MARKOWITZ MODEL AND EFFICIENT FRONTIER
The Markowitz Model of portfolio analysis generates
an efficient frontier, which is a set of efficient
Efficient portfolio is that portfolio which has no
The same expected return of portfolio
standard deviation of portfolio
The same standard deviation of portfolio and higher
expected return of portfolio
A higher expected return of portfolio and lower
FOR A TWO-SECURITY CASE
Coefficient of correlation
Proportion of A
Proportion of B
EFFICIENT FRONTIER FOR THE
return , E (Rp)
Standard deviation, σ p
ASSUMPTIONS OF MARKOWITZ MODEL
The Markowitz portfolio theory is based on a few
Investors are risk-averse and thus have a
preference for expected return and dislike for risk.
Investors act as if they make investment
decisions on the basis of the expected return and the
variance (or standard deviation) about security return
EVOLUTION OF CAPITAL ASSET PRICING MODEL
The CAPM was developed in the mid-1960s. The
model has generally been attributed to William
Sharpe, but John Lintner and Jan Mossin also made
similar independent derivations.
Consequently, the model is often referred to as
Sharpe-Lintner-Mossin (SLM) Capital Asset Pricing
The CAPM explains the relationship that should
exist between the securities’ expected returns and
their risks in terms of the means and standard
ASSUMPTIONS OF CAPM
The capital market theory is built on the basis of Markowitz’s
portfolio model. This theory is based on certain assumptions
All the investors are considered to be efficient investors who like to
position themselves on the efficient frontier.
Investors are free to borrow or lend any amount of money at the
Risk-Free Rate of Return (RFR).
All investors are expected to have homogeneous expectations.
All investors have same investment time horizons.
All investments are assumed to be infinitely divisible making it
possible to even buy or sell fractional shares of any portfolio.
The process of buying or selling of assets does not involve any
It is assumed that the inflation rate is fully anticipated, or in other
situation it may totally be absent thus resulting in no changes in the
Another assumption of the theory is the equilibrium in the capital
markets, that is, all the investments are correctly priced on par with
their risk levels.
STANDARD DEVIATION Vs. BETA
Standard deviation is a measure of the dispersion of a
set of data from its mean. The more spread apart the
data, the higher the deviation. Standard deviation is
calculated as the square root of variance.
Beta is a measure of the volatility, or systematic
risk, of a security or a portfolio in comparison to the
market as a whole.
one holding multiple assets, the contribution of any one
of the assets to the riskiness of the portfolio is its
systematic or non-diversifiable risk. Thus, for a welldiversified portfolio, the appropriate measure of risk
would be beta.
In simple words, a risky asset is one which gives
uncertain future returns whereas a risk-free
asset whose expected return is fully certain
and thus the standard deviation of such
expected returns comes to zero, i.e., σf=0.
Thus the rate of return earned on such assets
should be the risk-free rate of return (rf).
COVARIANCE OF RISK-FREE ASSET WITH A RISKY
The covariance between two sets of returns, A and B where
asset A is a risk free asset.
= Σ[rA- E(rA)] [ rB - E(rB)]/n
The uncertainty for a risk-free asset is known, so σA=0,
which implies that rA = E(rA) for all the periods. Thus, rA E(rA) =0, which further leads to the facts that the product of
any other expressions with this expression will be zero. This
will result in the covariance of the risk-free asset with any
risky asset or portfolio to be also zero. Similarly, the
correlation between any risky asset and risk-free asset, will
COMBINING A RISK-FREE ASSET WITH A
Any portfolio that combines a risk free asset with
any risky asset, the standard deviation is the
linear proportion of the standard deviation of the
risky asset portfolio.
RISK-RETURN POSSIBILITIES WITH LEVERAGE
An investor always wants to increase his expected
returns. Say, a person has borrowed an amount
which is 50 percent of his original wealth, the effect of
this on the expected return for the portfolio would be:
= Wf (rf) + (1 - Wf) E(rk) where k is the
risky assets portfolio
= – 0.50 (rf) + [1 – (–0.50)] E(rk)
= – 0.50 (rf) + 1.50 E(rk)
Thus, we see that the return increases in a linear fashion
along the line of risk-free rate (rf) and ‘k’.
Now, suppose E(rf) = 0.10
And E(rk) = 0.24
E(ri) = = – 0.50 (0.10) + 1.5 (0.24) = 0.31 or 31%
Similar is the effect of standard deviation of the
E(σi) = (1 – Wf) σk
= [1 – (–0.5)] σk = 1.50 σk
Portfolio Possibilities Combining the Risk-Free
Asset and Risky Portfolios on the Efficient Frontier
E(R port )
E(σ port )
Risk Return Possibilities with Leverage in the
To attain a higher expected return than is
available at point M (in exchange for accepting
Either invest along the efficient frontier beyond
point M, such as point D
Or, add leverage to the portfolio by borrowing
money at the risk-free rate and investing in the
risky portfolio at point M
Lending and Borrowing at the Riskfree rate
The portfolio expected return for any portfolio i that combines f and M is
E(ri) = WfRf + (1 – Wf) E (rM)
Wf = The percentage of the portfolio invested in the riskless security f
1 – Wf = The percentage of the portfolio invested in the risky portfolio M.
The portfolio variance for portfolio i is:
σi² =W²f σ²f + (1 - Wf)² σ²M + 2 Wf (1 - Wf) σf, M
By definition, σ²f = 0. Thus,
σ²i = (1 - Wf)² σ²M (or)
σi = (1 - Wf) σM
Borrowing And Lending At Riskfee Rate Rf
And Investing In The Risky Portfolio
Dominant Portfolio “M”
E(R port )
E(σ port )
RISKLESS LENDING AND
return , E (Rp)
• •F Y
Standard deviation, σ p
Thus, with the opportunity of lending and borrowing, the efficient frontier changes. It is no
longer AFX. Rather, it becomes Rf SG as it domniates AFX.
• Since Rf SG dominates AFX, every investor would do
well to choose some combination of Rf and S. A
conservative investor may choose a point like U,
whereas an aggressive investor may choose a point like
• Thus, the task of portfolio selection can be separated into
Identification of S, the optimal portfolio of risky
Choice of a combination of Rf and S, depending on
one’s risk attitude.
This is the import of the celebrated separation
Explanation : SEPARATION THEOREM
According to the portfolio theory, each investor should
choose an appropriate portfolio along the efficient
The particular portfolio chosen may or may not involve
borrowing or the use of leveraged, or short positions.
The investment decision (to choose efficient portfolio
from among others) and the financing decision
(whether or not their portfolio involved borrowing, or
short sales) are determined simultaneously in
accordance with the risk level, identified by the
investor at an acceptable level.
A portfolio that contains all securities is called the Market Portfolio.
Because all investors should choose the market portfolio, it
should contain all available securities.
THE MARKET PORTFOLIO IN PRACTICE
It is a known fact that the market portfolio has all different kinds
of stocks included, and when in equilibrium, the different assets
included in the portfolio are in proportion to their market value.
Ideally, the market portfolio should not only contain stocks and
bonds but also other assets such as coins, stamps, real estate
and options, each assigned a different weight in proportion to their
respective market values.
THE CAPITAL MARKET LINE (CML)
A line used in the capital asset pricing model to illustrate the
rates of return for efficient portfolios depending on the risk-free
rate of return and the level of risk (standard deviation) for a
The CML is derived by drawing a tangent line from the intercept
point on the efficient frontier to the point where the expected
return equals the risk-free rate of return.
The CML is considered to be superior to the efficient frontier
since it takes into account the inclusion of a risk-free asset in
Thus the CML not only represents the new efficient frontier, but
it also expresses the equilibrium pricing relationship between
E(r) and σ for all efficient portfolios lying along the line.
CAPITAL MARKET LINE
Rf + λ σj
E(RM) - Rf
STANDARD DEVIATION, σ p
THE CAPITAL ASSET PRICING MODEL (CAPM)
A model that describes the relationship between risk and
expected return and that is used in the pricing of risky
The general idea behind CAPM is that investors need to be
compensated in two ways: time value of money and risk.
The time value of money is represented by the risk-free (rf)
rate in the formula and compensates the investors for placing
money in any investment over a period of time.
The other half of the formula represents risk and calculates
the amount of compensation the investor needs for taking
on additional risk. This is calculated by taking a risk measure
(beta) that compares the returns of the asset to the market
over a period of time and to the market premium (Rm-rf).
The CAPM states that the expected return of a security
or a portfolio equals the rate on a risk-free security plus
a risk premium. If this expected return does not meet or
beat the required return, then the investment should not
be undertaken. The security market line plots the results
of the CAPM for all different risks (betas).
thus the CML is important in describing the
equilibrium relationship between expected return and
risk for efficient portfolios that contains no unsystematic
Thus as per the CAPM model, the expected return of
any asset is given by a formula of the form:
E[ri] = rf + [Number of Units of Risk][Risk Premium per
SECURITY MARKET LINE (SML)
The SML essentially graphs the results from the capital
asset pricing model (CAPM) formula. The x-axis
represents the risk (beta), and the y-axis represents
the expected return. The market risk premium is
determined from the slope of the SML.
E (R i ) = R f + [ E (R M) - R f ] β i
RELATIONSHIP BETWEEN SML AND CML
E(RM ) - Rf
E(Ri) = Rf +
ρ iM σi σM
E(RM ) - Rf
= Rf +
ρ iM σi
IF i AND M ARE PERFECTLY CORRELATED ρ iM = 1. SO
E(RM ) - Rf
= Rf +
APPLICATION OF CML AND CAPM
There are a number of applications of ex post SML in
security analysis and portfolio management. Among
(a) Evaluating the performance of portfolios;
(b) Tests of asset pricing theories; and
(c) Tests of market efficiency Ex ante SML can be used
to identify mispriced securities. It represents the liner
relationship between the expected rates of return for
securities and their expected betas.
PERFORMANCE EVALUATION OF PORTFOLIOS
The performance of portfolio is frequently evaluated
based on the security market line criterion –
A large positive alpha being taken as an indicator of
superior (above-normal) performance and
A large negative alpha being taken as an indicator of
inferior (below-normal) performance.
TESTS OF ASSET PRICING THEORIES
The CAPM pricing model is given by the
E(ri) = rf + [E(rM) – rf] βi
According to the theory, the expected return on
security i, E(ri), is related to the risk-free rate, rf,
plus a risk premium, [E(rM) – rf] βi ,which
includes the expected return on the market
TESTS OF MARKET EFFICIENCY
SML can also be used for testing market efficiency. As
we know, when markets are efficient, the scope for
abnormal return will not be there and returns on all
securities will be commensurate with the underlying
risk. That is, all assets are correctly priced and
provide a normal return for their level of risk and the
difference between return earned on the asset and
required rate of return on the asset should be
statistically insignificant if markets are efficient.
IDENTIFYING MISPRICED SECURITIES
The ex ante SML can be used for identifying mispriced
(under and overvalued) securities.