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
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