The
Markowitz
Portfolio
Theory
Theory and
Applications
Shafin Shabir Naik
AAA1325
Contents
1. Introduction
2. Portfolio Expected Value and Variance
3. Diversification
4. Mean Variance Optimization
5. Efficient Frontier
6. Efficient Frontier in Excel
7. Bibliography
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Introduction
People invest with the aim of earning returns on their investments. But these
returns are uncertain which creates an element of risk for the investors.
Nevertheless, investor is also interested in the total return and rate of return
that he gets from his investment. The formulas for calculating them for a
single asset are given below
Total Return, R =
amount received
amount invested
Rate of return, r = amount received – amount invested
Amount invested
i.e,
R=
r=
where,
r = rate of return
X1 = amount received
X0 = amount invested
But normally people invest in more than one asset. Suppose an investor
invests in n asset where n= 1,2,3...,n . His total budget is X0. We form a
portfolio using this information. This can be done by associating a weight for
each asset such that the sum of weights is 1, i.e,
wi can take negative values as well. Whenever wi is negative the concerned
asset is said to be shorted. This process is called short selling.
The total return(R) for the portfolio is
where, Ri denotes the total return of asset i.
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Also, the rate of return of the portfolio is
where ri denotes the rate of return of asset i.
Portfolio Expected Value and Variance
When an asset is originally acquired, its rate of return(r) is usually uncertain.
Therefore we consider r to be a random variable. Accordingly we calculate the
expected value and the variance of the portfolio.
Expected ValueTake n assets with random rates of return r1 , r2, ...,rn each having expected
values E(r1)= µ 1,E(r2) = µ 2,..., E(rn) = µ n.
The expected rate of return in terms of portfolio will be
VarianceVariance represents the risk of investing in an asset. It is denoted by .
Covariance is denoted by
for two assets i and j. Variance of a portfolio is
given by the following formula
Diversification
Investors are well aware of the adage “Don’t put all your eggs in the same
basket.” This is the essence of diversification. It is the systematic process by
which variance of a portfolio can be decreased by including additional assets in
the portfolio.
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Diversification can be mathematically stated using the formulas of combining
variances. Suppose n assets of equal weights are mutually uncorrelated in a
portfolio. Therefore, wi = 1/n for each i. Let expected rate of return be m and
variance . The overall rate of return will be
The corresponding variance is
The graph above shows effect of diversification on the non-market risk. As we
increase the number of securities the overall risk decreases.
So far we have formula for calculating variance of a portfolio with uncorrelated
assets. For correlated securities variance is calculated in the following ways. Let
the covariance of cov(ri,rj) = k for i≠j. Therefore variance
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Thus we have a formula for calculating risk in all cases. We can now prove the
claims that diversification really leads to lower risk. However we have assumed
expected rates of returns to be fixed. In general, as our variance decreases due to
diversification so does our expected rates of return. Therefore, blind
diversification may decrease our returns. This is where Markowitz Portfolio
Theory comes handy. Harry Markowitz solved this problem in a systematic
way and helped investors to make rational decisions regarding how much to
invest.
Mean Variance Optimization
Harry Markowitz was the first economist who formalized the trade-off between
higher returns and lower risk. He proposed the following approach: for a given
level of expected returns, find the portfolio allocation with smallest risk. His
theory is summarized in the diagram given below.
Mathematically, the optimization problem can be solved as follows.
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Assume that there are n assets. The expected rates of return are µ 1,µ 2,…,µn and
the covariances are
for i,j = 1,2,3,…,n. A portfolio is defined by a set of n
weights wi = 1,…,n that sum to 1. To find minimum-variance, we fix the
expected value at µ. Therefore the problem will be
minimize
subject to
The factor ½ is for convenience only.
To solve this problem we use Lagrange multipliers λ and µ.
Since we need the minimum, we differentiate the Lagrangian with respect to
each variable and set the derivative to zero.
This comes out to be equal to
Therefore we have n equations plus the two equations of constraints, which
implies that we have a total of n+2 equations. Correspondingly, there are n+2
unknowns- wi’s, µ and λ. The solution of these equations will produce the
weights for an efficient portfolio with mean R.1
Efficient Frontier
Portfolio is represented on a mean-standard deviation diagram. Now if we fix
the level of risk at , we have two possible returns- µ 1 and µ 2 where µ 1<µ 2. A
rational investor will prefer µ 2 over µ 1. We assume that investor shows
monotonic preferences. He will do whenever µ<µ i where i=1,2,3,...,n.
Therefore, he will always lie above µ in the mean-standard deviation diagram.
1
Investment Science by David G. Luenberger
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µ2
µ
µ1
Diagram- Portfolio Frontier
This portion of the diagram is called the efficient frontier. The shaded green
region is the efficient frontier for the above diagram. Any point in the efficient
frontier corresponds to portfolio allocation w* such that for any other portfolio
allocation w´ yielding return µ´ and standard deviation we have the following:
If
< , then µ´<µ*, and if µ´ > µ* then > . In simple terms, no other
portfolio allocation can achieve a higher expected return than µ* with smaller
risk than
.2
Efficient Frontier in Excel
Due to its importance, I will illustrate a way to plot the efficient frontier in
excel. Take two securities, ONGC and BPCL. The expected return and
corresponding risk of these two securities are shown below.
Expected Returns
Standard Deviation
ONGC
BPCL
0.15
0.34
0.3
0.4
I have assumed that these assets are negatively correlated with correlation equal
to -0.3. Now take different possible portfolios that you can create from these
two assets. (For simplicity, I have made short-selling illegal in this example. So
w can never be less than 0).
Kai Lai Chu g, Farid Aitsahlia, Ele e tary Pro a ility Theory ith “to hasti Pro ess a d a I trodu tio to
Mathe ati al Fia a e 00 .
2
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Portfolio
A
B
C
D
E
ONGC
BPCL
1
0.8
0.5
0.3
0
0
0.2
0.5
0.7
1
Expected Return
Standard Deviation
0.15
0.3
0.188
0.229085137
0.245
0.210950231
0.283
0.267170358
0.34
0.4
I have calculated the expected returns and standard deviation using the formulas
given in this paper. After that we plot the Mean-Standard deviation diagram.
Mean-Standard Deviation Diagram
0.4
0.35
0.3
0.25
0.2
Efficient Frontier
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
Now the efficient frontier will be the increasing portion of graph with 0.25 as
the lowest return. This helps an investor to make a decision, on whether how
much return he should expect given the risk of acquiring it.
For example, corresponding to 34% return investor has to face 40% risk and
corresponding to 25% return he faces 20% risk.
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Conclusion
Markowitz Portfolio Theory plays an important role for an investor. It allows
him to take informed decision about his investment. This theory also explains
the trade-off between maximizing returns and minimizing the associated risk
with the return. According to Bartvold and Begg, “The basic premise of portfolio
theory is that the variance of returns for a portfolio of risky assets is a function not
only of the variance of each individual asset, but also of the covariance [or
correlation] between each asset (variance of return, or standard deviation of return,
can be considered a measure of economic risk). When multiple risky assets are held
within a portfolio, it can be expected that some properties will increase in value while
at the same time others will decrease in value. By holding risky assets in groups,
some of the risk of each asset may be reduced or eliminated through the process of
diversification.
Additional potential arises from understanding the relationship among the respective
projects relative to success or failure. If two projects are negatively correlated, i.e. if
success in one project is associated with failure of the other, and failure in one is
associated with success in the other, this significantly reduces the risk of double
failure across the two projects. If independent, the full value of diversification is
achieved; if negatively correlated, natural hedges reduce the risk of complete loss
potentially to zero.”3
3
Brat old, R. B.; Begg, “. H. E e Opti ists “hould Opti ize , “PE
, prepared for the “PE A
Technical Conference and Exhibition held in Denver, Colorado, 5—8 October 2003.
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