Asset Pricing Chapter VI. Risk Aversion and Investment Decisions, - - PowerPoint PPT Presentation

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions

Asset Pricing

Chapter VI. Risk Aversion and Investment Decisions, Part II: Modern Portfolio Theory June 20, 2006

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions

max

{a1,a2,...,aN} EU(Y0(1 + rf) +

N

i=1 ai(˜

ri − rf)) = max

{w1,w2,...,wN} EU(Y0(1 + rf) +

N

i=1 wiY0(˜

ri − rf)) (1)

max {w1,w2,...,wN} EU n Y0 h (1 + rf ) + XN

i=1 wi (˜

ri − rf ) io = EU {Y0 [1 + ˜ rP]} = EU n ˜ Y1

• (2)

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Backward Induction

More About Utility Functions

Step 1: The consumption-savings decision

C0 + S0 = Y0

Step 2: The Portfolio Problem

N risky assets with (1 − N

i=1 wi)(Y0 − C0)

(w1(Y0 − C0), w2(Y0 − C0), .., wN(Y0 − C0))

Step 3: Tomorrow’s Consumption Choice

U(c1, c2, .....cn) p1θc1θ + .. + pmθcmθ ≤ Yθ. Yθ = (Y0 − C0)

• (1 + rf) +

N

i=1 wi(riθ − rf)

• Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Backward Induction

Backward Induction

U(Yθ) ≡def max

(c1θ,...,cmθ)u (c1θ, ..., cmθ)

s.t. p1θc1θ + .. + pmθcmθ ≤ Yθ. max {w1,w2,...wN} EU( ˜ Y) =

• θ

πθU (Yθ).

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Backward Induction

From Utility on wealth to utility on return: max EU( ˜ Y) = max EU ((Y0 − C0)(1 + ˜ rP)) =def max E ˆ U(˜ rP) "Mean-Variance Utility Function"

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Backward Induction

Taylor series approximation: EU

• ˜

Y

• = U
• E
• ˜

Y

• + U′

E

• ˜

Y ˜ Y − E

• ˜

Y

• + 1

2U′′ E

• ˜

Y ˜ Y − E

• ˜

Y 2 + H3 (3) Computing expected utility using this approximation: EU( ˜ Y) = U

• E
• ˜

Y

• + U′

E

• ˜

Y E( ˜ Y) − E

• ˜

Y

• =0

+ 1 2U′′ E

• ˜

Y

• E
• ˜

Y − E

• ˜

Y 2

• =σ2(˜

w)

+EH3 = U

• E
• ˜

Y

• + 1

2U′′ E

• ˜

Y

• σ2

˜ Y

• + EH3

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

Description of the Opportunity Set in the Mean-Variance Space: The Gains form Diversification and the Efficient Frontier

" The main idea of this section is the following: The expected return to a portfolio is the weighted average of the expected returns of the assets composing the portfolio. The same result is not generally true for the variance: the variance of a portfolio is generally smaller than the weighted average of the variances

• f individual asset returns corresponding to this portfolio.

Therein lies the gain from diversification".

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

The typical investor likes expected return µR and dislikes standard deviation σR Recall that an asset (or portfolio) A is said to mean-variance dominate an asset (or portfolio B) if µA ≥ µB while σA ≥ σB Define the efficient frontier as the locus of all non-dominated portfolios in the mean-standard deviation space. By definition, no ("rational") mean-variance investor would choose to hold a portfolio not located on the efficient frontier.

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

The Efficient Frontier: Two perfect correlated risky assets

σR = w1σ1 + (1 − w1)σ2. µR = ¯ r1 + ¯ r2 − ¯ r1 σ2 − σ1 (σR − σ1) ,

s1 s sR s2 m r1 mR

– –

r2

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

The Efficient Frontier: Two imperfectly correlated risky assets

sR mR B A C D Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

The Efficient Frontier: Two perfectly negative correlated risky assets

s m s1 s2 r1

– –

r2

–

r2 s1 + s2 s2 r1

–

s1 + s2 s1 +

–

r2 s1 + s2 s2 r1

–

s1 + s2 s1 + sR + s1 + s2 r1 – r2

– – –

r2 s1 + s2 s2 r1

–

s1 + s2 s1 + sR – s1 + s2 r1 – r2

– –

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

One Risky and One Risk Free Asset

s m s2 r1

– –

r2 s2 r2 – r1

– –

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

One Risk Free Asset and "N" Risky Assets

s m rf T F E Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions Two perfect correlated risky assets Two imperfectly correlated risky assets Two perfectly negative correlated risky assets One Risky and One Risk Free Asset One Risk Free Asset and "N" Risky Assets Portfolio optimization problem

The portfolio optimization problem:

(QP) min

wi′s

• i
• j

wiwjσij s.t.

i

wi¯ ri = µ

• i

wi = 1

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions

A Separation Theorem

The Optimal Portfolios of Two Differently Risk Aversion Investors

s m rf T Agent 1 Agent 2 Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions

6.5 Conclusions

MPT: A normative interpretation Spelling out the information requirements for portfolio analysis Using historical returns for asset allocation purposes Static vs. Dynamic portfolio allocation

Asset Pricing

6.1 Introduction 6.2 More About Utility Functions 6.3 Opportunity Set in the Mean-Variance Space 6.4 The Optimal Portfolio: A Separation Theorem 6.5 Conclusions

Key concepts

Backward induction; indirect utility Gains from diversification Efficient Frontier Optimal Portfolio Two fund or separation theorem

Asset Pricing