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57043bos46238cp5 PDF
57043bos46238cp5 PDF
PORTFOLIO MANAGEMENT
LEARNING OUTCOMES
After going through the chapter student shall be able to understand
Activities in Portfolio Management
Objectives of Portfolio Management
Phases of Portfolio Management
(1) Security Analysis
(2) Portfolio Analysis
(3) Portfolio Selection
(4) Portfolio Revision
(5) Portfolio Evaluation
Portfolio Theories
(1) Traditional Approach
(2) Modern Approach (Markowitz Model or Risk-Return Optimization)
Risk Analysis
(1) Elements of Risk
(2) Diversion of Risk
(3) Risk & Return
1. INTRODUCTION
Investment in the securities such as bonds, debentures and shares etc. is lucrative as well as
exciting for the investors. Though investment in these securities may be rewarding, it is also fraught
with risk. Therefore, investment in these securities requires a good amount of scientific and analytical
skill. As per the famous principle of not putting all eggs in the same basket, an investor never invests
his entire investable funds in one security. He invests in a well diversified portfolio of a number of
securities which will optimize the overall risk-return profile. Investment in a portfolio can reduce risk
without diluting the returns. An investor, who is expert in portfolio analysis, may be able to generate
trading profits on a sustained basis.
Every investment is characterized by return and risk. The concept of risk is intuitively understood by
investors. In general, it refers to the possibility of the rate of return from a security or a portfolio of
securities deviating from the corresponding expected/average rate and can be measured by the
standard deviation/variance of the rate of return.
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(ii) Stability of Income: To facilitate planning more accurately and systematically the
reinvestment or consumption of income.
(iii) Capital Growth: It can be attained by reinvesting in growth securities or through purchase of
growth securities.
(iv) Marketability i.e. the case with which a security can be bought or sold: This is essential
for providing flexibility to investment portfolio.
(v) Liquidity i.e. nearness to money: It is desirable for the investor so as to take advantage of
attractive opportunities upcoming in the market.
(vi) Diversification: The basic objective of building a portfolio is to reduce the risk of loss of
capital and/or income by investing in various types of securities and over a wide range of
industries.
(vii) Favourable Tax Status: The effective yield an investor gets from his investment depends on
tax to which it is subjected to. By minimising the tax burden, yield can be effectively improved.
and follow different techniques. Fundamental analysis, the older of the two approaches,
concentrates on the fundamental factors affecting the company such as
the EPS of the company,
the dividend pay-out ratio,
the competition faced by the company,
the market share, quality of management, etc.
fundamental factors affecting the industry to which the company belongs.
The fundamental analyst compares this intrinsic value (true worth of a security based on its
fundamentals) with the current market price. If the current market price is higher than the intrinsic
value, the share is said to be overpriced and vice versa. This mispricing of securities gives an
opportunity to the investor to acquire the share or sell off the share profitably. An intelligent investor
would buy those securities which are underpriced and sell those securities which are overpriced.
Thus it can be said that fundamental analysis helps to identify fundamentally strong companies
whose shares are worthy to be included in the investor's portfolio.
The second approach to security analysis is ‘Technical Analysis’. As per this approach the share
price movements are systematic and exhibit certain consistent patterns. Therefore, properly studied
past movements in the prices of shares help to identify trends and patterns in security prices and
efforts are made to predict the future price movements by looking at the patterns of the immediate
past. Thus Technical analyst concentrates more on price movements and ignores the fundamentals
of the shares.
In order to construct well diversified portfolios, so that Unsystematic Risk can be eliminated or
substantially mitigated, an investor will like to select securities across diverse industry sectors which
should not have strong positive correlation among themselves.
The efficient market hypothesis holds that-share price movements are random and not systematic.
Consequently, neither fundamental analysis nor technical analysis is of value in generating trading
gains on a sustained basis. The EMH thus does not subscribe to the belief that it is possible to book
gains in the long term on a sustained basis from trading in the stock market. Markets, though
becoming increasingly efficient everywhere with the passage of time, are never perfectly efficient.
So, there are opportunities all the time although their durations are decreasing and only the smart
investors can look forward to booking gains consistently out of stock market deals.
2.2 Portfolio Analysis
Once the securities for investment have been identified, the next step is to combine these to form a
suitable portfolio. Each such portfolio has its own specific risk and return characteristics which are
not just the aggregates of the characteristics of the individual securities constituting it. The return
and risk of each portfolio can be computed mathematically based on the risk-return profiles for the
constituent securities and the pair-wise correlations among them.
From any chosen set of securities, an indefinitely large number of portfolios can be constructed by
varying the fractions of the total investable resources allocated to each one of them. All such
portfolios that can be constructed out of the set of chosen securities are termed as Feasible
Portfolios.
2.3 Portfolio Selection
The goal of a rational investor is to identify the Efficient Portfolios out of the whole set of Feasible
Portfolios mentioned above and then to zero in on the Optimal Portfolio suiting his risk appetite. An
Efficient Portfolio has the highest return among all Feasible Portfolios having identical Risk and has
the lowest Risk among all Feasible Portfolios having identical Return. Harry Markowitz’s portfolio
theory (Modern Portfolio Theory) outlines the methodology for locating the Optimal Portfolio for an
investor (unlike the CAPM, the Optimal Portfolio as per Markowitz Theory is investor specific).
2.4 Portfolio Revision
Once an optimal portfolio has been constructed, it becomes necessary for the investor to constantly
monitor the portfolio to ensure that it does not lose it optimality. Since the economy and financial
markets are dynamic in nature, changes take place in these variables almost on a daily basis and
securities which were once attractive may cease to be so with the passage of time. New securities
with expectations of high returns and low risk may emerge. In light of these developments in the
market, the investor now has to revise his portfolio. This revision leads to addition (purchase) of
some new securities and deletion (sale) of some of the existing securities from the portfolio. The
nature of securities and their proportion in the portfolio changes as a result of the revision.
This portfolio revision may also be necessitated by some investor-related changes such as
availability of additional funds for investment, change in risk appetite, need of cash for other
alternative use, etc.
Portfolio revision is not a casual process to be taken lightly and needs to be carried out with care,
scientifically and objectively so as to ensure the optimality of the revised portfolio. Hence, in the
entire process of portfolio management, portfolio revision is as important as portfolio analysis and
selection.
2.5 Portfolio Evaluation
This process is concerned with assessing the performance of the portfolio over a selected period of
time in terms of return and risk and it involves quantitative measurement of actual return realized
and the risk borne by the portfolio over the period of investment. The objective of constructing a
portfolio and revising it periodically is to maintain its optimal risk return characteristics. Various types
of alternative measures of performance evaluation have been developed for use by investors and
portfolio managers.
This step provides a mechanism for identifying weaknesses in the investment process and for
improving these deficient areas.
It should however be noted that the portfolio management process is an ongoing process. It starts
with security analysis, proceeds to portfolio construction, and continues with portfolio -revision and
end with portfolio evaluation. Superior performance is achieved through continual refinement of
portfolio management skill.
Following three ratios are used to evaluate the portfolio:
2.5.1 Sharpe Ratio
Sharpe Ratio measures the Risk Premium per unit of Total Risk for a security or a portfolio of
securities. The formula is as follows:
Ri - R f
σi
Example: Let’s assume that we look at a one year period of time where an index fund earned 11%
Treasury bills earned 6%
The standard deviation of the index fund was 20%
Therefore S = 11-6/.20 = 25%
The Sharpe ratio is an appropriate measure of performance for an overall portfolio particularly when
it is compared to another portfolio, or another index such as the S&P 500, Small Cap index, etc.
That said however, it is not often provided in most rating services.
Example: Consider two Portfolios A and B. Let return of A be 30% and that of B be 25%. On the
outset, it appears that A has performed better than B. Let us now incorporate the risk factor and find
out the Sharpe ratios for the portfolios. Let risk of A and B be 11% and 5% respectively. This means
that the standard deviation of returns - or the volatility of returns of A is much higher than that of B.
If risk free rate is assumed to be 8%,
Sharpe ratio for portfolio A= (30-8)/11=2% and
Sharpe ratio for portfolio B= (25-8)/5=3.4%
Higher the Sharpe Ratio, better is the portfolio on a risk adjusted return metric. Hence, our primary
judgment based solely on returns was erroneous. Portfolio B provides better risk adjusted returns
than Portfolio A and hence is the preferred investment. Producing healthy returns with low volatility
is generally preferred by most investors to high returns with high volatility. Sharpe ratio is a good
tool to use to determine a portfolio that is suitable to such investors.
On the contrary, if we consider diversified equity funds, the element of unsystematic risk would be
very negligible as these funds are expected to be well diversified by virtue of their nature. Hence,
Treynor ratio would me more apt here.
It is widely found that both ratios usually give similar rankings. This is based on the fact that most of
the portfolios are fully diversified. To summarize, we can say that when the fund is not fully
diversified, Sharpe ratio would be a better measure of performance and when the portfolio is fully
diversified, Treynor ratio would better justify the performance of a fund.
Example: In 2005 - 06 where Fidelity Magellan had earned about 18%. Many bond funds had earned
13 %. Which is better? In absolute numbers, 18% beats 13%. But if we then state that the bond
funds had about half the market risk, now which is better? We don’t even need to do the formula for
that analysis. But that is missing in almost all reviews by all brokers. For clarification, we do not
suggest they put all the money into either one- just that they need to be aware of the implications.
2.5.3 Jensen Alpha
This is the difference between a portfolio’s actual return and those that could have been made on a
benchmark portfolio with the same risk- i.e. beta. It measures the ability of active management to
increase returns above those that are purely a reward for bearing market risk. Caveats apply
however since it will only produce meaningful results if it is used to compare two portfolios which
have similar betas.
Assume Two Portfolios
A B Market Return
Return 12 14 12
Beta 0.7 1.2 1.0
Risk Free Rate = 9%
The return expected = Risk Free Return + Beta portfolio (Return of Market - Risk Free Return)
Using Portfolio A, the expected return = 0 .09 + 0.7 (0.12 - 0.09) = 0.09 + 0.021 = 0.111
3. PORTFOLIO THEORIES
Portfolio theory forms the basis for portfolio management. Portfolio management deals with the
selection of securities and their continuous shifting in the portfolio to optimise returns to suit the
objectives of an investor. This, however, requires financial expertise in selecting the right mix of
securities in changing market conditions to get the best out of the stock market. In India as well as
in a number of Western countries, portfolio management service has assumed the role of a
specialised service and a number of professional investment bankers/fund managers compete
aggressively to provide the best options to high net-worth clients, who have little time to manage
their own investments. The idea is catching on with the growth of the capital market and an
increasing number of people want to earn profits by investing their hard-earned savings in a planned
manner.
A portfolio theory guides investors about the method of selecting and combining securities that will
provide the highest expected rate of return for any given degree of risk or that will expose the investor
to the lowest degree of risk for a given expected rate of return. Portfolio theory can be discussed
under the following heads:
3.1 Traditional Approach
The traditional approach to portfolio management concerns itself with the investor, definition of
portfolio objectives, investment strategy, diversification and selection of individual investment as
detailed below:
(i) Investor's study includes an insight into his – (a) age, health, responsibilities, other assets,
portfolio needs; (b) need for income, capital maintenance, liquidity; (c) attitude towards risk;
and (d) taxation status;
(ii) Portfolio objectives are defined with reference to maximising the investors' wealth which is
subject to risk. The higher the level of risk borne, the more the expected returns.
(iii) Investment strategy covers examining a number of aspects including:
(a) Balancing fixed interest securities against equities;
(b) Balancing high dividend payout companies against high earning growth companies as
required by investor;
(c) Finding the income of the growth portfolio;
(d) Balancing income tax payable against capital gains tax;
(e) Balancing transaction cost against capital gains from rapid switching; and
(f) Retaining some liquidity to seize upon bargains.
(iv) Diversification reduces volatility of returns and risks and thus adequate equity diversification
is sought. Balancing of equities against fixed interest bearing securities is also sought.
(v) Selection of individual investments is made on the basis of the following principles:
(a) Methods for selecting sound investments by calculating the true or intrinsic value of a
share and comparing that value with the current market value (i.e. by following the
fundamental analysis) or trying to predict future share prices from past price
movements (i.e., following the technical analysis);
(b) Expert advice is sought besides study of published accounts to predict intrinsic value;
(c) Inside information is sought and relied upon to move to diversified growth companies,
switch quickly to winners than loser companies;
(d) Newspaper tipsters about good track record of companies are followed closely;
(e) Companies with good asset backing, dividend growth, good earning record, high
quality management with appropriate dividend paying policies and leverage policies
are traced out constantly for making selection of portfolio holdings.
In India, most of the share and stock brokers follow the above traditional approach for selecting a
portfolio for their clients.
3.2 Modern Approach (Markowitz Model or Risk-Return Optimization)
Originally developed by Harry Markowitz in the early 1950's, Portfolio Theory - sometimes referred
to as Modern Portfolio Theory - provides a logical/mathematical framework in which investors can
optimise their risk and return. The central plank of the theory is that diversification through portfolio
formation can reduce risk, and return is a function of expected risk.
Harry Markowitz is regarded as the father of Modern Portfolio Theory. According to him, investors
are mainly concerned with two properties of an asset: risk and return. The essence of his theory is
that risk of an individual asset hardly matters to an investor. What really matters is the contribution
it makes to the investor's overall risk. By turning his principle into a useful technique for selecting
the right portfolio from a range of different assets, he developed the 'Mean Variance Analysis' in
1952.
We shall discuss this theory in greater detail later in this chapter.
4. RISK ANALYSIS
Before proceeding further it will be better if the concept of risk and return is discussed. A person
makes an investment in the expectation of getting some return in the future. But, the future is
uncertain and so is the future expected return. It is this uncertainty associated with the returns from
an investment that introduces risk for an investor.
It is important here to distinguish between the expected return and the realized return from an
investment. The expected future return is what an investor expects to get from his investment and
is uncertain. On the other hand, the realized return is what an investor actually obtains from his
investment at the end of the investment period. The investor makes the investment decision based
on the expected return from the investment. However, the actual return realized from the investment
may not correspond to the expected return. This possible variation of the actual return from the
expected return is termed as risk. If actual realizations correspond to expectations exactly, there
would be no risk. Risk arises where there is a possibility of variation between expectations and
realizations with regard to an investment.
Thus, risk arises from the variability in returns. An investment whose returns are fairly stable is
considered to be a low-risk investment, whereas an investment whose returns fluctuate significantly
is considered to be a highly risky investment. Government securities whose returns are fairly stable
and which are free from default are considered to possess low risk whereas equity shares whose
returns are likely to fluctuate widely around their mean are considered risky investments.
The essence of risk in an investment is the variation in its returns. This variation in returns is caused
by a number of factors. These factors which produce variations in the returns from an investment
constitute the elements of risk.
4.1 Elements of Risk
Let us consider the risk in holding securities, such as shares, debentures, etc. The elements of risk
may be broadly classified into two groups as shown in the following diagram.
The first group i.e. systematic risk comprises factors that are external to a company (macro in nature)
and affect a large number of securities simultaneously. These are mostly uncontrollable in nature.
The second group i.e. unsystematic risk includes those factors which are internal to companies
(micro in nature) and affect only those particular companies. These are controllable to a great extent.
The total variability in returns of a security is due to the total risk of that security. Hence,
Total risk = Systematic risk + Unsystematic risk
affecting all the companies. Unsystematic risk can be further subdivided into business risk and
financial risk.
(i) Business Risk: Business risk emanates from sale and purchase of securities affected by
business cycles, technological changes etc. Business cycles affect all types of securities viz. there
is cheerful movement in boom due to bullish trend in stock prices whereas bearish trend in
depression brings down fall in the prices of all types of securities. Flexible income securities are
more affected than fixed rate securities during depression due to decline in their market price.
(ii) Financial Risk: It arises due to changes in the capital structure of the company. It is also
known as leveraged risk and expressed in terms of debt-equity ratio. Excess of debt vis-à-vis equity
in the capital structure indicates that the company is highly geared. Although a leveraged company's
earnings per share are more but dependence on borrowings exposes it to the risk of winding-up for
its inability to honour its commitments towards lenders/creditors. This risk is known as leveraged or
financial risk of which investors should be aware of and portfolio managers should be very careful.
4.2 Diversion of Risk
As discussed above the total risk of an individual security consists of two risks systematic risk and
unsystematic risk. It should be noted that by combining many securities in a portfolio the
unsystematic risk can be avoided or cancelled out which is attached to any particular security. The
following diagram depicts how the risk can be reduced with the increase in the number of securities.
From the above diagram it can be seen that total risk is reducing with the increase in the number of
securities in the portfolio. However, ultimately when the size of the portfolio reaches certain limit, it
will contain only the systematic risk of securities included in the portfolio.
The above table gives the probability distribution of possible returns from an investment in shares.
Such distribution can be developed by the investor with the help of analysis of past data and
modifying it appropriately for the changes he expects to occur in a future period of time.
With the help of available probability distribution two statistical measures one expected return and
the other risk of the investment can be calculated.
4.3.1 Expected Return
The expected return of the investment is the probability weighted average of all the possible returns.
If the possible returns are denoted by Xi and the related probabilities are p(Xi) the expected return
may be represented as X and can be calculated as:
n
X= ∑x i p(X i)
i=1
It is the sum of the products of possible returns with their respective probabilities.
The expected return of the share in the example given above can be calculated as shown below:
Calculation of Expected Return
∑x i p(X i) 37.00
i=1
Variance of each security is generally denoted by σ2 and is calculated by using the following formula:
n
∑ [(X - X) i
2
p(X i )]
i=1
Continuing our earlier example the following table provides calculations required to calculate the risk
i.e. Variance or Standard Deviation (SD).
Possible Probability Deviation Deviation squared Product
returns Xi p(Xj) (Xi - X ) (Xi - X )2 (Xi - X )2 p(Xj)
(%)
20 0.20 -17.00 289.00 57.80
30 0.20 -7.00 49.00 9.80
40 0.40 3.00 9.00 3.60
50 0.10 13.00 169.00 16.90
60 0.10 23.00 529.00 52.90
Var (σ2) 141.00
because it is not diversifiable. Hence, the main interest of the investor lies in the measurement of
systematic risk of a security.
4.3.3 Measurement of Systematic Risk
As discussed earlier, systematic risk is the variability in security returns caused by changes in the
economy or the market and all securities are affected by such changes to some extent. Some
securities exhibit greater variability in response to market changes and some may exhibit less
response. Securities that are more sensitive to changes in factors are said to have higher systematic
risk. The average effect of a change in the economy can be represented by the change in the stock
market index. The systematic risk of a security can be measured by relating that security’s variability
vis-à-vis variability in the stock market index. A higher variability would indicate higher systematic
risk and vice versa.
The systematic risk of a security is measured by a statistical measure which is called Beta The main
input data required for the calculation of beta of any security are the historical data of returns of the
individual security and corresponding return of a representative market return (stock market index).
There are two statistical methods i.e. correlation method and the regression method, which can be
used for the calculation of Beta.
4.3.3.1 Correlation Method : Using this method beta (β) can be calculated from the historical data
of returns by the following formula:
rim σ i σ m
βi =
2
σm
Where
rim = Correlation coefficient between the returns of the stock i and the returns of the market
index.
σ i = Standard deviation of returns of stock i
4.3.3.2 Regression Method : The regression model is based on the postulation that there exists a
linear relationship between a dependent variable and an independent variable. The model helps to
calculate the values of two constants, namely alfa (α) and beta (β). β measures the change in the
dependent variable in response to unit change in the independent variable, while α measures the
value of the dependent variable even when the independent variable has zero value. The formula of
the regression equation is as follows:
Y = α + βX
where
Y = Dependent variable
X = Independent variable
α and β are constants.
α = Y - βX
The formula used for the calculation of α and β are given below.
n∑ XY - (∑ X)(∑ Y)
β=
n∑ X ( ∑ X)
2 2
where
n = Number of items.
Y = Dependent variable scores.
X = Independent variable scores.
For the purpose of calculation of β, the return of the individual security is taken as the dependent
variable and the return of the market index is taken as the independent variable. The regression
equation is represented as follows:
R i = α + βi Rm
where
Ri = Return of the individual security.
Rm = Retum of the market index.
α = Estimated return of the security when the market is stationary.
βi = Change in the return of the individual security in response to unit change in
the return of the market index. It is, thus, the measure of systematic risk of a security.
Here it is very important to note that a security can have betas that are positive, negative or zero.
• Positive Beta- indicates that security’s return is dependent on the market return and moves
in the direction in which market moves.
• Negative Beta- indicates that security’s return is dependent on the market return but moves
in the opposite direction in which market moves.
• Zero Beta- indicates that security’s return is independent of the market return.
Further as beta measures the volatility of a security’s returns relative to the market, the larger the
beta, the more volatile the security. A beta of 1.0 indicates a security of average risk. A stock with
beta greater than 1.0 has above average risk i.e. its returns would be more volatile than the market
returns. For example, when market returns move up by 6%, a stock with beta of 2 would find its
returns moving up by 12% (i.e. 6% x 2). Similarly, decline in market returns by 6% would produce a
decline of 12% (i.e. 6% x 2) in the return of that security.
A stock with beta less than 1.0 would have below average risk. Variability in its returns would be
less than the market variability.
Beta is calculated from historical data of returns to measure the systematic risk of a security. It is a
historical measure of systematic risk. In using this beta for investment decision making, the investor
is assuming that the relationship between the security variability and market variability will continue
to remain the same in future also.
4.4 Portfolio Analysis
Till now we have discussed the risk and return of a single security. Let us now discuss the return
and risk of a portfolio of securities.
4.4.1 Portfolio Return
For a portfolio analysis an investor first needs to specify the list of securities eligible for selection or
inclusion in the portfolio. Then he has to generate the risk-return expectations for these securities.
The expected return for the portfolio is expressed as the mean of its rates of return over the time
horizon under consideration and risk for the portfolio is the variance or standard deviation of these
rates of return around the mean return.
The expected return of a portfolio of assets is simply the weighted average of the returns of the
individual securities constituting the portfolio. The weights to be applied for calculation of the portfolio
return are the fractions of the portfolio invested in such securities.
Let us consider a portfolio of two equity shares A and B with expected returns of 16 per cent and 22
per cent respectively.
The formula for the calculation of expected portfolio return may be expressed as shown below:
n
rp = ∑ xi r i
i=1
r p = Expected return of the portfolio.
If 40 per cent of the total funds is invested in share A and the remaining 60 per cent in share B, then
the expected portfolio return will be:
(0.40 x 16) + (0.60 x 22) = 19.6 per cent
4.4.2 Portfolio Risk
As discussed earlier, the variance of return and standard deviation of return are statistical measures
that are used for measuring risk in investment. The variance of a portfolio can be written down as
the sum of 2 terms, one containing the aggregate of the weighted variances of the constituent
securities and the other containing the weighted co-variances among different pairs of securities.
Covariance (a statistical measure) between two securities or two portfolios or a security and a portfolio
indicates how the rates of return for the two concerned entities behave relative to each other.
The covariance between two securities A and B may be calculated using the following formula:
COVAB =
∑ [R A - R A ][RB - RB ]
N At the beginning please add the summation sign in the numerator
where
COVAB = Covariance between x and y.
RA = Return of security x.
RB = Return of security y.
R A = Expected or mean return of security x.
N = Number of observations.
The calculation of covariance can be understood with the help of following table:
Calculation of Covariance
Deviation Deviation
Year RX Rx - R x RY RY - R Y [R x - R x ][R y - R y ]
1 11 -4 18 5 -20
2 13 -2 14 1 -2
3 17 2 11 -2 -4
4 19 4 9 -4 -16
Rx = 15 Ry =13 -42
∑ [R x - R x ] [R y - R y ]
- 42
i=1
Cov xy = = = -10.5
n 4
From the above table it can be seen that the covariance is a measure of how returns of two securities
move together. In case the returns of the two securities move in the same direction consistently the
covariance is said to be positive (+). Contrarily, if the returns of the two securities move in opposite
directions consistently the covariance would be negative (-). If the movements of returns are
independent of each other, covariance would be close to zero (0).
The coefficient of correlation is expressed as:
Cov AB
rAB =
σ A σB
where
rAB = Coefficient of correlation between x and y.
CovAB = Covariance between A and B.
σA= Standard deviation of A.
σB = Standard deviation of B.
It may be noted on the basis of above formula the covariance can be expressed as the product of
correlation between the securities and the standard deviation of each of the securities as shown
below:
CovAB = σA σB rAB
It is very important to note that the correlation coefficients may range from -1 to 1. A value of -1
indicates perfect negative correlation between the two securities’ returns, while a value of +1
indicates a perfect positive correlation between them. A value of zero indicates that the returns are
independent.
The calculation of the variance (or risk) of a portfolio is not simply a weighted average of the
variances of the individual securities in the portfolio as in the calculation of the return of portfolio.
The variance of a portfolio with only two securities in it can be calculated with the following formula.
σ p2 = x 12 σ 12 + x 22 σ 22 + 2x 1 x 2 (r12 σ1σ 2 )
where
σp2 = Portfolio variance.
r= 20% 25%
σ= 50% 30%
rab= -0.60
Now suppose a portfolio is constructed with 40 per cent of funds invested in A and the remaining 60
per cent of funds in B (i.e. P = 0.4A + 0.6B).
Using the formula of portfolio return the expected return of the portfolio will be:
RP= (0.40 x 20) + (0.60 x 25) = 23%
And the Variance and Standard Deviation of the portfolio will be:
Variance
σp 2 = (0.40)2 (50)2 + (0.60)2 (30)2 + 2(0.40)(0.60)(- 0.60)(50)(30) = 400 + 324 - 432 = 292
Standard deviation
As the first set of factors is parametric in nature for the investor in the sense that he has no control
over the returns, risks and co-variances of individual securities. The second set of factors is choice
factor or variable for the investors in the sense that they can choose the proportions of each security
in the portfolio.
4.4.3 Reduction or dilution of Portfolio Risk through Diversification
The process of combining more than one security in to a portfolio is known as diversification. The
main purpose of this diversification is to reduce the total risk by eliminating or substantially mitigating
the unsystematic risk, without sacrificing portfolio return. As shown in the example mentioned above,
diversification has helped to reduce risk. The portfolio standard deviation of 17.09 is lower than the
standard deviation of either of the two securities taken separately which were 50 and 30 respectively.
Incidentally, such risk reduction is possible even when the two constituent securities are
uncorrelated. In case, however, these have the maximum positive correlation between them, no
reduction of risk can be achieved.
In order to understand the mechanism and power of diversification, it is necessary to consider the
impact of covariance or correlation on portfolio risk more closely. We shall discuss following three
cases taking two securities in the portfolio:
(a) Securities’ returns are perfectly positively correlated,
(b) Securities’ returns are perfectly negatively correlated, and
(c) Securities’ returns are not correlated i.e. they are independent.
4.4.3.1 Perfectly Positively Correlated : In case two securities returns are perfectly positively
correlated the correlation coefficient between these securities will be +1 and the returns of these
securities then move up or down together.
The variance of such portfolio can be calculated by using the following formula:
σ p2 = x 12 σ 12 + x 22 σ 22 + 2x 1 x 2 r12 σ1σ 2
or
σ = (x 1σ1 + x 2 σ 2 )
2 2
p
Taking the above example we shall now calculate the portfolio standard deviation when correlation
coefficient is +1.
Standard deviation of security A = 40
Standard deviation of security B = 25
Proportion of investment in A = 0.4
Proportion of investment in B = 0.6
Correlation coefficient = +1.0
Portfolio standard deviation maybe calculated as:
σp = (0.4) (40) + (0.6) (25) = 31
Thus it can be seen that the portfolio standard deviation will lie between the standard deviations of
the two individual securities. It will vary between 40 and 25 as the proportion of investment in each
security changes.
Now suppose, if the proportion of investment in A and B are changed to 0.75 and 0.25 respectively;
portfolio standard deviation of the portfolio will become:
σp = (0.75) (40)+ (0.25) (25) = 36.25
It is important to note that when the security returns are perfectly positively correlated, diversification
provides only risk averaging and no risk reduction because the portfolio risk cannot be reduced
below the individual security risk. Hence, reduction of risk is not achieved when the constituent
securities’ returns are perfectly positively correlated.
4.4.3.2 Perfectly Negatively Correlated : When two securities’ returns are perfectly negatively
correlated, two returns always move in exactly opposite directions and correlation coefficient
between them becomes -1. The variance of such negatively correlated portfolio may be calculated
as:
σp2 = x12 σ12 + x 22 σ22 − 2x1x 2 ( r12 σ1σ2 )
σ p2 = (x1σ1 - x 2σ 2 )2
Taking the above example we shall now calculate the portfolio standard deviation when correlation
coefficient is -1.
σp = (0.4)(40) - (0.6)(25) =1
Thus from above it can be seen that the portfolio risk has become very low in comparison of risk of
individual securities. By changing the weights it can even be reduced to zero. For example, if the
proportion of investment in A and B are 0.3846 and 0.6154 respectively, portfolio standard deviation
becomes:
= (0.3846)(40) - (0.6154)(25) = 0
Although in above example the portfolio contains two risky assets, the portfolio has no risk at all.
Thus, the portfolio may become entirely risk-free when security returns are perfectly negatively
correlated. Therefore, diversification can substantially reduce or even eliminate risk when securities
are perfectly negatively correlated, . However, in real life it is very rare to find securities that are
perfectly negatively correlated.
4.4.3.3 Returns are uncorrelated or independent : When the returns of two securities are entirely
uncorrelated, the coefficient of correlation of these two securities would be zero and the formula for
portfolio variance will be as follows:
σ p2 = x 12 σ12 + x 22 σ 22 + 2x 1 x 2 r12 σ1σ 2
σ p2 = x12σ12 + x 22σ 22
σ p = x 12 σ 12 + x 22 σ 22
Taking the above example we shall now calculate the portfolio standard deviation when correlation
coefficient is 0.
Correlation Portfolio
coefficient Standard Deviation
1.00 31
0.60 27.73
0 21.93
-0.60 13.89
-1.00 1.00
Summarily it can be concluded that diversification reduces risk in all cases except when the security
returns are perfectly positively correlated. With the decline of correlation coefficient from +1 to -1,
the portfolio standard deviation also declines. But the risk reduction is greater when the security
returns are negatively correlated.
4.4.4 Portfolio with more than two securities
So far we have considered a portfolio with only two securities. The benefits from diversification
increase as more and more securities with less than perfectly positively correlated returns are
included in the portfolio. As the number of securities added to a portfolio increases, the standard
deviation of the portfolio becomes smaller and smaller. Hence, an investor can make the portfolio
risk arbitrarily small by including a large number of securities with negative or zero correlation in the
portfolio.
But, in reality, no securities show negative or even zero correlation. Typically, securities show some
positive correlation, that is above zero but less than the perfectly positive value (+1). As a result,
diversification (that is, adding securities to a portfolio) results in some reduction in total portfolio risk
but not in complete elimination of risk. Moreover, the effects of diversification are exhausted fairly
rapidly. That is, most of the reduction in portfolio standard deviation occurs by the time the portfolio
size increases to 25 or 30 securities. Adding securities beyond this size brings about only marginal
reduction in portfolio standard deviation.
Adding securities to a portfolio reduces risk because securities are not perfectly positively correlated.
But the effects of diversification are exhausted rapidly because the securities are still positively
correlated to each other though not perfectly correlated. Had they been negatively correlated, the
portfolio risk would have continued to decline as portfolio size increased. Thus, in practice, the
benefits of diversification are limited.
The total risk of an individual security comprises two components, the market related risk called
systematic risk and the unique risk of that particular security called unsystematic risk. By combining
securities into a portfolio the unsystematic risk specific to different securities is cancelled out.
Consequently, the risk of the portfolio as a whole is reduced as the size of the portfolio increases.
Ultimately when the size of the portfolio reaches a certain limit, it will contain only the systematic
risk of securities included in the portfolio. The systematic risk, however, cannot be eliminated. Thus,
a fairly large portfolio has only systematic risk and has relatively little unsystematic risk. That is why
there is no gain in adding securities to a portfolio beyond a certain portfolio size. Following figure
depicts the diversification of risk in a portfolio.
The figure shows the portfolio risk declining as the number of securities in the portfolio increases,
but the risk reduction ceases when the unsystematic risk is eliminated.
4.4.5 Calculation of Return and Risk of Portfolio with more than two securities
The expected return of a portfolio is the weighted average of the returns of individual securities in
the portfolio, the weights being the proportion of investment in each security. The formula for
calculation of expected portfolio return is the same for a portfolio with two securities and for portfolios
with more than two securities. The formula is:
n
rp = ∑ x i ri
i=1
Where
rp = Expected return of portfolio.
Q 16 0.2
R 22 0.1
S 20 0.4
The expected return of this portfolio may be calculated using the formula:
rp = (0.3)(11) + (0.2)(16) + (0.1)(22) + (0.4)(20)
where
σp2 = Portfolio variance.
xi = Proportion of funds invested in security i (the first of a pair of securities).
xj = Proportion of funds invested in security j (the second of a pair of securities).
σij = The covariance between the pair of securities i and j
n = Total number of securities in the portfolio.
or
n n
2
σ =
p ∑∑ x i x j σ i σ jri j
i=1 i=1
where
σp2 = Portfolio variance.
σi = Standard Deviation of security i
σj = Standard Deviation of security j
rij = The co-efficient of correlation between the pair of securities i and j
Let us take the following example to understand how we can compute the risk of multiple asset
portfolio.
X Y Z
0.25 X 1 x 16 x 16 0.7 x 16 x 7 0.3 x 16 x 9
0.35 Y 0.7 x 7 x 16 1x7x7 0.4 x 7 x 9
0.40 Z 0.3 x 9 x 16 0.4 x 9 x 7 1x9x9
Once the variance-covariance matrix is set up, the computation of portfolio variance is a
comparatively simple operation. Each cell in the matrix represents a pair of two securities.
When all these products are summed up, the resulting figure is the portfolio variance. The square
root of this figure gives the portfolio standard deviation.
Thus the variance of the portfolio given in the example above can now be calculated.
σ p2 = (0.25 x 0.25 x 1 x 16 x 16) + (0.25 x 0.35 x 0.7 x 16 x 7) + (0.25 x 0.40 x 0.3 x 16 x 9) +
(0.35 x 0.25 x 0.7 x 7 x 16) + (0.35 x 0.35 x 1 x 7 x 7) + (0.35 x 0.40 x 0.4 x7 x 9) + (0.40 x
0.25 x 0.3 x 9 x 16) + (0.40 x 0.35 x 0.4 x 9 x 7) + (0.40 x 0.40 x 1 x 9 x 9)
= 16+6.86+4.32+6.86+6.0025+3.528+4.32+3.528+12.96 = 64.3785
The portfolio standard deviation is:
σp= 64.3785 =8.0236
Hence, the formula for computing portfolio variance may also be stated as follows:
n n
σ = ∑∑ x i x j ri j σ i σ j
2
p
i =1 j =1
Thus from above discussion it can be said that a portfolio is a combination of assets. From a given
set of 'n' securities, any number of portfolios can be created. These portfolios may comprise of two
securities, three securities, all the way up to 'n' securities. A portfolio may contain the same securities
as another portfolio but with different weights. A new portfolios can be created either by changing
the securities in the portfolio or by changing the proportion of investment in the existing securities.
Thus summarily it can be concluded that each portfolio is characterized by its expected return and
risk. Determination of expected return and risk (variance or standard deviation) of each portfolio that
can be used to create a set of selected securities which is the first step in portfolio management and
called portfolio analysis.
(v) All investors are risk averse. For a given expected return he prefers to take minimum risk, for
a given level of risk the investor prefers to get maximum expected return.
(vi) Investors are assumed to be rational in so far as they would prefer greater returns to lesser
ones given equal or smaller risk and are risk averse. Risk aversion in this context means
merely that, as between two investments with equal expected returns, the investment with
the smaller risk would be preferred.
(vii) ‘Return’ could be any suitable measure of monetary inflows like NPV but yield has been the
most commonly used measure of return, so that where the standard deviation of returns is
referred to it is meant the standard deviation of yield about its expected value.
5.2 Efficient Frontier
Markowitz has formalised the risk return relationship and developed the concept of efficient frontier.
For selection of a portfolio, comparison between combinations of portfolios is essential. As a rule, a
portfolio is not efficient if there is another portfolio with:
(a) A higher expected value of return and a lower standard deviation (risk).
(b) A higher expected value of return and the same standard deviation (risk)
(c ) The same expected value but a lower standard deviation (risk)
Markowitz has defined the diversification as the process of combining assets that are less than
perfectly positively correlated in order to reduce portfolio risk without sacrificing any portfolio returns.
If an investors’ portfolio is not efficient he may:
(i) Increase the expected value of return without increasing the risk.
(ii) Decrease the risk without decreasing the expected value of return, or
(iii) Obtain some combination of increase of expected return and decrease risk.
This is possible by switching to a portfolio on the efficient frontier.
Portfolio has the highest return among all portfolios with identical risk and the lowest risk among all
portfolios with identical return). Fig – 1 depicts the boundary of possible investments in securities,
A, B, C, D, E and F; and B, C, D, are lying on the efficient frontier.
The best combination of expected value of return and risk (standard deviation) depends upon the
investors’ utility function. The individual investor will want to hold that portfolio of securities which
places him on the highest indifference curve, choosing from the set of available portfolios. The dark
line at the top of the set is the line of efficient combinations, or the efficient frontier. The optimal
portfolio for an investor lies at the point where the indifference curve for the concerned investor
touches the efficient frontier. This point reflects the risk level acceptable to the investor in order to
achieve a desired return and provide maximum return for the bearable level of risk. The concept of
efficient frontier and the location of the optimal portfolio are explained with help of Fig-2.
The shaded area represents all attainable or feasible portfolios, that is all the combinations of risk
and expected return which may be achieved with the available securities. The efficient frontier
contains all possible efficient portfolios and any point on the frontier dominates any point to the right
of it or below it.
Consider the portfolios represented by points B and E. B and E promise the same expected return
E (R1) but the risk associated with B is σ (R1) whereas the associated with E is σ (R2). Investors,
therefore, prefer portfolios on the efficient frontier rather than interior portfolios given the assumption
of risk aversion; obviously, point A on the frontier represents the portfolio with the least possible risk,
whilst D represents the portfolio with the highest possible rate of return with highest risk.
The investor has to select a portfolio from the set of efficient portfolios lying on the efficient frontier.
This will depend upon his risk-return preference. As different investors have different preferences,
the optimal portfolio of securities will vary from one investor to another.
CAPM provides a conceptual framework for evaluating any investment decision where capital is
committed with a goal of producing future returns. CAPM is based on certain assumptions to provide
conceptual framework for evaluating risk and return. Some of the important assumptions are
discussed below:
(i) Efficient market: It is the first assumption of CAPM. Efficient market refers to the existence
of competitive market where financial securities and capital assets are bought and sold with
full information of risk and return available to all participants. In an efficient market, the price
of individual assets will reflect a real or intrinsic value of a share as the market prices will
adjust quickly to any new situation, John J. Hampton has remarked in “Financial decision
making” that although efficient capital market is not much relevant to capital budgeting
decisions, but CAPM would be useful to evaluate capital budgeting proposal because the
company can compare risk and return to be obtained by investment in machinery with risk
and return from investment in securities.
(ii) Rational investment goals: Investors desire higher return for any acceptable level of risk or
the lowest risk for any desired level of return. Such a rational choice is made on logical and
consistent ranking of proposals in order of preference for higher good to lower good and this
is the scale of the marginal efficiency of capital. Beside, transactive preferences and certainty
equivalents are other parameters of rational choice.
(iii) Risk aversion in efficient market is adhered to although at times risk seeking behaviour is
adopted for gains.
(iv) CAPM assumes that all assets are divisible and liquid assets.
(v) Investors are able to borrow freely at a risk less rate of interest i.e. borrowings can fetch equal
return by investing in safe Government securities.
(vi) Securities can be exchanged without payment of brokerage, commissions or taxes and
without any transaction cost.
(vii) Securities or capital assets face no bankruptcy or insolvency.
Based on above assumptions the CAPM is developed with the main goal to formulate the return
required by investors from a single investment or a portfolio of assets. The required rate of return is
defined as the minimum expected return needed so that investors will purchase and hold an asset.
Risk and return relationship in this model stipulates higher return for higher level of risk and vice
versa. However, there may be exception to this general rule where markets are not efficient.
Three aspects are worth consideration:
(a) Stock market is not concerned with diversifiable risk
(b) It is not concerned with an investor having a diversified portfolio
(c) Compensation paid is restricted to non-diversifiable risk.
Thus an investor has to look into the non-diversifiable portion of risk on one side and returns on the
other side. To establish a link between the two, the required return one expects to get for a given
level of risk has been mandated by the Capital Asset Pricing Model.
If the risk free investment Rf is 5%, an investor can earn this return of 5% by investing in risk free
investment. Again if the stock market earns a rate of return Rm which is 15% then an investor
investing in stocks constituting the stock market index will earn also 15%. Thus the excess return
earned over and above the risk free return is called the risk premium (Rm – Rf) ie (15% - 5%) = 10%
which is the reward for undertaking risk, So, if an investment is as risky as the stock market, the risk
premium to be earned is 10%.
If an investment is 30% riskier than the stock market, it would carry risk premium i.e. 30% more than
the risk premium of the stock market i.e. 10% + 30% of 10% = 10% + 3% = 13%. β identifies how
much more risky is an investment with reference to the stock market. Hence the risk premium that a
stock should earn is β times the risk premium from the market [β × (Rm – Rf)]. The total return from
an investment is the risk free rate of return plus the risk premium. So the required return from a stock
would be Rj = Rf + [β × (Rm – Rf)]. In the above example 5% + 1.3 × (15-5) = 18%
The risk premium on a stock varies in direct proportion to its Beta. If the market risk premium is 6%
and β of a stock is 1.2 then the risk premium for that stock is 7.2% (6% × 1.2) where (Rm – Rf) = 6%
and β =1.2
Illustration 1
A company’s beta is 1.40. The market return is 14%. The risk free rate is 10% (i) What is the
expected return based on CAPM (ii) If the risk premium on the market goes up by 2.5% points, what
would be the revised expected return on this stock?
Solution
(i) Computation of expected return based on CAPM
Rj = Rf + β (Rm – Rf) = 10% + 1.40 (14% - 10%) = 10% + 5.6% = 15.6%
(ii) Computation of risk premium if the market goes up by 2.5 points
The return from the market goes up by 2.5% i.e. 14% + 2.5% = 16.5%
Expected Return based on CAPM is given by
Rj = 10% + 1.40 (16.5% - 10%) = 10% + 1.40 × 6.5% = 10% + 9.1% = 19.1%
7.1 Security Market Line
A graphical representation of CAPM is the Security Market Line, (SML). This line indicates the rate
of return required to compensate at a given level of risk. Plotting required return on Y axis and Beta
on the X-axis we get an upward sloping line which is given by (Rm – Rf), the risk premium.
The higher the Beta value of a security, higher would be the risk premium relative to the market.
This upward sloping line is called the Security Market Line. It measures the relationship between
systematic risk and return.
Illustration 2
The risk premium for the market is 10%. Assuming Beta values of 0, 0.25, 0.42, 1.00 and 1.67.
Compute the risk premium on Security K.
Solution
Market Risk Premium is 10%
0.00 0%
0.25 2.50%
0.42 4.20%
1.00 10.00%
1.67 16.70%
Illustration 3
Treasury Bills give a return of 5%. Market Return is 13% (i) What is the market risk premium (ii)
Compute the β Value and required returns for the following combination of investments.
Market 0 30 70 100
Solution
Risk Premium Rm – Rf = 13% - 5% = 8%
β is the weighted average investing in portfolio consisting of market β = 1 and treasury bills (β = 0)
Treasury Bills:
Portfolio β Rj = Rf + β × (Rm – Rf)
Market
1 100:0 0 5% + 0(13%-5%)=5%
2 70:30 0.7(0)+0.3(1)=0.3 5%+0.3(13%-5%)=7.40%
3 30:70 0.3(0)+0.7(1)=0.7 5%+0.7(13%-5%)=10.60%
4 0:100 1 5%+1.0(13%-5%)=13%
Year Returns
25 + ( 279 - 242 )
2002 – 2003 ×100 = 25.62%
242
30 + ( 305 - 279 )
2003 – 2004 ×100 = 20.07%
279
35 + ( 322 - 305 )
2004 – 2005 ×100 = 17.05%
305
Year X Y XY Y2
2002-2003 25.62 12.62 323.32 159.26
2003-2004 20.07 21.79 437.33 474.80
2004-2005 17.05 5.32 90.71 28.30
62.74 39.73 851.36 662.36
62.74 39.73
=X = 20.91,
= Y = 13.24
3 3
∑ XY − nXY
β=
2
∑ Y 2 − nY
851.36 - 3(20.91)(13.24)
=
662.36 - 3(13.24)2
851.36 - 830.55 20.81
= = = 0.15
662.36 - 525.89 136.47
Stock A B C
Expected Return (%) 18 11 15
Beta Factor 1.7 0.6 1.2
If the risk free rate is 9% and the expected rate of return on the market portfolio is 14% which of the
above stocks are over, under or correctly valued in the market? What shall be the strategy?
Solution
Required Rate of Return is given by
Rj = Rf + β (Rm-Rf)
For Stock A, Rj = 9 + 1.7 (14 - 9) = 17.50%
Stock B, Rj = 9 + 0.6 (14-9) = 12.00%
Stock C, Rj = 9 + 1.2 (14-9) = 15.00%
Illustration 7
Information about return on an investment is as follows:
Risk free rate 10% (b) Market Return is 15% (c) Beta is 1.2
(i) What would be the return from this investment?
(ii) If the projected return is 18%, is the investment rightly valued?
(iii) What is your strategy?
Solution
Required rate of Return as per CAPM is given by
Rj = Rf + β (Rm-Rf)
= 10 +1.2 (15-10) = 16%
If projected return is 18%, the stock is undervalued as CAPM < Expected Return .The Decision
should be BUY.
7.4 Modification for leverage
The above mentioned discussions have assumed all equity financing and that the beta used in the
equations is an unlevered beta. However, the beta is actually a function of the leverage as well as
the business risk .As a company increases the proportion of debt capital in its capital structure, both
its beta and the required return increase in a linear manner. Hence in case one wishes to use the
CAPM as a model for valuing cost of equity in order to determine financially feasible investments,
one needs to take into account the difference of leverage in the proxy company/project and the
company/project whose required return is to be computed.
Mathematically
D
β j= β uj 1 + (1- T) where β j & β uj are the levered and unlevered betas respectively., D/S is the debt
S
to equity ratio in market value terms and T is the corporate tax rate.
7.5 Advantages and Limitations of CAPM
The advantages of CAPM can be listed as:
(i) Risk Adjusted Return: It provides a reasonable basis for estimating the required return on an
investment which has risk in built into it. Hence it can be used as Risk Adjusted Discount
Rate in Capital Budgeting.
(ii) No Dividend Company: It is useful in computing the cost of equity of a company which does
not declare dividend.
There are certain limitations of CAPM as well, which are discussed as follows:
(a) Reliability of Beta: Statistically reliable Beta might not exist for shares of many firms. It may
not be possible to determine the cost of equity of all firms using CAPM. All shortcomings that
apply to Beta value applies to CAPM too.
(b) Other Risks: By emphasing on systematic risk only, unsystematic risks are of importance to
share holders who do not possess a diversified portfolio.
(c) Information Available: It is extremely difficult to obtain important information on risk free
interest rate and expected return on market portfolio as there is multiple risk free rates for
one while for another, markets being volatile it varies over time period.
Where, λ 1, λ 2 , λ 3 , λ 4 are average risk premium for each of the four factors in the model and
βi1 , βi 2 , βi 3 , βi 4 are measures of sensitivity of the particular security i to each of the four factors.
Where,
Ri = expected return on security i
α i = intercept of the straight line or alpha co-efficient
β i = slope of straight line or beta co-efficient
R m = the rate of return on market index
∈i = error term.
According to the equation, the return of a stock can be divided into two components, the return due
to the market and the return independent of the market. βi indicates the sensitiveness of the stock
return to the changes in the market return. For example, βi of 1.5 means that the stock return is
expected to increase by 1.5 % when the market index return increases by 1 % and vice-versa.
Likewise, βi of 0.5 expresses that the individual stock return would change by 0.5 per cent when
there is a change of 1 per cent in the market return. β i of 1 indicates that the market return and the
security return are moving in tandem. The estimates of βi and αi are obtained from regression
analysis.
The single index model is based on the assumption that stocks vary together because of the common
movement in the stock market and there are no effects beyond the market (i.e. any fundamental
factor effects) that account the stocks co-movement. The expected return, standard deviation and
co-variance of the single index model represent the joint movement of securities. The mean return
is:
R i = α i + β i R m + ∈i
The variance of security’s return:
σ 2 = β2 i σ 2 m + σ 2 ∈i
The covariance of returns between securities i and j is:
σ ij = β iβ j σ 2m
The variance of the security has two components namely, systematic risk or market risk and
unsystematic risk or unique risk. The variance explained by the index is referred to systematic risk.
The unexplained variance is called residual variance or unsystematic risk.
The systematic risk can be calculated by using following formula:
Systematic risk = β 2 i × variance of market index
2 2
= β iσ m
Unsystematic risk = Total variance - Systematic risk.
∈i 2 =
σi2 - Systematic risk.
Thus, the total risk = Systematic risk + Unsystematic risk.
= β2i σ 2m + ∈2i .
From this, the portfolio variance can be derived
N 2
2
N
σ p = ∑ X i β i σ m + ∑ Xi 2 ∈i2
2
i=1 i =1
Where,
σ p2
= variance of portfolio
Likewise expected return on the portfolio also can be estimated. For each security αi and βi should
be estimated.
N
R P = ∑ x i (α i + β i R m )
i =1
Where,
Solution
The co-efficient of determination (r2) gives the percentage of the variation in the security’s return that is
explained by the variation of the market index return. In the X company stock return, 18 per cent of
variation is explained by the variation of the index and 82 per cent is not explained by the index.
According to Sharpe, the variance explained by the index is the systematic risk. The unexplained
variance or the residual variance is the unsystematic risk.
Company X:
Unsystematic risk(
∈2i ) = Total variance of security return - systematic risk
= 6.3 – 1.134
= 5.166 or
= Variance of Security Return (1-r2)
= 6.3 X (1-0.18) = 6.3 X 0.82 = 5.166
Total risk = β 2ι x σ m2 + ∈2ι
= 1.134 + 5.166 = 6.3
Company Y:
Systematic risk = βi2 x σ m2
= (0.685)2 x 2.25 = 1.056
Unsystematic risk = Total variance of the security return - systematic risk.
= 5.86-1.056 = 4.804
2
N 2 N 2 2
σ p = ∑ X i β i σ m + ∑ X i ∈i
2
i=1 i=1
(
N R - R f βi
σ2m ∑ i
)
i=1 2
σei
Ci =
N β2
1 + σ2m ∑ i
2
i = 1 σei
Where,
σ m2 = variance of the market index
σ ∈i2 = variance of a stock’s movement that is not associated with the movement of
market index i.e. stock’s unsystematic risk.
(d) Compute the cut-off point which the highest value of Ci and is taken as C*. The stock whose
excess-return to risk ratio is above the cut-off ratio are selected and all whose ratios are
below are rejected. The main reason for this selection is that since securities are ranked from
highest excess return to Beta to lowest, and if particular security belongs to optional portfolio
all higher ranked securities also belong to optimal portfolio.
(e) Once we came to know which securities are to be included in the optimum portfolio, we shall
calculate the percent to be invested in each security by using the following formula:
Zi
Xio =
N
∑ Zi
j=1
where
Bi Ri - Ro
Zi = - C*
σ2ei Bi
The first portion determines the weight each stock and total comes to 1 to ensure that all funds are
invested and second portion determines the relative investment in each security.
rise in the interest rate one may shift for long term bonds to medium and short term. A long term
bond is more sensitive to interest rate variation compared to a short term bond.
(c) Security Selection: Security selection involves a search for under price security. If one has to
resort to active stock selection he may employ fundamental / technical analysis to identify stocks
which seems to promise superior return and concentrate the stock components of portfolio on them.
Such stock will be over weighted relative to their position in the market portfolio. Like wise stock
which are perceived to be unattractive will be under weighted relative to their position in the market
portfolio.
As far as bonds are concerned security selection calls for choosing bonds which offer the highest
yields to maturity and at a given level of risk.
(d) Use of Specialised Investment Concept: To achieve superior return, one has to employ a
specialised concept/philosophy particularly with respect to investment in stocks. The concept which
have been exploited successfully are growth stock, neglected or out of favour stocks, asset stocks,
technology stocks and cyclical stocks.
The advantage of cultivating a specialized investment concept is that it helps to:
(i) Focus one’s effort on a certain kind of investment that reflects one’s ability and talent.
(ii) Avoid the distraction of perusing other alternatives.
(iii) Master an approach or style through sustained practice and continual self criticism.
The greatest disadvantage of focusing exclusively on a specialized concept is that it may become
obsolete. The changes in the market risk may cast a shadow over the validity of the basic premise
underlying the investor philosophy.
10.2 Passive Portfolio Strategy
Active strategy was based on the premise that the capital market is characterized by efficiency which
can be exploited by resorting to market timing or sector rotation or security selection or use of special
concept or some combination of these sectors.
Passive strategy, on the other hand, rests on the tenet that the capital market is fairly efficient with
respect to the available information. Hence they search for superior return. Basically, passive
strategy involves adhering to two guidelines. They are:
(a) Create a well diversified portfolio at a predetermined level of risk.
(b) Hold the portfolio relatively unchanged over time unless it became adequately diversified or
inconsistent with the investor risk return preference.
A fund which is passively managed are called index funds. An Index fund is a mutual fund scheme
that invests in the securities of the target Index in the same proportion or weightage. Though it is
designed to provide returns that closely track the benchmark Index, an Index Fund carries all the
risks normally associated with the type of asset the fund holds. So, when the overall stock market
rises/falls, you can expect the price of shares in the index fund to rise/fall, too. In short, an index
fund does not mitigate market risks. Indexing merely ensures that your returns will not stray far from
the returns on the Index that the fund mimics. In other words, an index fund is a fund whose daily
returns are the same as the daily returns obtained from an index. Thus, it is passively managed in
the sense that an index fund manager invests in a portfolio which is exactly the same as the portfolio
which makes up an index. For instance, the NSE-50 index (Nifty) is a market index which is made
up of 50 companies. A Nifty index fund has all its money invested in the Nifty fifty companies, held
in the same weights of the companies which are held in the index.
10.3 Selection of Securities
There are certain criteria which must be kept in mind while selecting securities. The selection criteria
for both bonds and equity shares are given as following:
10.3.1 Selection of Bonds
Bonds are fixed income avenues. The following factors have to be evaluated in selecting fixed
income avenues:
(a) Yield to maturity: The yield to maturity for a fixed income avenues represent the rate of return
earned by the investor, if he invests in the fixed income avenues and holds it till its maturity.
(b) Risk of Default: To assess such risk on a bond, one has to look at the credit rating of the
bond. If no credit rating is available relevant financial ratios of the firm have to be examined
such as debt equity, interest coverage, earning power etc and the general prospect of the
industry to which the firm belongs have to be assessed.
(c) Tax Shield: In the past, several fixed income avenues offers tax shields but at present only a
few of them do so.
(d) Liquidity: If the fixed income avenues can be converted wholly or substantially into cash at a
fairly short notice it possesses a liquidity of a high order.
10.3.2 Selection of Stock (Equity Share)
Three approaches are applied for selection of equity shares- Technical analysis, Fundamental
analysis and Random selection analysis.
(a) Technical analysis looks at price behaviours and volume data to determine whether the share
will move up or down or remain trend less.
(b) Fundamental analysis focuses on fundamental factors like earning level, growth prospects
and risk exposure to establish intrinsic value of a share. The recommendation to buy hold or
sell is based on comparison of intrinsic value and prevailing market price.
(c) Random selection analysis is based on the premise that the market is efficient and security
is properly priced.
Levels of Market Efficiency And Approach To Security Selection
This policy is suitable for the investor whose risk tolerance is positively related to portfolio and stock
market return but drops to zero of below floor value.
Concluding, it can be said that following are main features of this policy:
(a) The value of portfolio is positively related and linearly dependent on the value of the stock.
(b) The value of portfolio cannot fall below the floor value i.e. investment in Bonds.
(c) This policy performs better if initial percentage is higher in stock and stock outperform the
bond. Reverse will happen if stock under perform in comparison of bond or their prices goes
down.
(b) Constant Mix Policy: Contrary to above policy this policy is a ‘do something policy’. Under
this policy investor maintains an exposure to stock at a constant percentage of total portfolio. This
strategy involves periodic rebalancing to required (desired) proportion by purchasing and selling
stocks as and when their prices goes down and up respectively. In other words this plan specifies
that value of aggressive portfolio to the value of conservative portfolio will be held constant at a pre-
determined ratio. However, it is important to this action is taken only there is change in the prices of
share at a predetermined percentage.
For example if an investor decided his portfolio shall consist of 60% in equity shares and balance
40% in bonds on upward or downward of 10% in share prices he will strike a balance.
In such situation if the price of share goes down by 10% or more, he will sell the bonds and invest
money in equities so that the proportion among the portfolio i.e. 60:40 remains the same. According
if the prices of share goes up by 10% or more he will sell equity shares and shall in bonds so that
the ratio remains the same i.e. 60:40. This strategy is suitable for the investor whose tolerance varies
proportionally with the level of wealth and such investor holds equity at all levels.
The pay-off diagram of this policy shall be as follows:
Accordingly, it gives a concave pay off, tends to do well in flat but fluctuating market.
Continuing above example let us how investor shall rebalance his portfolio under different scenarios
as follows:
(a) If price decreases
Share Value Value Total Stock to Bond to
Price of of Bond Stock
Shares Bonds Switching Switching
100 Starting Level 50,000 50,000 1,00,000 - -
80 Before 40,000 50,000 90,000 - -
Rebalancing
After 45,000 45,000 90,000 - 5,000
Rebalancing
60 Before 33,750 45,000 78,750 - -
Rebalancing
After 39,360 39,390 78,750 - 5,610
Rebalancing
(c) Constant Proportion Insurance Policy : Under this strategy investor sets a floor below which
he does not wish his asset to fall called floor, which is invested in some non-fluctuating assets such
as Treasury Bills, Bonds etc. The value of portfolio under this strategy shall not fall below this
specified floor under normal market conditions. This strategy performs well especially in bull market
as the value of shares purchased as cushion increases. In contrast in bearish market losses are
avoided by sale of shares. It should however be noted that this strategy performs very poorly in the
market hurt by sharp reversals. The following equation is used tp determine equity allocation:
Target Investment in Shares = multiplier (Portfolio Value – Floor Value)
Basis Buy & Hold Policy Constant Mix Policy Constant Proportion
Portfolio Insurance
Pay-off Line Straight Concave Convex
Protection in Definite in Down Not much in Down Good in Down
Down/Up Markets market market but relatively market and performs
poor in Up market well in Up market
(c) International Spread Swap – In this swap portfolio manager is of the belief that yield spreads
between two sectors is temporarily out of line and he tries to take benefit of this mismatch.
Since the spread depends on many factor and a portfolio manager can anticipate appropriate
strategy and can profit from these expected differentials.
(d) Tax Swap – This is based on taking tax advantage by selling existing bond whose price
decreased at capital loss and set it off against capital gain in other securities and buying
another security which has features like that of disposed one.
(3) Interest Rate Swap: Interest Rate Swap is another technique that is used by Portfolio Manager.
This technique has been discussed in greater details in the chapter on Derivative.
5. Hedge Funds
6. Closely Held Companies
7. Distressed Securities
8. Commodities
9. Managed Futures
10. Mezzanine Finance
Since, some of the above terms have been covered under the respective chapter in this study, we
shall cover other terms hereunder.
14.1 Real Estates
As opposed to financial claims in the form of paper or a dematerialized mode, real estate is a tangible
form of assets which can be seen or touched. Real Assets consists of land, buildings, offices,
warehouses, shops etc.
Although real investment is like any other investment but it has some special features as every
country has their own laws and paper works which makes investment in foreign properties less
attractive. However, in recent time due to globalization investment in foreign real estate has been
increased.
14.1.1 Valuation Approaches
Comparing to financial instrument the valuation of Real Estate is quite complex as number of
transactions or dealings comparing to financial instruments are very small.
Following are some characteristics that make valuation of Real Estate quite complex:
(i) Inefficient market: Information as may not be freely available as in case of financial securities.
(ii) Illiquidity: Real Estates are not as liquid as that of financial instruments.
(iii) Comparison: Real estates are only approximately comparable to other properties.
(iv) High Transaction cost: In comparison to financial instruments, the transaction and
management cost of Real Estate is quite high.
(v) No Organized market: There is no such organized exchange or market as for equity shares
and bonds.
14.1.2 Valuation of Real Estates
Generally, following four approaches are used in valuation of Real estates:
(1) Sales Comparison Approach – It is like Price Earning Multiplier as in case of equity shares.
Benchmark value of similar type of property can be used to value Real Estate.
(2) Income Approach – This approach like value of Perpetual Debenture or unredeemable
Preference Shares. In this approach the perpetual cash flow of potential net income (after deducting
expense) is discounted at market required rate of return.
(3) Cost Approach – In this approach, the cost is estimated to replace the building in its present
form plus estimated value of land. However, adjustment of other factors such as good location,
neighborhood is also made in it.
(4) Discounted After Tax Cash Flow Approach – In comparison to NPV technique, PV of expected
inflows at required rate of return is reduced by amount of investment.
14.2 Private Equity
Following 3 types of private equity investment shall be discussed here:
14.2.1 Mezzanine Finance
It is a blend or hybrid of long term debt and equity share. It is a kind of equity funding combined with
the characteristics of conventional lending as well as equity. This is a highly risky investment and
hence mezzanine financer receives higher return.
This type of financing enhances the base of equity as in case of default the debt is converted into
equity. Mezzanine financing can be used for financing heavy investments, buyout, temporary
arrangement between sanction of heavy loan and its disbursement. However, compared to western
world, this type of financing is not so popular in India.
14.2.2 Venture Capital
The History of Venture Capital in India can be traced back to the 70’s, when the Government of
India, getting aware that an inadequate funding and financial structure was hampering
entrepreneurialism and start-ups, appointed a committee to tackle the issue. Approximately ten
years later, the first three all- Indian funds were standing: IDBI, ICICI and IFCI.
With the institutionalization of the industry in November 1988, the government announced its
guidelines in the “CCI” (Controller of Capital Issues). These focused on a very narrow description of
Venture Capital and proved tobe extremely restrictive and encumbering, requiring investment in
«innovative technologies started by first generation entrepreneur. This made investment in VC highly
risky and unattractive.
At about the same time, the World Bank organized a VC awareness seminar, giving birth to players
like: TDICICI, GVFL, Canbank and Pathfinder. Along with the other reforms the government decided
to liberalize the VC Industry and abolish the “CCI”, while in 1995 Foreign Finance companies were
allowed to invest in the country.
Nevertheless, the liberalization was short-spanned,with new calls for regulation being made in 1996.
The new guidelines’ loopholes created an unequal playing ground that favoured the foreign players
and gave no incentives to domestic high net worth individuals to invest in this industry.
VC investing got considerably boosted by the IT revolution in 1997, as the venture capitalists became
prominent founders of the growing IT and telecom industry.
Many of these investors later floundered during the dotcom bust and most of the surviving ones
shifted their attention to later stage financing, leaving the risky seed and start-up financing to a few
daring funds.
14.2.2.1 Structure of fund in India : Three main types of fund structure exist: one for domestic
funds and two for offshore ones:
(a) Domestic Funds : Domestic Funds (i.e. one which raises funds domestically) are usually
structured as:
i) a domestic vehicle for the pooling of funds from the investor, and
ii) a separate investment adviser that carries those duties of asset manager.
The choice of entity for the pooling vehicle falls between a trust and a company, (India, unlike most
developed countries does not recognize a limited partnership), with the trust form prevailing due to
its operational flexibility.
(b) Offshore Funds : Two common alternatives available to offshore investors are: the “offshore
structure” and the “unified structure”.
Offshore structure
Under this structure, an investment vehicle (an LLC or an LP organized in a jurisdiction outside
India) makes investments directly into Indian portfolio companies. Typically, the assets are managed
by an offshore manager, while the investment advisor in India carries out the due diligence and
identifies deals.
Unified Structure
When domestic investors are expected to participate in the fund, a unified structure is used.
Overseas investors pool their assets in an offshore vehicle that invests in a locally managed trust,
whereas domestic investors directly contribute to the trust. This is later device used to make the
local portfolio investments.
Venture capital means funds made available for startup firms and small businesses with exceptional
growth potential. Venture capital is money provided by professionals who alongside management
invest in young, rapidly growing companies that have the potential to develop into significant
economic contributors.
Venture Capitalists generally:
• Finance new and rapidly growing companies
• Purchase equity securities
• Assist in the development of new products or services
• Add value to the company through active participation.
14.2.2.2 Characteristics : Venture capital follows the following characteristics:
Long time horizon: The fund would invest with a long time horizon in mind. Minimum period of
investment would be 3 years and maximum period can be 10 years.
Lack of liquidity: When VC invests, it takes into account the liquidity factor. It assumes that there
would be less liquidity on the equity it gets and accordingly it would be investing in that format. they
adjust this liquidity premium against the price and required return.
High Risk: VC would not hesitate to take risk. It works on principle of high risk and high return. So
higher riskiness would not eliminate the investment choice for a venture capital.
Equity Participation: Most of the time, VC would be investing in the form of equity of a company.
This would help the VC participate in the management and help the company grow.
Besides, a lot of board decisions can be supervised by the VC if they participate in the equity of a
company.
14.2.2.3 Advantages: Advantages of brining VC in the company:
It injects long- term equity finance which provides a solid capital base for future growth.
The venture capitalist is a business partner, sharing both the risks and rewards. Venture
capitalists are rewarded with business success and capital gain.
The venture capitalist is able to provide practical advice and assistance to the company based
on past experience with other companies which were in similar situations.
The venture capitalist also has a network ofcontacts in many areas that can add value to the
company.
The venture capitalist may be capable of providing additional rounds of funding should it be
required to finance growth.
Venture capitalists are experienced in the process of preparing a company for an initial public
offering (IPO) of its shares onto the stock exchanges or overseas stock exchange such as
NASDAQ.
They can also facilitate a trade sale.
14.2.2.4 Stages of funding: Stages of funding for VC:
1. Seed Money: Low level financing needed to prove a new idea.
2. Start-up: Early stage firms that need funding for expenses associated with marketing and
product development.
3. First-Round: Early sales and manufacturing funds.
4. Second-Round: Working capital for early stage companies that are selling product, but not
yet turning in a profit.
5. Third Round: Also called Mezzanine financing, this is expansion money for a newly profitable
company
6. Fourth-Round: Also called bridge financing, it is intended to finance the "going public" process
14.2.2.5 Risk matrix : Risk in each stage is different. An indicative Risk matrix is given below:
14.2.2.6 VC Investment Process: The entire VC Investment process can be segregated into the
following steps:
1. Deal Origination: VC operates directly or through intermediaries. Mainly many practicing
Chartered Accountants would work as intermediary and through them VC gets the deal.
Before sourcing the deal, the VC would inform the intermediary or its employees about the following
so that the sourcing entity does not waste time:
• Sector focus
• Stages of business focus
• Promoter focus
• Turn over focus
Here the company would give a detailed business plan which consists of business model, financial
plan and exit plan . All these aspects are covered in a document which is called Investment
Memorandum (IM). A tentative valuation is also carried out in the IM.
2. Screening: Once the deal is sourced the same would be sent for screening by the VC. The
screening is generally carried out by a committee consisting of senior level people of the VC. Once
the screening happens, it would select the company for further processing.
3 Due Diligence:. The screening decision would take place based on the information provided by
the company. Once the decision is taken to proceed further, the VC would now carry out due
diligence. This is mainly the process by which the VC would try to verify the veracity of the documents
taken. This is generally handled by external bodies, mainly renowned consultants. The fees of due
diligence are generally paid by the VC .
However, in many case this can be shared between the investor (VC) and Investee (the company)
depending on the veracity of the document agreement.
4. Deal Structuring: Once the case passes through the due diligence it would now go through the
deal structuring. The deal is structured in such a way that both parties win. In many cases, the
convertible structure is brought in to ensure that the promoter retains the right to buy back the share.
Besides, in many structures to facilitate the exit, the VC may put a condition that promoter has also
to sell part of its stake along with the VC . Such a clause is called tag- along clause.
5. Post Investment Activity: In this section, the VC nominates its nominee in the board of the
company. The company has to adhere to certain guidelines like strong MIS, strong budgeting
system, strong corporate governance and other covenants of the VC and periodically keep the VC
updated about certain mile-stones. If milestone has not been met the company has to give
explanation to the VC. Besides, VC would also ensure that professional management is set up in
the company.
6. Exit plan: At the time of investing , the VC would ask the promoter or company to spell out in
detail the exit plan. Mainly, exit happens in two ways: one way is ‘sell to third paty(ies)’ . This sale
can be in the form of IPO or Private Placement to other VCs. The second way to exit is that promoter
would give a buy back commitment at a pre- agreed rate (generally between IRR of 18% to 25%) .
In case the exit is not happening in the form of IPO or third party sell, the promoter would buy back.
In many deals, the promoter buyback is the first refusal method adopted i.e. the promoter would get
the first right of buyback.
14.2.3 Distressed securities
It is a kind of purchasing the securities of companies that are in or near bankruptcy. Since these
securities are available at very low price, the main purpose of buying such securities is to make
efforts to revive the sick company. Further, these securities are suitable for those investors who
cannot participate in the market and those who wants to avoid due diligence.
Now, question arises how profit can be earned from distressed securities. We can see by taking long
position in debt and short position in equity, how investor can earn arbitrage profit.
(i) In case company’s condition improves because of priority, the investor will get his interest
payment which shall be more than the dividend on his short position in equity shares.
(ii) If company is condition further deteriorates the value of both share and debenture goes down.
He will make good profit from his short position.
Risks Analysis of Investment in Distressed Securities : On the face, investment in distressed
securities appears to be a good proposition but following types of risks are need to be analyzed.
Price Probability
115 0.1
120 0.1
125 0.2
130 0.3
135 0.2
140 0.1
Required:
(i) Calculate the expected return.
(ii) Calculate the Standard deviation of returns.
2. Mr. A is interested to invest ` 1,00,000 in the securities market. He selected two securities B
and D for this purpose. The risk return profile of these securities are as follows :
You are required to calculate the portfolio return of the following portfolios of B and D to be
considered by A for his investment.
(i) 100 percent investment in B only;
(ii) 50 percent of the fund in B and the rest 50 percent in D;
(iii) 75 percent of the fund in B and the rest 25 percent in D; and
(iv) 100 percent investment in D only.
Also indicate that which portfolio is best for him from risk as well as return point of view?
3. Consider the following information on two stocks, A and B :
A B C D E F
Return (%) 8 8 12 4 9 8
Risk (Standard deviation) 4 5 12 4 5 6
(i) Assuming three will have to be selected, state which ones will be picked.
(ii) Assuming perfect correlation, show whether it is preferable to invest 75% in A and
25% in C or to invest 100% in E
5. The distribution of return of security ‘F’ and the market portfolio ‘P’ is given below:
Probability Return %
F P
0.30 30 -10
0.40 20 20
0.30 0 30
You are required to calculate the expected return of security ‘F’ and the market portfolio ‘P’,
the covariance between the market portfolio and security and beta for the security.
6. The rates of return on the security of Company X and market portfolio for 10 periods
are given below:
Period Return of Security X (%) Return on Market Portfolio (%)
1 20 22
2 22 20
3 25 18
4 21 16
5 18 20
6 −5 8
7 17 −6
8 19 5
9 −7 6
10 20 11
(i) What is the beta of Security X?
(ii) What is the characteristic line for Security X?
7. XYZ Ltd. has substantial cash flow and until the surplus funds are utilised to meet the future
capital expenditure, likely to happen after several months, are invested in a portfolio of short-
term equity investments, details for which are given below:
You are required to compute Reward to Volatility Ratio and rank these portfolio using:
♦ Sharpe method and
♦ Treynor's method
assuming the risk free rate is 6%.
ANSWERS/ SOLUTIONS
Answers to Theoretical Questions
1. Please refer paragraph 10.3
2. Please refer paragraph 1.2
3. Please refer paragraph 7
Answers to the Practical Questions
1. Here, the probable returns have to be calculated using the formula
D P1 − P0
R
= +
P0 P0
Calculation of Probable Returns
n n
σ2p = ∑∑ w i w jρij σ i σ j
i=1 j=1
(ii) Stock A:
Variance = 0.5 (10 – 13)² + 0.5 (16 – 13) ² = 9
Standard deviation = 9 = 3%
Stock B:
Variance = 0.5 (12 – 15) ² + 0.5 (18 – 15) ² = 9
Standard deviation = 3%
(iii) Covariance of stocks A and B
CovAB = 0.5 (10 – 13) (12 – 15) + 0.5 (16 – 13) (18 – 15) = 9
(iv) Correlation of coefficient
Cov AB 9
rAB = = =1
σ A σB 3 × 3
σP = X 2 A σ2 A + X 2Bσ2B + 2X A X B (σ A σBσ AB )
Co Var PM − 168
Beta= = = − .636
σ M2 264
6. (i)
Period R X RM R X − R X RM − RM (R X )(
− R X RM − RM ) (R M − RM )2
1 20 22 5 10 50 100
2 22 20 7 8 56 64
3 25 18 10 6 60 36
4 21 16 6 4 24 16
5 18 20 3 8 24 64
6 -5 8 -20 -4 80 16
7 17 -6 2 -18 -36 324
8 19 5 4 -7 -28 49
9 -7 6 -22 -6 132 36
10 20 11 5 -1 -5 1
150 120 357 706
ΣRX ΣRM ∑ (R X − R X )(R M − R M ) ∑ (R M − R M )
2
R X = 15 R M = 12
2
−
∑ RM − RM
= 706 = 70.60
σ2 M =
n 10
−
−
∑ R X − R X R M − R M
= 357
CovX M= = 35.70
n 10
Cov X M m 35.70
Betax = = = 0.505
σ 2M 70.60
Alternative Solution
Period X Y Y2 XY
1 20 22 484 440
2 22 20 400 440
3 25 18 324 450
4 21 16 256 336
5 18 20 400 360
6 -5 8 64 -40
7 17 -6 36 -102
8 19 5 25 95
9 -7 6 36 -42
10 20 11 121 220
150 120 2146 2157
X = 15 Y = 12
ΣXY - n X Y
=
ΣX 2 - n(X)2
2157 - 10 × 15 × 12 357
= = = 0.506
2146 - 10 × 12 × 12 706
(ii) R X = 15 R M = 12
y = α + βx
15 = α + 0.505 × 12
Alpha (α) = 15 – (0.505 × 12) = 8.94%
Characteristic line for security X = α + β × RM
Where, RM = Expected return on Market Index
∴Characteristic line for security X = 8.94 + 0.505 RM
7. (i) Computation of Beta of Portfolio
Investment No. of Market Market Dividend Dividend Composition β Weighted
shares Price Value Yield β
I. 60,000 4.29 2,57,400 19.50% 50,193 0.2339 1.16 0.27
II. 80,000 2.92 2,33,600 24.00% 56,064 0.2123 2.28 0.48
III. 1,00,000 2.17 2,17,000 17.50% 37,975 0.1972 0.90 0.18
IV. 1,25,000 3.14 3,92,500 26.00% 1,02,050 0.3566 1.50 0.53
11,00,500 2,46,282 1.0000 1.46
2,46,282
Return of the Portfolio = 0.2238
11,00,500
Beta of Port Folio 1.46
β2A × σ M
2
= (0.40)2(0.01) = 0.0016
βB2 × σ M
2
= (0.50)2(0.01) = 0.0025
β2C × σ M
2
= (1.10)2(0.01) = 0.0121
Residual Variance
A 0.015 – 0.0016 = 0.0134
B 0.025 – 0.0025 = 0.0225
C 0.100 – 0.0121 = 0.0879
(iii) Portfolio variance using Sharpe Index Model
Systematic Variance of Portfolio = (0.10)2 x (0.66)2 = 0.004356
Unsystematic Variance of Portfolio = 0.0134 x (0.20)2 + 0.0225 x (0.50)2 + 0.0879 x
(0.30)2 = 0.014072
Total Variance = 0.004356 + 0.014072 = 0.018428
(iii) Portfolio variance on the basis of Markowitz Theory
2
= (wA x wAx σ A ) + (wA x wBxCovAB) + (wA x wCxCovAC) + (wB x wAxCovAB) + (wB x wBx
σ B2 ) + (wB x wCxCovBC) + (wC x wAxCovCA) + (wC x wBxCovCB) + (wC x wCx σ 2c )
= (0.20 x 0.20 x 0.015) + (0.20 x 0.50 x 0.030) + (0.20 x 0.30 x 0.020) + (0.20 x 0.50
x 0.030) + (0.50 x 0.50 x 0.025) + (0.50 x 0.30 x 0.040) + (0.30 x 0.20 x 0.020) + (0.30
x 0.50 x 0.040) + (0.30 x 0.30 x 0.10)
= 0.0006 + 0.0030 + 0.0012 + 0.0030 + 0.00625 + 0.0060 + 0.0012 + 0.0060 + 0.0090
= 0.0363
11. Return of the stock under APT
Factor Actual Expected Difference Beta Diff. х
value in % value in % Beta
GNP 7.70 7.70 0.00 1.20 0.00