# Risk Static Portfolio Choices

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```					               Static Portfolio Choices (L4)
• Following topics are covered:
– Mean and Variance as Choice Criteria – an example
– Insurance
• Optimal coinsurance
• Optimality of deductible insurance
– 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
• Materials covered in Chapter 5, CWS
• Efficient frontier

Materials from CWS3&5 and EGS3&4
L4: Static Portfolio Choice               1
Using Mean and Variance as Choice Criteria:
An Example
• If the distribution of return offered by assets is jointly normal, then we can
maximize expected simply by selecting the best combination of mean and
variance.
• Assuming the return on an asset is normal distributed with mean E and
variance σ2, we can write the utility function as: U=U(R, E, σ)
• Then the expected utility is:

E (U )   U ( R) f ( R; E;  )dR


• We want to show that the marginal rate of substitution between return and
risk is positive and that the indifference curves are convex.

L4: Static Portfolio Choice                         2
Positive Slope in Indifference Curve

• Converting R to a standard normal variable, Z=(R-E)/σ.
– then f(R;E;σ)=(1/σ)f(Z;0;1)

E (U )   U ( E  Z ) f ( Z ;0;1)dZ


• An indifference curve is defined as the locus of points where the change
in expected utility is zero – dE(U)=0.

dE    U ' ( E  Z )Zf (Z ;0;1)dZ  0
 
d

U ' ( E  Z ) f ( Z ;0;1)dZ


• The denominator must be positive because U’(.) must be positive. The
numerator is negative (why?)

L4: Static Portfolio Choice                  3
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

L4: Static Portfolio Choice                          4
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
by the transaction cost and risk reduction.

L4: Static Portfolio Choice                    5

•   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?

L4: Static Portfolio Choice                 6
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
max H ( )  Eu(w0  P0  (1   ) x)


Eu ( y )
H ' ( )               E[( x  P0 )u ' ( y )]

 2 Eu( y)
H ' ' ( )               E[(x  P0 ) 2 u' ' ( y)]  0
 2
EU would be hump-shaped w. r. t. β. Thus β* is determined by:
Eu ( y )
H ' ( )               E[( x  P0 )u ' ( y )]

=   0

L4: Static Portfolio Choice   7
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
– 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*
cov(x, u ' ( w  x)
where *  ExEu ' ( w 0 x)
0
– Undiversified risk is an alternative form of transaction cost (page 52)

L4: Static Portfolio Choice                      8
Comparative Statics in Coinsurance Problem
• The effect of risk aversion
• 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

L4: Static Portfolio Choice      9
Comparative Statics in Coinsurance Problem

•   Effect of initial wealth:
•   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

 *                    H ' (  * )
The sign of              is same as
w                        w

L4: Static Portfolio Choice                        10
Proof of Proposition 3.3

• The sign of әβ*/ әw is same as әH’(β)/ әw
H ' (  )
 E[( x  P0 )u yy ' ' ( y )]
w
where P0=(1+λ)E(x), y=w0-βP0-(1-β)x, and H’(β)=E[(x                                         -   P
0)u’(y)]
DARA implies -u’ is more concave than u.
E[u ' ( y )]
Note that                        E[( x  P0 )u yy ' ' ( y )]

 E[( x  P0 )u yy ' ' ( y )] >0 at β=
β   *   .   T   h   u   s   p   r   o   v   e   d   .

L4: Static Portfolio Choice                                                         11
Comparative Statics in Coinsurance Problem

(i) decrease if u exhibits constant or increasing absolute risk aversion
(ii) either increase or decrease if u exhibits decreasing absolute risk aversion

 *                 H ' (  *)
Look at      
To evaluate


L4: Static Portfolio Choice                        12
Deductible Insurance

• Deductible provide the best compromise between the
willingness to cover the risk and the limitation of the insurance
• 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)

L4: Static Portfolio Choice             13
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

L4: Static Portfolio Choice                    14
Optimal Investment in Risky Assets
a*  arg max Eu( w  y)

Assume that H=Eu(w+αy). H’(α)=E[u’                                                                                                                                                                                              (       w           +                   α       y               )       *       y       ]   ; H’’(α)=E(u’’*y2)≤0
The optimal α                       *       f   o       l   l       o   w       s       H       ’           (       α           *           )       =           E       [           u           ’           (       w       +               α           *           y           )       *           y       ]

If α*   =   0   ,   t   h   e   n   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               i       s           e       q           u           i           v       a       l   e       n       t           t       o           t       a       k           i   n       g       c   o   i   n   s   u   r   a   n   c   e   β   *   <   1

Other results in proposition 4.1 should also follow.
What can we learn from the above condition?

L4: Static Portfolio Choice                                                                                                                                                                                                                                     15
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 (CRRA), the demand for
stocks is proportional to wealth: a*=kw. More specifically,
we have
*          y        1

w         y  uy2
2
R( w)

• 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)

L4: Static Portfolio Choice                   16
The Effect of Background Risk
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.
~   ~
  arg max Eu(w     y)
**

We want to compare α** with α*, the demand for risky asset when there is no
background risk.                               ~
 *  arg max Eu(w   y)

Assuming v(z)=Eu(z+ε), we have
~
  arg max Ev(w   y)
**


We just need to check if v is more concave than u, utility function
corresponding to y without background risk.

L4: Static Portfolio Choice                    17
v and u

• To show v is more concave than u, we need show
v' ' ( z )    Eu ' ' ( z   )    u' ' ( z)
                          
v' ( z )     Eu ' ( z   )      u' ( z)
for all ε such that Eε=0.
T       h       e       a   b   o   v   e   c   o   n   d   i   t   i   o   n   i   s   e   q       u       i       v       a       l       e       n           t           t   o           s       h   o   w   i   n   g   E   h   (   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.
u' ' ( z)    u' ' ' ' ( z)
I   
.   e   .     ,                                                                           f       o       r           a       l       l           z           .

u' ( z)      u' ' ' ( z)

L4: Static Portfolio Choice                                                                                                     18
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).

L4: Static Portfolio Choice             19
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

L4: Static Portfolio Choice                 20
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

L4: Static Portfolio Choice             21
Diversification in Expected-Utility Model
n     n                n
z  1  x0 (1   ai )   ai xi  1  x0   ai ( xi  x0 )
i 1   i               i 1
A person maximizes the certainty equivalent of final wealth Ez-1/2Avar[z].
n
Ez  1  x0   ai (  i  x0 )
i 1
n       n
Var[ z ]   ai a j  ij
i 1 j 1

Differentiating the certainty equivalent wealth wrt ai and setting it to 0:
n
 i  x0  A a *j  ij  0
j 1

In a matrix form, i  x0  Aa * 

L4: Static Portfolio Choice                22
Interpretation
1 1
The investment in risky asset i: a*        (   x0 )
A
n
The investment in risk-free asset: a0 *  1   a *j
j 1

When returns are independently distributed, we have:
1  i  x0
a i* 
A  ii
• Investment Implication: all investors, whatever their attitude to risk,
should purchase the same portfolio of risky assets.
• Two-fund Separation

L4: Static Portfolio Choice                    23
Frontier Portfolios
• A portfolio is a frontier portfolio if it has the minimum
variance among portfolios that have the same expected rate of
return. A portfolio p is a frontier portfolio if and only if wp, the
N-vector portfolio weights of p, is the solution to the quadratic
program:
1 T
min w Vw
{ w} 2

s.t.,         wT e  E[rp ]
wT 1  1

See HL, page 63-65.

L4: Static Portfolio Choice           24
Frontier in Mean-Variance Space

See HL, page 66 (3.11.2a)

L4: Static Portfolio Choice   25
Minimum Variance Portfolio
• Minimum variance portfolio (mvp): expected return=A/C;
var=1/C
• The covariance of the minimum variance portfolio and any
portfolio (not only those on the frontier) is always equal to the
variance of the rate of return on the minimum variance
portfolio

L4: Static Portfolio Choice              26
Efficient Portfolios
• Those frontier portfolios which have expected rates of return
strictly higher than that of the minimum variance portfolios
• Inefficient portfolios
• Any combination of efficient portfolios will be an efficient
portfolio.
• For any portfolio p on the frontier, expect for mvp, there exists
a unique frontier portfolio, denoted by zc(p), which has a zero
covariance with p. (HL, page 70)

L4: Static Portfolio Choice              27
Portfolios with Risk Free Assets
1 T
min       w Vw
{ w}   2

s.t.    wT e  (1  wT 1)rf  E[rp ]

Solution see HL, page 76-80.

How does it differ from EGS’s derivations?

L4: Static Portfolio Choice   28
Materials covered in Ch5, CWS
• Two-asset portfolios
– Minimum variance portfolio
– Minimum variance opportunity set
– Efficient set
• Efficent set with one risky and one risk-free asset – page 126
• Many assets
– CML

L4: Static Portfolio Choice              29
Technical Note on Comparative Statics
• Comparative statics: how the equilibrium condition changes when an
exogenous variable changes

Assume: g=g(y,α). y is endogenous, i.e., y=y(α), α is exogenous.
y
Deriving comparative statics is to see the sign of    .

g ( y,  )
The FOC of g is              g y ( y,  )  0
y
g 2 ( y,  )
The SOC of g is                     0 if g is concave.
y 2
If g is concave, the optimal y* must satisfy the condition g y ( y,  )  0
When α changes, it will affect y and thus the equilibrium condtion
differentiate g y ( y,  )  0 in both sides, we have
dy *    g y          dy *                                       g y ( y ,  )     dy *
       sign (      )  sign( g y ) =>we need to look at                for
      g yy                                                                    

L4: Static Portfolio Choice                                30
Technical Note on Differentiation
• Leibnitz’s Rule
d b ( )                           db( )            da( ) b ( ) f ( x, )
d a( ) f ( x, )dx  f (b( ), ) d  f (a( ), ) d  a( )  dx
Special case, if b(θ)=b and a(θ)=a, then only last term remains.

L4: Static Portfolio Choice                          31
Example

Assumptions:
(1) x=end of period value before tax, which follows a distribution of f(x);
(2) y = amount promised to debt holders
(3) interest rate = 0
(4) τ = tax rate
(5) k=proportional bankruptcy cost = k*x
The firm picks up appropriate debt payment level y which maximize the firm’s value.
Derive the comparative statics: ∂y*/∂ τ

L4: Static Portfolio Choice                        32
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
– Setup the model

L4: Static Portfolio Choice                       33

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