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

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

Jan 24, 2024 524 likes •670 views

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

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SLIDE 1

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

Asset Pricing

Chapter V. Risk Aversion and Investment Decisions, Part I June 20, 2006

Asset Pricing

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SLIDE 2

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results The Canonical Portfolio Problem

The various problems considered in this chapter (and the next)

max

a

EU( ˜ Y1) = max EU (Y0 (1 + rf) + a (˜ r − rf)) , (1) Consider first an agent solving the following two period consumption-savings problem: max

s E{U(Y0 − s) + δU(s ˜

R)}, s.t. Y0 ≥ s ≥ 0 (2) max

{a,s} U(Y0 − s) + δEU(s(1 + rf) + a(˜

r − rf)), (3)

Asset Pricing

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SLIDE 3

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results The Canonical Portfolio Problem

max

a

EU( ˜ Y1) = max EU (Y0 (1 + rf) + a (˜ r − rf)) , (4) First order condition (FOC): E

  • U′ (Y0 (1 + rf) + a (˜

r − rf)) (˜ r − rf)

  • = 0

(5) Theorem (Theorem 5.1:) Assume U

′( ) > 0, and U′′( ) < 0 and let ˆ

a denote the solution to problem (1). Then ˆ a > 0 ⇔ E˜ r > rf ˆ a = 0 ⇔ E˜ r = rf ˆ a < 0 ⇔ E˜ r < rf

Asset Pricing

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SLIDE 4

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results The Canonical Portfolio Problem

Theorem (Proof of Theorem 5.1:)

Define W (a) = E {U (Y0 (1 + rf) + a (˜ r − rf))}. The FOC (5) can then be written W ′ (a) = E [U′ (Y0 (1 + rf) + a (˜ r − rf)) (˜ r − rf)] = 0 . By risk aversion (U′′ < 0), W ′′ (a) = E

  • U′′ (Y0 (1 + rf) + a (˜

r − rf)) (˜ r − rf)2 < 0, that is, W ′(a) is everywhere decreasing. It follows that ˆ a will be positive if and only if W ′ (0) = U′ (Y0 (1 + rf)) E (˜ r − rf) > 0 (since then a will have to be increased from the value of 0 to achieve equality in the FOC). Since U′ is always strictly positive, this implies ˆ a > 0 if and only if E (˜ r − rf) > 0. The other assertion follows similarly.

Asset Pricing

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SLIDE 5

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results The Canonical Portfolio Problem

U(Y) = ln Y a Y0 = −(1 + rf)[E˜ r − rf] (r1 − rf)(r2 − rf) > 0. (6)

Asset Pricing

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5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

Theorem (5.2:) Suppose, for all wealth levels Y, R1

A(Y) > R2 A(Y) where Ri A (Y)

is the measure of absolute risk aversion of investor i, i = 1, 2. Then ˆ a1(Y) < ˆ a2(Y) Theorem ( 5.3:) Suppose, for all wealth levels Y > 0, R1

R(Y) > R2 R(Y) where

Ri

R(Y) is the measure of relative risk aversion of investor

i, i = 1, 2. Then ˆ a1(Y) < ˆ a2(Y).

Asset Pricing

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SLIDE 7

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

Theorem (5.4 (Arrow, 1971)) Let ˆ a = ˆ a (Y0) be the solution to problem (1) above; then: (i) R′

A(Y)

< 0 ⇔ ˆ a′(Y0) > 0 (ii) R′

A(Y)

= 0 ⇔ ˆ a′(Y0) = 0 (iii) R′

A(Y)

> 0 ⇔ ˆ a′(Y0) < 0.

Asset Pricing

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SLIDE 8

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

η(Y, ˆ a) = dˆ

a/ˆ a dY/Y = Y ˆ a dˆ a dY

Theorem (5.5 (Arrow, 1971):) If, for all wealth levels Y, (i) R′

R(Y)

= 0 (CRRA) then η = 1 (ii) R′

R(Y)

< 0 (DRRA) then η > 1 (iii) R′

R(Y)

> 0 (IRRA) then η < 1

Asset Pricing

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SLIDE 9

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

Theorem (5.6 (Cass and Stiglitz, 1970):) Let the vector     ˆ a1 (Y0) . . ˆ aJ (Y0)     denote the amount optimally invested in the J risky assets if the wealth level is Y0. Then     ˆ a1 (Y0) . . ˆ aJ (Y0)     =     a1 . . aJ     f (Y0) (for some arbitrary function f(·)) if and only if either (i) U′(Y0) = (θY0 + κ)∆ or (ii) U′(Y0) = ξe−vY0

Asset Pricing

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5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

max

s E{U(Y0 − s) + δU(s ˜

R)}, s.t. Y0 ≥ s ≥ 0 (7) U′(Y0 − s) = δE{U′(s ˜ R)˜ R} (8) Theorem (5.7 (Rothschild and Stiglitz,1971):) Let ˜ RA , ˜ RB be two return distributions with identical means such that ˜ RA SSD ˜ RB , and let sA and sB be, respectively, the savings out of Y0 corresponding to the return distributions ˜ RA and ˜ RB. If R′

R(Y) ≤ 0 and RR(Y) > 1, then sA < sB ;

If R′

R(Y) ≥ 0 and RR(Y) < 1, then sA > sB .

Asset Pricing

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5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

P(c) = −U′′′(c)

U′′(c)

P(c)c = −cU′′′(c)

U′′(c)

Theorem (5.8) Let ˜ RA, ˜ RB be two return distributions such that ˜ RA SSD ˜ RB, and let sA and sB be, respectively, the savings out of Y0 corresponding to the return distributions ˜ RA and ˜

  • RB. Then,

sA ≥ sB iff c P(c) ≤ 2, and conversely, sA < sB iff c P(c) > 2

Asset Pricing

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5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

max

{a,s} U(Y0 − s) + δEU(s(1 + rf) + a(˜

r − rf)), (9) Assume CRRA s : (Y0 − s)−γ(−1) + δE

  • [s(1 + rf) + a(˜

r − rf)]−γ(1 + rf)

  • = 0

a : E

  • (s(1 + rf) + a(˜

r − rf))−γ(˜ r − rf)

  • = 0

Solution: a/s independent of s

Asset Pricing

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SLIDE 13

5.2 Risk Aversion and Portfolio Allocation; Risk Free vs. Risky Assets 5.3 Portfolio Composition, Risk Aversion and Wealth 5.4 Risk Aversion and Risky Portfolio Composition 5.5 Risk Aversion and Saving Behavior 5.6 Key Concepts and Results

Increasing, decreasing, constant abosolute/relative risk aversion and their effects on portfolio composition ( risk free asset vs. Risky portfolio) Risk aversion and the composition of the optimal risky portfolio (Cass-Stiglitz) Prudence and Savings behavior

Asset Pricing