CHAPTER 6

Risk Aversion and Capital Allocation to Risky Assets

INVESTMENTS | BODIE, KANE, MARCUS

McGraw-Hill/Irwin Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved. Allocation to Risky Assets

• will avoid risk unless there is a reward.

• The utility model allows optimal allocation between a risky portfolio and a risk-free asset.

INVESTMENTS | BODIE, KANE, MARCUS 6-2 Risk and Risk Aversion

Speculation Gamble Taking considerable Bet on an uncertain risk for a outcome for enjoyment commensurate gain • Parties (should) assign • Parties have the same probabilities heterogeneous to the possible expectations outcomes • So there are trades • Gambling happens among rational parties because it is fun!

INVESTMENTS | BODIE, KANE, MARCUS Let’s play a game

• I will toss a coin and pay you some money X if heads and nothing if tails • How much are you willing to pay to play this game? – For X = $1 – For X = $2 – For X = $10 – For larger X?

INVESTMENTS | BODIE, KANE, MARCUS 6-4 Let’s flip the game

• I will toss a coin and you pay me some money X if heads and nothing if tails • How much are you asking me to play this game? – For X = $1 – For X = $2 – For X = $10 – For larger X?

INVESTMENTS | BODIE, KANE, MARCUS 6-5 Risk Aversion and Utility Values

• Investors are willing to consider: – risk-free assets – speculative positions with positive risk premia • Investors will reject fair games or worse • Portfolio attractiveness increases with and decreases with risk. • What happens when return increases with risk?

INVESTMENTS | BODIE, KANE, MARCUS 6-6 Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%)

How to compare? Each portfolio receives a utility score to assess the ’s risk/return trade off

INVESTMENTS | BODIE, KANE, MARCUS 1-7 A Utility Function 1 U  Er A 2 2 푈 = utility or welfare (a measure of happiness) 퐸[푟] = expected return on the asset or portfolio 퐴 = coefficient of risk aversion 휎2 = variance of returns ½ = a scaling factor (we’ll see later why ½ turn useful)

There are other utility functions out there: must increase with E[r] and decrease with 휎2

INVESTMENTS | BODIE, KANE, MARCUS 6-8 A Utility Function – Meaning of A 1 U  Er A 2 2 A = coefficient of risk aversion. Interpretation: A>0 - Risk Averse. Penalizes risk. Will want a larger risk premium for riskier investments. A=0 - Risk Neutral. A pure trader, only concerned about expectation. Indifferent to a fair game. A<0 - Risk Lover. Adjusts utility up for risk because enjoys the risk. A gambler, bored with risk-free, will prefer for riskier investments.

INVESTMENTS | BODIE, KANE, MARCUS 6-9 Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion

1 U  Er A 2 2

Three investors with A= 2.0, 3.5 and 5.0 Q. What portfolio will each choose?

INVESTMENTS | BODIE, KANE, MARCUS 1-10 Risk-Return Trade-off

INVESTMENTS | BODIE, KANE, MARCUS 6-11 Mean-Variance (M-V) Criterion

• Portfolio A dominates portfolio B if:

• And ErA   ErB 

 A   B Q. How do you find a family of portfolios you are indifferent to?

INVESTMENTS | BODIE, KANE, MARCUS 6-12 Utility Indifference Curve

INVESTMENTS | BODIE, KANE, MARCUS 6-13 How Do You Estimate Risk Aversion?

• Use questionnaires • Observe individuals’ decisions when confronted with risk • Observe how much people are willing to pay to avoid risk • Use common sense

INVESTMENTS | BODIE, KANE, MARCUS 6-14 Capital Allocation Across Risky and Risk-Free Portfolios Asset Allocation: Controlling Risk: • Is a very important •Simplest way: adjust part of portfolio the fraction of the construction. portfolio invested in risk-free assets • Refers to the choice versus the portion among broad asset invested in the risky classes. assets

INVESTMENTS | BODIE, KANE, MARCUS 6-15 Basic Asset Allocation

Portfolio Total Market Value $300,000 Risk-free money market fund $90,000 Equities $113,400 Bonds (-term) $96,600 Total risk assets $210,000 $113,400 $96,600 w   0.54 w   0.46 E $210,000 B $210,00 These weights are within the risky portfolio Q. What is the risk-free vs risky composition?

INVESTMENTS | BODIE, KANE, MARCUS 6-16 Basic Asset Allocation

Let y = weight of the risky portfolio P, in the complete portfolio; (1-y) = weight of risk-free assets:

$210,000 $90,000 y   0.7 1 y   0.3 $300,000 $300,000

$113,400 $96,600 E :  0.378 B :  0.322 $300,000 $300,000 These weights are within the entire portfolio

INVESTMENTS | BODIE, KANE, MARCUS 6-17 The Risk-Free Asset

• Only the government can issue default-free bonds (caveats). – Risk-free in real terms only if price indexed and maturity equal to investor’s holding period. • T-bills viewed as “the” risk-free asset • Money market funds also considered risk-free in practice (caveat, remember fall 2008?)

INVESTMENTS | BODIE, KANE, MARCUS 6-18 Figure 6.3 Spread Between 3-Month CD and T-bill Rates

INVESTMENTS | BODIE, KANE, MARCUS 1-19 Portfolios of One Risky Asset and a Risk-Free Asset You can create a complete portfolio by splitting funds between safe and risky assets. Let: • y = portion allocated to the risky portfolio, P • (1-y) = portion to invest in risk-free asset, F.

Build a complete portfolio C: rC  yrp  1 yrf

Take expectations: ErC   yErp  1 yrf Rearrange terms: Er   r  yEr  r  C f pf risk premium Q. What’s the porfolio’s c?

INVESTMENTS | BODIE, KANE, MARCUS 6-20 Example Using Chapter 6.4 Numbers

Risky Risk-free

E(rp) = 15% rf = 7%

p = 22% rf = 0%

y = % in p (1-y) = % in rf

INVESTMENTS | BODIE, KANE, MARCUS 6-21 Example (Ctd.)

• The expected return on the complete portfolio is the risk-free rate plus the weight of P times the risk premium of P Er   r  yE r  r  C f pf risk premium

Erc   7  y15 7

INVESTMENTS | BODIE, KANE, MARCUS 6-22 Example (Ctd.)

• The risk of the complete portfolio is the weight of P times the risk of P because the risk free asset has zero standard deviation:

 C  y P  22y

INVESTMENTS | BODIE, KANE, MARCUS 6-23 Example (Ctd.)

Place the two portfolios P and F on the {r,} plane. Varying y from 0 to 1 describes a line between F and P, what is the slope?

Rearrange and substitute y=C / P:

 C 8 ErC   rf  ErP  rf  7   C  P 22

ErP  rf 8 Slope   Intercept  rf  7  P 22

INVESTMENTS | BODIE, KANE, MARCUS 6-24 Fig. 6.4 The Opportunity Set

y =1 Q. What’s the value of y here? What does it mean?

y =0

INVESTMENTS | BODIE, KANE, MARCUS 1-25 Capital Allocation Line with Leverage

• y>1 means borrow money to lever up your investment (e.g. buy on )

• There is asymmetry: lend (invest) at rf=7% but borrow at rf=9% – Lending range slope = 8/22 = 0.36 – Borrowing range slope = 6/22 = 0.27

• CAL kinks at P

INVESTMENTS | BODIE, KANE, MARCUS 6-26 Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates

You borrow

You lend

INVESTMENTS | BODIE, KANE, MARCUS 1-27 Risk Tolerance and Asset Allocation

• The investor must choose one optimal portfolio, C, from the set of feasible choices (by changing y) – Expected return of the complete portfolio:

퐸[푟퐶] = 푟푓 + 푦(퐸[푟푃] − 푟푓)

– Variance: 2 2 2  C  y  P

INVESTMENTS | BODIE, KANE, MARCUS 6-28 Utility Function depending on y

• Express U as a function of y 1 U  Er   A 2 C 2 C 1 U  r  yEr  r  Ay 2 2  f p f 2 P • U is a quadratic function of y U  ay 2  yb  c

INVESTMENTS | BODIE, KANE, MARCUS 6-29 Table 6.4 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4

Example:

rf = 7%

E(rp) = 15%

p = 22%

INVESTMENTS | BODIE, KANE, MARCUS 6-30 Figure 6.6 Chart Utility as a Function of the Allocation to the Risky Asset (y)

INVESTMENTS | BODIE, KANE, MARCUS 1-31 Maximize Utility Function w.r.t. y

• Express U as a function of y 1 U  Er   A 2 C 2 C 1 U  r  yEr  r  Ay 2 2  f p f 2 P • The maximize w.r.t. y Erp  rf ymaxU  2 A P

INVESTMENTS | BODIE, KANE, MARCUS 6-32 Utility Indifference curves Utility Indifference Levels 1 U  Er A 2  const 2 For example: U  0.05, 0.09

INVESTMENTS | BODIE, KANE, MARCUS 6-33 Table 6.5 Spreadsheet Calculations of Indifference Curves

INVESTMENTS | BODIE, KANE, MARCUS 1-34 Table 6.6 Expected Returns on Four Indifference Curves and the CAL

Risk aversion coefficient A=4

INVESTMENTS | BODIE, KANE, MARCUS 1-35 Put it together and find your optimal allocation 6-36 2. Map your Utility 3. Find optimal y indifference to maximize your curves U along Capital Alllocation Line

1. Draw the Capital Alllocation Line by varying y

INVESTMENTS | BODIE, KANE, MARCUS Passive Strategies: The Line (CML) • Capital allocation line formed investment in two passive portfolios: 1. “risk-free” -term T-bills (or equivalent) 2. asset that mimics a broad market index

• The passive strategy avoids any direct or indirect security analysis

• Supply and demand forces may make such a strategy a reasonable choice for many investors INVESTMENTS | BODIE, KANE, MARCUS 6-37 Passive Strategies: The • From 1926 to 2012, the passive risky portfolio offered an average risk premium of 8.1% with a standard deviation of 20.48%, resulting in a reward-to- ratio of 0.40

INVESTMENTS | BODIE, KANE, MARCUS 6-38