fnce30001 optimal risky portfolios

Investment Management Semester 1, 2012

S_2: Optimal Risky Portfolios

Seminar Overview

1. Diversification and Portfolio Risk 2. Covariance and Correlation in Portfolio Construction 3. Minimum-variance Portfolio 4. Portfolio Opportunity Set 5. Efficient Frontier 6. Separation Property

2

Portfolio Construction

Portfolio construction generally has three steps:

Capital allocation between the risky portfolio and risk-free assets (what we did last lesson).

The optimal capital allocation depends on risk aversion and the risk-return trade-off of the optimal risky portfolio.

Asset allocation across wide asset classes (stocks, international stocks, long-term bonds).

Security selection.

In this lesson, we will examine the risky portfolio and discuss the power of diversification.

3

Diversification and Portfolio Risk

There are two main sources of risks in a portfolio: Firm-specific risks Economy (business cycle, inflation, interest rates, etc.)

Consider a portfolio that has only one stock: BHP Now consider another portfolio that has two stocks: BHP and Coles

With diversification (adding more securities to our portfolio), we can reduce our exposure to firm-specific risks (portfolio volatility is reduced).

However, we will still be exposed to the common economic factors.

4

5

Two Types of Risk

Market risk (systematic risk): the risk that is attributable to

economy wide risk sources Unique risk (nonsystematic risk): the risk that can be eliminated

by diversification

Diversification and Portfolio Risk

6

Whats on the x-axis??

Portfolios of Two Risky Assets

An efficient diversification aims to reduce portfolio risk for any given level of expected return.

Consider a portfolio with wD invested in a bond fund and 1-wD invested in a stock fund.

The expected return of the portfolio is:

)()()( EEDDP rEwrEwrE +=

7

8


The variance of the portfolio is: We can express the portfolio variance as:

remember that Thus, the variance of a portfolio is a weighted sum of covariances

and each weight is the product of the portfolio proportions of the pair of assets in the covariance term.

),(222222 EDEDEEDDP rrCovwwww ++=

),(2),(),( 222 EDEDEEEDDDP rrCovwwrrCovwrrCovw ++=

2),( DDD rrCov =

( )

( ) ( )

2 2 2 2 2

22

2 ,1

P D D E E D E D E D E

P D D E E P D D E E

w w w w r rif

w w w w

= + +

=

= + = +


We can reduce the portfolio variance if the covariances are negative.

Note that even if the covariance term is positive, the portfolio standard deviation is less than the weighted average of the individual security standard deviations (except where correlation coefficient = 1).

The portfolio return is not affected by correlation between returns

Thus, investors should always prefer to add to their portfolios assets with low or negative correlation.

( ) ( ) ( )P D D E EE r w E r w E r= +

9

Example

We will consider the following two portfolios throughout this seminar

Debt Equity

Expected Return 8% 13%

Standard Deviation 12% 20%

Covariance 0.0072

Correlation Coefficient 0.30

10

Less Than Perfect Correlation

Range of values for correlation coefficient

If correlation coefficient = 1.0, the securities would be perfectly positively correlated

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

But, how low can portfolio standard deviation be? If we have perfectly negatively correlated assets in our portfolios, then

the portfolio standard deviation can be reduced to zero by choosing appropriate weights.

11 , + ED

11

Example: Less Than Perfect Correlation

Correlation coefficient < 1

3.004.0014.0

007.0),(0072.0),(,04.0,60.0,014.0,40.0 22

===

=====

ED

ED

EDEEDD

rrCovrrCovww

( )

168.004.0)60.0(014.0)40.0()(

142.002.004.0014.0)3.0()60.0)(40.0(2)04.0()60.0()014(.)40.0( 222

=+=

=

=++=

weightedP

P

12

Example: Less Than Perfect Correlation

Correlation coefficient = 1

Therefore, when the correlation coefficient is less than 1, the portfolio standard deviation is less than the weighted average of the individual standard deviations.

( )

168.004.0)60.0(014.0)40.0()(

168.0028.004.0014.0)0.1()60.0)(40.0(2)040(.)60.0()014.0()40.0( 222

=+=

=

=++=

weightedP

P

13

Portfolio Expected Returns

14

Correlation Coefficient and Weights

correlation coefficient < 1 the portfolio standard deviation first falls as the weight of the equity fund increases from zero to 1, but then rises again as the portfolio becomes heavily invested in stocks and becomes undiversified again.

correlation coefficient = 1

portfolio standard deviation will increase monotonically from the low-risk asset (debt portfolio) to the high-risk asset (equity portfolio).

minimum-variance portfolio

has a standard deviation smaller than that of either of the individual component assets.

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Portfolio Expected Return and Standard Deviation

Previous two figures can be combined to show the relationship between portfolio standard deviation and expected return.

No-benefit from

diversification

Benefit of diversification

Portfolio opportunity

set

16

Portfolio Expected Return and Standard Deviation

Even though the expected return of a portfolio is simply the weighted average of the asset expected returns, this is not the case for standard deviation.

Investors benefit from diversification when the correlation coefficient

is less than perfectly positive. The lower the correlation, the greater the potential benefit from diversification.

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Minimum Variance Portfolio

The standard deviation of the minimum-variance portfolio will be smaller than the standard deviations of the individual component assets.

For a two-asset portfolio, the portfolio weights that will minimize

portfolio standard deviation are given as: ( )

( )

( ) ( )( ) ( )

2

2 2

cov ,( )

2cov ,

( ) 1 ( )

1 cov 1 0.12 0.20 0.024 0.65, 0.35

( ) 0.65 8% 0.35 13% 9.75%

E D EMin

D E D E

Min Min

Min Min

P

r rw D

r r

w E w D

at w D w E

E r

=

+

=

= = = = =

= + =

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Example: Portfolio Opportunity Set

Consider the following data for Stocks AA and BB: Compute and draw the portfolio opportunity set for the two stocks.

AA BB

Expected Return 6% 16% Standard Deviation

25% 29%

Correlation -0.15 Covariance -0.01088

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Example: Portfolio Opportunity Set (contd)

Lets first generate the covariance matrix.

The minimum-variance portfolio weights are:

AA BB

AA 0.0625 -0.01088

BB -0.01088 0.0841

( )( )

2

2 2

0.0841 0.01088( , )( ) 0.562 ( , ) 0.0625 0.0841 2 0.01088

( ) 1 ( ) 1 0.56 0.44

BB AA BBMin

AA BB AA BB

Min Min

Cov r rw AACov r r

w BB w AA

= = =

+ +

= = =

20


Expected return and the standard deviation of the minimum variance portfolio are:

( )0.52 2

( ) (0.56)(0.06) (0.44)(0.16) 0.104

(0.56 )(0.0625) (0.44 )(0.0841) 2(0.56)(0.44)( 0.01088) 0.1747

P

P

E r

= + =

= + + =

21

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Finally, by repeating this exercise for different weights of AA and BB, we obtain the portfolio opportunity set:

Portfolio Opportunity Set

0.000

0.020

0.040

0.060

0.080

0.100

0.120

0.140

0.160

0.180

0.000 0.050 0.100 0.150 0.200 0.250 0.300 0.350

Standard Deviation

Expe

cted

Ret

urn

The Optimal Risky Portfolio (with Two Risky Assets and a Risk-Free Asset)

How does an investor chooses the optimal portfolio from the opportunity set?

The choice depends on the risk aversion of the investor. Investors with lower risk aversions will prefer portfolios to the

northeast (higher expected return and risk). Investors with greater risk aversions will prefer portfolios to the

southwest (lower expected return and risk).

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Asset Allocation: Risky and Risk-Free Assets

In chapter 6, we looked at how investors can allocate their funds between risky and risk-free securities.

In our discussion so far, we have considered a risky portfolio

comprising a stock and a bond fund. But, we still need to decide the proportion of the portfolio to be

allocated between the stock and bond funds. To do this, we will introduce a risk-free asset (T-bills) to the portfolio

allocation problem. So, we now have three assets: the stock fund, bond fund, and risk-

free asset.

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38.0)(

=

=

B

fB rrEoSharpeRati

34.0)(

=

=

A

fA rrEoSharpeRati

25 Would you want CAL(A) or CAL(B)?


The objective is to find the weights wD and wE that result in the highest slope of the CAL.

So, the objective function is to maximize:

where subject to

P

fP rrE

)(

[ ] 2/12222 ),(2)()()(

EDEDEEDDP

EEDDP

rrCovwwww

rEwrEwrE

++=

+=

1=+ ED ww

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Example: Asset Weights

Lets find wD and wE , using equation 13 in the text,

thus,

[ ]42.0

142.005.011.0

142.0)0072.0)(60.0)(40.0(2)2.0)(60.0()12.0)(40.0(

11.013)60.0(8)40.0()(2/12222

=

=

=++=

=+=

P

P

P

S

rE

( ) ( )( ) ( ) ( )

DE

EDfEfDDfEEfD

EDfEEfDD

wwrrCovrrErrErrErrE

rrCovrrErrEw

=

++

=

1

),()()()()(),()()(

22

2

( ) ( )( ) ( ) ( )

60.040.01

40.0)0072.0(05.013.005.008.0)0144.0(05.013.0)04.0(05.008.0

)0072.0(05.013.0)04.0(05.008.0

==

=++

=

E

D

w

w

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Optimal Risky Portfolio

42.0)(

=

=

P

fP rrEoSharpeRati

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Optimal Complete Portfolio

In the last lesson, we examined the optimal complete portfolio given the optimal risky portfolio and the risk-free asset.

Now that we know how to find the optimal risk portfolio and we can

include the individual investors risk aversion (risk preferences) to find the optimal complete portfolio.

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Example: Optimal Complete Portfolio With the utility function:

y is the proportion of the portfolio in the risky assets (maximizing U)

If A = 4, E(rP)=0.11, and variance = 0.142, then y is

The investor will put 74.39% of funds in portfolio P and the rest (25.61%) in

the risk-free asset (T-bills) Remember, portfolio P consists of stocks and bonds. Thus,

221)( ArEU =

2

)(

P

fp

ArrE

y

=

2 2

( ) 0.11 0.05 0.7439 0.06974(0.142 )

p f

P

E r ry U

A

= = = =

4463.0)60.0)(7439.0(2976.0)40.0)(7439.0(

==

==

E

D

ywyw

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Optimal Complete Portfolio

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How to construct the Complete Portfolio?

Find the return characteristics of all securities Establish the risky portfolio

Calculate the optimal risky portfolio (equation 13) Calculate the expected return and standard deviation of the

optimal risky portfolio using the weights obtained from the previous step.

Allocate funds between the risky portfolio and the risk-free asset

Calculate the proportion of the complete portfolio allocated to the optimal risk portfolio and to the risk-free asset (T-bills).

Calculate the share of the complete portfolio invested in each risky asset and in the risk-free asset (T-Bill).

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The Efficient Frontier

We can increase the number of securities in the complete portfolio. The process is same as in the two risky assets example.

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The Efficient Frontier

All the portfolios that lie on the minimum-variance frontier from the global minimum-variance portfolio and upward provide the best risk-return combinations (thus, candidates for the optimal risky portfolio).

The part above the global minimum-variance point is called efficient frontier.

The bottom part of the minimum-variance frontier is not efficient Because there is always another portfolio with the same

standard deviation, but higher expected return.

34

Efficient Frontier

The next step in the optimal complete portfolio is to find the steepest CAL that is tangent to the efficient frontier.

35

Separation Property

A portfolio manager will offer the same risky portfolio to all clients regardless of their degree of risk aversion.

The degree of risk aversion becomes important only in choosing the

desired point along the CAL. A less (more) risk-averse investor will invest less (more) in the risk-free

asset and more (less) in the risky portfolio.

Separation property states that the asset allocation problem can be divided into two independent steps: Determination of the optimal risky portfolio

This portfolio will be same for all clients. Determination of the complete portfolio that also includes the risk-free

asset.

36

Separation Property

Thus, fund managers can serve many customers by only offering at least one portfolio.

Different managers will have different inputs for the optimal risky portfolio and will come up with different optimal portfolios.

37

Separation Property Without any restrictions on the security selection problem, the

portfolio P is the optimal risky portfolio. However, once we start to introduce restrictions or constraints (for

example, taxes), different investors might choose portfolio A as the optimal risky portfolio.

Example: Morningstar Vanguard Funds 38

Seminar Summary

The expected return of a portfolio is the weighted average of the component security expected returns with the investment proportions as weights.

The variance of a portfolio is the weighted sum of the elements of

the covariance matrix with the product of the investment proportions as weights.

Even if the covariances are positive, the portfolio standard

deviation is less than the weighted average of the component standard deviations, as long as the assets are not perfectly positively correlated. Thus portfolio diversification is of value as long as assets are less than perfectly correlated.

39

40

Seminar Summary

The efficient frontier is the graphical representation of a set of portfolios that maximize expected return for each level of portfolio risk. Rational investors will choose a portfolio on the efficient frontier.

In general, portfolio managers will arrive at different efficient

portfolios because of differences in methods and quality of security analysis.

If a risk-free asset is available and input lists are identical, all

investors will choose the same portfolio on the efficient frontier of risky assets: the portfolio tangent to the CAL. All investors with identical input lists will hold an identical risky portfolio, differing only in how much each allocates to this optimal portfolio and to the risk-free asset. This result is characterized as the separation principle of portfolio construction.

fnce30001 optimal risky portfolios

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