Portfolio Theory
The Benefits of Diversification
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Overview
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Originally proposed by Harry Markowitz in
1950s, was the first formal attempt to quantify the
risk of a portfolio and develop a methodology for
determining the optimal portfolio.
Harry Markowitz was the first person to show
quantitatively why and how diversification
reduces risk.
In recognition of his seminal contributions in this
field, he was awarded the Nobel Prize in
Economics in 1990.
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Portfolio Expected Return
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The expected return of a portfolio is a weighted average
of the expected returns of the individual securities in the
portfolio:
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E RP    wi E ( Ri )
i 1
where ,
E RP   expected return on portfolio
wi  weight of security i in the portfolio
E ( Ri )  expected return on security i
n  number of securities in portfolio
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Note that the weight of a security represents the
proportion of portfolio vvalue.
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Portfolio Risk
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Measured by Standard Deviation or Variance as in case
n
of individual securities.
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   wi 2 i 2
i 1
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The standard deviation of a portfolio is not a weighted
average of the standard deviations of the individual
securities.
The riskiness of a portfolio depends on both the riskiness
of the securities, and the way that they move together
over time (correlation)
This is because the riskiness of one asset may tend to be
canceled by that of another asset
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Measurement of Co-movements in
security returns
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Co-movements between the returns of securities are
measured by covariance (an absolute measure) and
coefficient of correlation (a relative measure)
Covariance reflects the degree to which the returns of
two securities vary or change together.
A positive covariance means that the returns of the two
securities move in the same direction
A negative covariance implies that the returns of the two
securities move in opposite direction.
Correlation coefficient vary between -1.0 and + 1.0
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Measurement of Portfolio Risk – 2
securities case
 P  w1212  w22 22  2r1, 21 2 w1w2
 P  w1212  w22 22  2COV1, 2 w1w2

Ri1  E ( R1 )Ri 2  E ( R2 )

COV1, 2  r1, 2 1 2 
(n - 1)
  pi Ri1  E ( R1 )Ri 2  E ( R2 )
Coefficien t of Correlatio n :
COV1,2
r1,2  1, 2 
 1 2
Remember, r1,2  r2,1
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Matrix Method
Security 1
Security 2
Security 1
Security 2
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Matrix Method
Security 1
Security 2
Security 1
w12 12
r1,21 2 w1w2
Security 2
r2,11 2 w1w2
w22 22
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Portfolio Risk – n security case
 P   r1, 21 2 w1w2
2
P 
 r1,21 2 w1w2
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Matrix Method – n securities
Securities
1
2
3
……….
……….
2
………..
3
………..
……..
……..
n
……….
………..
…….. ……..
1
n
……….
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Matrix Method – n securities
1
1
w12 12
2
r2,11 2 w1w2
3
r1,3 1 3w1w3 r3, 2 3 2 w3w2
3
……….
n
r1, 21 2 w1w2
r1,3 1 3w1w3
……….
r1, n1 n w1wn
w22 22
r2,3 3 2 w3w2
………..
r2, n n 2 wn w2
w32 32
………..
………..
……….
……..
……..
n
2
r1, n1 n w1wn
……….
…….. ……..
Securities
wn2 n2
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Efficient Frontier – 2 Security Case
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Suppose an investor is evaluating two securities A
and B,
Security A
Security B
Expected Return
12%
20%
Standard Deviation
20%
40%
Coef. Of Correlation
- 0.20
The investor can combine securities A and B in a
portfolio in a number of ways by changing the
proportion of funds allocated to them.
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Portfolios of Investor – Infinite Choices
Portfolio
Proportion of A, Proportion of B,
wA
wB
Expected
Return, E(Rp)
Standard
Deviation, σp
1 (A)
1.00
0.00
12.00%
20.00%
2
0.90
0.10
12.80%
17.64%
3
0.759
0.241
13.93%
16.27%
4
0.50
0.50
16.00%
20.49%
5
0.25
0.75
18.00%
29.41%
6 (B)
0.00
1.00
20.00%
40.00%
Plotting the options described above, we get the following
exhibit:
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v
Portfolio Options
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The
effect
of
diversification can be
seen by comparing the
curved line between
points A and B with the
straight line between A
and B.
The
straight
line
represents the risk
return possibilities by
combining A and B if
the correlation coef.
Between two had been
1.
Since the curved line is
always to the left of the
straight
line,
the
diversification effect is
illustrated in the figure.
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v
Portfolio Options……..
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Portfolio 3 represents
the minimum variance
portfolio (MVP) or
more accurately, the
minimum std.
deviation portfolio.
The investor
considering portfolio of
A and B faces an
opportunity set or
feasible set
represented by curved
line AB.
By choosing an
appropriate mix
between the two
securities, the investor
can achieve any point
on the curved line.
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v
Portfolio Options……..
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Technically, a backward bend occurs when r
≤ 0; it may or may not occur when r > 0
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The curve bends
backwards between
points A and 3 (MVP),
meaning that, for a
portion of the feasible
set, the std. deviation
decreases although
expected return
increases.
This happens due to
diversification effect of
including security B in
the portfolio (corr. Coef
being negative)
For some length,
increasing the
proportion of B, curve
bends backwards.
After that, std deviation
increases with return.
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Portfolio Options……..
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No investor would like
to invest in a portfolio
whose expected return
is less than that of MVP.
Although, the entire
curve from A to B is
feasible, investors
would consider only the
segment from 3 to B.
This is called the
efficient set or the
efficient frontier.
Points lying along the
efficient frontier are
called efficient
portfolios.
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Efficient frontier for diff degrees of r…..
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Risk can even be reduced to zero, by choosing
the weights (wA and wB ) suitably in case
where r = -1.0
This can be done by replacing r = -1.0 in
portfolio variance equation and equating the
same to zero.
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Efficient frontier for diff degrees of r
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The feasible
frontier under
various degree of
Coefficient of
Correlation is
delineated in the
figure
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Efficient Frontier for the n-Security case
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In a multi-security
case, the collection of
all the possible
portfolios is
represented by the
broken egg shaped
region, referred to as
the feasible region,
shown in the figure.
The number of
possible portfolios in
that region is
virtually endless.
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Efficient Frontier for the n-Security case…..
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What really matters
to the investor is the
northwest boundary
of the feasible region
(AFX)
Referred
to
as
efficient frontier, this
boundary contains
all
the
efficient
portfolios.
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Efficient portfolio:
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A portfolio is efficient if and only if there is no
alternative with:
i. The same E(Rp) and a lower σp
ii. The same σp and higher E(Rp)
iii. A higher E(Rp) and a lower σp
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Efficient Frontier for the n-Security case…..
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A portfolio like Z is
inefficient because
portfolio like B and
D, among others,
dominate it.
The efficient frontier
is the same for all
investors because
portfolio theory is
based on the
assumption that
investors have
homogeneous
expectations.
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How can this set of efficient portfolios
be actually obtained from the
innumerable portfolio possibilities that
lie before the investor ?
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The set of efficient portfolios may be
determined with the help of graphical
analysis, or calculus analysis, or quadratic
programming analysis.
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Risk-Return Indifference Curve
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• The indifference cureves Ip and Iq
represent the risk return trade offs of
two investors P and Q (both are risk
averse)
• Q is more risk averse than P
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All points lying on an
indifference curve
provide the same level
of satisfaction.
In general, the steeper
the slope of the
indifference curve, the
greater the degree of
risk aversion.
Each person has a map
of indifference curves.
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Utility Indifference Curves
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Points A and B, offer the
same level of satisfaction
Points R and S offer the
same level of satisfaction.
The level of satisfaction
increases as one moves
leftward in the indifference
curve map.
The indifference curve Ip2
represents a higher level of
satisfaction as compared to
the indifference curve
Ip1…….
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Optimal Portfolio
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Given the efficient frontier and the risk-return
indifference curve, the optimal portfolio is
found at the point of tangency between the
efficient frontier and utility indifference
curve.
This point represents the highest level of
utility the investor can reach.
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Optimal Portfolio….
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Two investors P
and Q, confronted
with the same
efficient frontier,
but having different
indifference curves
are shown to
achieve their
highest utility at
points P* and Q*
respectively
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Riskless Lending and Borrowing
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So far we have assumed that all the securities on
the efficient set are risky.
Suppose that investors can also lend and borrow
money at risk free rate Rf
Risk-free asset has zero correlation with all the
points in the feasible region of risky portfolios.
So a combination of Rf and any point in the
feasible region of risky securities will be
represented by a straight line.
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Lending and Borrowing Opportunity
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Consider point Y, a
portfolio of risky
securities.
Investors can combine Rf
and Y and reach any
point along straight line
from Rf to Y and even
beyond – to go beyond
they have to leverage.
Lets refer this line as I
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Contd…
Although the investor can reach any point
on line I, no point on this line is optimal.
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Consider line II which
runs from Rf to S and
beyond.
Line II is tangent to the
efficient set of risky
securities, so it provides
the investor the best
possible opportunities.
Line II dominates line I –
for that matter, it
dominates any other line
between Rf and any point
in the feasible region of
risky securities.
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Contd…
With the opportunity of lending and
borrowing, the efficient frontier changes.
It is no longer AFX, it becomes Rf SG
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For every point on AFX,
there is at least one point
on Rf SG which is superior.
Compared to point C on
AFX, D on Rf SG offers
higher expected return for
the same standard
deviation.
Compared to point B on
AFX, E on Rf SG offers
higher expected return for
the same standard
deviation.
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Contd…
Every investor would do well to choose
some combination of Rf and S.
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A conservative investor
may choose point like U –
weighing Rf more in his
portfolio (lending)
An aggressive investor
may choose point like V –
weighing S more in his
portfolio (borrowing)
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Contd….
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Steps: Portfolio Selection
1.
2.
Identification of S, the optimal portfolio of
risky securities.
Choice of a combination of Rf and S,
depending on one’s risk attitude.
This is the import of celebrated Separation
Theorem, first enunciated by James Tobin, a
Nobel Laureate in Economics.
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The Single Index Model
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Markowitz model is highly information sensitive.
If there are n securities, the Markowitz model requires n
expected returns, n variance terms and n(n-1)/2
covariance terms. [Total inputs required = n(n+3)/2]
The problem of estimating a large number of covariance
terms becomes intractable, particularly for the institutional
investors.
Until the Markowitz model is simplified in terms of
covariance inputs, it can scarcely become an operational
tool.
In his seminal contribution, Markowitz had suggested
that an index, to which securities are related, may be used
for the purpose of generating the covariance terms.
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Contd…
Taking a cue from Markowitz, William Sharpe
developed the single index model, which expresses the
returns on each security as a function of the return on a
broad market index as follows:
Ri = ai + bi RM + ei
Where,
Ri = Return on security I
RM = Return on market index
ai = constant return
bi = measure of the sensitivity of the security i’s
return to the return on the market index (popularly
called beta)
ei = error term
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Contd…
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The estiamtes of ai (the constant term) and bi (the slope term) may be
obtained by regressing the return on security i on the corresponding
return on the market index.
Note that the single-index model is based on the following
assumptions:
• The error term ei has an expected value of zero and a finite
variance.
• The error term is not correlated with the return on the market
portfolio:
COV (ei , RM ) = 0
• Securities are related only through their common response to the
return on the market index. This implies that the error term for
security i is not correlated with the error term of any other
security:
COV (ei , ej ) = 0
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Generating the inputs to the Markowitz Model
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The single index model helps immensely in
obtaining the following inputs required for
applying the Markowitz model:
i. The expected return of each security
ii. The variance of return on each security, and
iii. The covariance of returns between each pair of
securities.
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The following equations may be used for the
purpose:
E(Ri ) = ai + bi E(RM )
Var (Ri ) = bi 2 [Var(RM )] + Var (ei )
Cov (Ri , Rj ) = bi bj Var (RM )
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Calculation of the Single Index Model
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Single index model requires only (3n+2) number of
inputs. [ai , bi , Var (ei ) for each security and E(RM ) and
Var (RM )]
William Sharpe found considerable similarity between
the efficient portfolios generated by single index model
and the Markowitz model.
Subsequent studies have also found that the single index
model performs well.
Since the single index model simplifies considerably the
input requirements and performs fairly well, it represents
a major practical advance in portfolio analysis.
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