Financial Economics Bocconi Lecture4

7/30/2019 Financial Economics Bocconi Lecture4

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Lecture 4

Risk Aversion and Capital

Allocation to Risky Assets



Portfolio Choice Under Risk Aversion:How SHOULD People Choose Portfolios?

• Capital Allocation Between a Risky and Risk-Free Asset(Chapter 6 BKM) – Capital Allocation Line

– Leveraged Investment

• Capital Allocation Among Risky Assets (Chapter 7 BKM) – Diversification

– Efficient Frontier

• Capital Allocation Among Risky Assets and a Risk-Free

Asset (Chapter 7 BKM)

• What if Everyone Chooses This Way? Market Equilibrium – CAPM (Chapter 9 BKM)



6-3

Allocation to Risky Assets

• Risk Aversion:

– Intuitively, risk aversion means that

investors will avoid risk unless there is a

reward.

• The utility model gives the optimal

allocation between a risky portfolio and

a risk-free asset.



6-4

Risk and Risk Aversion

• Speculation

– Taking considerable risk for a commensurate gain

• Risk: material enough to affect the decision

• Commensurate gain: positive risk premium (expected profit is greater

than the risk-free alternative) – Risk aversion and speculation are not inconsistent

• Gamble

– Bet or wager on an uncertain outcome for enjoyment

– Fair game: risky investment with a risk premium of zero. Will be

rejected by a risk averse investor

• When two parties enter into a transaction (e.g. a futures

contract on the euro), can they both be speculating?



6-5

Risk Aversion and Utility Values

• Risk-averse investors are willing to consider:

– risk-free assets

– speculative positions with positive risk

premiums

• Risk-neutral investors care only about expected

returns. They would take on fair games.

• Portfolio attractiveness increases with expected

return and decreases with risk.



6-6

Mean-Variance (M-V) Criterion • Portfolio A dominates portfolio B if:

And

What happens

when return

increases withrisk?

( ) ( ) A B E r E r B A



6-7

How to choose between these portfolios?

Each portfolio receives a utility score to

assess the investor’s risk/return trade off



6-8

Utility Function

U = utility

E ( r ) = expected

return on the asset

or portfolio

A = coefficient of risk

aversion

2 = variance of returns

½ = a scaling factor

21( )2

U E r A



6-9

Utility Scores of Alternative Portfolios for Investors with Varying Degree of Risk Aversion

The utility score of a risky portfolio can be interpreted as the

certainty equivalent rate of return (rate which risk-free

investments would need to offer to provide the same utility score as

the risky portfolio.)

A portfolio is desirable only if its certainty equivalent rate of return

exceeds the risk-free rate.



6-10

Indifference Curves

If we plot all equally-preferred

portfolios in the mean-

standard deviation plane, we

get the investor’s indifference

curve.

More risk-averse investors

have steeper indifference

curves than less risk averse

investors.



6-11

Estimating Risk Aversion

• Use questionnaires

• Observe individuals’ decisions whenconfronted with risk

• Observe how much people are willing to

pay to avoid risk



Capital Allocation Between a Risky andRisk-Free Asset

• An Example: Risk-Free Asset Risky Asset (Portfolio)

– Expected return 7% 15%

– Standard Deviation 0% 22%

– Portfolio share (1-y) y

• Risk Premium: E{r p} - E{r

f } = 8%

– Extra return required to make people indifferent between risky and risk-free assets

• Two Critical Rules for a portfolio with one risky and one risk-free asset:

– Expected Return is weighted average of expected returns to each security:

• E{r c} = (1-y) E{r

f } + y E{r

p}

• For example, if y = 0.7, E{r c} = 12.6 = 0.3*7% + 0.7*15%

– Standard Deviation is weight on risky asset times standard deviation of riskyasset:

c= y

p= (0.7) (22%) = 15.4%

• The rest of the portfolio-risk formula drops out (mathematically)

c2 = y2

p2 + (1-y)2

f 2 + 2y(1-y)

p

f Corr(p,f)

0 0



6-13

The Investment Opportunity Set

Possible combinationsof risk and return forma straight line

– At y = 0, invest all in zero-riskasset: E{r

c} = 7%,

c = 0

– At y = 1, invest all in riskyasset: E{r

c} = 15%,

c= 22%

– At y = 0.7, invest 30% in zero-risk asset, 70% in riskyasset: E{r

c} = 12.6%,

c=

15.4%



Capital Allocation Between a Risky and Risk-Free Asset

• Capital Allocation Line

– Set of feasible combinations of E{r c} and c

– When considering investor portfolio choices, CAL serves as a

budget line

• How to Choose the optimal allocation between the risk-free asset

and the risky asset ?

– Maximize utility subject to CAL

E{r}

7%

0

15%

22%

.y=0

Capital Allocation LineU

1

U2

U3 More utility




• How to Choose?

– Maximize utility subject to CAL

– Portfolio with 55% in risky asset provides highest possibleutility, U2 for this investor (A=3)

• At y = 55%, what do we know about portfolio?

– E{r*c} = 0.45 E{r

f } + 0.55 E{r

p} = 11.4%

*c

= 0.55 p

= 12.1%

E{r}

7%

0

15%

22%

Capital Allocation LineU1

U2

U3

12.1%

11.4% .y=0.55



● Finding the best allocation, y*: Two approaches:

1. Numerical approximation:

● Calculate U(y) for all feasible values of y.

● Find y that gives highest value of U(y)

2. Calculus, to get it exactly right

This makes sense:

● Best share of risky asset y* grows with the risk premium on risky assets, E{rp} - rf

● Best share of risky asset falls with risk aversion, A

● Best share of risky asset falls with that asset's own risk, 2


2

{ }* p f

p

E r r y

A



6-17

Illustration of numerical optimization



6-18

Utility as a Function of Allocation to theRisky Asset, y



Risk Aversion and Portfolio Allocations

• Lower risk aversion leads to higher share in risky asset (A=1.94)

– Higher expected return, higher risk

E{r}

7%

0

15%

22%

Capital Allocation Line

.y=0.55

.y=0.85

18.7%

13.8%

Lower risk aversion

7%

Higher risk aversion

E{U(y)}

Share in risky asset, y1.y=0.55 .y=0.85




• Why does the CAL continue up and to the right? Potential to borrow – If you borrow 50% of wealth y = 1.5

– "Investment" in zero-risk asset is negative: (1-y) = - 0.5

– E{r c} = -0.5*7% + 1.5*15% = 18.5%,

c= 1.5*22% = 33%.

E{r}

7%

0

15%

22%

.y=0

.y=1

33%

18.5% .y=1.5




• Leveraged Investing: With even lower risk aversion, investor could choose to

borrow (A=1.32)

• y* = 1.25,

• E{r*c} = - 0.25 E{r f } + 1.25 E{r p} = 17% *

c= 1.25

p= 27.5%

E{r}

7%

0

Capital Allocation Line

.y=0.55

Even lower risk aversion

Higher risk aversion

.y=1.2517%

22% 27.5%

15%

6 22



6-22

• Leveraged Investing: Borrowed funds typically

cost more than the risk-free rate:

• Suppose borrowed funds cost 9% (r f = 7%).

• CAL now has a kink at P where y = 1 – Slope for y > 1 is lower than slope for y ≤ 1

– Lending range slope = 8/22 = 0.36

– Borrowing range slope = 6/22 = 0.27

Capital Allocation Line with Leverage

6 23



6-23

The Opportunity Set with DifferentialBorrowing and Lending Rates

6 24



6-24

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).

Financial Economics Bocconi Lecture4

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