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PORTFOLIO
MANAGEMENT Asset Allocation, Diversification, Rebalancing
ABSTRACT Portfolio management is the art and
science of making decisions about
investment mix and policy,
matching investments to objectives,
asset allocation for individuals and
institutions, and balancing risk
against performance.
Sangapu Pranathi CA FINAL SFM
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Contents Portfolio Management ........................................................................................................................... 3
Phases in Portfolio Management .......................................................................................................... 4
Portfolio Manager .................................................................................................................................. 5
Factors affecting Investment Decisions in Portfolio Management ...................................................... 5
Risk in Portfolio Theory .......................................................................................................................... 6
Risk associated with Securities is affected by Government Policy ....................................................... 7
Interest Rate Risk, Re-Investment Risk, Default Risk ............................................................................ 8
Beta as a measure of Risk ...................................................................................................................... 8
Measurement of Risk using Standard Deviation and Variance ............................................................ 9
Computation of Beta ............................................................................................................................ 10
Capital Asset Pricing Model [CAPM] .................................................................................................... 11
Arbitrage Pricing Theory (APT) Model ................................................................................................. 13
Hedge Risks using Risk Free Investments ............................................................................................ 13
Illustrations ........................................................................................................................................... 15
Security Analysis – Expected Return and Standard Deviation........................................................ 15
Calculation of Beta – Variance Approach ........................................................................................ 15
Market Sensitivity Index (Beta) and Expected Return .................................................................... 16
Covariance and Correlation Co-efficient ......................................................................................... 17
Covariance and Expected Return ..................................................................................................... 18
Average Return and Standard Deviation ......................................................................................... 19
Co-efficient of Variation ................................................................................................................... 20
Systematic and Unsystematic Risk and Characteristic Line ............................................................ 21
Systematic and Unsystematic Risk .................................................................................................. 22
Average Return on Portfolio ............................................................................................................ 22
Expected Rate of Return .................................................................................................................. 23
Portfolio Beta and Return – Effect of Change in Portfolio .............................................................. 24
Risk and Return Comparison ............................................................................................................ 24
Portfolio Risk and Return ................................................................................................................. 25
CAPM – Evaluation of Securities ...................................................................................................... 26
Portfolio of Investment in Mutual Funds ........................................................................................ 27
Return and Risk of a Portfolio, Proportion of Investment .............................................................. 28
Portfolio Management – CPPI Model .............................................................................................. 29
CAPM – Investing Decisions ............................................................................................................. 30
CAPM – Overvaluation vs Undervaluation ...................................................................................... 30
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Expected Return on Stocks, Alpha and SML .................................................................................... 31
Portfolio Analysis – Two Factor Model ............................................................................................ 32
Portfolio Returns – Arbitrage Pricing Theory .................................................................................. 33
Beta of Company’s Assets ................................................................................................................ 33
Project Beta – Unlevered Firm ......................................................................................................... 34
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Portfolio Management 1. A portfolio refers to a collection of investment tools such as stocks, shares, mutual funds,
bonds, and cash and so on depending on the investor’s income, budget and convenient time
frame.
2. Investment in Securities requires a good amount of scientific and analytical skills
3. The art of selecting the right investment policy for the individuals in terms of minimum risk
and maximum return is called as portfolio management.
4. Portfolio management refers to managing an individual’s investments in the form of bonds,
shares, cash, mutual funds etc. so that he earns the maximum profits within the stipulated
time frame.
5. Portfolio management refers to managing money of an individual under the expert guidance
of portfolio managers.
6. In a layman’s language, the art of managing an individual’s investment is called as portfolio
management.
7. An Investor has to follow the famous principle – “Never put all eggs in one basket”, an
Investor never invests his entire investable funds in one security
8. He has to invest in a Well Diversified Portfolio of number of securities which will optimise
the overall-risk return
Need for Portfolio Management
1. Portfolio management presents the best investment plan to the individuals as per their
income, budget, age and ability to undertake risks.
2. Portfolio management minimizes the risks involved in investing and also increases the
chance of making profits.
3. Portfolio managers understand the client’s financial needs and suggest the best and unique
investment policy for them with minimum risks involved.
4. Portfolio management enables the portfolio managers to provide customized investment
solutions to clients as per their needs and requirements.
Activities in Portfolio Management
1. Selection of Securities
2. Construction of all Feasible Portfolios with the help of the selected securities
3. Selecting an Optimal Portfolio for the concerned investor, based on the comparison of all
feasible Portfolios
Objectives of Portfolio Management
1. Safety of Principal amount and also keeping its purchasing power intact
2. Accurate and systematically planning of Reinvestment and Consumption of Income
3. Attainment of Capital Growth by reinvesting in Growth securities
4. Providing flexibility of investment portfolio by making the security marketable
5. Securities in the portfolio should be liquid, so that the investor can take advantage of the
market
6. Basic Objective of Portfolio management is to reduce risk of loss of capital and income by
investing in various types of securities
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Phases in Portfolio Management 1. Security Analysis
a. There are many types of securities available in the market including equity shares,
preference shares, debentures and bonds
b. Apart from it, there are many new securities that are issued by companies such as
Convertible Debentures, Deep Discount Bonds, Floating Rate bonds, flexi bonds, zero
coupon bonds, global depository receipts, etc.
c. It forms the initial phase of the portfolio management process and involves the evaluation
and analysis of risk return features of individual securities
d. The basic approach for investing in securities is to sell the overpriced securities and
purchase under-priced securities.
e. The Security analysis comprises of Fundamental Analysis and technical analysis
2. Portfolio Analysis
a. A portfolio refers to a group of securities that are kept together as an investment.
b. Investors make investment in various securities to diversify the investment to make it risk
averse.
c. A large number of portfolios can be created by using the securities from desired set of
securities obtained from initial phase of security analysis.
d. By selecting the different sets of securities and varying the amount of investments in each
security, various portfolios are designed.
e. After identifying the range of possible portfolios, the risk-return characteristics are
measured and expressed quantitatively.
f. It involves the mathematically calculation of return and risk of each portfolio.
3. Portfolio Selection
a. During this phase, portfolio is selected on the basis of input from previous phase Portfolio
Analysis.
b. The main target of the portfolio selection is to build a portfolio that offer highest returns at
a given risk.
c. The portfolios that yield good returns at a level of risk are called as efficient portfolios.
d. The set of efficient portfolios is formed and from this set of efficient portfolios, the optimal
portfolio is chosen for investment.
e. The optimal portfolio is determined in an objective and disciplined way by using the
analytical tools and conceptual framework provided by Markowitz’s portfolio theory.
4. Portfolio Revision
a. After selecting the optimal portfolio, investor is required to monitor it constantly to ensure
that the portfolio remains optimal with passage of time.
b. Due to dynamic changes in the economy and financial markets, the attractive securities
may cease to provide profitable returns.
c. These market changes result in new securities that promises high returns at low risks.
d. In such conditions, investor needs to do portfolio revision by buying new securities and
selling the existing securities.
e. As a result of portfolio revision, the mix and proportion of securities in the portfolio
changes.
5. Portfolio Evaluation
a. This phase involves the regular analysis and assessment of portfolio performances in terms
of risk and returns over a period of time.
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b. During this phase, the returns are measured quantitatively along with risk born over a
period of time by a portfolio.
c. The performance of the portfolio is compared with the objective norms.
d. Moreover, this procedure assists in identifying the weaknesses in the investment
processes.
Portfolio Manager 1. A Portfolio Manager is a person or group of people responsible for investing a mutual, exchange-
traded or closed-end fund’s assets, implementing its investment strategy and managing day-to-
day portfolio trading
2. A Portfolio Manager is one of the most important factors to consider when looking at fund
investing
3. Portfolio Management can be active or passive, and historical performance records indicate that
only a minority of active fund managers consistently beat the market
4. Discretionary Portfolio Manager
a. He exercises a full degree of discretion and freedom, in respect of the investments or
management of the portfolio of securities or the funds of the client
b. He manages the funds of each client individually and independently, in accordance with
the needs of the Client, in a manner which does not resemble a Mutual Funds
5. Non-Discretionary Portfolio Manager
a. He manages the funds in accordance with the directions and instructions of the Client.
Degree of freedom is comparatively less
b. Instead of making changes to the portfolio at their own discretion, the Portfolio
Managers refer relevant advice and information to the Client, who then makes the
actual investment decision
Factors affecting Investment Decisions in Portfolio Management
1. Selection of Type of Securities
a. What type of securities are to be chosen?
2. Proportion of Investment
a. What should be the proportion of investment in Fixed Interest/Dividend Securities
and variable interest/dividends bearing securities
3. Identification of Industry
a. In case investments are to be made in the shares or debentures of companies, which
particular industry shows potential of growth?
4. Identification of Company
a. After identifying industries with high growth potential, selection of the Company, in
whose shares or securities investments are to be made
5. Objectives of portfolio
a. If the portfolio is to have a safe and steady returns, then securities with low risk
would be selected.
b. In case of portfolios which are floated for high returns, then risk investments which
carry a higher rate of return will be selected
6. Timing of purchase
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a. At what price the share is acquired for the Portfolio, depends entirely on the timing
decisions
b. If a person wishes to make any gains, he should buy when the shares are selling at a
low price and sell when they are at a high price
Risk in Portfolio Theory 1. Risk refers to the possibility of the rate of return from a security or a portfolio of securities
deviating from the corresponding expected/average rate
2. The essence of risk in an investment is the variation in its return, which is caused by a
number of factors
3. Risk Aversion is an intrinsic attribute of Investors that leads to the tendency to avoid risk
unless adequately compensated. Thus, Risk Aversion is the degree to which investors abhor
uncertainty surrounding their investment
4. Risk Appetite is the willingness to bear risk. It consists of two components – Degree to which
Investors dislike the associated uncertainty and the level of that uncertainty
5. Risk Premium is the reward for holding a risky investment rather than a risk-free
investment. Thus, Risk Premium measures the additional returns that Investors require to
hold assets whose returns are more variable than those of low risk ones
Systematic Risk [Non-Diversifiable Risk] Unsystematic Risk [Diversifiable Risk]
These arise out of external and uncontrollable factors, which are not specific to a security or industry to which such security belongs. They arise out of general and system-wide factors, like economic, Political and social changes
These are risks that emanate from known and controllable factors, which are unique and/or related to a particular security or industry. These are in addition to Systematic Risk that affects that particular security/industry
These risks affect a large number of securities simultaneously and are considered macro in nature
These are internal/specific to particular security/industry and are considered micro in nature
These Risks are absolute, i.e. they cannot be eliminated by diversification
These risks can be eliminated by diversification of portfolio. As the number of securities in the portfolio increases, Unsystematic Risk is eliminated and only Systematic Risk of those securities remains
These are further sub-classified into – 1. Market Risk 2. Interest Rate Risk 3. Purchasing Power Risk
These are further sub-classified into – 1. Business Risk 2. Financial Risk 3. Default Risk
Classification of Systematic Risk:
1. Market Risk –
a. These are the risks that are triggered due to social, political and economic events
b. These Risks arises due to changes in demand and supply, expectations if the
investors, information flow, investor’s risk perception, etc. consequent to the social,
political and economic events
2. Interest Rate Risk –
a. Uncertainty of Future Market values and extent of income in the future, due to
fluctuations in the general level of interest, is known as Interest Rate Risk
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b. These are risks arising due to fluctuating rates of interest and cost of corporate debt.
The cost of Corporate Debt depends on the interest rates prevailing, maturity
periods, credit worthiness of the Borrowers, monetary and credit policy of RBI, etc.
3. Purchasing Power Risk or Inflation Risk –
a. Purchasing Power Risk is the erosion in the value of money due to the effects of
inflation
b. In inflationary conditions, Purchasing Power Risk is felt more in Bonds and Fixed
Income Securities. This Risk is however less in Flexible Income Securities like Equity
Shares or Common Stock where rise in dividend income off-sets increase in the rate
of inflation and provides advantage of Capital Gains
Kinds of Unsystematic Risk:
1. Business Risk –
a. It is the volatility in revenues and profits of particular company due to its market
conditions, product mix, competition, etc.
b. It may arise due to external reasons or internal reasons
c. Business Risks also emanates from sale and purchase of securities affected by
business cycles, technological changes, etc.
d. Flexible Income Securities are more affected than Fixed Rate Securities during
depression due to decline in their Market Price
2. Financial Risk or Leverage Risk –
a. These are risks that are associated with the Capital Structure of a Company. A
Company with no Debt Financing, has no Financial Risk
b. Higher the Financial Leverage, higher the Financial Risk
c. These may also arise due to short-term liquidity problems, shortage in Working
Capital due to funds locked in Working Capital and Receivables, etc.
3. Default Risk or Maturity Risk –
a. These arise due to default in meeting the financial obligations on time. Non-
payment of financial dues on time increases the insolvency and bankruptcy costs
b. Maturity Risk is the risk associated with the likelihood of Issuer/Government issuing
a new security in place of redeeming the existing security.
c. In case of Corporate Securities, it is called as Credit Risks
Risk associated with Securities is affected by Government Policy Risks due to Government Policies will affect the entire economy as such and therefore, are classified
generally as Systematic Risk. Examples of areas in which government policies and the impact on the
securities is as follows –
1. Tax Policies – Both direct and Indirect Tax policies have an impact on the business
environment and economy in general. Specific tax advantages to certain sectors/businesses
have pronounced impact on that sector
2. Industrial Policy – Relaxation of restrictions, simplification of compliance aspects,
permission for expansion into multiple business activities, FDI, etc. will favourably impact
the business entities. Restrictive economic and industrial policies will adversely impact the
economy
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3. Incentives and Exim Policy – Exports are generally supported with incentives by the
Government. Withdrawal of the same can adversely affect earnings of Export
Houses/Entities
4. Change in Planned and Unplanned Expenditure – Generally, increase in unplanned
expenditure results in creation of new infrastructure and increasing the scale of economic
activities. As a result, in general, revenue and operating income of various sectors will
improve
Interest Rate Risk, Re-Investment Risk, Default Risk Interest Rate Risk:
1. Interest Rate Risk arise on account of inverse relationship of price and interest. These are
typical of any fixed coupon security with a fixed period to maturity
2. This risk can be completely eliminated in case an investor’s investment horizon identically
matches the term of security
Re-Investment Risk:
1. Re-Investment Risk is the risk that the rate at which the interim cash flows are re-invested
may fall thereby affecting the returns
2. The most prevalent tool deployed to measure returns over a period of time is the Yield to
Maturity method which assumes that the Cash Flows generated during the life of a security
is reinvested at the rate of YTM
Default Risk:
1. These arise due to default in meeting the financial obligations on time
2. A variant of this is the maturity risk, i.e. the possibility of issuing a new security in place of
redeeming the existing security
Beta as a measure of Risk 1. Beta of a security measures the sensitivity of the security with reference to a broad based
Market Index like BSE Sensex, NIFTY
2. Beta measures the Systematic Risk, i.e. that which affects the market as a whole, and hence
cannot be eliminated through diversification
3. Beta is a factor of the following –
a. Standard Deviation (Risk) of the Security or Portfolio
b. Standard Deviation (Risk) of the market
c. Correlation between the Security and the Market
4. It is applicable for the Invertor who invests his money in a portfolio of securities, i.e.
Portfolio Investor
5. A Portfolio Investor would look into eliminating the Diversifiable Risk, and evaluate the exact
extent of Systematic or Non-Diversifiable Risk
6. A Rational Investor views the Beta of a Security as the Security’s proper measure of risk. He
understands that the market does not reward for Diversifiable Risk, since the Investor
himself is expected to diversify the risk himself
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Measurement of Risk using Standard Deviation and Variance Total Risk = Systematic Risk + Unsystematic Risk
Systematic Risk: It represents that portion of Total Risk which is attributable to factors that affect
the market as a whole. Beta is a measure of Systematic Risk
Unsystematic Risk: It is the residual risk or balancing figure; Total Risk Less Systematic Risk
When Standard Deviation is taken as Total Risk:
𝜎𝑆 = 𝜎𝑆 × 𝜌𝑆𝑀 + 𝜎𝑆 × (1 − 𝜌𝑆𝑀)
𝜎𝑆 = 𝛽𝑆𝑀 × 𝜎𝑀 + 𝜎𝑆 × (1 − 𝜌𝑆𝑀)
Systematic Risk + Unsystematic Risk
Where, S = Standard Deviation of the Returns from Security S
SM = Correlation Co-efficient between returns from Security S and Market Portfolio
ΒSM = Beta of Security S with reference to Market Returns
When Variance is taken as Total Risk:
𝜎𝑆2 = 𝛽𝑆
2 × 𝜎𝑀2 + 𝜎𝑆
2 × (1 − 𝜌𝑆𝑀2 )
Systematic Risk + Unsystematic Risk
Where, 𝜎𝑆2 = Variance of the Returns from Security S
𝜌𝑆𝑀2 = Square of Correlation Co-efficient between Return from Security S and Market [Co-efficient of
Determination]
Unsystematic Risk is computed only as the balancing figure, and not as a separate item
Variance
𝜎2 = ∑[𝑅𝑆 − �̅�𝑆]2
𝑁
Standard Deviation
𝜎 = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒
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Computation of Beta Beta = Expected Movement: It gives the expected movement in the Return of a Security (or Market
Price of the Security) per unit of movement in the Market Portfolio Return
Beta as a measure of Systematic Risk: The relationship is explained as follows –
Beta of Security S (βS) = 𝜎𝑆
𝜎𝑀× 𝜌𝑆𝑀
[Movement is Security S per unit of Movement in Market Portfolio] x [Extent of Correlation between
Security S and Market Portfolio (= Probability of such movement)]
[Average Movement] x [Probability of the Average Movement]
Inferences:
1. Based on Relationship between a Security’s Risk and Market Risk
If Beta is less than 1 Security is Less Risky than the Market Portfolio
If Beta is Equal to 1 Security is as risky as the Market Portfolio
If Beta is More than 1 Security is more risky than the Market Portfolio
2. Based on dependency between Security’s Return and Market Money
If Beta value is Negative Security’s Return is Dependent on the Market Return, but moves in
the opposite direction in which market moves
If Beta value is Zero Security’s Return is Independent of Market Return
If Beta value is Positive Security’s Return is Dependent on the Market Return, and moves in
the same direction in which market moves
Computation:
Using Standard Deviation and Correlation: Beta of Security (βS) = 𝜎𝑆
𝜎𝑀× 𝜌𝑆𝑀
Using Covariance and Market Variance: Beta of a Security (βS) = 𝐶𝑜𝑣𝑆𝑀
𝜎𝑀2 =
𝜌𝑆𝑀×𝜎𝑆×𝜎𝑀
𝜎𝑀2
From Basic Data: Beta of a Security (βS) = ∑ 𝑅𝑀𝑅𝑆− 𝑛�̅�𝑀�̅�𝑆
∑ 𝑅𝑀2 − 𝑛�̅�𝑀
2
Where,
n = No. of pairs of observations considered (generally, the number of years/months/days)
ΣRMRS = Aggregate of Product
ΣRM2 = Aggregate of Return Square
ṜM = Mean of Market Return = [Aggregate of Market Returns]/[Number of years]
ṜS = Mean of Security Return = [Aggregate of Security Returns]/[Number of Years]
COVSM = Covariance between Security and Market, computed as ∑[𝑅𝑀− �̅�𝑀]×[𝑅𝑆− �̅�𝑆]
𝑛
Co-Variance is an absolute measure of co-movement between two variables, i.e. the extent to which
they are generally above their means or below their means at the same time.
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Covariance between M and S = COVMS = ∑[𝐷𝑀×𝐷𝑆]
𝑁
Where, DM represents the Deviation of Return from the Mean Return of Portfolio M
DS represents the Deviation of Return from the Mean Return of Portfolio S
Since Covariance is an absolute measure of relationship between two securities, its value will range
between +∞ to -∞
Correlation of Co-efficient is a measure of closeness of the relationship between two random
variables and is bounded by the values +1 and -1
It can be equated to probability of movement. A Correlation value of 0.70 can be inferred as a 70%
movement in values of two variables in the same direction. A negative of 0.70 can be inferred as a
70% movement in values of two variables in opposite direction
Based on Covariance and Standard Deviation: XY = 𝐶𝑂𝑉𝑋𝑌
𝜎𝑋×𝜎𝑌
Based on Probability Distribution of Future Returns: XY = ∑[𝑋𝑖− 𝐸(𝑋𝑖)] × [𝑌𝑖− 𝐸(𝑌𝑖)]
𝜎𝑋𝜎𝑌
Based on Historical Realized Returns: XY = 𝑛 ∑ 𝑋𝑖𝑌𝑖− ∑ 𝑋𝑖 ∑ 𝑌𝑖
√[𝑛 ∑ 𝑋𝑖2− (∑ 𝑋𝑖)2]− [𝑛 ∑ 𝑌𝑖
2− (∑ 𝑌𝑖)2]
Valuation and Inference: Portfolio Risks will be –
1. Maximum when two components of a portfolio stand perfectly positively correlated
2. Minimum when two components of a portfolio stand perfectly negatively correlated
Capital Asset Pricing Model [CAPM] Assumptions:
1. Efficient Market
Efficient Market is characterized by free flow of information on risk and return, to all
participants, no dominance by a single investor, prices of individual assets reflect their real
or intrinsic value, and financial assets and capital assets are bought and sold freely without
restrictions/market imperfections
2. No Transaction Costs
Securities can be exchanged without payment of brokerage, commission or taxes and
without any Transaction costs
3. Rational Investors
Rational Behaviour of Investors is characterized by their desire for higher return for any
acceptable level of risk, and lower risk for any desired level of return. Features of rationally
also include – logical and consistent ranking of proposals, trans-active preferences, certainty
equivalents
4. Risk Aversion
Generally, Risk Aversion is efficient market is adhered to. Sometimes, risk seeking behaviour
is adopted for gains
5. Asset Nature
Total Asset Quantity is fixed, All Assets are divisible and liquid, Securities or Capital Assets
face no bankruptcy or insolvency
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6. Borrowings
Investors can borrow and lend unlimited amount at the risk-free rate
Formula:
Expected Return on Portfolio E(RP) = RF + [βP x (RM – RF)]
RF = Risk Free Rate of Interest or Return
βP = Portfolio Beta
RM = Expected Return on Market Portfolio
Valuation:
1. CAPM is essentially a model for determining the Intrinsic Value or Equilibrium Price of an
Asset
2. Equilibrium or Intrinsic Price of an Asset is determined using the Expected Return as arrived
at using CAPM
3. This Expected Return is the minimum return that the investors require from the asset in
relation to the relative systematic risk of the asset
4. The price of an asset is the Present Value of the Future Cash Flows generated by the Asset as
discounted by the Expected Return as determined using the CAPM.
Inference:
Situation Inference Action
CAPM Return < Estimated Return Undervalued Security BUY
CAPM Return = Estimated Return Correctly Valued Security HOLD
CAPM Return > Estimated Return Overvalued Security SELL
Advantages of CAPM:
1. Use in Capital Budgeting: CAPM Provides a reasonable basis for estimating the required
return on an Investment which has risk in built into it. So, it can be used as Risk Adjusted
Discount Rate in Capital Budgeting
2. No Dividend Company: CAPM is useful in computing the cost of equity of a company which
does not declare dividend
3. Linkage: CAPM provides a logical linkage between the activities of a Company, and its Cost
of Capital
Limitations of CAPM:
1. Lack of Information: It is difficult to obtain information on Risk Free Interest Rate and
Expected Return on Market Portfolio, since there are multiple Risk Free Rates, and Market
Returns always vary over time period, since markets are volatile
2. Unreliability 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
3. Other Risks: By emphasing on systematic risk only, unsystematic risks are of importance to
shareholders who do not possess a diversified portfolio
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Arbitrage Pricing Theory (APT) Model Arbitrage pricing theory (APT) is a well-known method of estimating the price of an asset. The theory
assumes an asset's return is dependent on various macroeconomic, market and security-specific
factors
APT is an alternative to the capital asset pricing model (CAPM). Stephen Ross developed the theory
in 1976.
The APT formula is:
E(rj) = rf + bj1RP1 + bj2RP2 + bj3RP3 + bj4RP4 + ... + bjnRPn
Where:
E(rj) = the asset's expected rate of return
rf = the risk-free rate
bj = the sensitivity of the asset's return to the particular factor
RP = the risk premium associated with the particular factor
The general idea behind APT is that two things can explain the expected return on a financial asset:
1) macroeconomic/security-specific influences and 2) the asset's sensitivity to those influences. This
relationship takes the form of the linear regression formula above.
There are an infinite number of security-specific influences for any given security including inflation,
production measures, investor confidence, exchange rates, market indices or changes in interest
rates. It is up to the analyst to decide which influences are relevant to the asset being analysed.
Once the analyst derives the asset's expected rate of return from the APT model, he or she can
determine what the "correct" price of the asset should be by plugging the rate into a discounted
cash flow model.
Note that APT can be applied to portfolios as well as individual securities. After all, a portfolio can
have exposures and sensitivities to certain kinds of risk factors as well.
Hedge Risks using Risk Free Investments Hedging using Risk Free Investments to increase Risk [Increase Portfolio Value]
1. Object is to increase the Beta value of Portfolio
2. Buy Stock and Sell Risk Free Investments
3. Value of Risk Free Investments to be bought – Portfolio value x [Desired Beta – Present Beta
of Portfolio]
4. Value of Risk Free Investments = [Portfolio Value x Desired beta] Less [Portfolio Value x
Present Beta]
5. Desired Beta is the Weighted Average Beta of the Risk Free Investments and the Beta of the
remaining investments. Risk-Free Investments do not carry any Beta. By selling Risk-Free
investments and investing the same in the Portfolio, risk attached to the Portfolio increases,
and there by Portfolio Risk increases.
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Hedging using Risk Free Investments to reduce Risk [Reduce Erosion in Value]
1. Object is to reduce Beta value of Portfolio
2. Sell Stock and Buy Risk Free Investments
3. Value of Risk Free Investments to be bought – Portfolio Value x [Present Beta of the Portfolio
– Desired Beta]
4. Risk Free Investments do not carry any Beta. By selling the portfolio stock, and buying Risk-
Free Investments, Risk attached to the portfolio gets reduced, and thereby Portfolio Risk
reduces
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Illustrations Security Analysis – Expected Return and Standard Deviation A Stock costing Rs. 120 pas no dividends. The possible prices that the Stock might sell for at the
end of the year with the respective probabilities are given below. Compute the Expected Return
and its Standard Deviation
Price 115 120 125 130 135 140
Probability 0.1 0.1 0.2 0.3 0.2 0.1
Price Return (R) = 120 – P
Probability P Expected Return (PXR) D = R - Ṝ
D2 P x D2
115 (5) 0.1 (0.5) (13.5) 182.25 18.225
120 0 0.1 0.0 (8.5) 72.25 7.225
125 5 0.2 1.0 (3.5) 12.25 2.450
130 10 0.3 3.0 1.5 2.25 0.675
135 15 0.2 3.0 6.5 42.25 8.450
140 20 0.1 2.0 11.5 132.25 13.225
Total Ṝ = 8.5 50.250
Expected Return on Security = Rs. 8.5
Risk of Security = √𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = √50.25 = 𝑅𝑠. 7.09
Calculation of Beta – Variance Approach From the following information pertaining to returns of Shares of Companies A, B and the Market
for the past 5 Years, calculate Beta (β) of A and B –
Year 1 2 3 4 5
Market 12% 14% 13% 12% 14%
Company A 16% 8% 13% 14% 19%
Company B 14% 17% 15% 20% 19%
Computation of Factors:
Year RM RA RB DM = RM - ṜM DA = RA - ṜA DB = RB - ṜB DM2 DM x DA DM x DB
1 2 3 4 5 = [2-3] 6 = [3 – 14] 7 = [4 – 17] 8 = [5]2 11 = 5 x 6 12 = 5 x 7
1 12 16 14 -1 2 -3 1 -2 3
2 14 8 17 1 -6 0 1 -6 0
3 13 13 15 0 -1 -2 0 0 0
4 12 14 20 -1 0 3 1 0 -3
5 14 19 19 1 5 2 1 5 2
65 70 85 4 -3 2
16 | P a g e
Market Portfolio Shares of Company A Shares of Company B
Mean �̅�𝑀 =
∑ �̅�𝑀
𝑛=
65
5= 13 �̅�𝐴 =
∑ �̅�𝐴
𝑛=
70
5= 14 �̅�𝐵 =
∑ �̅�𝐵
𝑛=
85
5= 17
Variance 𝜎𝑀2 =
∑ 𝐷𝑀2
𝑛=
4
5= 0.80 - -
Covariance and Correlation:
Combination Market and A Market and B
Covariance 𝐶𝑂𝑉𝑀𝐴 =
∑[𝐷𝑀 × 𝐷𝐴]
𝑛=
3
5= −0.60 𝐶𝑂𝑉𝑀𝐵 =
∑[𝐷𝑀 × 𝐷𝐵]
𝑛=
2
5= 0.40
Computation of Beta:
a. Security A (βA) = COVMA/M2 = -0.60/0.80 = -0.75
b. Security B (βB) = COVMB/M2 = 0.40/0.80 = 0.50
Market Sensitivity Index (Beta) and Expected Return Calculate the Market Sensitivity Index, and the Expected Return on the Portfolio from the
following data:
Particulars % Particulars %
Standard Deviation of an Asset 2.5% Risk Free Rate of Return 13.0%
Market Standard Deviation 2.0% Expected Return on Market Portfolio 15.0%
What will be the Expected Return on the Portfolio, if Portfolio Beta is 0.5, Risk Free Return is 10%
and PM is 0.8
Basic Data for Computation of Expected Return
Notation Particulars Case (a) Case (b)
P Standard Deviation of Asset 2.5% 2.5%
M Market Standard Deviation 2.0% 2.0%
MP Correlation Co-efficient of Portfolio with Market 0.80 0.80
RF Risk Free Rate of Return 13% 10%
RM Expected Return on Market Portfolio 15% 15%
ΒA Portfolio Beta To be ascertained 0.5
Computation of Expected Return:
Case (a) Case (b)
Portfolio Beta 𝛽𝐴 = 𝜎𝑃
𝜎𝜎𝑀 × 𝜌𝑀𝑃 2.5/2.0 x 0.8 = 1.00 0.5
Expected Return = RF + [βP x (RM – RF) 0.13 + [1 x (0.15 – 0.13)] = 15% 0.10+[0.5x(0.15-0.10)] = 12.5%
17 | P a g e
Covariance and Correlation Co-efficient The historical rates of Return of two Securities over the past 10 years are given:
Calculate the Covariance and the Correlation Co-efficient of the two securities:
Years 1 2 3 4 5 6 7 8 9 10
Sec 1 (Ret %)
12 8 7 14 16 15 18 20 16 22
Sec 1 (Ret %)
20 22 24 18 15 20 24 25 22 20
Computation of Factors (R1 = Return of Security 1, R2 = Return of Security 2)
Year R1 R2 D1 = R1 - Ṝ2 D2 = R2 - Ṝ2 D12 D2
2 D1 x D2
1 12 20 -2.8 -1 7.84 1 2.8
2 8 22 -6.8 1 46.24 1 -6.8
3 7 24 -7.8 3 60.84 9 -23.4
4 14 18 -0.8 -3 0.64 9 2.4
5 16 15 1.2 -6 1.44 36 -7.2
6 15 20 0.2 -1 0.04 1 -0.2
7 18 24 3.2 3 10.24 9 9.6
8 20 25 5.2 4 27.04 16 20.8
9 16 22 1.2 1 1.44 1 1.2
10 22 20 7.2 -1 51.84 1 -7.2
207.6 84 -8
Security 1 Security 2
Mean �̅�1 =
∑ �̅�1
𝑛=
148
10= 14.8 �̅�2 =
∑ �̅�2
𝑛=
210
10= 21
Variance 𝜎12 =
∑ 𝐷12
𝑛=
207.6
10= 20.76 𝜎22 =
∑ 𝐷22
𝑛=
84
10= 8.4
Standard Deviation 1 = 20.76 = 4.56 2 = 8.4 = 2.90
Combination Security 1 and 2
Covariance 𝐶𝑂𝑉1,2 =
∑[𝐷1 × 𝐷2]
𝑛=
−8
10= −0.80
Correlation 𝜌1,2 =
𝐶𝑂𝑉1,2
𝜎1 × 𝜎2=
−0.8
4.56 × 2.89= −0.06
18 | P a g e
Covariance and Expected Return The distribution of Return of Security “F” and the Market Portfolio “P” is given below:
Probability 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
Expected Return and Risks of Security “F”
Scenario Probability Return % Exp Return % Deviation % D2 Variance [P x D2]
1 2 3 4 = 2x3 5 = 3 – Σ4 6 = 52 7 = 2x6
1 0.30 30 9 13 169 50.7
2 0.40 20 8 3 9 3.6
3 0.30 0 0 (17) 289 86.7
Σ = 17.00% 141
Expected Return on Security F = 17.00%
Expected Return and Risks of Market Portfolio P
Scenario Probability Return % Expected Return %
Deviation % D2 Variance [PxD2]
1 2 3 4 = 2x3 5 = Σ4 – 3 6 = 52 7 = 2x6
1 0.30 (10) (3) (24) 576 172.8
2 0.40 20 8 6 36 14.4
3 0.30 30 9 16 256 76.8
14.00% 264
Expected Return on Market Portfolio P = 14.00%
Computation of Covariance of Securities F and Market Portfolio P
Scenario Probability P Deviation DF
from Mean for F%
Deviation (DP) from Mean
for P%
Deviation Product (DFP)
= DF x DP
Covariance (P x DFP)
1 2 3 4 5 = 3 x 4 6 = 2 x 5
1 0.30 13 (24) (312) (93.6)
2 0.40 3 6 18 7.2
3 0.30 (17) 16 (272) (81.6)
(168)
Covariance of Securities F and Market Portfolio (P) [COVFP] = (168.00)
Beta = CPVFP/P2 = -168/264 = -0.636
19 | P a g e
Average Return and Standard Deviation Shanthanu Co. Ltd. invested on 01/04/2012 in certain Equity Shares as below:
Name of the Company No. of Shares Cost (Rs.)
Mayank Ltd 1000 [Rs. 100 each] 200000
Nupur Ltd 500 [Rs. 10 each] 150000
In September, 2012, 10% Dividend was paid out by Mayank Ltd. and in October 2012, 30%
dividend paid out by Nupur Ltd. On 31/03/2013 market quotations showed a value of Rs. 220 and
Rs. 290 per share for Mayank Ltd and Nupur Ltd respectively.
On 01.04.2013, investment advisors indicate (a) that the dividends from Mayank Ltd and Nupur
Ltd for the year ending 31.03.2014 are likely to be 20% and 35% respectively and (b) that the
probabilities of market quotations on 31.03.2014 are as below:
Probability Price per Share of Mayank Ltd Price per Share of Nupur Ltd
0.2 220 290
0.5 250 310
0.3 280 330
You are required to –
1. Calculate the Average Return from the Portfolio for the year ending 31.03.2013
2. Calculate the Expected Average Return from the Portfolio for the year 2013-14; and
3. Advise X Ltd of the comparative risk in the two investments by calculating the Standard
Deviation in each case.
Calculation of Return on Portfolio for 2012-13 -
Particulars Mayank Ltd Nupur Ltd
Return % = 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑+ [𝑃𝑟𝑖𝑐𝑒𝑇1− 𝑃𝑟𝑖𝑐𝑒𝑇0]
𝑃𝑟𝑖𝑐𝑒𝑇0
10 + [220 − 200]
200
= 15%
3 + [290 − 300]
300
= (2.33%) Weighted Average (Expected) Return = [15% x 2/3.50] – [2.33% x 1.5/3.50]
7.57%
Calculate of Expected Return for 2006-07
Particulars Mayank Ltd Nupur Ltd
Expected Price at T1 (220 x 0.2) + (250 x 0.5) + (280 x 0.3) (290 x 0.2) + (310 x 0.5) + (330 x 0.3)
253
-
-
312
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 + [𝑃𝑟𝑖𝑐𝑒𝑇2− 𝑃𝑟𝑖𝑐𝑒𝑇1
]
𝑃𝑟𝑖𝑐𝑒𝑇1
20 + [253 − 220]
220
24.09%
3.50 + [312 − 290]
290
8.79%
Weighted Average (Expected) Return = [2.49% × 220000
220000+145000] + [8.79% ×
145000
220000+145000]
= 18.01%
20 | P a g e
Standard Deviation Mayank Ltd
Exp MV Exp Gain
Exp Dividend
Exp Yield
Mean D (4) – 53
D2 Probability PD2
(1) (2) (3) (4) (5) (6) (7) (8) (9)
220 0 20 20 4 -33 1089 0.2 217.8
250 30 20 50 25 -3 9 0.5 4.50
280 60 20 80 24 27 729 0.3 218.70
Σ = 53 441.00
Standard Deviation = PD2 = 441 = 21
Standard Deviation Nupur Ltd
Exp MV Exp Gain
Exp Dividend
Exp Yield
Mean D (4) – 53
D2 Probability PD2
(1) (2) (3) (4) (5) (6) (7) (8) (9)
290 0 3.5 3.5 0.70 -22 484 0.2 96.8
310 20 3.5 23.5 11.75 -2 7 0.5 2.00
330 40 3.5 43.5 13.05 18 324 0.3 97.2
Σ = 25.50
196.00
Standard Deviation = PD2 = 196 = 14
Inference – Shares of Company Mayank Ltd is more risky as the Standard Deviation of Mayank Ltd. is
more than the Standard Deviation of Nupur Ltd.
Co-efficient of Variation A company is considering Projects X and Y with following information:
Project X: Expected NPV – Rs. 122000 and Standard Deviation – Rs. 90000
Project Y: Expected NPV – Rs. 225000 and Standard Deviation – Rs. 120000
Which Project will you recommend based on the above data?
Co-efficient of Variation = Standard Deviation/Expected NPV
CVX = 90000/122000 = 0.74
CVY = 120000/225000 = 0.53
Conclusions:
1. Co-efficient of Variation (Risk per unit of Return) of Project X is more than that of Project Y
2. Hence, Project X appears to be more risky
21 | P a g e
Systematic and Unsystematic Risk and Characteristic Line The Return on Stock A and Market Portfolio for a period of 6 years are as follows -
Year Return on A (%) Return on Market Portfolio
1 12 8
2 15 12
3 11 11
4 2 -4
5 10 9.5
6 -12 -2
You are required to determine –
1. Characteristic Line for Stock A
2. The Systematic and Unsystematic Risk of Stock A
Computation of Beta of Stock
Years RM RA DM2 = (RM - ṜM) DA
2 = (RA - ṜA) DM2 DA
2 DM x DM
(1) (2) (3) (4) = [(2) – 5.75] (5) = [(3) – 6.33] (6) = (4)2 (7) = (5)2 (8) = (4x5)
1 8 12 2.25 5.67 5.06 32.15 12.76
2 12 15 6.25 8.67 39.06 75.17 54.19
3 11 11 5.25 4.67 27.56 21.81 24.52
4 -4 2 (9.75) (4.33) 95.06 18.75 42.22
5 9.5 10 3.75 3.67 14.06 13.47 13.76
6 -2 -12 (7.75) (18.33) 60.06 335.00 142.10
34.5 38 240.86 497.33 289.5
Market Portfolio Stock A
Mean ṜM = ∑ �̅�𝑀
𝑛=
34.5
6= 5.75 ṜA =
∑ �̅�𝐴
𝑛=
38
6= 6.33
Variance 𝜎𝑀
2 = ∑ 𝐷𝑀
2
𝑛=
240.86
6= 40.14 𝜎𝐴
2 = ∑ 𝐷𝐴
2
𝑛=
497.33
6= 82.89
Standard Deviation 𝜎𝑀 = √40.14 = 6.34 𝜎𝐴 = √82.89 = 9.10
Beta βA = COVM, A/M2 = 48.25/40.14 = 1.20
Computation of Characteristic Line for Stock A
Particulars Value
Y = ṜA 6.33
Β 1.20
X = ṜM (Expected Return on Market Index) 5.75
Characteristic Line for Stock A = y = a + βx, a = 6.33 – 6.90 = -0.57%
6.33 = a + 1.20x5.75
Characteristic Line for Stock A = -0.58 + 1.2019 RM
Analysis of Risk into Systematic Risk and Unsystematic Risk
Particulars Standard Deviation Approach Variance Approach
Total Risk 9.10% 82.89%
Systematic Risk Total Risk x MA (or) β x m = 9.10 x 0.8363 = 7.61%
Total Risk x MA2 (or) β2 x m
2= = 2.89 x 0.83632 = 57.9731%
Unsystematic Risk [Total Risk – Systematic Risk]
9.10 – 7.61 = 1.49% 82.89 – 57.9731 = 24.9169%
22 | P a g e
Systematic and Unsystematic Risk Security β Random Error ei Weight
L M N K
1.60 1.15 1.40 1.00
7 11 3 9
0.25 0.30 0.25 0.20
You are required to find out the risk of the portfolio If the Standard Deviation of the Market Index
(m) is 18%.
Computation of Risk of Portfolio
Security β Weight Product Unsystematic
Risk (SD) = ei
Unsystematic Risk (Variance Approach)
Product = Unsys. Risk x (Weight)2(1)
(1) (2) (3) (4) = (2x3) (5) (6) = (5)2 (7) = (6) x (3)2
L M N K
1.60 1.15 1.40 1.00
0.25 0.30 0.25 0.20
0.40 0.345 0.35 0.20
7 11 3 9
49 121
9 81
49x0.25x0.25 = 3.06 121x0.3x0.30 = 10.89
9x0.25x0.25 = 0.56 81x0.20x0.20 = 3.24
Total 1.00 1.295 Σ Unsystematic Risk = 17.75
1. Beta of the Product β = 1.295
2. Systematic Risk (Variance Approach) of the Portfolio = β2 x M2 = (1.295)2 x (18)2 = 543.35
3. Total Risk (Variance Approach) = Systematic Risk 543.35 + Unsystematic Risk 17.75 = 561.11
Average Return on Portfolio Stock A and Stock B have the following historical returns –
1 2 3 4 5
A’s Return (KA) -12.24 23.67 34.45 5.82 28.30
B’s Return (KB) -5.00 19.55 44.09 1.20 21.16
You are required to calculate the average rate of return for each Stock during the period. Assume
that someone held a Portfolio consisting 50% of Stock A and 50% of Stock B.
What would have been the realized rate of return on the Portfolio in each year Period? What
would have been the average return on the Portfolio during the period? [You may assume that
year ended on 31st March].
Calculation of Average Rate of Return on Portfolio during the period
Period Stock A’s Return % Stock B’s Return %
1 2 3 4 5
-12.24 23.67 34.45 5.82
28.30
-5.00 19.55 44.09 1.20
21.16
Total 80.00 81.00
Average Rate of Return 80/5 Years = 16% 81/5 Years = 16.20%
23 | P a g e
Calculation of realized Rate of Return on Portfolio during the Period
Period Stock A Stock B Total Net
Return Proportion Return Net Return Proportion Return Net Return
1 2 3 4 5
0.50 0.50 0.50 0.50 0.50
-12.24 23.67 35.45 5.82
28.30
-6.12 11.84 17.73 2.91
14.15
0.50 0.50 0.50 0.50 0.50
-5.00 19.55 44.09 1.20
21.16
-2.50 9.78
22.05 0.60
10.58
-8.62 21.62 39.78 3.51
24.73
40.51 40.51 81.02
Average Rate of Return = 81.02/5 = 16.20%
Expected Rate of Return Compute Return under CAPM and the Average Return of the Portfolio from the following
information –
Investment Initial Price Dividends Market Price at the end of the Year Β Risk Factor
Cement Ltd Steel Ltd
Liquor Ltd
25 35 45
2 2 2
50 60
135
0.80 0.70 0.50
GOI Bonds 1000 140 1005 0.99
Risk Free Return = 14%
Computation of Expected Return and Average Return
Securities Cost Dividend Capital Gain Expected Return = Rf + β(Rm – Rf)
Cement Ltd 25 2 [50-25] = 25 [14 + 0.80(23.66 – 14)] = 23.86%
Steel Ltd 35 2 [60-35] = 25 [14 + 0.70(26.33 – 14)] = 22.63%
Liquor Ltd 45 2 [135-45] = 90 [14 + 0.50(26.33 – 14)] = 20.17%
GOI Bonds 1000 140 [1005-1000] = 5 [14 + 0.99(26.33 – 14)] = 26.21%
Total 1105 146 145
Notes:
Return on Market Portfolio – Expected Return on Market Portfolio [Rm]
= Dividends + Capital Gains
Cost of the Total Investment=
146 + 145
1105 × 100= 26.33%
In the absence of Return of a Market Portfolio, it is assumed that portfolio containing one unit of the
four securities listed above would result in a completely diversified portfolio, and therefore represent
the Market Portfolio.
Portfolio’s Expected Return based on CAPM –
1. If the Portfolio contains the above securities in equal proportion in terms of value –
Expected Return = [23.86% + 22.63% + 20.17% + 26.21%] ÷ 4 = 23.22%
2. If the Portfolio contains one unit of the above securities, then –
Securities Cost Expected Return Product
Cement Limited 25 23.86% 25 x 23.86 = 596.25
Steel Limited 35 22.63% 35 x 22.63 = 792.05
Liquor Limited 45 20.17% 45 x 20.17 = 907.65
GOI Bonds 1000 26.21% 1000 x 26.21 = 26210
Total 1105 28505.95
Weighted Return 28505.95/1105 = 25.79%
Therefore, Expected Return from Portfolio (based on CAPM) = 25.79%
24 | P a g e
Portfolio Beta and Return – Effect of Change in Portfolio Sadie has invested in four Securities A, B, C and D, the Particulars of which are as follows –
Security A B C D
Amount Invested 125000 150000 80000 145000
Beta (β) 0.60 1.50 0.90 1.30
If RBI Bonds carries an Interest Rate of 5% and NIFTY yields 12%, what is the expected return on
Portfolio? If Investment in Security C is replaced by Investment in RBI Bonds, what is the
corresponding change in Portfolio Beta and Expected Return?
Computation of Expected Return on Portfolio [under CAPM]
Computation of Weighted Beta [Beta of the Portfolio]
Security Amount Invested
Proportion of Investment to Total Investment
Beta of Investment
Weighted Beta
A B C D
125000 150000 80000
145000
125000 ÷ 500000 = 0.25 150000 ÷ 500000 = 0.30 80000 ÷ 500000 = 0.16
145000 ÷ 500000 = 0.29
0.60 1.50 0.90 1.30
0.25 x 0.60 = 0.150 0.30 x 1.50 = 0.450 0.16 x 0.90 = 0.144 0.29 x 1.30 = 0.377
Total 500000 1.00 1.121
Computation of Expected Return on Portfolio
Expected Return [E(RP)] = RF + β(RM + RF) = 5% + [1.121 x (12% - 5%)] = 5% + [1.121 x 7%]
= 5% + 7.847% = 12.847%
Computation of Expected Return [Investment in C, replaced by RBI Bonds] (CAPM)
Computation of Weighted Beta [Beta of the Portfolio]
Since β of Risk Free Investments [RBI Bonds] is 0, the Weighted Average Beta will be [1.121 – 0.144]
= 0.977
Computation of Expected Return on Portfolio
Expected Return = RF + β(RM + RF)
= 5% + 0.977(12% - 5%)
= 5% + 0.977(7%)
= 5% + 6.839% = 11.839%
Risk and Return Comparison Consider the following information on two Stocks A and B:
Year Return A (%) Return B (%)
1 2
10 16
12 18
You are required to determine:
1. The Expected Return on a Portfolio containing A and B in the proportion of 40% and 60%
respectively
2. The Standard Deviation of Return from each of the two stocks
3. The Covariance of Returns from the two stocks
4. Correlation Co-efficient between the Returns of the two stocks
5. The Risk of a Portfolio containing A and B in the proportion of 40% and 60%
The Expected Return on Stock A = 10+16
2= 13%
The Expected Return on Stock B = 12+18
2= 15%
25 | P a g e
The Expected Return on the Portfolio consisting of A and B = (0.4 x 13) + (0.6 x 15) = 14.2%
Year R1 R2 D1 = R1 - Ṝ1 D2 = R2 = Ṝ2 D12 D22 D1 x D2
(1) (2) (3) (4) = [(2)-13] (5) = [(3)-15] (6) = (4)2 (7) = (5)2 (8) = (4)x(5)
1 2
10 16
12 18
-3 3
-3 3
9 9
9 9
9 9
ΣR1 = 26 ΣR2=30 18 18 18
Stock A Stock B
Mean Ṝ1 =
𝛴Ṝ1
𝑛=
26
2= 13 Ṝ2 =
𝛴Ṝ2
𝑛=
30
2= 15
Variance 𝜎𝑅12 =
∑ 𝐷12
𝑛=
18
2= 9 𝜎𝑅22 =
∑ 𝐷22
𝑛=
18
2= 9
Standard Deviation R1 = 9 = 3 R2 = 9 = 3
Covariance and Correlation
Combination Stock A and B
Covariance 𝐶𝑂𝑉𝐴𝐵 =
∑(𝐷𝐴 × 𝐷𝐵)
𝑛=
18
2= 9
Correlation 𝜌𝐴𝐵 =
𝐶𝑂𝑉𝐴𝐵
𝜎𝐴 × 𝜎𝐵=
9
3 × 3= 1
Portfolio Risk 𝜎𝑃 = √(𝜎𝐴2 × 𝑊𝐴2) + (𝜎𝐵2 × 𝑊𝐵2) + 2(𝜎𝐴 × 𝑊𝐴 × 𝜎𝐵 × 𝑊𝐵 × 𝜌𝐴𝐵)
= √(0.42 × 32) + (0.62 × 32) + 2(0.4 × 0.6 × 3 × 3 × 1)
= √1.44 + 3.24 + 4.32 = √9 = 3
Portfolio Risk and Return An Investor has decided to Invest Rs. 100000 in the Shares of two Companies, namely ABC and XYZ.
The projections of Returns from the Shares of the two Companies along with their Probabilities
are as follows:
Probability ABC (%) XYZ (%)
0.20 0.25 0.25 0.30
12 14 -7 28
16 10 28 -2
You are required to –
1. Comment on Return and Risk of Investment in Individual Shares
2. Compare the Risk and Return of these two Shares with a portfolio of these Shares in equal
proportions
3. Find out the proportion of each of the above shares to formulate a minimum Risk Portfolio
Computation of Expected Returns and Risk of the Individual Shares
P RA RB PxRA PxRB DA = RA - ṜA
DA = RB - ṜB
P x (DA)2 P x (DB)2 P(DAxDB)
0.20 0.25 0.25 0.20
12 14 -7 28
16 10 28 -2
2.40 3.50 -1.75 8.40
3.20 2.50 7.00 -0.60
0.55 1.45
-19.55 15.45
3.90 -2.10 15.90 -14.10
0.06 0.53
95.55 71.61
3.04 1.10
63.20 59.64
0.43 -0.76
-77.71 -65.35
ṜA=12.55% ṜB=12.1% A2=167.75 B2=126.98 -143.39
Standard Deviation = A = 167.75 = 12.95%, Standard Deviation = B = 126.98 = 11.27%
26 | P a g e
Computation of Co-efficient of Variation [Risk per unit of Return]
ABC = 12.95/12.55 = 1.03
XYZ = 11.27/12.10 = 0.93
Hence, based on Risk, XYZ is more preferable.
Risk and Return of the Portfolio (50% : 50% mix)
Return on Portfolio
Return % = (12.55 x 0.50) + (12.10 x 0.50) = 12.325%
Returns in Amount = 1000 x 12.325 = Rs. 12325
Risk of Portfolio – Standard Deviation of the Portfolio [Matrix Approach]
WABC (0.50) WXYZ (0.50)
WABC
(0.50) = 0.5 x 0.5 x ABC
2
= 0.25 x (12.95)2 = 41.9256
=0.5 x 0.5 x COV(XYZ, ABC) = 0.25 x (-144.255) = -36.0638
BABC XYZ2 =
41.9256+31.7532x(-36.0638) = 1.5512%
WXYZ
(0.5) 0.5 x 0.5 x COV(XYZ, ABC)
= 0.25 x (-144.255) = -36.0638
0.5 x 0.5 x XYZ2
= 0.25 x (11.27)2 = 31.7532
ABC XYZ2 = 1.245%
Minimum Risk Portfolio = 𝜎𝑋𝑌𝑍
2 − 𝐶𝑂𝑉(𝐴𝐵𝐶,𝑋𝑌𝑍)
𝜎𝐴𝐵𝐶2 + 𝜎𝑋𝑌𝑍
2 − 2𝐶𝑂𝑉(𝐴𝐵𝐶,𝑋𝑌𝑍)
=(11.27)2 − (−143.39)
(12.95)2 + (11.27)2 − 2(−143.39)= 46.5%(Proportion of Investments in ABC Share)
WABC = 100% - WABC = 100% - 46.5% = 53.5% (Proportion of Investments in XYZ Share)
CAPM – Evaluation of Securities Following is the data regarding six Securities –
Securities A B C D E F
Return (%) 8 8 12 4 9 8
Risk (%) (Standard Deviation) 4 5 12 4 5 6
1. Which of the Securities will be selected?
2. Assuming perfect correlation, whether it is preferable to Invest 75% in Security A and 25%
Security C
Selection of Securities:
1. Securities A, B and F have identical return at 8%. However, Security A has a risk of 4% only
(least among A, B and F). Therefore, A should be selected (as it is the Security with the least
risk and highest return in its risk category)
2. Securities B and E have identical risk factor at 5%. However, return on Security E is more
than B. Therefore, E should be preferred over B
Selection – A and E may be selected
Security C and B may also be selected on grounds of higher return
Investment in A and C
Since there is a perfect correlation between A and C, risk and return can be averaged with
proportion.
1. Return on Portfolio A and C – 75% of Return on Security A + 25% of Return on Security C i.e.
75% x 8 + 25% x 12% = 6% + 3% = 9% (Risk on Portfolio)
2. Risk on the Portfolio of A and C – 75% of risk of Security A + 25% of Risk of Security C i.e. 75%
x 4 + 25% x 12% = 3% + 3% = 6% (Risk on Portfolio)
Compared to Investment in Securities A and C, investment E is better. This is because, for the same
return (i.e. 9%), Security E has a lower risk factor (at 5% against 6% for the portfolio of A and C)
27 | P a g e
Portfolio of Investment in Mutual Funds Company has a choice of Investments between several different Equity Oriented Funds. The
company has an amount of Rs. 1 Crore to invest. The details of the Mutual Funds are follows –
Mutual Funds A B C D E
Beta β 1.6 1.0 0.9 2.0 0.6
Required –
1. If the Company invests 20% of its Investments in the first two Mutual Funds and an equal
amount in the Mutual Funds C, D and E what is Beta of the Portfolio?
2. If the Company invests 15% of its investments in C, 15% in A, 10% in E and the Balance in
equal amount in the other two Mutual Funds, what is the Beta of the Portfolio?
3. If the Expected Return of the market portfolio is 12% at a Beta Factor of 1.0, what will be
the Portfolio’s Expected Return in both the situations given above?
Situation A – Investment in A and B at 20% each, equal proportion in C, D and E
Mutual Funds Proportion of Investment Beta of the Fund Proportion x Fund Beta
A B C D E
0.2 0.2 0.2 0.2 0.2
1.6 1.0 0.9 2.0 0.6
0.2 x 1.6 = 0.32 0.2 x 1.0 = 0.20 0.2 x 0.9 = 0.18 0.2 x 2.0 = 0.40 0.2 x 0.6 = 0.12
Portfolio Beta 1.22
Investment in C, D, E = [1 – Investment in A and B]/3 = [1-0.2-0.2]/3 = 0.6/3 = 0.2 or 20%
Situation B – Investment in A at 15^%, C at 15% and E at 10%, equal proportion in B and D
Mutual Fund Proportion on Investment Beta of the fund Proportion x Fund Beta
A B C D E
0.15 0.30 0.15 0.30 0.10
1.6 1.0 0.9 2.0 0.6
0.15 x 1.6 = 0.24 0.30 x 1.0 = 0.30
0.15 x 0.9 = 0.135 0.30 x 2.0 = 0.60 0.10 x 0.6 = 0.06
Portfolio Beta 1.335
Investment in B and D = [1 – Investment A,C and E]/2 = [1 – 0.15 – 0.15]/2 = 0.6/2 = 0.3 or 30%
Expected Return from Portfolio
In the absence of Risk Free Rate of Return (RF), it is assumed that expected return from portfolio is to
be computed using Market Model i.e. there is no risk free return, and the entire Fund Return moves
in line with the Market Return. CAPM is not applicable.
Expected Return = Market Return x Portfolio Beta
Situation A –
Return % = 12% x 1.22 = 14.64%
Return in INR = Rs. 1 Crore x 14.64% = Rs. 14.64 Lakhs
Situation B –
Return % = 12% x 1.335 = 16.02%
Return in INR = Rs. 1 Crore x 16.02% = Rs. 16.02 Lakhs
28 | P a g e
Return and Risk of a Portfolio, Proportion of Investment Kevin wants to invest in Stock market. He has got the following information about individual
securities –
Security Expected Return Beta ei2
A B C D E F
15 12 10 09 08 14
1.5 2
2.5 1
1.2 1.5
40 20 30 10 20 30
Market Index Variance is 10% and the Risk Free Return is 7%. What should be the optimum
portfolio assuming no short sales?
Ranking based on Trey-nor Ratio
Security Expected Return Risk Premium Beta Trey-nor Ratio Rank
A B C D E F
15 12 10 09 08 14
15-7 = 8 12-7 = 5 10-7 = 3 09-7 = 2
08 – 7 = 1 14-7 = 07
1.5 2
2.5 1
1.2 1.5
8÷1.5 = 5.33 5÷2 = 2.50
3÷2.5 = 1.20 2÷1 = 2
1÷1.2 = 0.83 7÷1.5 = 4.67
1 3 5 4 6 2
Computation of Zi values Security
ei2 𝑅𝑖 − 𝑅𝑓
𝜎𝑒𝑖2
× 𝛽
Cum. Values
𝛽𝑖2
𝜎𝑒𝑖2
Cum.
Values 𝐶𝑖 =𝜎𝑚2 ∑
𝑅𝑖 − 𝑅𝑓𝜎𝑒𝑖2
1 + 𝜎𝑚2 ∑𝛽2
𝜎𝑚2
𝑍𝑖
= 𝛽1
𝜎𝑒𝑖2(
𝑅𝑖 − 𝑅𝑓
𝛽𝑖
− 𝐶𝑀𝑎𝑥)
A 8/40 x1.5 = 0.3
0.3 1.52÷40 = 0.056
0.056 [10x0.3]/[1+0.1x0.056] = 1.923
0.0375 x (5.33 – 2.814) = 0.09435
F 7/30x1.5 = 0.350
0.650 1.52÷30 = 0.075
0.131 [10x0.650]/[1+10x0.131] = 2.814
0.050 x [4.667 – 2.814] = 0.09280
B 5/20 x 2 = 0.5
1.150 22 ÷ 20 = 0.2
0.331 10 × 1.150
1 + 10 × 0.331= 2.668
Ci is Decreasing Not Applicable
D 2/10x1 = 0.2
1.350 12 ÷ 10 = 0.1
0.431 10 × 1.350
1 + 10 × 0.431= 2.542
Not Applicable
C 3/30x2.5 = 0.250
1.6 2.52÷30 = 0.208
0.639 10 × 1.6
1 + 10 × 0.639= 2.165
Not Applicable
E 1/20x1.2 = 0.06
1.66 1.22÷20 = 0.072
0.711 10 × 1.66
1 + 10 × 0.711= 2.047
Not Applicable
Since Ci [Confidence Index/Cut-off Point] values is maximum after considering Security A and F, and
starts coming down upon inclusion of any further security, the portfolio should be made up of only
Security A and F.
The proportion of A and F in the portfolio should be based on Zi values i.e., weight of investment is
as follows –
A = 0.09435 ÷ (0.09435 + 0.09280) = 0.5041 or 50.41%
F = 0.09280 ÷ (0.09435 + 0.09280) = 0.4959 or 49.59%
29 | P a g e
Portfolio Management – CPPI Model Valarie has a fund of Rs. 3 Lakhs which she wants to invest in Share Market with rebalancing
target after every 10 days to start with for a period of one month from now. The Present NIFTY is
5326. The minimum NIFTY within a month can at most be 4793.4. She wants to know as to how
she would rebalance her portfolio under the following situations, according to the theory of
constant proportion portfolio insurance policy, using “2” as the multiplier:
1. Immediately to start with
2. 10 days later-being the 1st day of rebalancing if NIFTY falls to 5122.96
3. 10 days further from the above data if the NIFTY touches 5539.05
For the sake of simplicity, assume that the value of her equity component will change in tandem
with that of the NIFTY and the Risk Free Securities in which she is going to invest will have no Beta.
Computation of Initial Investment, Floor and Multiplier
Fund Value = 300000
Floor = Lower value expected in terms of Fund Value
Lowest Value expected = Lowest NIFTY 4793.40/Current NIFTY 5326 = 90%
Therefore, Floor = Fund Value x 90% = 270000 [Assuming this constant across the time zone]
Multiplier = 2
Acceptable Loss at the beginning = Fund value or Amount available Rs. 300000 Less Floor Rs. 270000
= Rs. 30000
Investment Position at the beginning (at T.0)
Amount invested in Risky Security (NIFTY) = Multiplier x Acceptable Loss in Value = 2 x 30000 = Rs.
60000
Therefore, amount to be invested in Risk Free Security = Fund value 300000 Less Risky Investment
60000 = Rs. 240000
Position at T.10 and Rebalancing Decision (at T.10)
Value of Risky Investment (NIFTY) [60000 x Closing Index 5122.96/Opening Index 5326] 57712
Value of Risk Free Investment (NO Change in Value since it is risk free) 240000
Total Fund Value at T.10 297712
Acceptable Loss at T.10 = Fund Value 297712 Less Floor 270000 27712
Therefore, Investment in Risky Securities (NIFTY) should be = 2 x Acceptable Loss 27712 55424
Therefore, revised Portfolio Structure
Investment in Nifty should be Rs. 55424, and investment in Risk Free Investment in Risk Free should
be Rs. 242288 [i.e. Fund Value Rs. 297712 – NIFTY Investment Rs. 55424]
Therefore, Value of Risky Investments to be sold and reinvested in Risk Free = 57712 – 55424 = Rs.
2288
Position at T.20 and Rebalancing Decision (at T.20)
Value of Risky Investment (NIFTY) [55424 x Closing Index 5539.04/Opn Index 5122.96] 59925
Value of Risk Free Investment [No Change in Value since it is risk free] 242288
Total Fund Value at T.20 302213
Acceptable Loss at T.20 = Fund Value 302213 – Floor 270000 32213
Therefore, Investment in Risky Securities should be = 2 x Accptble Loss 32213 64426
Therefore, revised Portfolio structure
Investment in NIFTY should be Rs. 62426
Investment in Risk Free should be Rs. 237787 [302213 – 64426]
Therefore value of Risk Free Investment to be sold and reinvested in Risky Investment = 242288 –
237787 = 4501
30 | P a g e
CAPM – Investing Decisions An Investor is holding 1000 shares of Flatlass Company. Presently the Dividend being paid by the
Company is Rs. 2 per share and the share is being sold at Rs. 25 per Share in the Market.
However several factors are likely to change during the course of the year as indicated below –
Risk Free Rate Market Risk Premium Beta Value Expected Growth Rate
Existing 12% 6% 1.4 5%
Revised 10% 4% 1.25 9%
In View of the above factors whether the investor should buy, hold or sell the Shares? Why?
Existing Revised
Rate of Return = Rf + β(Rm – Rf) 12% + 1.4(6%) = 20.4% = 10% + 1.25(4%) = 15%
Price of Share P0 = 𝐷0(1+𝑔)
𝐾𝑒−𝑔
2 × (1.05)
0.204 − 0.05=
2.10
0.154= 𝑅𝑠. 13.63
2 × 1.09
0.15 − 0.09=
2.18
0.06= 𝑅𝑠. 36.33
Current Market Price Rs. 25 Rs. 25
Inference Over-Priced Under-Priced
Decision Sell Buy
CAPM – Overvaluation vs Undervaluation An Investor holds two Stocks A and B. An Analyst prepared ex-ante probability distribution for the
possible Economic Scenarios and the conditional Returns for the two Stocks and the Market Index
as shown below:
Economic Scenario Probability Conditional Returns %
A B Market
(G) (S) (R)
0.40 0.30 0.30
25 10 -5
20 15 -8
18 13 -3
The Risk Free Rate during the next year is expected to be around 11%. Determine whether the
investor should liquidate his holdings in Stocks A and B or on the contrary make fresh investments
in them. CAPM assumptions are holding true.
Computation of Expected Returns, Standard Deviation
Scenario Prob. Ret A
Mean Ret B Mean Mkt
Return Mean DM =
RM–0.2 DM
2 P x DM2
P RA P x RA RB P x RB RM P x RM
G 0.4 25 10 20 8.0 1.8 7.2 7.8 60.84 24.34
S 0.3 10 3 15 4.5 13 3.9 2.8 7.84 2.35
R 0.3 -5 -1.5 -8 -2.4 -3 -0.9 -13.2 174.24 52.27
Estimated Returns 11.5 10.1 10.2 Market Variance 78.96
Standard Deviation of the Market = 78.96 = 8.89%
Computation of Covariance
P DA DB DM DA x DM P x (DA x DM) DB x DM P (DB x DM)
RA – 11.5 RB – 10.1 RM – 0.2
0.4 0.3 0.3
13.5 -1.5
-16.5
9.9 4.9
-18.1
7.8 2.8
-13.2
105.3 -4.2
217.8
42.12 -1.26 65.34
77.22 13.72
238.92
30.89 4.12 7.17
106.20 106.68
COVAM = 106.20, COVBM = 106.68
31 | P a g e
Computation of CAPM Return
Beta - β
ΒA = COVAM/M2 = 106.20/78.96 = 1.34
βB = COVBM/M2 = 106.68/78.96 = 1.35
Under CAPM, Equilibrium Return = Rf + β(RM – Rf)
Expected Return of Security A = 11% + 1.34(10.2 – 11) = 9.93%
Expected Return of Security B = 11% + 1.35(10.2 – 11) = 9.92%
Conclusion and Recommendation
Security A Security B
Estimated Returns Expected Return under CAPM Estimated Ret vs Expected Ret
11.50 9.93
Exp Ret is Lower Stock A is Under Priced
10.10 9.92
Expected Return is Lower Stock B is under-priced
Recommendation Buy/Hold Buy/Hold
Expected Return on Stocks, Alpha and SML Expected Returns on two Stocks for particular Market Returns are given in the following table –
Market Reduction Aggressive Defensive
7% 25%
4% 40%
9% 18%
You are required to calculate:
1. Beta of the two Stocks
2. Expected Return of each Stock, if the Market Return is equally likely to be 7% or 25%
3. The Security Market Line (SML), if the Risk Free Rate is 7.5% and Market Return is equally
likely to be 7% or 25%
4. The Alpha of the Two Stocks
Assuming perfect Correlation, the Beta of the two Stocks:
Aggressive Stock = [40% - 4%]/[25% - 7%] = 2
Defensive Stock = [18% - 9%]/ [25% - 7%] = 0.50
Expected Return of the Two Stocks:
Aggressive Stock = [0.5 x 4%] + [0.5 x 40%] = 22%
Defensive Stock = [0.5 x 9%] + [0.5 x 18%] = 13.5%
Security Market Line (SML):
0.5 x 7% + 0.5% x 25% = 16%
Market Risk Premium = (Rm – Rf) = 16% - 7.5% = 8.5%
SML is required return = 7.5% + β x 8.5%
Alphas of Stocks:
Alpha for Stock A = [Actual Returns] – [Rf + βA(Rm – Rf)] = 0.22 – [0.075 + 2 x 0.085] = -2.5%
Alpha for Stock B = [Actual Returns] – [Rf + βB(Rm – Rf)] = 0.135 – [0.075 + 0.5 x 0.085] = 1.75%
32 | P a g e
Portfolio Analysis – Two Factor Model Mr. X owns a portfolio with the following characteristics –
Security A Security B Risk Free Security
Factor 1 Sensitivity Factor 2 Sensitivity
Expected Return
0.80 0.60 15%
1.50 1.20 20%
0 0
10%
It is assumed that Security Returns are generated by a two-factor model –
1. If Mr. X has Rs. 100000 to invest and sells short Rs. 50000 of Security B and Purchases Rs.
150000 of Security A what is the sensitivity of Mr. X’s portfolio to the two factors?
2. If Mr. X borrows Rs. 100000 at the risk free rate and invests the amount he borrows along
with the original amount of Rs. 100000 in Security A and B in the same proportion as
described in Part (a), what is the sensitivity of the portfolio to the two factors?
3. What is the expected Return Premium of Factor 2?
Sale of Security B and Investment in Security A
Security Portfolio Value (Weights)
Sensitivity (Factor 1)
Product (Factor 1)
Sensitivity (Factor 2)
Product (Factor 2)
A (Invested)
150000 0.80 120000 0.60 90000
B (Sold)
(50000) 1.50 (75000) 1.20 (60000)
100000 30000
Portfolio Sensitivity – Products/Weights for –
Factor 1 = 45000/100000 = 0.45
Factor 2 = 30000/100000 = 0.30
Borrowing at Risk Free Return, Investment in Security A and Security B
Security Portfolio Value (Weights)
Sensitivity (Factor 1)
Product (Factor 1)
Sensitivity (Factor 2)
Product (Factor 2)
A (Invested)
300000 0.80 240000 0.60 180000
B (Invested)
(100000) 1.50 (150000) 1.20 (120000)
Risk Free (Sold)
(100000) 0.00 NIL 0.00 NIL
100000 90000 60000
Portfolio Sensitivity – Products/Weights for –
Factor 1 = 90000/100000 = 0.90
Factor 2 = 60000/100000 = 0.60
Return Premium of Factor 2
Since security returns are generated by a two factor model, it assumed that the model is linear
equation in two variables –
Rs = Rf + βF1X + βF2Y, where,
Rs = Return of the Security
RF = Risk Free Return
βF1 = Factor 1 Sensitivity
βF2 = Factor 2 Sensitivity
X = Return Premium for Factor 1
Y = Return Premium for Factor 2
33 | P a g e
Therefore, RA = 15% = 10% + 0.8x + 0.6y = 0.8x + 0.6y = 5
RB = 20% = 10% + 1.5x + 1.2y = 1.5x + 1.2y = 10
From First Equation, x = [5 – 0.6y]/0.8 = 6.25 – 0.75y
Substituting for x in second equation,
1.5 x (6.25 – 0.75y 0 + 1.2y = 10
9.375 – 1.125y + 1.2y = 10
0.625 = 0.075y
Y = 0.625\0.075 = 8.33%
Therefore, Expected Return Premium for Factor 2 is 8.33%
Portfolio Returns – Arbitrage Pricing Theory Mr. Kevin intends to invest in Equity Shares of a Company the value of which depends upon
various parameters as mentioned below –
Factor Beta Expected Value in % Actual Value in %
GNP Inflation
Interest Rate Stock Market Index
Industrial Production
1.20 1.75 1.30 1.70 1.00
7.70 5.50 7.75
10.00 7.00
7.70 7.00 9.00
12.00 7.50
If the Risk Free Rate of Interest be 9.25%, how much is the return of the Share under Arbitrage
Pricing Theory
Factor Actual
Value % Expected Value %
Difference Beta Difference x Beta
GNP Inflation
Interest Rate Stock Market Index
Industrial Production
7.70 7.00 9.00
12.00 7.50
7.70 5.50 7.75
10.00 7.00
0.00 1.50 1.25 2.00 0.50
1.20 1.75 1.30 1.70 1.00
0.00 2.63 1.63 3.40 0.50
Total 8.16
Return under Arbitrage Pricing Theory = 8.16% + 9.25% (Risk Free Return) = 17.41%
Beta of Company’s Assets The Total Market Value of Equity Share of Sun Company is Rs. 6000000 and the Total Value of the
Debt is Rs. 4000000. The Treasurer estimate that the Beta of the Stocks is currently 1.5 and that
the expected Risk Premium on the Market is 10%. The Treasury Bill Rate is 8%.
Required –
1. What is the Beta of the Company’s existing Portfolio of Assets?
2. Estimate the Company’s Cost of Capital and the Discount Rate for an expansion of the
Company’s Present business
Beta of Company’s existing Portfolio of Assets
Notation Value
βE
βD E D βA
RM - RF RF
Beta of Equity Beta of Debt (since Company’s Debt Capital is risk less, its beta is Zero)
Value of Equity Value of Debt
Beta of Company Assets (Weighted Average Beta) Risk Premium
Risk Free Rate of Return
1.5 0
6000000 4000000
To Calculate 10% 8%
34 | P a g e
𝛽𝐴 =[𝛽𝐸 × 𝐸𝑞𝑢𝑖𝑡𝑦] + [𝛽𝐷 × (𝐷𝑒𝑏𝑡 × (1 − 𝑇𝑎𝑥))]
[𝐸𝑞𝑢𝑖𝑡𝑦] + [𝐷𝑒𝑏𝑡 (1 − 𝑇𝑎𝑥)]
= [1.50 × 60𝐿] + [0 × 40𝐿]
60𝐿 + 40𝐿=
90
100= 0.90
Estimation of Company’s Cost of Capital
Cost of Capital = Ke = Rf + βp x [Risk Premium] = 8 + [0.9 x 10] = 8+9 = 17%
Discount Rate for an expansion of the Company’s present business
In case of expansion plan, 17% can be used as discount factor
In case of diversification plan, a different discount factor would be used depending on its risk profile
Project Beta – Unlevered Firm The Capital of Jazz Ltd, an exclusive software service provider to Bazz Ltd. is made up of 40%
Equity Share Capital, 60% Accumulated Profits and Reserves. Jazz does not have any other clients.
The Sensex yields a Return of 14%. The Risk-less Return is measured at 6.75%.
1. If the Shares of Jazz Ltd carry a Beta (βJAZZ) of 1.6, compute Cost of Capital, and also the
Beta of activity support service to Bazz Ltd
2. If there is another client, Kraze Ltd, accounting for 35% of Assets of Jazz Ltd, with a Beta of
1.40, what should be the Beta of Bazz Ltd, so that the Equity Beta of 1.60 in not affected?
In such a case, what should be expected Return from Bazz Ltd and Kraze Ltd?
Beta of Services to Bazz Ltd [Single Project Model]
Description of Factor Measure
Capital Structure of Jazz Ltd Nature of Capital Structure of Jazz Beta of Equity of Jazz Ltd [βU] Project Status [Multiple or Single] Project Beta [Beta of Service to Bazz = βB] Rule for Unlevered Firm with Single Project Therefore, Beta of Software Services to Bazz Ltd [β Firm = β Assets]
All Equity Unlevered
1.60 Single
To be Ascertained βU = βJ
1.60
Cost of Capital
= Return Expected on Shares of Jazz
= Expected Return on Jazz under CAPM
= Rf + βJAZZ x (Rm – Rf)
= 6.75% + [1.60 x (14% - 6.75%)]
= 18.35
Beta of Services of Bazz Ltd [Multiple Project Model]
Beta of Jazz Shares Ltd (βJAZZ) under Multiple Project Scenario = Weighted Average of Betas of
Projects
βJazz = WJAZZ x βBAzz + Wkraze x βkraze
1.60 = [(1 – 35%) x βBazz] + [35% x 1.4]
1.60 = 0.65 x βBAZZ + 0.49
βBazz = 1.708
Beta of Bazz Ltd should be 1.708
35 | P a g e
Expected Return on Project Bazz and Project KRaze
Expected Return on Project Bazz
Rf + βbazz (Rm – Rf)
6.75% + [1.708{14% - 6.75%)] = 6.75% + 12.383% = 19.133%
Expected Return on Project KRaze
Rf + [β(Rm – Rf)]
= 6.75% + 1.40[14% - 6.75%] = 16.90%
