Lecture 4 Portfolio Tools. Portfolio Weights Financial Markets and Corporate Strategy, David...
-
Upload
spencer-horton -
Category
Documents
-
view
218 -
download
0
Transcript of Lecture 4 Portfolio Tools. Portfolio Weights Financial Markets and Corporate Strategy, David...
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.
Financial Markets and Corporate Strategy, David Hillier
Short Sales and Portfolio Weights
Financial Markets and Corporate Strategy, David Hillier
Feasible Portfolios
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
Portfolio Returns
Financial Markets and Corporate Strategy, David Hillier
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.
Financial Markets and Corporate Strategy, David Hillier
The Portfolio Return Formula
1 1 2 2 …+ p N NR x r x r x r
1
N
i i
i
x r
Financial Markets and Corporate Strategy, David Hillier
Expected Portfolio Returns
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
Variance
2( ) [( ) ]var r E r r
Financial Markets and Corporate Strategy, David Hillier
Computing Variances
Financial Markets and Corporate Strategy, David Hillier
Estimating Variances with Historical Data
Financial Markets and Corporate Strategy, David Hillier
Standard Deviation
( ) ( )xr x r
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
Result 4.2
Given positive portfolio weights on two assets, the lower the correlation, the lower the variance of the portfolio.
Financial Markets and Corporate Strategy, David Hillier
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.
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
Computing the Variance of a Portfolio of Three Assets
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
The Mean-Standard Deviation Diagram
Financial Markets and Corporate Strategy, David Hillier
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.
Financial Markets and Corporate Strategy, David Hillier
Exhibit 4.3
Financial Markets and Corporate Strategy, David Hillier
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
Financial Markets and Corporate Strategy, David Hillier
When the Risky Investment has a Negative Portfolio Weight
p 2 2x
p2
2x
p
12
1x
2 fp pf
2
r rR r
Financial Markets and Corporate Strategy, David Hillier
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.
Financial Markets and Corporate Strategy, David Hillier
Portfolios of Two Perfectly Positively Correlated Assets
Financial Markets and Corporate Strategy, David Hillier
Forming a Riskless Portfolio with Two Perfectly Negatively Correlated Assets
Financial Markets and Corporate Strategy, David Hillier
Portfolios of Two Assets with Arbitrary Correlation
Financial Markets and Corporate Strategy, David Hillier
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.
Financial Markets and Corporate Strategy, David Hillier
How to Use Covariances Alone to Reduce Portfolio Variance
Financial Markets and Corporate Strategy, David Hillier
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.
Financial Markets and Corporate Strategy, David Hillier
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.
Financial Markets and Corporate Strategy, David Hillier
Forming a Minimum Variance Portfolio for Asset Allocation
Financial Markets and Corporate Strategy, David Hillier
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:
Financial Markets and Corporate Strategy, David Hillier
Financial Markets and Corporate Strategy, David Hillier
Thank You