for quadratic utility, A(x) = c/(1− cx). Hence,c = A/(1 + Ax) = 1/(A−1 + x).

Given x , a higher A corresponds to a higher c. Therefore, we canstudy the effect of risk aversion on the optimal portfolio through c.

Clearly, for either θ∗s or θ∗b, only the first term is affected by c, and onlythe 2nd term is affected by W0.




Consider the realistic case of µ > Rf . We have

1∂θ∗s∂c = − µ−Rf

c2(σ2+(µ−Rf )2)< 0, ∂θ∗b

∂c = −∂θ∗s∂c > 0.

� Intuition: An investor armed with a quadratic utility reduces herinvestment in the risky asset and increases her investment in thesafe asset, when she becomes more risk-averse.

2∂θ∗s∂W0

= − (µ−Rf )Rfσ2+(µ−Rf )2 < 0, ∂θ∗b

∂W0= 1− ∂θ∗s

∂W0> 1.

� This result is a bit counterintuitive. We may expect θ∗s not todecrease for a higher W0.

� Since A(x) = c/(1− cx), an investor armed with a quadratic utilitybecomes more risk averse when she gets wealthier (i.e., a higherx). This effect dominates and diminishes her risky position.



A Numerical Example

Choose the following baseline parameters: c = 0.75, µ = 1.12,σ = 0.2, Rf = 1.04, and W0 = 1.

0.5 0.6 0.7 0.8 0.9 1

-0.5

0

0.5

1

1.5

0.5 1 1.5

W0

-0.5

0

0.5

1

1.5




How about the other three parameters, µ, σ, and Rf ? We rewrite θ∗s as

θ∗s =1

σ2

µ−Rf+ (µ− Rf )

× (1/c − Rf W0). (32)

� Given σ, the left term is maximized when µ− Rf = σ, and itdecreases when µ− Rf moves away from σ. We expect anonmonotonic relation between θ∗s and µ, with a turning point atRf + σ. If the sign of the right term is positive, θ∗s is first increasingand then decreasing w.r.t. µ.� When the sign of the right term is positive, we can easily see thatθ∗s is decreasing w.r.t. σ.� For σ > µ− Rf > 0, the left term is decreasing in Rf . Ignoring the

effect of Rf in the right term , we find that θ∗s is decreasing in Rfwhen the right term is positive.



A Numerical Example

Choose the following baseline parameters: c = 0.75, µ = 1.12,σ = 0.2, Rf = 1.04, and W0 = 1.

1.1 1.2 1.3 1.4 1.50

0.2

0.4

0.6

0.8

1

1.02 1.04 1.06 1.08 1.1

0.2

0.4

0.6

0.8

0.05 0.1 0.15 0.2 0.25

-1

0

1

2




Intuition for µ:� Given W0, we may expect the investor to increase her investment

in the risky asset when the risky asset becomes more attractive –providing a higher expected return.� However, we find a nonmonotonic relation between θ∗s and µ. The

reason is that µ also brings about a wealth effect which in turnimpacts the investor’s risk aversion and exerts an oppositeinfluence on the optimal investments. The ultimate outcome is atradeoff.� Specifically, a higher expected return of the risky asset increases

her wealth level on average, which simultaneously lift her riskaversion. This provides a motivation for her to reduce her riskyasset position. The total effect depends on the detailed parametervalues.



Optimal Portfolio under Power Utility

For the Power utility U(x) = x1−γ

1−γ , we make the followingsimplifications.

We write W1 = θbRf + θsRs = W0(φbRf + φsRs), where φb and φs arethe fractions of wealth invested in the riskfree and risky assets,respectively.

We now approximate2W1 by a lognormal distribution thusW1 ≈W0eφbrf +φs rs , where rf is the constant riskfree interest rate andrs ∼ N(µr , σr ) is the random return of the risky asset.

2In a continuous time model where time runs on through infinitesimal steps insteadof one lumpy period, the approximation will become simple truth. Let’s save this joy foryour further study. Heuristically, the sum of two lognormal random variables islognormally only approximately, that is,wex∆t + (1 − w)ey∆t ≈ 1 + (wx + (1 − w)y)∆t ≈ e(wx+(1−w)y)∆t . Meanwhile,wexdt + (1 − w)eydt = 1 + (wx + (1 − w)y)dt + o(dt) = e(wx+(1−w)y)dt .




The F.O.C.s are

∂L

∂φb= W−γ

0 rf E [e−γφ∗b rf−γφ∗s rs ]− λ = 0,

∂L

∂φs= W−γ

0 E [rse−γφ∗b rf−γφ∗s rs ]− λ = 0,

∂L

∂λ= φ∗b + φ∗s − 1 = 0.




W−γ0 rf e−γrfφ

∗b e−γµrφ∗s +0.5γ2σ2

r φ∗2s = λ, (33)

=⇒ W−γ0 (µr − γσ2

r φ∗s)e−γrfφ

∗b e−γµrφ∗s +0.5γ2σ2

r φ∗2s = λ, (34)

φ∗b + φ∗s = 1. (35)

(34)÷(33), we get

φ∗s =µr − rf

γσ2r, φ∗b = 1− µr − rf

γσ2r. (36)

Very intuitively, an investor increases her position in the risky assetwhen� its risk premium is higher (i.e., a higher µr and/or a lower rf );� its risk is lower (i.e., a smaller σr );� she is less risk averse (i.e., a lower A(x) = γ/x , or a lower γ given

x).QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 53 / 83


A Numerical Example

Choose the following baseline parameters: γ = 3, µr = 0.12, σr = 0.2,rf = 0.04, and W0 = 1.

1 2 3 4 5

-0.5

0

0.5

1

1.5

0.05 0.1 0.15 0.2

0

0.5

1

0.1 0.15 0.2 0.25 0.3

-1

0

1

2

0.02 0.04 0.06

0.2

0.4

0.6

0.8



The Effect of Initial Wealth

An interesting observation:

∂φ∗s∂W0

=∂φ∗b∂W0

= 0.

An investor armed with the power utility function invests a fixed fractionin the risky asset no matter of her initial wealth levels.

0.5 1 1.5

0.4

0.5

0.6

0.5 1 1.5

0.2

0.4

0.6

0.8



Relative Risk Aversion

The above results motivate us to define another measure of riskaversion: the Relative Risk Aversion Coefficient

R(x) = xA(x) = −xU ′′(x)

U ′(x).

According to our previous discussions, we can see that absolute riskaversion indicates how the investor’s dollar amount in the risky assetchanges with changes in initial wealth while relative risk aversionindicates how the investor’s portfolio proportion (or portfolio weight) inthe risky asset changes with changes in initial wealth.



Investment Horizon

We can also say something about investment horizon.� Power: φ∗s = µr−rf

γσ2r

.

For investment horizon of T instead of 1, we simply replace µr − rf with(µr − rf )T and σ2

r with σ2r T .

� Power: φ∗s(T ) = (µr−rf )Tγσ2

r T = µr−rfγσ2

r.

The optimal fraction of wealth in the risky asset does not increase withinvestment horizon. Inconsistent with conventional wisdom?



Expected Utility vs Mean-Variance Analysis

We have examined two approaches to the optimal portfolio choiceproblem, namely, Expected Utility and Mean-Variance Analysis.

Mean-Variance Analysis is intuitive hence popular both in practice andacademia, while Expected Utility Framework is axiomatic hence deeplyrooted in decision theory. How do the two relate to each other?

� Mean-Variance Analysis: Investors use variance to measure risk,and they trade off a higher expected return with a lower variance.� Expected Utility Framework: Investors generally care about higher

moments, like skewness and kurtosis, of returns.� However, under certain assumptions on preferences and asset

returns, both approaches lead to similar investment decisions.



Assumptions on Preferences and Asset Returns

In the one-period setup, we can approximate the utility function as

U(W1) = U(E(W1)) + U ′(E(W1))(W1 − E(W1))

+12

U ′′(E(W1))(W1 − E(W1))2 + ...

+1n!

U(n)(E(W1))(W1 − E(W1))n + ....



Assumptions on Preferences and Asset Returns

An investor’s expected utility depends only on the mean and varianceof her wealth/the portfolio return if� she has the quadratic utility hence all the derivatives of order 3 or

higher are equal to zero,� One undesirable property of quadratic utility: increasing absolute

risk aversion w.r.t. initial wealth.� or the return distribution is normal hence all the derivatives of

order 3 or higher can be expressed by the mean and variance.� Note that two-parameter distributions include gamma, normal,

lognormal, etc. The stable distributions allow a portfolio to have thesame distribution as those of its underlying assets – the so called“additivity”. And the normal is the only stable distribution that hasfinite variance.



Recent Studies on Higher Moments∗

The literature has accumulated empirical evidence on the effect ofhigher moments on asset pricing. For example,� Skewness and Kurtosis:

1 Harvey, Campbell R., and Akhtar Siddique. Conditional skewness in assetpricing tests. Journal of Finance 55.3 (2000): 1263-1295.

2 Conrad, Jennifer, Robert F. Dittmar, and Eric Ghysels. Ex ante skewnessand expected stock returns. Journal of Finance 68.1 (2013): 85-124.

3 Amaya, D., Christoffersen, P., Jacobs, K., and Vasquez, A. (2015). Doesrealized skewness predict the cross-section of equity returns? Journal ofFinancial Economics, 118(1), 135-167.

� Tail Risk:1 Bollerslev, Tim, and Viktor Todorov. Tails, fears, and risk premia. Journal of

Finance 66.6 (2011): 2165-2211.2 Kelly, Bryan, and Hao Jiang. Tail risk and asset prices. Review of Financial

Studies 27.10 (2014): 2841-2871.3 Bollerslev, Tim, Viktor Todorov, and Lai Xu. Tail risk premia and return

predictability. Journal of Financial Economics 118.1 (2015): 113-134.



Higher Order Derivatives of Utility Functions∗

Write [x , y ] to denote a lottery with a 50-50 chance of receiving eitheroutcome x or outcome y . We assume throughout that all individualsprefer more wealth to less. (The first derivative is positive, U ′ < 0.)� Risk aversion:

Consider an individual with an initial wealth W > 0. Let k1 > 0,k2 > 0, and W > k1 + k2. All variables are defined so as tomaintain a strictly positive total wealth.Consider two lotteries, A: [W − k1,W − k2] and B:[W ,W − k1 − k2].An individual is risk averse (U ′′ < 0) if and only if lottery A ispreferred to lottery B for all possible values of W , k1, and k2. Notethat A and B have the same expected final wealth, but B has ahigher variance than A.



Prudence∗

� Prudence:Consider an individual with an initial wealth W > 0. Let k > 0 andω be any zero-mean random variable.Consider two lotteries, A: [W − k ,W + ω] and B: [W ,W − k + ω].Eeckhoudt and Schlesinger (2006) define prudence as apreference for lottery A over lottery B for every arbitrary W , k , andω. They also show how this lottery preference is equivalent to aconvex marginal utility in expected-utility models, U ′′′ > 0.



Temperance∗

� Temperance:Consider an individual with an initial wealth W > 0. Let ω and δ betwo independent zero-mean random variables.Consider two lotteries, A: [W + ω,W + δ] and B: [W ,W + ω + δ].Eeckhoudt and Schlesinger (2006) define temperance as apreference for lottery A over lottery B for every arbitrary W , δ, andω. They also show how this lottery preference is equivalent to aconcave 2nd derivative in expected-utility models, U ′′′′ < 0.3

3For even higher order derivatives, see: Deck, Cary, and Harris Schlesinger.Consistency of higher order risk preferences. Econometrica 82.5 (2014): 1913-1943.



The Capital Asset Pricing Model

Portfolio choice theory provides a foundation for an asset pricingmodel.� Optimal portfolio choices of individual investors represent their

demands for assets.� Aggregate demands of assets are achieved through summing up

individual demands.� Equilibrium asset prices are determined by equating aggregate

demands to asset supplies.The Capital Asset Pricing Model (CAPM), was derived at about thesame time by four individuals: Jack Treynor (1961), William Sharpe4

(1964), John Lintner (1965), and Jan Mossin (1966).

4William Sharpe is a student of Harry Markowitz and shared the 1990 Nobel prizewith Markowitz and Merton Miller.



Assumptions

1 All investors are mean-variance optimizers a la Markowitz.2 All investors have a common time horizon and homogeneous

expectations.3 The riskless asset can be bought or sold in unlimited amounts.4 All investors are price takers, i.e., they cannot influence prices.5 All investors trade without transaction or taxation costs.6 All assets are perfectly divisible and liquid.



Model Setup

There are N ≥ 2 risky assets with no redundant securities.� A finite mean vector µ (N × 1).� And a symmetric positive-definite variance-covariance matrix Σ

(N × N).A portfolio a has a weight vector wa (N × 1).� Mean: µa = µ>wa.� Variance: σ2

a = w>a Σwa.� The covariance between the returns of portfolios a and b:σab = w>a Σwb.



The Mean-Variance Problem

We first find the minimum-variance portfolios in the absence of ariskless asset. The minimum-variance portfolio with expected return µpis the solution w∗ to

minw

12

w>Σw (37)

s.t . µ>w = µp, (38)

1>w = 1. (39)



Solution

We form the Lagrangian

L =12

w>Σw − λ(µ>w − µp)− ζ(1>w − 1).

The F.O.C.s are

∂L∂w

= Σw − λµ− ζ1 = 0, (40)

∂L∂λ

= µ>w − µp = 0, (41)

∂L∂ζ

= 1>w − 1 = 0. (42)



Solution

From (40),w∗ = λΣ−1µ + ζΣ−11. (43)

Take w∗ into the constraints (41) and (42), we determine themultipliers as follows.

λ =µpA− B

∆, ζ =

C − µpB∆

,

A ≡ 1>Σ−11 > 0, B ≡ 1>Σ−1µ,

C ≡ µ>Σ−1µ > 0, ∆ ≡ AC − B2 > 0.

(44)



Solution

A > 0 and C > 0 since Σ−1 is positive definite.∆ ≡ AC − B2 > 0 is from the Cauchy-Schwarz inequality

〈α,β〉2 ≤ 〈α,α〉〈β,β〉,

where equality is obtained iff α = kβ with a constant k .

Note that for a symmetric positive definite matrix, we can always writeit as Σ−1 = H>H, where H is a N × N matrix. Let α = H1 andβ = Hµ. We get

A ≡ 1>Σ−11 = 1>H>H1 = (H1)>(H1) = 〈α,α〉,C ≡ µ>Σ−1µ = µ>H>Hµ = (Hµ)>(Hµ) = 〈β,β〉,

B2 ≡ (1>Σ−1µ)2 = (1>H>Hµ)2 = ((H1)>(Hµ))2 = 〈α,β〉2.

We have assumed that not all assets have the same expected returns,hence µ 6= k1.



Efficient Frontier

The equation of the minimum-variance set is

σ2p = w∗>Σw∗

= w∗>Σ(λΣ−1µ + ζΣ−11)

= λw∗>µ + ζw∗>1= λµp + ζ

=Aµ2

p − 2Bµp + C∆

.

This equation is a parabola. In the mean-standard deviation space, thecurve is a hyperbola.The slope of the curve is

dµp

dσp=

σp∆

Aµp − B.

Since ∆ > 0, the efficient frontier corresponds to µp > B/A.QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 72 / 83


Two-Fund Separation

Replacing the multipliers in (43), we have

w∗ = f + gµp, (45)

where

f =CΣ−11− BΣ−1µ

∆, g =

AΣ−1µ− BΣ−11∆

.

Given any two distinct mean-variance efficient portfolios a and b.

w∗a = f + gµa, w∗b = f + gµb.

Then for any other mean-variance efficient portfolio with expectedreturn µc , it can be constructed by forming a portfolio with a and b, bychoosing weight θ = µc−µb

µa−µbin the first portfolio. To verify, we show that

w∗c = θw∗a + (1− θ)w∗b = θ(f + gµa) + (1− θ)(f + gµb)

= f + g(θ(µa − µb) + µb) = f + gµc .



Beta Representation of Expected Returns

For a portfolio with a weight w in the first asset and 1− w in thesecond asset, the portfolio return mean and standard deviation are

µp = wµ1 + (1− w)µ2,

σp =√

w2σ21 + 2w(1− w)σ1σ2Corr(R1,R2) + (1− w)2σ2

2.

Hence,∂µp

∂σp=∂µp/∂w∂σp/∂w

=µ1 − µ2

σ−1p (wσ2

1 + (1− 2w)σ1σ2Corr(R1,R2)− (1− w)σ22).

At w = 1, the slope simplifies to∂µp

∂σp=

µ1 − µ2

σ1 − σ2Corr(R1,R2)=

µ1 − µ2

σ1(1− Cov(R1,R2)/σ21). (46)




Let the first asset be a mean-variance efficient portfolio with return R1and the second asset any risky asset with return R2. According to thedefinition of the mean-variance frontier, the slope of the curve thatconnects assets 1 and 2 must be tangent to the mean-variance frontierat the point on the curve corresponding to asset 1. The slope of thetangent line of the mean-variance frontier at asset 1 is

∂µp

∂σp=µ1 − µ0

σ1,

where µ0 is the vertical intercept of the tangent line.Let the slope equal to that in (46) and define β ≡ Cov(R1,R2)

σ21

, we have

µ1 − µ0

σ1=

µ1 − µ2

σ1(1− β). (47)




Rearranging terms, we get

µ2 = µ0 + β(µ1 − µ0). (48)

� β is the coefficient of a population regression of R2 on R1.� µ0 depends on the choice of the mean-variance frontier portfolio

(asset 1).� This beta representation is just math of quadratic forms, no

economics has been employed yet.



Adding a Riskfree Asset

Now we have N + 1 assets with the last one being riskfree.

minw

12

w>Σw (49)

s.t . rf + (µ− rf 1)>w = µp. (50)(51)

where rf is the riskfree interest rate.We do not need the constraint 1>w = 1 here since we can support ourrisky asset portfolio through riskless lending/borrowing. Adding ariskfree asset great simplifies our work. Form the Lagrangian

L =12

w>Σw − λ((µ− rf 1)>w − µp + rf ).



Solution

Following similar steps (do them by yourself), we get

w∗ = λΣ−1(µ− rf 1), w∗N+1 = 1− 1>w∗, (52)

λ =µp − rf

C − 2Brf + Ar2f.

Hence, the optimal risky portfolio w∗ is a scalar λ, which depends onµp, times a portfolio vector w = Σ−1(µ− rf 1), which does not dependon µp.The equation of the minimum-variance set is

σ2p = (µp − rf )2(C − 2Brf + Ar2

f )−1.

In mean-standard deviation space the locus is a pair of rays withcommon intercepts at rf and slopes of ±(C − 2Brf + Ar2

f )1/2.



The Tangency Portfolio

Thus, with a risk-free asset, all MV portfolios are combinations of agiven risky asset portfolio, with weights proportional to w and therisk-free asset.

This portfolio of risky assets is called the tangency portfolio and hasweight vector 1

1>ww such that its elements sum to one – a portfolio

weight vector.

With a risk-free asset, all efficient portfolios lie on a line from rf thatruns through this tangent portfolio.




Consider the mean-variance efficient portfolio that consists entirely ofrisky assets. Denote this as asset 1. The relation between theexpected return to any other asset, say asset 2, and the expectedreturn to asset 1 is

µ2 = rf + β(µ1 − rf ). (53)

� β is still the coefficient of a population regression of R2 on R1.� rf does not depend on the choice of the mean-variance frontier

portfolio (asset 1).� This beta representation, like the version without a riskfree asset,

is simply an implication of mean-variance frontier math. It is notderived from economic principles.

The economic content of the CAPM is purely in arguing that, inequilibrium, the market portfolio is the tangency portfolio.



CAPM Equilibrium

For a portfolio p formed with the N + 1 assets with the last one beingriskfree,� Mean: µp = rf + (µ− rf 1)>w .� Variance: σ2

p = w>Σw .An investor i has quadratic utility

U i(x) = x − γ ix2.

Note that the investors could be heterogeneous with different γ i ’s.Investor i solves

maxw

E [U i(rp)] = E [rp]− γ iE [rp]2 − γ iVar(rp)

= µp − γ iµ2p − γ iσ2

p. (54)



CAPM Equilibrium

The F.O.C. is

∂E [U i(rp)]

∂w= (1− 2γ iµp)

∂µp

∂w− 2γ i ∂σ

2p

∂w= (1− 2γ iµp)(µ− rf 1)− 2γ iΣw = 0.

We get the solution

w∗ =1− 2γ iµp

2γ i Σ−1(µ− rf 1), w∗N+1 = 1− 1>w∗, (55)

Note that the optimal portfolio weights are proportional to the tangencyportfolio and the rest is invested in the risky asset.



CAPM Equilibrium

� Since all investors hold combinations of the tangency portfolio andthe risk-free asset, the aggregate demand for each risky assetmust be in proportion to its representation in the tangencyportfolio.� In equilibrium, demand equals supply so that the supply of each

risky asset must be in proportion to its weight in the tangencyportfolio: w∗ = wM , where the weight in the market portfolio wM isthe value of the risky asset divided by the total value of all riskyassets.� Beta Representation: Denote the market portfolio as asset 1. The

relation between the expected return to any other asset, say asset2, and the expected return to asset 1 is

µ2 = rf + β(µM − rf ), (56)

where β is the coefficient of a population regression of R2 on RM .QUANTITATIVE FINANCE SoF, SHUFE June 16, 2020 83 / 83

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Quantitative Finance - PART III - ASSET PRICING