lecture 4 portfolio tools. portfolio weights financial markets and corporate strategy, david hillier

Lecture 4

Portfolio Tools

Portfolio Weights

Money held in jMonetary value of the portfolioj

assetx

Financial Markets and Corporate Strategy, David Hillier

The Two-Stock Portfolio

A portfolio consists of £1 million in Vodafone equity and £3 million in British Airways equity. What are the portfolio weights of the two equities?

Answer: The portfolio has a total value of £4 million. The weight on Vodafone is £1,000,000/£4,000,000 .25 or 25 per cent, and the weight on British Airways is £3,000,000/£4,000,000 = .75 or 75 per cent.


Short Sales and Portfolio Weights


Feasible Portfolios


Computing Portfolio Weights for a Portfolio of many SecuritiesDescribe the weights of a €40,000 portfolio invested in four

securities. The amounts invested in each security are as follows:

Security: 1 2 3 4Amount: €20,000 €5,000 €0 €25,000


Portfolio Returns


Computing Portfolio Returns for a Two Security Portfolio

A £4,000,000 portfolio consists of £1,000,000 of Vodafone equity and £3,000,000 of British Airways equity. If Vodafone shares have a return of 10 per cent and British Airways shares have a return of 5 per cent, determine the portfolio return using both (1) the ratio method and (2) the portfolio-weighted average method.


The Portfolio Return Formula

1 1 2 2 …+ p N NR x r x r x r

1

N

i i

i

x r


Expected Portfolio Returns


Expected Return Properties

( ) ( )E xr xE r

1 2 1 2 1 2 1 2( ) ( ) ( ) and ( ) ( ) ( )E r r E r E r E r r E r E r

1 1 2 2 1 1 2 2( ) ( ) ( ) ( )pE R E x r x r x E r x E r


Expected Portfolio Returns and Arbitrary WeightsThe Palisades Quant Fund has a portfolio weight of x in the

Eurostoxx 50 index, which has an expected return of 11 per cent. The fund’s investment in Treasury bills, with a portfolio weight of 1 – x, earns 5 per cent. What is the expected return of the portfolio?

Answer:

( ) .11 .05(1 ) .05 .06pE R x x x


Result 4.1

The expected portfolio return is the portfolio-weighted average of the expected returns of the individual stocks in the portfolio:

1

N

p i i

i

R x r


Variance

2( ) [( ) ]var r E r r


Computing Variances


Estimating Variances with Historical Data


Standard Deviation

( ) ( )xr x r


Covariance and Correlation

12 1 1 2 2[( )( )]E r r r r

( , ) ( )cov r r var r1 2

1 21 2

( , )( , )

cov r rr r

1 21 2

1 2( , ) ,

r rr r cov

1 2 1 2( , ) ( , ) cov r r r r ij ij i j Financial Markets and Corporate Strategy, David Hillier

Variance of a two-asset portfolio

21 1 2 2 1 1 2 2 1 2 2 2( ) {[ ( )] } var x r x r E x r x r E x r x r

21 1 2 2 1 1 2 2 1 1 2 2

21 1 1 2 2 2

( ) {[ ( )] }{[ ( ) ( )] }

var x r x r E x r x r x r x rE x r r x r r

2 2 2 21 1 2 2 1 1 2 2 1 2 1 1 2 221

2 2 2 21 1 2 2 1 2 1 1 2 221

2 21 2 1 2 1 221

2 2 2 21 2 122 21 1

( ) [ ( ) ( ) 2 ( )( )][( ) ] [( ) ] 2 [( )( )]( ) ( ) 2 ( , )

2

var x r x r E x r r x r r x x r r r rx E r r x E r r x x E r r r rx var r x var r x x cov r rx x x x


Correlation, Diversification and Portfolio Variances

2 2 2 21 1 2 2 1 2 1 22 21 1( ) 2var x r x r x x x x


Result 4.2

Given positive portfolio weights on two assets, the lower the correlation, the lower the variance of the portfolio.


Result 4.3

The standard deviation of either (1) a portfolio of two investments where one of the investments is riskless, or (2) a portfolio of two investments that are perfectly positively correlated, is the absolute value of the portfolio-weighted average of the standard deviations of the two investments.


Result 4.4

The formula for the variance of a portfolio return is given by:

where ij covariance between the returns of assets i and j.

2p

1 1

:N N

i j ij

i j

x x


Computing the Variance of a Portfolio of Three Assets


Variance of a Portfolio

2 2 2p

1

2N

j j i j ij

j i j

x x x

2 2 2 2 21 2 122 2p 1 1 2x x x x

2

i 1 j 1

N N

i j ij i jp x x


Result 4.5

For any asset, indexed by k, the covariance of the return of a portfolio with the return of asset k is the portfolio-weighted average of the covariances of the returns of the investments in the portfolio with asset k’s return – that is:

pk i ik

i 1

N

x


The Mean-Standard Deviation Diagram


Result 4.6

Whenever the portfolio mean and standard deviations are portfolio-weighted averages of the means and standard deviations of two investments, the portfolio mean-standard deviation outcomes are graphed as a straight line connecting the two investments in the mean-standard deviation diagram.


Exhibit 4.3


When Both Portfolio Weights are Positive

1 f 2 2

p 2 2

pR x r x rx

p2

2x

p

12

1x

2 fp pf

2

r rR r


When the Risky Investment has a Negative Portfolio Weight

p 2 2x

p2

2x

p

12

1x

2 fp pf

2

r rR r


Result 4.7

When investors employ risk-free borrowing (that is, leverage) to increase their holdings in a risky investment, the risk of the portfolio increases.


Portfolios of Two Perfectly Positively Correlated Assets


Forming a Riskless Portfolio with Two Perfectly Negatively Correlated Assets


Portfolios of Two Assets with Arbitrary Correlation


Result 4.8

The covariance of an asset return with the return of a portfolio is proportional to the variance added to the portfolio return when the asset’s portfolio weight is increased by a small amount, keeping the weighting of other assets fixed by financing the additional holdings of the asset with another investment that has zero covariance with the portfolio.


How to Use Covariances Alone to Reduce Portfolio Variance


Result 4.9

If the difference between the covariances of the returns of assets A and B with the return of a portfolio is positive, slightly increasing the portfolio’s holding in asset A and reducing the position in asset B by the same amount increases the portfolio return variance. If the difference is negative, the change will decrease the portfolio return variance.


Result 4.10

The portfolio of a group of assets that minimizes return variance is the portfolio with a return that has an equal covariance with every asset return.


Forming a Minimum Variance Portfolio for Asset Allocation


Finding the Minimum Variance Portfolio

Suntharee Beers, a Thai firm, wants to branch out internationally. Recognizing that the markets in Japan and Europe are difficult to break into, it is contemplating capital investments to open franchises in India (investment 1), Russia (investment 2) and China (investment 3). Given that Suntharee has a fixed amount of capital to invest in foreign franchising, and recognizing that such investment is risky, Suntharee wants to find the minimum variance investment proportions for these three countries. Solve Suntharee’s portfolio problem. Assume that the returns of the franchise investments in the three countries have covariances as given in the next slide:


Thank You

lecture 4 portfolio tools. portfolio weights financial markets and corporate strategy, david hillier

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