l2: static portfolio choice1 lecture 2: static portfolio choices we cover the following topics in...
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L2: Static Portfolio Choice 1
Lecture 2: Static Portfolio Choices
• We cover the following topics in this part– Choice I: Insurance
• Optimal insurance with loading
• Optimal coinsurance
• Optimality of deductible insurance
– Choice II: Optimal Investment Portfolio
• Portfolio of single risky and risk-free assets
• The effect of background risk
• Portfolio of multiple risky assets and risk-free assets
L2: Static Portfolio Choice 2
Means in Managing Risks
• Insurance– Risky assets with a full insurance is like investing in a risk-free asset
– Partial insurance is like a combo of risk free assets (full insurance) and risky assets
• Diversified portfolio– Risk-free asset and single risky asset
– Multiple risky assets
– Risk-free asset and multiple risky assets
L2: Static Portfolio Choice 3
Static Portfolio Choice I: Insurance• Maximize an agent’s utility when there is costly or costless hedging contract
available • The case of actuarially fairly priced insurance• Assuming loss follows a distribution of x (where x>=0); premium = P;
Indemnity schedule = I(x)• Insurer reimburses policyholder for the full value of any loss, I(x)=x• When the premium is actuarially fair, P=EI(x)=E(x)• Suppose the insured is risk averse, how much is he willing to pay for the
insurance?– With insurance, his expected utility is u(E(x))– Without insurance, his expected utility is E(u(x))=u(E(x)-П)– Insurance increases the certainty equivalent by П
• As a result, risk-averse agents would take full insurance when insurance prices are actuarially fair
• When the insurance has loading, optimal insurance amount is determined by the transaction cost and risk reduction.
L2: Static Portfolio Choice 4
Insurance Loading
• Suppose the chance of the ship being sunk is ½.
• Insurer’s loading is 10% of the actual value of the policy
• Suppose the amount of insurance purchased is I, P(I)=(I/2)+0.1(I/2)=0.55I
• A square-root utility function
• Questions– What is the expected wealth?
– What is the expected utility?
– What is maximum utility?
L2: Static Portfolio Choice 5
Optimal Coinsurance• Definition: I(x)=βx
• Insurance pricing rule:– P(β)=(1+λ)EI(x)= βP0, where P0 =(1+ λ)Ex
• Random final wealth = w0- βP0-(1- β)x))1(()(max 00 xPwEuH
)](')[()(
)(' 0 yuPxEyEu
H
0)]('')[()(
)('' 202
2
yuPxE
yEuH
EU would be hump-shaped w. r. t. β. Thus β* is determined by:
)](')[()(
)(' 0 yuPxEyEu
H
=0
L2: Static Portfolio Choice 6
Mossin’s Theorem• Full insurance (β*=1) is optimal at an actuarially fair price,
λ=0, while partial coverage (β*<1) is optimal if the premium includes a positive loading, λ>0.– An intuition is that full insurance may be still optimal if the degree of
risk aversion of the policyholder is sufficiently high.
– This is not correct given that risk aversion is a second-order phenomenon. For a very small level of risk, individual behavior toward risk approaches risk neutrality. For risk neutral policyholder, any saving in transaction cost is good.
– Also may have corner solution of β*=0 when λ>0, occurring λ≥λ* where
– Undiversified risk is an alternative form of transaction cost (page 52))('
)(',cov(*
0
0
xwExEu
xwux
L2: Static Portfolio Choice 7
Comparative Statics in Coinsurance Problem
• Proposition 3.2: Consider two utility function u1 and u2 that are increasing and concave, and suppose that u1 is more risk averse than u2 in the sense of Arrow and Pratt (Equation 1.7, page 11). Then, β1*>β2*
– Sounds natural
• Proposition 3.3: An increase in initial wealth will (i) decrease the optimal rate of coinsurance β* if u exhibits decreasing absolute risk aversion (proof see page 54 and the next slide)(ii) increase the optimal rate of coinsurance β* if u exhibits increasing absolute risk aversion(iii) cause no change in optimal rate of coinsurance β* if u exhibits constant absolute risk aversion
• Proposition 3.4: An increase in the premium loading factor λ will cause β* to (i) decrease if u exhibits constant or increasing absolute risk aversion(ii) either increase or decrease if u exhibits decreasing absolute risk aversion
L2: Static Portfolio Choice 8
Proof of Proposition 3.3
• The sign of әβ*/ әw is same as әH’(β)/ әw
)]('')[()('
0 yuPxEw
Hyy
where P0=(1+λ)E(x), y=w0-βP0-(1-β)x, and H’(β)=E[(x-P0)u’(y)] DARA implies -u’ is more concave than u.
Note that )]('')[()]('[
0 yuPxEyuE
yy
)]('')[( 0 yuPxE yy >0 at β= β*. Thus proved.
L2: Static Portfolio Choice 9
Deductible Insurance
• Deductible provide the best compromise between the willingness to cover the risk and the limitation of the insurance deadweight cost.
• Suppose a risk-averse policyholder selects an insurance contract (P, I(.)) with P=(1+λ)EI(x) and I(x) nodecreasing and I(x)≥0 for all x. Then the optimal contract contains a straight deductbile D; that is I(x)=max(0, x-D)
L2: Static Portfolio Choice 10
Static Portfolio Choice II: Diversification
• Investors who consume their entire wealth at the end of the current period• The case containing a risk-free asset and a risky asset• The risk-free rate of return of the bond is r. the return of the stock is a
random variable x• Initial wealth w0
• α is invested in stock• Ending Portfolio Value=(w0- α)(1+r)+ α(1+x)=w0(1+r)+a(x-r)=w+ αy
L2: Static Portfolio Choice 11
Optimal Investment in Risky Assets)(maxarg* ywEua
Assume that H=Eu(w+αy). H’(α)=E[u’(w+ αy )*y]; H’’(α)=E(u’’*y2)≤0 The optimal α* follows H’(α*)=E[u’(w+ α*y )*y] If α*=0, then H’(0)=E[u’(w)*y]=u’(w)*y=0. As u’(w)>0 (increasing utility function), to have H’(0)=0, y=0. This leads to the first part of proposition 4.1 (p66): The optimal investment in the risky asset is positive iff the expected excess return is positive. Note, this is very similar to the optimal coinsurance problem in Ch3 (p50). Investing in risky asset α*>0 is equivalent to taking coinsurance β*<1 Other results in proposition 4.1 should also follow.
What can we learn from the above condition?
L2: Static Portfolio Choice 12
Further Thoughts• As long as there is a positive excess return y, investors should
invest in the stock market– Participation puzzle
• Under constant relative risk aversion, the demand for stocks is proportional to wealth: a*=kw. More specifically, we have
– Equity premium puzzle• Assuming a reasonable level of risk aversion lead to unreasonable shares
of investment in common stocks• Using historical data, μ=6%, and σ=16%. If R=2, the investment in equity
is 117%. Evan when R=10, equity investment would be 23%. (EGS page 66)
• Mehra and Prescott (85)
)(
1*22 wRuw yy
y
L2: Static Portfolio Choice 13
Background Risk
)(maxarg~
** ywEv
One way to explain the surprisingly large demand for stocks in the theoretical model is to recognize that there are other sources of risk on final wealth than the riskiness of assets returns.
We want to compare α** with α*, the demand for risky asset when there is no background risk. Assuming v(z)=Eu(z+ε), we have
We just need to check if v is more concave than u, utility function corresponding to y without background risk.
)(maxarg~~
** ywEu
L2: Static Portfolio Choice 14
v and u
• To show v is more concave than u, we need show
)('
)(''
)('
)(''
)('
)(''
zu
zu
zEu
zEu
zv
zv
for all ε such that Eε=0. The above condition is equivalent to showing Eh(z, ε)≤0, where h(z, ε)=u’’(z+ε)u’(z)-u’’(z)u’(z+ε) A necessary condition is h is concave in ε for all z.
I.e., )('''
)(''''
)('
)(''
zu
zu
zu
zu for all z.
L2: Static Portfolio Choice 15
Conditions regarding Background Risk
Consider the following three statements:
1. any zero-mean background risk reduces the demand for other independent risk
2. for all z, -u’’’’(z)/u’’’(z)>=-u’’(z)/u’(z)
3. absolute risk aversion is decreasing and convex.
Condition 2 is necessary for condition 1 under the assumption that u’’’ is positive. Condition 3 is sufficient for condition 1 and 2.
Power utility function satisfies (3).
L2: Static Portfolio Choice 16
Portfolio of Risky Assets
• Two assets following the same distributions of x1 and x2 that are independently and identically distributed
• Perform expected utility maximization
• If two assets are i.i.d., holding a balanced portfolio is optimal– Home bias
L2: Static Portfolio Choice 17
Diversification in Mean-Variance Model
• There are n risky assets, indexed by i = 1, 2, …, n
• The return of asset i is denoted by xi, whose mean is µi and covariance between returns of assets I and j is σij
• Risk free asset whose return r= x0
L2: Static Portfolio Choice 18
Diversification in Expected-Utility Model
n
iii
n
iii
n
ii xxaxxaaxz
100
10 )(1)1(1
A person maximizes the certainty equivalent of final wealth Ez-1/2Avar[z].
n
iii xaxEz
100 )(1
n
i
n
jijji aazVar
1 1
][
Differentiating the certainty equivalent wealth wrt ai and setting it to 0:
n
jijji aAx
1
*0 0
In a matrix form, *0 Aaxi
L2: Static Portfolio Choice 19
Interpretation
The investment in risky asset i: )(1
* 01 x
Aa
The investment in risk-free asset:
n
jjaa
1
*0 1*
When returns are independently distributed, we have:
ii
ii
x
Aa
0* 1
• Investment Implication: all investors, whatever their attitude to risk, should purchase the same portfolio of risky assets.
• Two-fund Separation
L2: Static Portfolio Choice 20
Technical Note on Comparative Statics• Comparative statics: how the equilibrium condition changes when an
exogenous variable changes
I.e., assuming g=g(y,α). y is endogenous, i.e., y=y(α) while α is exogenous.
Deriving comparative statics is to see the sign of y
.
The FOC of g is 0),(),(
yg
y
ygy
The SOC of g is 0),(
2
2
y
yg if g is concave.
If g is concave, the optimal y* must satisfy the condition 0),( yg y
When α changes, it will affect y and thus the equilibrium condtion differentiate 0),( yg y in both sides, we have
yy
y
g
gdy
*
sign )()*
( ygsigndy
=>we need to look at
gg
L2: Static Portfolio Choice 21
Technical Note on Differentiation
• Leibnitz’s Rule
dxxf
d
daaf
d
dbbfdxxf
d
d b
a
b
a
)(
)(
)(
)(
),()()),((
)()),((),(
Special case, if b(θ)=b and a(θ)=a, then only last term remains. Example: Yu, Lin, Oppenheimer and Chen 2007, on setting up a model
Other examples: comparative statics of option price
L2: Static Portfolio Choice 22
Example
Assumptions (1) x=end of period value before tax, following a distribution of f(x); (2) y = amount promised to debt holders (3) t = interest rate (4) k=proportional bankruptcy cost = k*x Derive the comparative statics: ∂y*/∂t (adapted from Bradley, Jarrell and Kim, 1984)
L2: Static Portfolio Choice 23
Exercises
• EGS, 3.2; 3.3; 4.2
• Set up a model for one of your ongoing projects– Provide a non-technical (i.e., no reference please) introduction of the
paper
– Identify the primary tradeoff in your story
– Setup the model
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