Portfolio management

Portfolio management
Portfolio Management

Market Secret www.marketsecret.net
Summary
Investment
 Investment is putting money into something with the expectation of gain on the
back of thorough analysis, has a high degree of security for the principal amount, as
well as security of return, within an expected period of time.
 In contrast putting money into something with an expectation of gain without
thorough analysis, without security of principal, and without security of return is
gambling.
 Putting money into something with an expectation of gain with thorough analysis,
without security of principal, and without security of return is speculation.
 There is a distinction between “good companies” and “good investments”
 The stock of a well-managed company may be too expensive
 The stock of a poorly-run company can be a great investment if it is cheap
enough
Portfolio management
 Portfolio is none other than Basket of Stocks. Portfolio Management is the
professional management of various securities (shares, bonds and other securities)
and assets (e.g., real estate) in order to meet specified investment goals for the
benefit of the investors.

Need for Portfolio Management
Why create a portfolio?
 To diversify/reduce/mitigate risk of a single security
 All securities in the portfolio may not move together
 If one goes down, others will go up and compensate for the loss of the
first one

Identified investment Objectives in Terms of Risk and Return
Investment objectives
 Capital Preservation –
 Capital Appreciation –
 Current Income –
 Total return
 Factors Affecting Risk Tolerance
 Age- an investor may have lower risk tolerance as they get older and financial
constraints are more prevalent.
 Family situation - an investor may have higher income needs if they are supporting
a child in college or an elderly relative.
 Wealth and income - an investor may have a greater ability to invest in a portfolio
if he or she has existing wealth or high income.
 Psychological - an investor may simply have a lower tolerance for risk based on his
personality.

Investment Constraints
 Investment Constraints
When creating a policy statement, it is important to consider an
investor's constraints. There are five types of constraints that
need to be considered when creating a policy statement. They are
as follows:
 Liquidity Constraints –
 Time Horizon
 Tax Concerns –
 Legal and Regulatory –
 Unique Circumstances –

Asset Allocation
 Asset Allocation is the process of dividing a portfolio among major asset categories such as bonds,
stocks or cash. The purpose of asset allocation is to reduce risk by diversifying the portfolio.
 The ideal asset allocation differs based on the risk tolerance of the investor. For example, a young
executive might have an asset allocation of 80% equity, 20% fixed income, while a retiree would be
more likely to have 80% in fixed income and 20% equities.
 cash and cash equivalents (e.g., deposit account, money market fund)
 fixed interest securities such as Bonds: investment-grade or junk (high-yield); government or
corporate; short-term, intermediate, long-term; domestic, foreign, emerging markets; or
Convertible security
 stocks: value, dividend, growth, sector specific or preferred (or a "blend" of any two or more of the
preceding); large-cap versus mid-cap, small-cap or micro-cap; public equities versus private
equities, domestic, foreign (developed), emerging or frontier markets
 Commodities: precious metals, broad basket, agriculture, energy, others
 commercial or residential real estate (also REITs)
 collectibles such as art, coins, or stamps
 insurance products (annuity, life settlements, catastrophe bonds, personal life insurance products,
etc.)
 derivatives such as long-short or market neutral strategies, options, collateralized debt and futures
 foreign currency
 venture capital, leveraged buyout, merger arbitrage or distressed securities

Type of Assets Allocation
 There are several types of asset allocation strategies based on investment goals, risk
tolerance, time frames and diversification: strategic, tactical, and core-satellite.
 Strategic Asset Allocation — the primary goal of a strategic asset allocation is to
create an asset mix that will provide the optimal balance between expected risk and
return for a long-term investment horizon.
 Tactical Asset Allocation — method in which an investor takes a more active
approach that tries to position a portfolio into those assets, sectors, or individual
stocks that show the most potential for gains.
 Core-Satellite Asset Allocation — is more or less a hybrid of both the strategic and
tactical allocations mentioned above.
Selection of right investment vehicle
 Market conditions
 Rate of Inflation
 Intrinsic valuation
Portfolio Monitoring

Defining Risk
 Risk refers to the chance that some unfavorable event will happen
 Investment risk is the probability that actual returns may deviate from expected
returns
 The chance that actual returns may be lower than expected return gives rise to
investment risk
 Higher the probability of actual returns being less than expected, higher will be
investment risk
Types of risk
 Systematic risk
 Unsystematic risk
Measuring Risk
 Risk is measured by standard deviation of possible returns
n
Variance (σ2
) = ∑ (ri – E(r))2
Pi
i=1
Standard Deviation (σ) = (σ2
)1/2

Nonsystematic/diversifiable risks
 In finance and economics, systematic risk (sometimes called aggregate risk,
market risk, or un-diversifiable risk) is vulnerability to events which affect
aggregate outcomes such as broad market returns, total economy-wide
resource holdings, or aggregate income. Systematic or aggregate risk arises
from market structure or dynamics which produce shocks or uncertainty faced
by all agents in the market; such shocks could arise from government policy,
international economic forces, or acts of nature, ITechnological innovations,
 The risk that is specific to an industry or firm. Examples of unsystematic risk
include losses caused by labor problems, nationalization of assets, or weather
conditions. This type of risk can be reduced by assembling a portfolio with
significant diversification so that a single event affects only a limited number of
the assets. Also called diversifiable risk

Variance, covariance
 Variance
 The average of the squared differences from the Mean.
 Standard Deviation
 The Standard Deviation is a measure of how spread out numbers is.
 Its symbol is σ (the greek letter sigma)
 The formula is easy: it is the square root of the Variance
n
Variance (σ2
) = ∑ (ri – E(r))2
Pi
i=1
Standard Deviation (σ) = (σ2
)1/2
 Covariance
 A measure of the degree to which returns on two risky assets move in tandem. A
positive covariance means that asset returns move together. A negative covariance
means returns move inversely.
 One method of calculating covariance is by looking at return surprises (deviations
from expected return) in each scenario. Another method is to multiply the correlation
between the two variables by the standard deviation of each variable.


Correlation coefficient
 Correlation coefficient measure that determines the degree to which two variable's
movements are associated.
The correlation coefficient is calculated as:
 The correlation coefficient will vary from -1 to +1. A -1 indicates perfect negative
correlation, and +1 indicates perfect positive correlation.
The correlation coefficient will vary from -1 to +1. A -1 indicates perfect negative
correlation, and +1 indicates perfect positive correlation.

12
Variance of A Linear Combination
 One measure of risk is the variance of return
 The variance of an n-security portfolio is:
2
1 1
where proportion of total investment in Security
correlation coefficient between
Security and Security
n n
p i j ij i j
i j
i
ij
x x
x i
i j
σ ρ σ σ
ρ
= =
=
=
=
∑∑

13
Variance of A Linear Combination (cont’d)
 The variance of a two-security portfolio is:
2 2 2 2 2
2p A A B B A B AB A Bx x x xσ σ σ ρ σ σ= + +

14
 Return variance is a security’s total risk
 Most investors want portfolio variance to be as low as possible without
having to give up any return
2 2 2 2 2
2p A A B B A B AB A Bx x x xσ σ σ ρ σ σ= + +
Total Risk Risk from A Risk from B Interactive Risk

15
 If two securities have low correlation, the interactive risk will be small
 If two securities are uncorrelated, the interactive risk drops out
 If two securities are negatively correlated, interactive risk would be
negative and would reduce total risk
 Various portfolio combinations may result in a given return
 The investor wants to choose the portfolio combination that provides the least
amount of variance

Measuring Systematic Risk
 How can we estimate the amount or proportion of an asset's risk that is diversifiable or
non-diversifiable?
 Systematic risk of a portfolio is measured by beta of a security
 Meaning of beta
 Tendency of a stock to move with the market
 Sensitivity of an asset’s price to the changes in the market
 Beta is a measure of sensitivity: it describes how strongly the stock return moves
with the market return.
)Var(R
)R,Cov(R
M
Mi
i =β

Returns
 Actual Return
 Realized return/historical return/return ex-post
 Expected Return
 Return ex-ante/anticipated return
 A weighted average of all possible returns, where weights represent probability
of each possible outcome
 Multiply each possible outcome with its probability and add them up over all
possible outcomes
 The table below provides a probability distribution for the returns on stocks A and B
State Probability Return On Return On
Stock A Stock B
1 20% 5% 50%
2 30% 10% 30%
3 30% 15% 10%
4 20% 20% -10%
 The state represents the state of the economy one period in the future i.e. state 1 could represent a
recession and state 2 a growth economy.
 The probability reflects how likely it is that the state will occur. The sum of the probabilities must equal
100%.
 The last two columns present the returns or outcomes for stocks A and B that will occur in each of the four
states.

Return
 Market” pays investors for two services they provide: (1) surrendering their capital
and forgoing current consumption and (2) risk
 The first gets you the time value of money.
 The second gets you a risk premium whose size depends on the share of total risk
you take on.
 Time has a value
 A dollar received today is worth more than a dollar received tomorrow
 This is because a dollar received today can be invested to earn interest
 The amount of interest earned depends on the rate of return that can be earned on
the investment
 Present value of a lump sum:
PV = CFt / (1+r)t
 Future value of a lump sum:
FVt = CF0 * (1+r)t
OR FVt = PV * (1+r)t

Measuring Expected Return
 E(r) = P1 r1 + P2r2 + … + Pnrn
n
= ∑ Pi ri
i=1
ri is the ith possible outcome and
Pi is the probability of ith outcome
Portfolio Expected Return
 A simple weighted average of the expected return of each security in the portfolio,
where weights represent the proportion of investment in each portfolio

 E(rp) = (w1× E(r1)) + (w2× E(r2)) + … +(wn× E(rn))
 n
 E(rp) = Σ wi E(ri)
 i=1

Return (%) Deviation from mean Squared Dev. from Mean
Probability A B P A B P A*B A B P
0.2 18 25 21.5 2 13 7.5 26 4 169 56.25
0.2 30 10 20 14 -2 6 -28 196 4 36
0.2 -10 10 0 -26 -2 -14 52 676 4 196
0.2 25 20 22.5 9 8 8.5 72 81 64 72.25
0.2 17 -5 6 1 -17 -8 -17 1 289 64
Mean 16 12 14 21 191.6 106 84.9
Risk and Return in Portfolios: Example
•Two Assets, A and B
•A portfolio, P, comprised of 50% of your total investment
invested in asset A and 50% in B.
•There are five equally probable future outcomes, see below.
In this case:
•VAR(RA) = 191.6, STD(RA) = 13.84, and E(RA) = 16%.
•VAR(RB) = 106.0, STD(RB) = 10.29, and E(RB) = 12%.
•COV(RA, RB) = 21
•CORR(RA, RB) = 21/(13.84*10.29) = .1475.
•VAR(RP)=84.9, STD(Rp)=9.21, E(Rp)=½ E(RA) + ½ E(RB)=14%
•Var(Rp) or STD(RP) is less than that of either component!

Risk and Return
Modern portfolio theory (MPT)

 Modern portfolio theory (MPT) is a theory of finance which attempts to
maximize portfolio expected return for a given amount of portfolio risk,
or equivalently minimize risk for a given level of expected return, by
carefully choosing the proportions of various assets. Although MPT is
widely used in practice in the financial industry and several of its
creators won a Nobel memorial prize for the theory,[1] in recent years
the basic assumptions of MPT have been widely challenged by fields
such as behavioral economics.
 MPT is a mathematical formulation of the concept of diversification in
investing, with the aim of selecting a collection of investment assets that
has collectively lower risk than any individual asset. That this is possible
can be seen intuitively because different types of assets often change in
value in opposite ways

Summary
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

Risk and Return
 When we are concerned with only one asset its risk and return can be
measured, as discussed, using expected return and variance of return.
 If there is more that one asset (so portfolios can be formed) risk
becomes more complex.
 Risky vs. Risk-free Assets
 The classifications of risky and risk-free assets are based on relative terms and
not on absolute terms. It is important to note that no financial asset can be
completely risk-free. A risk-free asset is defined as an asset that has the lowest
level of risk among all the available assets. In other words, it is “risk-free” relative
to the other assets
 Expected return of a risky asset as follows:
 E(Rr)=Rf+(E(Rr-Rf)
 = minimum compensation+ compensation for taking additional risk(risk premium)
 We will show there are two types of risk for individual assets:
 Diversifiable/nonsystematic/idiosyncratic risk
 Non-diversifiable/systematic/market risk

Risky vs. Risk-free Assets
 Suppose an investor is putting together a portfolio that contains both types of
assets: w proportion of the portfolio is made up of the risky asset and (1-w) is made
up of risk-free asset. As a result, the return of the portfolio
 Rp=w.Rr+(1-w)rf

Risky vs. Risk-free Assets
 We can then substitute the w as defined in the formula for the expected return of the
portfolio as follows
 The equation above represents the risk and return relationship of a portfolio with a
risky asset and a risk-free asset. It is also known as the Capital Allocation Line
(CAL), and the following is a graphical representation of the CAL:

CHAPTER 9 – The Capital Asset Pricing
Model (CAPM)
9 - 27
The Capital Asset Pricing Model
 Uses include:
 Determining the cost of equity capital.
 The relevant risk in the dividend discount model to estimate a
stock’s intrinsic (inherent economic worth) value.
Estimate
Investment’
s Risk (Beta
Coefficient)
Determine
Investment’s
Required Return
Estimate the
Investment’s
Intrinsic Value
Compare to the
actual stock price in
the market
2i
M
i,M
σ
COV
=β )( iMi RFERRFk β−+=
gk
D
P
c −
= 1
0
Is the
stock fairly
priced?

CHAPTER 9 – The Capital
Asset Pricing Model (CAPM)
9 - 28
Assumptions

CAPM is based on the following assumptions:
 All investors have identical expectations about expected returns,
standard deviations, and correlation coefficients for all securities.
 All investors have the same one-period investment time horizon.
 All investors can borrow or lend money at the risk-free rate of
return (RF).
 There are no transaction costs.
 There are no personal income taxes so that investors are
indifferent between capital gains an dividends.
 There are many investors, and no single investor can affect the
price of a stock through his or her buying and selling decisions.
Therefore, investors are price-takers.
 Capital markets are in equilibrium.

CHAPTER 9 – The Capital Asset Pricing
Model (CAPM)
9 - 29
 An hypothesis by Professor William Sharpe
 Hypothesizes that investors require higher rates of return for greater
levels of relevant risk.
 There are no prices on the model, instead it hypothesizes the
relationship between risk and return for individual securities.
 It is often used, however, the price securities and investments.
 Under Valued and Over Valued Stocks: The CAPM model can be practically used to buy, sell or hold
stocks. CAPM provides the required rate of return on a stock after considering the risk involved in an
investment. Based on current market price or any other judgmental factors (benchmark) one can
identify as to what would be the expected return over a period of time. By comparing the required
return with the expected return the following investment decisions are available
 (a)  When CAPM < Expected Return – Buy: This is due to the stock being undervalued i.e. the stock
gives more return than what it should give.
 (b)  When CAPM > Expected Return – Sell: This is due to the stock being overvalued i.e. the stock
gives less return than what it should give.
 (c)  When CAPM = Expected Return – Hold: This is due to the stock being correctly valued i.e. the
stock gives same return than what it should

9 - 30
Security market line
 Security market line or SML (also known as characteristic line) is
the graphical representation of Capital Asset Pricing Model or
CAPM. Know more about Capital Asset Pricing Model. Security
market line is a straight sloppy line which gives the relationship
between expected rate of return and market risk (or systematic
risk) of over all market.
 When used in portfolio management, the SML represents the
investment's opportunity cost (investing in a combination of the
market portfolio and the risk-free asset).

SML

9 - 32
The Capital Market Line
σρ
ER
RF
MERM
σM
P
M
M
P
RFER
RFk σ
σ 




 −
+=
CML
The CML is
that set of
achievable
portfolio
combinations
that are
possible when
investing in
only two
assets (the
market
portfolio and
the risk-free
asset (RF).
The market
portfolio is the
optimal risky
portfolio, it
contains all
risky securities
and lies
tangent (T) on
the efficient
frontier.
The CML has
standard
deviation of
portfolio
returns as the
independent
variable.

9 - 33
The Market Portfolio and the Capital Market Line (CML)
 line used in the capital asset pricing model to illustrate the rates of return
for efficient portfolios depending on the risk-free rate of return and the
level of risk (standard deviation) for a particular portfolio.
 CML measures the risk through standard deviation, or through a total risk
factor. On the other hand, the SML measures the risk through beta, which
helps to find the security’s risk contribution for the portfolio
 The slope of the CML is the incremental expected return divided by the
incremental risk.
 This is called the market price for risk. Or
 The equilibrium price of risk in the capital market.
RF-ER
CMLtheofSlope
M
M
σ
=
[9-4]

Sharpe Index Model
 William Sharpe has developed a simplified variant of Markowitz model that
reduces substantially its data and computational requirements. It is known
as Single index model or One- factor analysis.
 8.1 Single Index Model: This model assumes that co-movement
between stocks is due to change or movement in the market index. Casual
observation of the stock prices over a period
 of time reveals that most of the stock prices move with the market index.
When the Sensex increases, stock prices also tend to increase and vice-
versa. This indicates that some underlying factors affect the market index
as well as the stock prices. Stock prices are related to the market index
and this relationship could be used to estimate the return on stock.

NN
σσmm
22
(R(Rii R─ R─ ff)β)βii
σσeiei
22
i=1i=1
CCii = N= N
1 + σ1 + σmm
22
ββii
22
σσeiei
22
i =1

Where,
σm
2
= Variance of the Market Index
σei
2
= Variance of a stock’s movement
that is not associated with the
movement of Market Index i.e. stock’s
unsystematic risk.

EXAMPLE- 1:

SOLUTION OF EXAMPLE- 1:

SOLUTION OF EXAMPLE- 2:

 CAPM is criticized because of
 Many unrealistic assumptions
 Difficulties in selecting a proxy for the market
portfolio as a benchmark
 Alternative pricing theory with fewer
assumptions was developed:
 Arbitrage Pricing Theory (APT)
Arbitrage Pricing Theory

Three Major Assumptions:
1. Capital markets are perfectly competitive
2. Investors always prefer more wealth to less
wealth with certainty
3. The stochastic process generating asset
returns can be expressed as a linear function
of a set of K factors or indexes

Does not assume:
 Normally distributed security returns
 Quadratic utility function
 A mean-variance efficient market portfolio

 The APT Model
E(Ri)=λ0+ λ1bi1+ λ2bi2+…+ λkbik
where:
λ0=the expected return on an asset with zero
systematic risk
λj=the risk premium related to the j th common
risk factor
bij=the pricing relationship between the risk
premium and the asset; that is, how
responsive asset i is to the j th common
factor

Comparing the CAPM & APT Models
CAPM APT
Form of Equation Linear Linear
Number of Risk Factors 1 K (≥ 1)
Factor Risk Premium [E(RM) – RFR] {λj}
Factor Risk Sensitivity βi {bij}
“Zero-Beta” Return RFR λ0
Unlike CAPM that is a one-factor model,
APT is a multifactor pricing model

Comparing the CAPM & APT Models
 However, unlike CAPM that identifies the
market portfolio return as the factor, APT
model does not specifically identify these
risk factors in application
 These multiple factors include
 Inflation
 Growth in GNP
 Major political upheavals
 Changes in interest rates

Using the APT
 λ1: The risk premium related to the first risk
factor is 2 percent for every 1 percent change
in the rate (λ1=0.02)
 λ2: The average risk premium related to the
second risk factor is 3 percent for every 1
percent change in the rate of growth (λ2=0.03)
 λ0: The rate of return on a zero-systematic risk
asset (i.e., zero beta) is 4 percent (λ0=0.04

Determining Sensitivities for Assets
 bx1 = The response of asset x to changes in the
inflation factor is 0.50 (bx1 0.50)
 bx2 = The response of asset x to changes in the
GDP factor is 1.50 (bx2 1.50)
 by1 = The response of asset y to changes in the
inflation factor is 2.00 (by1 2.00)
 by2 = The response of asset y to changes in
the GDP factor is 1.75 (by2 1.75)

Using the APT to Estimate Expected Return
22110)( iii bbRE λλλ ++=
21 03.02.04.)( iii bbRE ++=

Asset X
E(Rx) = .04 + (.02)(0.50) + (.03)(1.50)
E(Rx) = .095 = 9.5%
Asset Y
E(Ry) = .04 + (.02)(2.00) + (.03)(1.75)
E(Ry)= .1325 = 13.25%
Using the APT to Estimate Expected Return

Valuing a Security Using the APT:
An Example
 Three stocks (A, B, C) and two common systematic
risk factors have the following relationship (Assume
λ0=0 )
E(RA)=(0.8) λ1 + (0.9) λ2
E(RB)=(-0.2) λ1 + (1.3) λ2
E(RC)=(1.8) λ1 + (0.5) λ2

Valuing a Security Using the APT:
An Example
 If λ1=4% and λ2=5%, then it is easy to compute the
expected returns for the stocks:
E(RA)=7.7%
E(RB)=5.7%
E(RC)=9.7%

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