Financial Economics Bocconi Lecture6

7/30/2019 Financial Economics Bocconi Lecture6

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INVESTMENTS| BODIE, KANE, MARCUSCopyri ght 2011 by The McGraw-H il l Companies, Inc. All ri ghts reserved.McGraw-Hill/Irwin

Optimal Risky Portfolios

CHAPTER 7: Part 2


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Starting Point

Recall from last lecture:

We are trying to build an optimal portfolio

We have access to only two assets: a bond

mutual fund and a stock mutual fund

D: debt (bond mutual fund), E: equity (stock

mutual fund)

Our question is: what is the optimal portfolio,given the investors preferences (i.e. the utility

function)


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Optimal Risky Portfolio (no risk-

free asset available)

),(2

),(

)],(222[

}{}{*

22

2

edCov

edCov

edCovA

REREe

ed

d

de

de

),(2),(22

2

varminedCov

edCovede

d

)],(222[

}{}{RelReturn

edCovA

RERE

ede

de

Minimum variancecomponent

Relative risk-

adjusted return

component

e* is the

fraction in

stocks


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Suppose now we have access to two risky assets (equity and bonds) and a risk-freeasset

Consider two possible risky portfolios, A (the minimum variance portfolio) and B.

Each one has associated "Capital Allocation Line," CALA and CALB

CALB dominates CALA

At S, you can experience the same risk on B as A

But expected return is higher on CALB

Capital Allocation Among Risky and Risk

Free Assets

E{r}

Standard deviation

Capital Allocation Lines

.rf

A

B

S

CALACALB


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Suppose two risky assets (equity and bonds) and a risk-free asset

Consider two possible risky portfolios, A and B

CALC dominates all other CAL's

The best CAL must be tangent to the efficient frontier

Key observation: best/tangent CAL has highest possible slope

Capital Allocation Among Risky and Risk

Free Assets

E{r}

Standard deviation

Capital Allocation Lines

.rf

A

B

S

CALACALB

Efficient Frontier

p

fp rrECALofSlope

}{..

CALC


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The Sharpe Ratio

Maximize the slope of the CAL for anypossible portfolio, P.

The objective function is the slope:

The slope is also the Sharpe ratio.

( )P fP

P

E r rS


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The Opportunity Set of the Debt and Equity Funds


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Determination of the Optimal Overall Portfolio


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With two risky assets, equities and bonds, allocation to bonds in risky portfolio:

d* = [E{rd} - rf]2e - [E{re} - rf]Cov(d,e) .[E{re} - rf]2d+ [E{rd} - rf]2e- [(E{re} - rf) + (E{rd} - rf)]Cov(d,e)

What a mess! However, it makes sense:

If E{rd} = E{re},2

e =2

d, and corr(d,e)2d, and corr(d,e) < 1, then d* > For imperfectly correlated assets with identical returns but differing risk, only goal is risk

minimization: Invest more than half in the asset with lower risk

If E{rd} < E{re}, but 2e = 2d, and corr(d,e) < 1, then d* < For imperfectly correlated assets with identical risk but differing returns, there's now a

risk-return trade-off: Invest less than half in the asset with lower returns

If E{rd} < E{re}, but 2e = 2d, and corr(d,e) 1, then d* = 0 For perfectly correlated assets with identical risk but differing returns, there is no risk-

return trade-off: Invest none in the asset with lower returns

Capital Allocation Among Risky and Risk Free Assets


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Optimal share of bonds in portfolio of risky assets, d*:

d* = [E{rd} - rf]2e - [E{re} - rf]Cov(d,e) )[E{re} - rf]2d + [E{rd} - rf]2e- [(E{re} - rf) + (E{rd} - rf)]Cov(d,e)

d* = 0.40 = [8 - 5]*202 - [13 - 5]*72 )[13 - 5]*122 + [8 - 5]*202 - [(13- 5) + (8 -5)]*72

Properties of optimal portfolio of risky assets:

E{rp(d*)} = 11% = 0.4* 8 + 0.6 * 13

p (d*) = 14.2% = [0.42

122

+ 0.62

202

+ 2*0.4*0.6*72]1/2

Slope of Capital Allocation Line = [E{rp(d*)}-rf]/p(d*) = [11 - 5]/14.2 =0.42



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An Example: Bonds Equities Risk Free

Expected return 8% 13% 5%

Standard Deviation 12 20 0

Correlation 0.30

Which portfolio of risky assets defines the (one and only) Capital Allocation Line?

At the minimum variance portfolio, share stocks= 0.20, E{RP

} = 0.09, Stdev(RP

) = 0.12,Slope = 0.349

At optimum portfolio, share stocks= 0.60, E{RP} = 0.11, Stdev(P) = 0.14, Slope = 0.423


Capital Allocation Line at Minimum Risk

Portfolio? No.

3%

5%

7%

9%

11%

13%

15%

0% 5% 10% 15% 20% 25%

Standard Deviation

ExpectedRetu

rn

The Right Capital Allocation Line

3%

5%

7%

9%

11%

13%

15%

0% 5% 10% 15% 20% 25%

Standard Deviation

Expected

Retu

rn


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Figure 7.9 The Proportions of the OptimalOverall Portfolio

Risk-free rate = 5%; A = 4

y*= (0.11-0.05)/(4*0.142^2)

y* = 0.7439e* = 0.6*0.7439 = 44.63%

d* = 0.4*0.7439 = 29.76%

T-bills = 1-y = 0.2561


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Markowitz Portfolio Selection Model

Security Selection The first step is to determine the risk-

return opportunities available.

All portfolios that lie on the minimum-variance frontier from the global

minimum-variance portfolio and upward

provide the best risk-return

combinations: efficient frontier


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The Minimum-Variance Frontier of RiskyAssets


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We now search for the CAL with the

highest reward-to-variability ratio

p

fp rrECALofSlope

}{..


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Figure 7.11 The Efficient Frontier of RiskyAssets with the Optimal CAL


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Everyone invests in P, regardless of their

degree of risk aversion. P must be the

market portfolio.

More risk averse investors put more in the

risk-free asset.

Less risk averse investors put more in P.


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The Power of Diversification

Remember:

Consider an equally-weighted portfolio. If we

define the average variance and averagecovariance of the securities as:

2

1 1

( , )n n

P i j i j

i j

w w Cov r r

2 2

1

1 1

1

1( , )

( 1)

n

i

i

n n

i j

j ij i

n

Cov Cov r r n n


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The Power of Diversification

We can then express portfolio variance as:

If all risk is firm-specific, the average covariance

is 0 and as n becomes large, the portfolio

variance converges to zero.

If there is non-diversifiable risk, the covariance

term is not zero, and the portfolio variance does

not converge to zero even as we add more and

more securities.

2 21 1P

nCov

n n


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Table 7.4 Risk Reduction of Equally WeightedPortfolios in Correlated and Uncorrelated Universes


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Optimal Portfolios and NonnormalReturns

Fat-tailed distributions can result in extreme

values of VaR (value at risk) and ES (expected

shortfall) and encourage smaller allocations to

the risky portfolio.

If other portfolios provide sufficiently better VaR

and ES values than the mean-variance efficient

portfolio, we may prefer these when faced with

fat-tailed distributions.

Financial Economics Bocconi Lecture6

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