One-Period Portfolio Choice Prof. Fernando Chague

2016

Definitions and Setup

There are i = [1, ..., n] risky assets Each cost pi at T = 0 but entitles the owner with a random payoff I I

˜i = R

x˜i pi

is the (gross) return ˜ i = x˜i −pi is the rate of return ˜ri = 1 − R pi

There is one risk-free asset with certain return at T = 1: Rf h i ˜ i − Rf Risk premium of a risky asset is E R

One-Period Problem The allocation problem: max

E [u (w ˜ )]

[θ1 ,...,θn ] n

s.t.

∑ θi pi = w0

i=1

n

s.t. w ˜ = y˜ + ∑ θi x˜i i=1

where: I I I

θi is the quantity of shares of asset i purchased w0 is the initial wealth y˜ is a source of random income at T = 1 (i.e. labor income)

Remarks: I I

No short-selling constraints No solvency constraints,w ˜ >0

One-Period Problem The allocation can be expressed in terms of the “volume” (of consumption good) invested φi = θi pi : max

E [u (w ˜ )]

[φ1 ,...,φn ] n

s.t.

∑ φi = w0

i=1

n

˜i s.t. w ˜ = y˜ + ∑ φi R i=1

or in portfolio weights πi = θi pi /w0 = φi /w0 : max

E [u (w ˜ )]

[π1 ,...,πn ] n

s.t.

∑ πi = 1

i=1

n

˜i s.t. w ˜ = y˜ + w0 ∑ πi R i=1

First-order Condition The allocation problem can be solved using the Lagrangean: !# " n # " n max [θ1 ,...,θn ,γ]

L = E u y˜ + ∑ θi x˜i

−γ

i=1

First-order condition:   E u 0 (w ˜ ) x˜i − γpi

∑ θi pi − w0

i=1

= 0

n

(abusing notation: w ˜ = y˜ + ∑i=1 θi∗ x˜i ) Which can be rewritten as:  0  u (w ˜) E x˜i = pi γ

First-order Condition

Most asset-pricing formulas are variations on the first-order condition:  u 0 (w ˜) E x˜i = pi γ 

This can be rewritten as: I I

I

E

h

˜) ˜ u 0 (w γ Ri

i

= 1, if pi > 0 (gross returns) i h  ˜i − R ˜ j = 0, if pi > 0 and pj > 0 (long-short portfolio, E u 0 (w ˜) R zero h cost portfolio,iexcess return) ˜ i − Rf ˜) R = 0, if pi > 0 and ∃ Rf E u 0 (w

Stochastic Discount Factor

Definition: A Stochastic Discount Factor (SDF) is any random variable m ˜ such that

E [m ˜ x˜i ] = pi

∀i

The proof of the existence of a SDF is also called the Fundamental Theorem of Asset Pricing

Stochastic Discount Factor If there are only k states of the nature ω1 , ..., ωk , then: k

∑ m˜ (ωj ) x˜i probj = pi

j=1

where probj is the probability of state j. Arrow-Debreu Security: I I I

Payoff: Pays 1 unit of consumption at state ωj and 0 units otherwise Price: qj If a SDF exists, then qj = m ˜ (ωj ) probj and m ˜ (ωj ) = qj /probj

m is the price of a unit of the consumption good in each state per unit of probability Other names for SDF: State Price Density, Pricing Kernel. Statistically, m ˜ is the Radon-Nikodym derivative of the set function that assigns prices to events (set of states) with respect to the probability of events.

Stochastic Discount Factor

A great portion of the asset-pricing literature is dedicated to finding SDFs Intersection with macroeconomics: impose economic structure and find equilibrium ˜ ) /γ We already imposed one such structure: m ˜ = u 0 (w I

From risk-aversion we have ∂ m/∂ ˜ w ˜ < 0: investors discount more (less) states with high (low) wealth.

Let’s solve for γ. For that, consider now the problem including c0 :

Stochastic Discount Factor The allocation problem including c0 : max

E [v (c0 , c˜1 )]

[θ1 ,...,θn ,c0 ]

n

n

s.t. c0 + ∑ θi pi = w0 and c˜1 = y˜ + ∑ θi x˜i i=1

i=1

The FOCs are: 

 ∂ E v (c0 , c˜1 ) = γ ∂ c0   ∂ E v (c0 , c˜1 ) x˜i = γpi ∀i ∂ c1 This implies a SDF: ∂ ˜1 ) ∂ c1 v (c0 , c

m ˜= E

h

∂ ˜1 ) ∂ c0 v (c0 , c

i

Stochastic Discount Factor Assuming time-additive utility v (c0 , c1 ) = u0 (c0 ) + u1 (c1 ) then:   E u10 (˜ c1 ) x˜i = pi u00 (c0 ) I

The expected marginal gain must be equal the marginal cost of saving at t = 0

If we simplify further: u0 (c0 ) = u (c0 ) and u1 (c1 ) = δ u (c1 ) then  0  δ u (˜ c1 ) E x˜i = pi u 0 (c0 ) This is the standard Consumption CAPM SDF: m ˜=

δ u 0 (˜ c1 ) u 0 (c0 )

Stochastic Discount Factor and Returns If pi > 0 then:

h i E m ˜ R˜i = 1

Note that by linearity of expectations and portfolio returns ˜ p = ∑ni=1 πi R ˜ i this is also true for portfolios (note πi is not random) R h i E m ˜ R˜P =

n

∑ πi = 1

i=1

Similarly, for zero-cost long-short portfolio it equals zero: i h  E m ˜ R˜i − R˜j = 0

Stochastic Discount Factor and Returns We can also relate SDF with risk-premia: h i 1 = E m ˜ R˜i   h i = cov m, ˜ R˜i + E [m] ˜ E R˜i Provided that E [m] ˜ > 0: h i E R˜i −

1 E [m] ˜

= −

  cov m, ˜ R˜i E [m] ˜

If a risk-free asset exists, then E [mR ˜ f ] = 1 implies E [m] ˜ Rf = 1 and: h i   E R˜i − Rf = −Rf cov m, ˜ R˜i

Stochastic Discount Factor and Returns

h i   E R˜i − Rf = −Rf cov m, ˜ R˜i Remarks: Risk premia has to do with covariances! I I I

“you discount more assets that pay off when wealth is high” “you dislike assets that covary with your wealth” “you require a higher risk-premium for assets that covary a lot with the SDF”

Likewise: I I I

“you discount less assets that pay off when wealth is low” “you like assets that offer you protection when wealth is lower” ...

Portfolio Allocation (Two Assets)
˜ ∼ µ, σ 2 and Rf . Assume no y˜ . R ˜ + φf Rf Allocation problem reduces to finding [φ , φf ]. Now w ˜ = φR Since w0 = φ + φf , set φf = w0 + φ , and the allocation problem simplifies to: h  i ˜ + (w0 − φ ) Rf max E u Rφ φ

FOC:

h  i ˜ − Rf = 0 E u 0 (w ˜) R

Remarks (when E [R] − Rf > 0 and risk aversion): I I

h i ˜ − Rf > 0 φf = w0 is never optimal, since u 0 (w ˜ )E R In fact, it can be shown that φ ∗ > 0 (this is only true when n = 1!)