Optimal portfolios and index model

Suppose your portfolio has only 1 stock, how many sources of risk can affect your portfolio?◦ Uncertainty at the market level◦ Uncertainty at the firm level

Market risk◦ Systematic or Nondiversifiable

Firm-specific risk◦ Diversifiable or nonsystematic

If your portfolio is not diversified, the total risk of portfolio will have both market risk and specific risk

If it is diversified, the total risk has only market risk

Why the std (total risk) decreases when more stocks are added to the portfolio?

The std of a portfolio depends on both standard deviation of each stock in the portfolio and the correlation between them

Example: return distribution of stock and bond, and a portfolio consists of 60% stock and 40% bond

state Prob. stock (%) Bond (%) Portfolio

Recession 0.3 -11 16

Normal 0.4 13 6

Boom 0.3 27 -4

What is the E(rs) and σs?

What is the E(rb) and σb?

What is the E(rp) and σp?

E(r) σBond 6 7.75Stock 10 14.92Portfolio 8.4 5.92

When combining the stocks into the portfolio, you get the average return but the std is less than the average of the std of the 2 stocks in the portfolio

Why? The risk of a portfolio also depends on the correlation between 2 stocks How to measure the correlation between the 2 stocks Covariance and correlation

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Prob rs E(rs) rb E(rb) P(rs- E(rs))(rb-

E(rb))0.3 -11 10 16 6 -630.4 13 10 6 6 00.3 27 10 -4 6 -51

Cov (rs, rb) = -114-114 The covariance tells the direction of the relationship between the 2 assets,

but it does not tell the whether the relationship is weak or strong Corr(rs, rb) = Cov (rs, rb)/ σs σb = -114/(14.92*7.75) = -0.99

Portfolio risk depends on the correlation between the returns of the assets in the portfolio

Covariance and the correlation coefficient provide a measure of the way returns two assets vary

Portfolio Return

Bond Weight

Bond Return

Equity Weight

Equity Return

p D ED E

P

D

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Another way to express variance of the portfolio:

2 ( , ) ( , ) 2 ( , )P D D D D E E E E D E D Ew w Cov r r w w Cov r r w w Cov r r

D,E = Correlation coefficient of returns

Cov(rD,rE) = DEDE

D = Standard deviation of returns for Security DE = Standard deviation of returns for Security E

Range of values for 1,2

+ 1.0 > > -1.0

If = 1.0, the securities would be perfectly positively correlated

If = - 1.0, the securities would be perfectly negatively correlated

2p = w1

212 + w2

222

+ 2w1w2 Cov(r1,r2)

+ w323

2

Cov(r1,r3)+ 2w1w3

Cov(r2,r3)+ 2w2w3

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Standard deviation is smaller than that of either of the individual component assets

Figure 7.3 and 7.4 combined demonstrate the relationship between portfolio risk

The relationship depends on the correlation coefficient

-1.0 < < +1.0 The smaller the correlation, the

greater the risk reduction potential If = +1.0, no risk reduction is

possible

Maximize the slope of the CAL for any possible portfolio, p

The objective function is the slope:

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The solution of the optimal portfolio is as follows

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An investor with risk-aversion coefficient A = 4 would take a position in a portfolio P

7439.0

142.4

05.11.22

p

fp

A

rrEy

The investor will invest 74.39% of wealth in portfolio P, 25.61% in T-bill. Portfolio P consists of 40% in bonds and 60% in stock, therefore, the percentage of wealth in stock =0.7349*0.6=44.63%, in bond = 0.7349*0.4=29.76%

Security Selection◦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

We now search for the CAL with the highest reward-to-variability ratio

Now the individual chooses the appropriate mix between the optimal risky portfolio P and T-bills as in Figure 7.8

2

1 1

( , )n n

P i j i ji j

ww Cov r r

The separation property tells us that the portfolio choice problem may be separated into two independent tasks◦Determination of the optimal risky portfolio is purely technical

◦Allocation of the complete portfolio to T-bills versus the risky portfolio depends on personal preference

Remember:

If we define the average variance and average covariance of the securities as:

We can then express portfolio variance as:

2

1 1

( , )n n

P i j i ji j

ww Cov r r

2 21 1P

nCov

n n

2 2

1

1 1

1

1( , )

( 1)

n

ii

n n

i jj ij i

n

Cov Cov r rn n

The efficient frontier was introduced by Markowitz (1952) and later earned him a Nobel prize in 1990.

However, the approach involved too many inputs, calculations◦ If a portfolio includes only 2 stocks, to calculate the variance of the

portfolio, how many variance and covariance you need?

◦ If a portfolio includes only 3 stocks, to calculate the variance of the portfolio, how many variance and covariance you need?

◦ If a portfolio includes only n stocks, to calculate the variance of the portfolio, how many variance and covariance you need? n variances n(n-1)/2 covariances

level firm at they uncertaint toduereturn ofcomponent :

levelmarket at they uncertaint toduereturn ofcomponent :

market the toistock of nessresponsive :

intercept :

market of premiumrisk :

istock of premiumrisk :

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Risk and covariance:◦ Total risk = Systematic risk + Firm-specific risk:

◦ Covariance = product of betas x market index risk:

◦ Correlation = product of correlations with the market index

2 2 2 2 ( )i i M ie

2( , )i j i j MCov r r

2 2 2

( , ) ( , ) ( , )i j M i M j Mi j i M j M

i j i M j M

Corr r r Corr r r xCorr r r

Portfolio’s variance:

Variance of the equally weighted portfolio of firm-specific components:

When n gets large, becomes negligible

222 2

1

1 1( ) ( ) ( )

n

P ii

e e en n

2 2 2 2 ( )P P M Pe

2 ( )Pe

risk specific :

componentrisk systematic :

risk Total:

2

22

2

2222

ei

mi

i

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2 .......... mnmp

When we diversify, all the specific risk will go away, the only risk left is systematic risk component

Now, all we need is to estimate beta1, beta2, ...., beta n, and the variance of the market. No need to calculate n variance, n(n-1)/2 covariances as before

Run a linear regression according to the index model, the slope is the beta

For simplicity, we assume beta is the measure for market risk Beta = 0 Beta = 1 Beta > 1 Beta < 1

Reduces the number of inputs for diversification

Easier for security analysts to specialize