# Ibs gurgaon-8 th ncm

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08-Jul-2015Category

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ENDOGENOUS BENCHMARKING OF SECTORAL MUTUAL FUNDS:A CASE STUDY

ENDOGENOUS BENCHMARKING OF MUTUAL FUNDS WITH BOOTSTRAP DEA IN R: SOME INDIAN EVIDENCEDR.RAM PRATAP SINHAASSOCIATE PROFESSOR OF ECONOMICSGOVERNMENT COLLEGE OF ENGINEERING AND LEATHER TECHNOLOGYBLOCK-LB,SECTOR-III,SALT LAKE,KOLKATA-700098E mail: [email protected], [email protected] NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20141IntroductionThe performance of mutual funds is generally evaluated in the context of a risk-return framework and the conceptual basis was provided by Markowitz (1952,1959) and Sharpe-Lintner (1964).

The M-V framework provided by Markowitz permitted to find out a set of minimum variance portfolios corresponding to a given/target rate of return. The CAPM framework , on the other hand, linked the excess return from a portfolio to the excess return available from the market portfolio thereby permitting exogenous benchmarking.EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20142Objective of the StudyThe study extends the traditional framework of mutual fund benchmarking in three directions:

(a) Use of multiple output indicators

(b) Incorporation of stochastic dominance indicators to apply in more general cases

(c) Use of bootstrap analysis to enable more robust evaluation of performanceEIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20143The Mean-Variance Criteria One of the earliest attempt towards portfolio benchmarking was by Markowitz (1952) and Tobin(1958) in the form of the mean-variance criterion.The basic idea behind the mean-variance approach is that the optimal portfolio for an investor is not simply any collection of securities but a balanced portfolio which provides the investor with the best combination of return and risk where return is measured by the expected value and risk is measured by the variance of the probability distribution of portfolio return.Given two discrete return distributions f(x) and g(x) , investors will prefer F(x) over F(G) if FG and VarF VarG (not both equalities holding simultaneously).EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20144The M-V Utility FunctionMarkowitz pointed out that in the context of risk aversion, a quadratic of the form a+bR+cR2 provides a close approximation of a smooth and concave utility function. In this case, maximization of expected utility implies:

Max E[U(R)]= Max [a+ b +c E(R2)] =Max [a +b +c (2+2)]

Where =expected value of R and 2 = variance of R. Therefore, this investor will choose his portfolio solely on the basis of the mean and variance of R.

EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20145The Mean -Variance Criteria In order to understand how the mean-variance criteria operates, let us consider the case of an n security portfolio where the returns from the n securities are denoted by r1,r2,---,rn and r2.the expected return from the portfolio is = r11+r2 2+..+rn n = rii = r and p2 = Tr2

Where r is a column vector of returns corresponding to the n securities and is a row vector of weights relating to the n securities included in the portfolio.

EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20146

Minimising Risk Relative to a Target Rate of Return

Suppose the portfolio manager/investor wants to minimize risk relative to a target rate of return T. Then the optimization program of the investor is:Min Tr2 Subject to r =T (the target rate of return) and eT =1 Where e is a row vector of whose all elements are unity.For solving the problem, we form the Lagrangean L= Tr2 +1(r- T)+ 2(eT -1)The first order conditions of minimization give us n+2 equations (including the two constraint equations) to solve for n unknowns (the n weights- 1, 2,-----, n).

EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20147Maximising Return Relating to a Target Level RiskThe optimization problem of the investor now becomesMax , where = r11+r2 2+..+rn n = r

Subject to, p2= T2 (the target level of variance) and eT =1

The problem can be solved as before by forming a Lagrangean function.

EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20148

Minimizing Risk and Maximising Return

The investor can incorporate the two objectives of maximizing return and minimizing risk in to a single objective function as:

Min ( Tr2 - r)Subject to eT =1

For >0, the term - r seeks to push r upwards to counterbalance the downward pull in respect of Tr2 .

EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-20149Extension to Non-normal CasesHadar and Russell (1969) pointed out that excepting some special cases (like the quadratic utility function), the specification of distributions in terms of their moments is not likely to yield strong results as information about the moments can not be used efficiently for the purpose of ordering uncertain prospects in a situation where the utility function is unknown.

In this context, Hadar and Russell proposed two decision rules based on stochastic dominance(ordering) which are stronger than the moment method.

In order to provide a very brief introduction to the concept of stochastic dominance, let us consider a random variable x taking the values xi. Let f and g denote the probability functions of x and F(xi) and G(xi) be the respective cumulative distributions.

EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-201410Concept of Stochastic DominanceFirst Order Stochastic Dominance (FSD): In our example elaborated above, f(x) dominates g(x) if F(x)G(x) for all xiX. Hadar and Russell proved that under this rule distributions may be ordered according to preference under any utility functions.Second Order Stochastic Dominance (SSD): The second rule is weaker than the first rule. In the discrete case second order stochastic dominance implies that f(x) dominates g(x) if rG(xi) xi rF(xi) xi for all r