Transcript Capital Allocation Line.
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CHAPTER 6
Risk Aversion and Capital Allocation to Risky Assets INVESTMENTS | BODIE, KANE, MARCUS
Allocation to Risky Assets
• Investors will avoid risk unless there is a reward.
– i.e. Risk Premium should be positive • Agents preference (taste) gives the optimal allocation between a risky portfolio and a risk-free asset.
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Speculation vs. Gamble
• Speculation – Taking considerable risk for a commensurate gain – Parties have heterogeneous expectations • Gamble – Bet or wager on an uncertain outcome for enjoyment – Parties assign the same probabilities to the possible outcomes INVESTMENTS | BODIE, KANE, MARCUS
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Table 6.1 Available Risky Portfolios (Risk free Rate = 5%) Each portfolio receives a utility score to assess the investor’s risk/return trade off INVESTMENTS | BODIE, KANE, MARCUS
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Utility Function
U
= utility of portfolio with return r
E ( r )
= expected return portfolio
A
= coefficient of risk s 2 aversion = variance of returns of portfolio ½ = a scaling factor
U
1 2
A
s 2 INVESTMENTS | BODIE, KANE, MARCUS
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Table 6.2 Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion
IN CLASS EXERCISE. Anwer:
How high the risk aversion coefficient (A) has to be so that L is preferred over M and H? INVESTMENTS | BODIE, KANE, MARCUS
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Mean-Variance (M-V) Criterion
• Portfolio A dominates portfolio B if: • And
E
A
s
A E
s
B
B
• As noted before: this does not determine the choice of
one
portfolio, but a whole set of efficient portfolios.
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Estimating Risk Aversion
• Use questionnaires • Observe individuals’ decisions when confronted with risk • Observe how much people are willing to pay to avoid risk INVESTMENTS | BODIE, KANE, MARCUS
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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.
– % of total Investment in risky vs. risk-free assets
Controlling Risk:
• Simplest way: Manipulate the fraction of the portfolio invested in risk-free assets versus the portion invested in the risky assets INVESTMENTS | BODIE, KANE, MARCUS
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Basic Asset Allocation Example
Total Amount Invested Risk-free money market fund Total risk assets Equities Bonds (long-term)
W E
$ 113 , 400 $ 210 , 000
Proportion of Risk
0 .
54
assets on Equities
$300,000 $90,000 $210,000 $113,400 $96,600
W B
$ 96 , 600 $ 210 , 00
Proportion of Risk
0 .
46
assets on Bonds
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Basic Asset Allocation
• P is the complete portfolio where we have
y
as the weight on the risky portfolio and (1-
y
) = weight of risk-free assets:
y
$ 210 , 000 $ 300 , 000 0 .
7 $ 113 , 400
E
: $ 300 , 000 .
378 • Complete Portfolio is: (0.3, 0.378, 0.322) 1
y
$ 90 , 000 $ 300 , 000 0 .
3 $ 96 , 600
B
: $ 300 , 000 .
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The Risk-Free Asset
• Only the government can issue default free bonds.
– 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 INVESTMENTS | BODIE, KANE, MARCUS
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Figure 6.3 Spread Between 3-Month CD and T-bill Rates
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Portfolios of One Risky Asset and a Risk Free Asset • It’s possible to create a complete portfolio by splitting investment funds between safe and risky assets.
– Let y=portion allocated to the risky portfolio, P – (1-y)=portion to be invested in risk-free asset, F.
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Example Using Chapter 6.4 Numbers
r
f
= 7%
E(r
p
) = 15% y = % in p
s
rf
= 0%
s
p
= 22% (1-y) = % in r
f
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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
c
r
f
( )
P
r
f E
c
7
y
15 7 INVESTMENTS | BODIE, KANE, MARCUS
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Example (Ctd.)
• The risk of the complete portfolio is the weight of P times the risk of P:
s
C
y
s
P
22
y
– This follows straight from the formulas we saw before and the fact that any constant random variable has zero variance.
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Feasible (var, mean)
• Taken together this determines the set of feasible (mean,variance) portfolio return:
E
c
s
C
7
y y
s
P
15 7 22
y
– This determines a straight line, which we call
Capital Allocation Line.
Next we derive it’s equation completely INVESTMENTS | BODIE, KANE, MARCUS
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Example (Ctd.)
• Rearrange and substitute y= s C / s P :
E
C
r f
s s
C
E
P
r f
7 8 22 s
C P
– The sub-index C is to stand for complete portfolio
Slope
E
P
s
P r f
8 22 – The slope has a special name: Sharpe ratio.
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Figure 6.4 The Investment Opportunity Set
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Capital Allocation Line with Leverage
• Lend at r f =7% and borrow at r f =9% – Lending range slope = 8/22 = 0.36
– Borrowing range slope = 6/22 = 0.27
• CAL kinks at P INVESTMENTS | BODIE, KANE, MARCUS
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Figure 6.5 The Opportunity Set with Differential Borrowing and Lending Rates
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Risk Tolerance and Asset Allocation
• The investor must choose one optimal portfolio,
C,
from the set of feasible choices – Expected return of the complete portfolio:
E r c
f
– Variance: ( )
P
r f
s
C
2
y
2 s
P
2 INVESTMENTS | BODIE, KANE, MARCUS
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Table 6.4 Utility Levels for Various Positions in Risky Assets (y) for an Investor with Risk Aversion A = 4
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Figure 6.6 Utility as a Function of Allocation to the Risky Asset, y
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Table 6.5 Spreadsheet Calculations of Indifference Curves
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Portfolio problem
• Agent’s problem with one risky and one risk-free asset is thus: • Pick portfolio (y, 1-y) to maximize utility U – U(y,1-y) = E(r_C) -0.5*A*Var(r_C) • Where r_C is the complete portfolio – This is the same as – r_f + y[E(r) – r_f] -0.5*A*y^2*Var(r) – Solution (take foc) is y (proportion on risky asset) as (1/A)*SharpeRatio INVESTMENTS | BODIE, KANE, MARCUS
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Figure 6.7 Indifference Curves for U = .05 and U = .09 with A = 2 and A = 4
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Figure 6.8 Finding the Optimal Complete Portfolio Using Indifference Curves
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Table 6.6 Expected Returns on Four Indifference Curves and the CAL
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Risk Tolerance and Asset Allocation
• The investor must choose one optimal portfolio,
C,
from the set of feasible choices – Expected return of the complete portfolio:
E r c
f
– Variance: ( )
P
r f
s
C
2
y
2 s
P
2 INVESTMENTS | BODIE, KANE, MARCUS
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One word on Indifference Curves
• If you see the IC curves over (mean,st. dev) you will note that these are all nice smooth concave curves. – This is an assumption.
– Note that agents have preference over random variables (representing payoff/return). A random variable, in general, is not completely described by (mean, variance). • That is, in general, we can have X and Y with mean(X) < mean (Y) and var(X)=var(Y) BUT X is ranked better than Y nonetheless. – IF agents have expected utility, we can solve this issue in two ways. INVESTMENTS | BODIE, KANE, MARCUS
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One word on Indifference Curves
• First method is: • Assumption 1: all random variables are normally distributed • Assumption 2: agents have expected utility with Bernoulli given by u(x)= a*x^2 + bx + c – BLACKBOARD (Expected utility computation) INVESTMENTS | BODIE, KANE, MARCUS
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Word on our Portfolio problem • So far we saw how to solver for
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Passive Strategies: The Capital Market Line • A natural candidate for a passively held risky asset would be a well-diversified portfolio of common stocks such as the S&P 500.
• The capital market line (CML) is the capital allocation line formed from 1-month T-bills and a broad index of common stocks (e.g. the S&P 500).
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Passive Strategies: The Capital Market Line • The CML is given by a strategy that involves investment in two passive portfolios: 1. virtually risk-free short-term T-bills (or a money market fund) 2. a fund of common stocks that mimics a broad market index.
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Passive Strategies: The Capital Market Line • From 1926 to 2009, the passive risky portfolio offered an average risk premium of 7.9% with a standard deviation of 20.8%, resulting in a reward-to-volatility ratio of .38.
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