Portfolio Choice Notes 1

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Ec 310: Portfolio Choice. Introduction. Portfolio Choice Plan. Two risky assets. Many risky assets. Riskless rate of return. Market equilibrium ...
Ec 310: Portfolio Choice Introduction

Portfolio Choice Plan

Two risky assets Many risky assets Riskless rate of return Market equilibrium

Ec 310: Portfolio Choice Introduction

Portfolio Choice: Decisions made uncertainty, where the uncertainty matters Two Steps: 1

Predict the distribution of returns for assets over the period of interest

2

Use these distributions of returns to make a portfolio choice

Ec 310: Portfolio Choice Introduction

1

Distribution of returns Prediction: Single number, e.g. HD will go up 1% next year Distribution, e.g. HD will go up between .5% - 2.5% with lower returns more likely according to: r .005 .01 .015 .02 .025

P(RHD = r) .3 .3 .2 .1 .1

RHD = random variable denoting per dollar return on HD

Ec 310: Portfolio Choice Two risky assets

Two Asset Problem Suppose you have $1000 to invest. How to split between two risky investment assets? Notation: Ri = return on asset i (r.v.) = amount earned on $1 in asset i e.g. Invest in Asset #1:

Return = 1000R1

Invest in Asset #2:

Return = 1000R2

Ec 310: Portfolio Choice Two risky assets

Notation Portfolio Choice: pi = fraction of $1000 invested in asset i p1 + p2 = 1 Traits: Expected Returns: E(R1 ) = µ1 , E(R2 ) = µ2 Variances: Var(R1 ) = σ21 , Var(R2 ) = σ22 Covariance: Cov(R1 , R2 ) = σ12 Correlation: Corr(R1 , R2 ) = ρ12 Risky Assets =⇒ Var(Ri ) > 0

 =

σ12 σ1 σ2



Ec 310: Portfolio Choice Two risky assets

Portfolio Return: Given shares p1 and p2 : (1000p1 )R1 + (1000p2 )R2 = 1000 (p1 R1 + p2 R2 ) | {z } Rp

Rp = Portfolio Return Traits: µp = E(Rp ) = p1 µ1 + p2 µ2 σ2p = Var(Rp ) = p21 σ21 + p22 σ22 + 2p1 p2 σ12

Ec 310: Portfolio Choice Two risky assets

Preferences (Mean-Variance)

Investor prefers:

• higher E(Rp ) • lower Var(Rp )

=⇒ Mean - Variance trade-off e.g. $1000 preferred to coin flip for $0 and $2000.

Ec 310: Portfolio Choice Two risky assets

Example. GM & MSFT E(RGM ) = .1112

E(RMSFT ) = .4082

Var(RGM ) = .0958

Var(RMSFT ) = .2772

SD(RGM ) = .3096

SD(RMSFT ) = .5265

Cov(RGM , RMSFT ) = .04337

Rp = p1 RGM + p2 RMSFT e.g. p1 = .25, p2 = .75 µp = (.25)(.1112) + (.75)(.4082) = .3340 q σp = (.25)2 (.0958) + (.75)2 (.2772) + 2(.25)(.75)(.04337) √ = .1782 = .4221

Ec 310: Portfolio Choice Two risky assets

Feasible (µp , σp ) points Describes mean and standard deviation of all possible portfolios (see twoassets.xls)

Ec 310: Portfolio Choice Two risky assets

Minimum Variance The portfolio with the smallest variance/standard deviation is called the minimum variance portfolio. min

06p1 61

FOC: 0 =

σ2p = p21 σ21 + (1 − p1 )2 σ22 + 2p1 (1 − p1 )σ12

d 2 dp1 σp

2 MV 2 MV = 2pMV 1 σ1 − 2(1 − p1 )σ2 + 2(1 − 2p1 )σ12

=⇒

2 MV 2 MV 2 2pMV 1 σ1 + 2p1 σ2 − 4p1 σ12 = 2σ2 − 2σ12

=⇒

2 2 2 pMV 1 2(σ1 + σ2 − 2σ12 ) = 2(σ2 − σ12 )

Solution: MV

p

=

MV (pMV 1 , p2 )

 =

σ21 − σ12 σ22 − σ12 , σ21 + σ22 − 2σ12 σ21 + σ22 − 2σ12



Ec 310: Portfolio Choice Two risky assets

Def n : Portfolio p dominates portfolio pˆ if E(Rp ) > E(Rpˆ )

SD(Rp ) 6 SD(Rpˆ )

and at least one of these inequalities is strict.

µp

p

pˆ (0, 0)

σp

Ec 310: Portfolio Choice Two risky assets

Efficiency

Use dominance to eliminate portfolios. Def n : The set of undominated feasible points is called the efficient frontier and corresponding portfolios are efficient portfolios. e.g. Dominated: p1 ∈ (.817, 1] Efficient: p1 ∈ [0, .817]

(GM - MV segment)

(MV - MSFT segment)

Ec 310: Portfolio Choice Two risky assets

Backward Bending Result: If σ12 < min{σ21 , σ22 }, then for some value of p1 ∈ (0, 1), then SD(Rp ) < min{SD(R1 ), SD(R2 )}

(i.e., σp < min{σ1 , σ2 })

 Why? Suppose σ12 < σ22 6 σ21 . d 2 σ = 2(σ12 − σ22 ) < 0 dp1 p p1 =0

Ec 310: Portfolio Choice Two risky assets

Diversification Consider the following extreme special case. Two stocks have identical traits: µ 1 = µ2 σ1 = σ2 Would you prefer a portfolio to A) Invest $1000 in Asset #1 OR B) Invest $500 in Asset #1 and $500 in Asset #2 (Or does it matter?)

?

Ec 310: Portfolio Choice Two risky assets

A) Invest $1000 in Asset #1 Per dollar return R1 with E(R1 ) = µ1 and Var(R1 ) = σ21 So total return is T1 = $1000R1 with E(T1 ) = 1000µ1 and Var(T1 ) = (1000)2 σ21 B) Invest $500 in Asset #1 and $500 in Asset #2 Per dollar return Rp = .5R1 + .5R2 with E(Rp ) = .5µ1 + .5µ2 = µ1 , σ12 σ1 σ2 σ1 σ2 = .5(1 + ρ12 )σ21

Var(Rp ) = (.5)2 σ21 + (.5)2 σ22 + 2(.5)2 = (.25 + .25 + .5ρ12 )σ21

So total return is Tp =  $1000R  p with E(Tp ) = 1000µ1 and 1+ρ12 2 2 Var(Tp ) = (1000) σ1 2

Ec 310: Portfolio Choice Two risky assets

Diversification Example Notes: Conclusion: E(T1 ) = E(Tp ) but Var(T1 ) > Var(Tp ) Moreover, Var(T1 ) > Var(Tp ) for all ρ12 < 1 Even positive correlation (ρ12 ) leads to improvement via diversification Conclusion also does not depend on 50/50 split; (Exercise: Make the same comparison for a portfolio p = (p1 , p2 ) with 0 < p1 < 1) ρ12 = 1: no improvement; ρ12 = 0: variance reduced by half ρ = −1: Var(Rp ) = Var(Tp ) = 0 !!

(Riskless asset)