CHAPTER 6. Risk Aversion and Capital Allocation to Risky Assets INVESTMENTS BODIE, KANE, MARCUS

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1 CHAPTER 6 Risk Aversion and Capital Allocation to Risky Assets INVESTMENTS BODIE, KANE, MARCUS McGraw-Hill/Irwin Copyright 011 by The McGraw-Hill Companies, Inc. All rights reserved.

2 6- Allocation to Risky Assets Investors will avoid risk unless there is a reward. The utility model allows optimal allocation between a risky portfolio and a risk-free asset.

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

4 6-4 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 = $ For X = $10 For larger X?

5 6-5 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 = $ For X = $10 For larger X?

6 6-6 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 expected return and decreases with risk. What happens when return increases with risk?

7 1-7 Table 6.1 Available Risky Portfolios (Risk-free Rate = 5%) How to compare? Each portfolio receives a utility score to assess the investor s risk/return trade off

8 A Utility Function U 1 E r A U = utility or welfare (a measure of happiness) E[r] = expected return on the asset or portfolio A = coefficient of risk aversion σ = 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 σ 6-8

9 6-9 A Utility Function Meaning of A U 1 E r A 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.

10 Table 6. Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion INVESTMENTS BODIE, KANE, MARCUS 1-10 U 1 A E r Three investors with A=.0, 3.5 and 5.0 Q. What portfolio will each choose?

11 6-11 Risk-Return Trade-off

12 6-1 Mean-Variance (M-V) Criterion Portfolio A dominates portfolio B if: And E r Er A B A B Q. How do you find a family of portfolios you are indifferent to?

13 6-13 Utility Indifference Curve

14 6-14 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

15 6-15 Capital Allocation Across Risky and Risk-Free Portfolios Asset Allocation: Is a very important part of portfolio construction. Refers to the choice among broad asset classes. Controlling Risk: Simplest way: adjust the fraction of the portfolio invested in risk-free assets versus the portion invested in the risky assets

16 6-16 Basic Asset Allocation Portfolio Total Market Value $300,000 Risk-free money market fund $90,000 Equities $113,400 Bonds (long-term) $96,600 Total risk assets $10,000 $113,400 $96,600 we 0.54 w $10,000 B $10,00 These weights are within the risky portfolio Q. What is the risk-free vs risky composition?

17 6-17 Basic Asset Allocation Let y = weight of the risky portfolio P, in the complete portfolio; (1-y) = weight of risk-free assets: $10,000 $90,000 y y 0. 3 $300,000 $300,000 $113,400 $96,600 E : B : 0. 3 $300,000 $300,000 These weights are within the entire portfolio

18 6-18 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 008?)

19 1-19 Figure 6.3 Spread Between 3-Month CD and T-bill Rates

20 Portfolios of One Risky Asset and a Risk-Free Asset INVESTMENTS BODIE, KANE, MARCUS 6-0 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: Take expectations: Rearrange terms: E E Q. What s the porfolio s c? r C p yr 1 r C ye r p 1 y r f r r yer r C f y p f risk premium r f

21 6-1 Example Using Chapter 6.4 Numbers Risky E(r p ) = 15% Risk-free r f = 7% p = % rf = 0% y = % in p (1-y) = % in r f

22 6- 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 r 7 y15 7 E c

23 6-3 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: y C P y

24 6-4 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 : E C r r Er C Slope f E P r P r f P 8 P rf 7 C 8 Intercept r f 7

25 1-5 Fig. 6.4 The Investment Opportunity Set y =1 Q. What s the value of y here? What does it mean? y =0

26 6-6 Capital Allocation Line with Leverage y>1 means borrow money to lever up your investment (e.g. buy on margin) There is asymmetry: lend (invest) at r f =7% but borrow at r f =9% Lending range slope = 8/ = 0.36 Borrowing range slope = 6/ = 0.7 CAL kinks at P

27 Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates INVESTMENTS BODIE, KANE, MARCUS 1-7 You borrow You lend

28 6-8 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: E[r C ] = r f + y(e[r P ] r f ) Variance: y C P

29 Utility Function depending on y Express U as a function of y P f p f C C y A r r E y r U A r E U U is a quadratic function of y c yb ay U

30 Table 6.4 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4 INVESTMENTS BODIE, KANE, MARCUS 6-30 Example: r f = 7% E(r p ) = 15% p = %

31 1-31 Figure 6.6 Chart Utility as a Function of the Allocation to the Risky Asset (y)

32 Maximize Utility Function w.r.t. y Express U as a function of y C r C A E U The maximize w.r.t. y max P f p U A r r E y 1 P f p f y A r r E y r U

33 6-33 Utility Indifference curves Utility Indifference Levels r 1 U E A For example: U 0.05, 0.09 const

34 1-34 Table 6.5 Spreadsheet Calculations of Indifference Curves

35 1-35 Table 6.6 Expected Returns on Four Indifference Curves and the CAL Risk aversion coefficient A=4

36 Put it together and find your optimal allocation 3. Find optimal y to maximize your U along Capital Alllocation Line. Map your Utility indifference curves Draw the Capital Alllocation Line by varying y

37 Passive Strategies: The Capital Market Line (CML) Capital allocation line formed investment in two passive portfolios: 1. risk-free short-term T-bills (or equivalent). 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

38 6-38 Passive Strategies: The Capital Market Line From 196 to 01, the passive risky portfolio offered an average risk premium of 8.1% with a standard deviation of 0.48%, resulting in a reward-to-volatility ratio of 0.40