CHAPTER SEVEN

PORTFOLIO ANALYSIS

THE EFFICIENT SET THEOREM THE THEOREM

An investor will choose his optimal portfolio from the set of portfolios that offermaximum expected returns for varying levels of

risk, andminimum risk for varying levels of returns

The set of portfolios meeting these two conditions is know as the efficient set(or the efficient frontier).

THE EFFICIENT SET THEOREM THE FEASIBLE SET

DEFINITION: represents all portfolios that could be formed from a group of N securities

All possible portfolios that could be formed from the n securities lie either on or within the boundary of the feasible set.

The set will have an umbrella-type shape.

THE EFFICIENT SET THEOREMTHE FEASIBLE SETrP

P0

THE EFFICIENT SET THEOREM EFFICIENT SET THEOREM APPLIED TO

THE FEASIBLE SETApply the efficient set theorem to the

feasible setthe set of portfolios that meet first conditions of

efficient set theorem must be identifiedconsider 2nd condition set offering minimum risk

for varying levels of expected return lies on the “western” boundary

remember both conditions: “northwest” set meets the requirements

THE EFFICIENT SET THEOREM Selection of the optimal portfolio

the investor plots indifference curves on the same figure as the efficient set and then proceed to choose the portfolio that is on the indifference curve that is farthest northwest.

The portfolio will correspond to the point at which an indifference curve is just tangent to the efficient set.

THE EFFICIENT SET THEOREM

THE OPTIMAL PORTFOLIO

E

rP

P0

THE EFFICIENT SET THEOREM

Indifference curves for the risk-averse investor is positively sloped and convex.

The efficient set is generally positively sloped and concave,meaning that if a straight line is drawn between any two points on the efficient set, the straight line will lie below the efficient set.

There will be only one tangency point between the investor’s indifference curves and the efficient set.

CONCAVITY OF THE EFFICIENT SET WHY IS THE EFFICIENT SET

CONCAVE? BOUNDS ON THE LOCATION OF

PORFOLIOS EXAMPLE:

Consider two securities Ark Shipping Company

E(r) = 5% = 20% Gold Jewelry Company

E(r) = 15% = 40%

CONCAVITY OF THE EFFICIENT SET

P

rP

A

G

rA = 5

A=20

rG=15

G=40

CONCAVITY OF THE EFFICIENT SET ALL POSSIBLE COMBINATIONS

RELIE ON THE WEIGHTS (X1 , X 2)X 2 = 1 - X 1

Consider 7 weighting combinations

using the formula

22111

rXrXrXrN

iiiP

BOUNDS ON THE LOCATION OF PORFOLIOS

A B C D E F G X1 1.00 0.83 0.67 0.50 0.33 0.17 0.00 X2 0.00 0.17 0.33 0.50 0.67 0.83 1.00

CONCAVITY OF THE EFFICIENT SETPortfolio return

A 5B 6.7C 8.3D 10E 11.7F 13.3G 15

CONCAVITY OF THE EFFICIENT SET USING THE FORMULA

we can derive the following:

2/1

1 1

N

i

N

jijjiP XX

CONCAVITY OF THE EFFICIENT SET

rP P=+1 P=-1A 5 20 20

B 6.7 10 23.33C 8.3 0 26.67D 10 10 30.00E 11.7 20 33.33F 13.3 30 36.67G 15 40 40.00

CONCAVITY OF THE EFFICIENT SET For any given set of weights, the lower

and upper bounds will occur when the correlation between the two securities is –1 and +1, respectively.

UPPER BOUNDSlie on a straight line connecting A and G

i.e. all must lie on or to the left of the straight line

which implies that diversification generally leads to risk reduction

CONCAVITY OF THE EFFICIENT SET LOWER BOUNDS

all lie on two line segmentsone connecting A to the vertical axisthe other connecting the vertical axis to

point G

any portfolio of A and G cannot plot to the left of the two line segments

which implies that any portfolio lies within the boundary of the triangle

CONCAVITY OF THE EFFICIENT SET

G

upper bound

lower bound

rP

P

CONCAVITY OF THE EFFICIENT SET SUMMARY

For any given set of weights ,the lower and upper bounds will occur when the correlation between the two securities is –1 and +1.

Any portfolio consisting of securities a and g will lie within or on the boundary of the triangle , with its actual location depending on the magnitude of the correlation coefficient between the two securities.

CONCAVITY OF THE EFFICIENT SET ACTUAL LOCATIONS OF THE PORTFOL

IO What if correlation coefficient (ij ) is zer

o?

CONCAVITY OF THE EFFICIENT SET

RESULTS:B = 17.94%B = 18.81%

B = 22.36%

B = 27.60%

B = 33.37%

CONCAVITY OF THE EFFICIENT SETACTUAL PORTFOLIO LOCATIONS

CD

F

CONCAVITY OF THE EFFICIENT SET The portfolio, consisting of two securiti

es, lie on a line that is curved, or bowed, to the left.

IMPLICATION: If ij < 0 line curves more to left If ij = 0 line curves to left If ij > 0 line curves less to left

CONCAVITY OF THE EFFICIENT SET KEY POINT

As long as -1 < the portfolio line curves to the left and the northwest portion is concave

i.e. the efficient set is concave

THE MARKET MODEL

A RELATIONSHIP MAY EXIST BETWEEN A STOCK’S RETURN AN THE MARKET INDEX RETURN

where intercept term ri = return on security rI = return on market index I

slope term random error term

iIIiiIi rr 1

THE MARKET MODEL

THE RANDOM ERROR TERMS i, I shows that the market model cannot exp

lain perfectly the difference between what the actual r

eturn value is and what the model expects it to be is attributable to i, I

THE MARKET MODEL

i, I CAN BE CONSIDERED A RANDOM VARIABLE DISTRIBUTION:

MEAN = 0

VARIANCE = i

THE MARKET MODEL Graphical representation of the market

model: The vertical axis measures the return on the

particular security The horizontal axis measures the return on

the market index The line goes through the point on the

vertical axis corresponding to the value of alpha.

The line has a slope equal to beta.

THE MARKET MODEL Beta

The slope in a security’s market model measures the sensitivity of the security’s returns to the market index’s returns

2I

iIiI

THE MARKET MODEL Beta

Betas greater than 1 are more volatile than the market index and are known as aggressive stocks.

Stocks with betas less than one are less volatile than the market index and are known as defensive stocks.

DIVERSIFICATION

PORTFOLIO RISK TOTAL SECURITY RISK:

i

has two parts:

where = the market risk of security i= the unique variance

of security i returns

2222iIiIi

22IiI2i

DIVERSIFICATION PORTFOLIO RISK and return

N

iiIipI

N

iiIipI

N

iiIipI

pIIpIpIp

X

X

X

rr

1

1

1

DIVERSIFICATION PORTFOLIO RISK and return

N

iiip

N

iiIipI

pIpIp

X

X

1

222

2

1

2

2222

DIVERSIFICATION

TOTAL PORTFOLIO RISK also has two parts: market and

unique Market Risk

diversification leads to an averaging of market risk

Unique Risk as a portfolio becomes more diversified, the

smaller will be its unique risk

DIVERSIFICATION

Unique Risk mathematically can be expressed as

N

iiP N1

22

2 1

NNN22

221 ...1

END OF CHAPTER 7