What and why l Portfolio management is actually the science which involves artful application of ultimate knowledge of finance, human behaviour, statistics.

What and whyWhat and whyWhat and whyWhat and why Portfolio management is actually the science

which involves artful application of ultimate knowledge of finance, human behaviour, statistics and economics.

Portfolio Management is actually a process…

Identification need of the

investor

Identify matching

instruments

Investment in various

securities

Active/passive Management of Portfolio Revision / Evaluation

What and why (Contd…)What and why (Contd…)

Human nature to reduce risk

‘Hot Coffee Vs. Ice-Cream’ Philosophy

if it rains Hot Coffee will sell - if it does not than Ice Cream.

So why not to invest in both - at least one would fetch some profit.

Process of Diversification

Human nature of diversifying because of fear of unknown and tendency of alternatives to behave inversely has given rise to the concept of Portfolio

The Big Picture - Portfolio ManagementThe Big Picture - Portfolio Management

What do you need to know?What do you need to know?

Coefficient of correlation

Standard Deviation

Co-Variance

all of them are tools used to measure the risk involved in a particular security.

Calculation of Risk, in the light of expected return ……… is the gist of Portfolio Management.

One should never forget - what determines the return is the total portfolio risk and not risks involved in individual securities.

Impact of DiversificationImpact of Diversification First question - Is a less risky portfolio possible?

Combination of two securites will be less risky so long as r(xy) <

Second Question - How to derive a ‘No-Risk’ Portfolio?

Theory of Calculus

at the end of the calculations, we get……..

Thus, there are two things which are important in determination of any portfolio

CO-Variance

Weight of an individual security

Insights Insights

In portfolio creation relationship of assets under consideration with one another is critical to risk minimization.

If two assets are perfectly negatively co-related - than it is always possible to derive Zero - Risk combination.

Lower the correlation - higher the gain from diversification

unless two assets are perfectly positively correlated

Efficiency FrontierEfficiency Frontier Assumption : An investor will choose that portfolio which

offers ……

maximum return for the same level of risk.

Minimum risk for the same level of return.

E

SD

Feasible Set

c

a

d

An envelop curve of all portfolios that lie between the global minimum variance portfolio and the maximum return portfolio is the Efficiency Frontier

Efficiency FrontierEfficiency Frontier

E

SD

Feasible Set

a

d

Generation Efficient PortfoliosGeneration Efficient Portfolios

Two constraints

Risk

Return

Two Approaches

Calculus

Quadratic Programming

formulation of Problem

Optimization under constraints - Optimization under constraints - Use of Lagrangean MultiplierUse of Lagrangean Multiplier

Suppose there are 3-securities A, B and C each offering 8%, 10% and 12% returns respectively.

Constraints:

We want 11% return from our total investment.

We want to invest in the given 3 securities only. (this constraint may be assumed - if not given explicitly)


Step - I : Problem formulation

our objective function should be the one that aims at minimizing the risk.

Objective function : Minimize Portfolio Variance

Data Requirement : for this we need data on independent and interactive risks of the securities.

In other words we need Variance and Co-variances of securities.


A B CA 0.15 -0.30 -0.40B -0.30 0.20 -0.20C -0.40 -0.20 0.25

Variance - Co-Variance MatrixA B C

A 0.15 -0.30 -0.40B -0.30 0.20 -0.20C -0.40 -0.20 0.25

Variance - Co-Variance Matrix

Objective Function:

Minimize V =

0.15a2 + 0.2b2 + 0.25c2 +2ab(-0.3) + 2ac(-0.4) + 2bc(-0.2)


Step - II : Formulation of Constraint Equations

Return Constraint

0.08a+0.10b+0.12c = 0.11

or

0.08a+0.10b+0.12c-0.11 = 0

Investment Constraint

a+b+c = 1

or

a+b+c-1 = 0

Application of LM:

Construct a function L which is … ..

L = Objective Function + (Constraint Functions) + (Constraint Functions)

here…

L = 0.15a2 + 0.2b2 + 0.25c2 +2ab(-0.3) + 2ac(-0.4) + 2bc(-0.2) + ( 0.08+0.10b+0.12c-0.11 ) + (a+b+c-1)

find out Partial Derivatives of L with respect to a,b,c, and . Set them equal to Zero.

Solve all the 5 equations.


Limitation of Efficiency Frontier Limitation of Efficiency Frontier ApproachApproach

Efficiency Frontier or Efficiency Locus can be traced with N-Securities also with the same kind of inputs.However volume of data required would be very large.

The amount of information needed in N-Securities is equal to

N - Expected Returns

N - Variance of Returns

(N2 - N) / 2 - Covariances

Total Data Requirement =

N (N + 3) / 2

Another limitation : Excessively Wide Scope

The Sharpe Index ModelThe Sharpe Index Model Assumption : Relative fluctuations in two securities are

not attributable to two securities only - they rather reflect their response to general business conditions. - which might be reflected by a single Index.

Advantage : Reduction in data requirement : no need for Co-variance Data

Model :

this can also be expressed as……

Wherein, ai is broken into (Expected value) and e (Random Value)

Rm ai Ri

ei Rm i i Ri

The Sharpe Index ModelThe Sharpe Index Model

Following are the results that one derives using Sharpe’s Single Index Model

E(Rm) i i E(Ri)

eimi (Ri)Var 222

m j i Rj)Cov(Ri, 2

Some other viewsSome other views Many researchers have put emphasis on diversification

as a tool to reduce risk - most of them stressed not the number of securities but right kinds of securities to reduce risk

King observed that in a typical stock half the variance results from elements that affect the whole market - that means one half of the total risk can never be reduced…

Evans and Archer suggest that unsystematic risk can be reduced naively by holding as few as 10-15 stocks - and infact it can be increased by duplicating the sector….

Security Market LineSecurity Market Line SML depicts the linear relationship between systematic risk and

expected return of individual securities and portfolios.

Remember : the linear relationship between Total Risk and return is depicted by Characteristic Line or Capital Market Line.

Application of SML

performance evaluation of portfolios

test and development of new asset-pricing theories

tests of market efficiency

identification of mis-priced securities

Ex - Post Security Market LineEx - Post Security Market Line

N(ri) is Normal Return that a security earns given a particular level of systematic risk

remember, in CML Beta is the slope of the line. Here Beta is not slope, it is one of the variables.

Yo and Y1 are regression co-efficients.

* Y YN(ri) 10

Ex - Post Security Market LineEx - Post Security Market Line

the difference between expected return and required return is called Alpha ()

a positive alpha implies underpricing of security

a negative alpha implies overpricing of security

β * λλ N(ri) 10

Examples of SMLExamples of SML

/ Ri`

Alpha on SMLAlpha on SML

Tax - Adjusted CAPMTax - Adjusted CAPM

The tax differential between Capital Gain and Dividend - a shortcoming of CAPM as it assumes no taxes

Michael Brennan came out with the concept of Tax Adjusted CAPM

Dm = Dividend Yield on market portfolio

Di = Dividend Yield on Stock

Td = Tax Rate on Dividend

Tg = Tax Rate On Capital Gain

T = Tax Factor = (Td-Tg)/(1-Tg)

TDi TDm] - T)-Rf(1 - β[E(Rm) T)-Rf(1 E(Ri)

Arbitrage Pricing Theory Arbitrage Pricing Theory

APT propounded by Stephen Ross recognizes that several systematic factors affect security returns. - not just one factor (Beta)

Two types of factors

Anticipated : incorporated by investors in into the prices

Unanticipated : source of most of the returns.

Movement of Unanticipated or Unsystematic factors cannot be predicted but responsiveness of asset prices to them can definitely be predicted.

Arbitrage Pricing Theory Arbitrage Pricing Theory

Systematic factors are primary sources of risk - principal determinants of risk.

Thus, entire APT can be divided into 3 discussion phases.

Return - Generating process

Risk-Return Relationship

Arbitrage Mechanism

Arbitrage Pricing TheoryArbitrage Pricing TheoryReturn - Generating ProcessReturn - Generating Process

Stock returns are generated as a function of responsiveness of assest to various factors

Portfolio SelectionPortfolio Selection

Risk and investor preference : Portfolio selection is a function of investor’s risk appetite and availability of optimum portfolio.

Indifference Indifference CurvesCurves

Optimum Optimum PortfolioPortfolio

E

SD


Simple rule : try to earn maximum Risk-adjusted Return

Risk PenaltyRisk Penalty

assumption : the more risk one bears the more undesirable is an additional unit of risk

Risk Penalty = (Risk Squared / Risk Tolerance)

Risk Squared = Variance of the Portfolio

Risk Tolerance = a number between 0 and 100 that shows the willingness to bear risk.


Utility : A concept derived on the basis of Risk Penalty

Utility = Expected Return - Risk Penalty

consider following example

Asset allocation frameworkAsset allocation framework

The process of creation of a portfolio across assets

Step - I : Determination of asset class

on the basis of maturity, form of return, certainty of return, tax status.

Step - 2: Estimation of Risk and Return

What and why l Portfolio management is actually the science which involves artful application of ultimate knowledge of finance, human behaviour, statistics.

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risk portfolio

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