Ibs gurgaon-8 th ncm

ENDOGENOUS BENCHMARKING OF MUTUAL FUNDS WITH BOOTSTRAP DEA IN ‘R’: SOME INDIAN EVIDENCE

DR.RAM PRATAP SINHA

ASSOCIATE PROFESSOR OF ECONOMICS

GOVERNMENT COLLEGE OF ENGINEERING AND LEATHER TECHNOLOGY

BLOCK-LB,SECTOR-III,SALT LAKE,KOLKATA-700098

E mail: [email protected], [email protected]

EIGHTH NATIONAL CONFERENCE ON INDIAN CAPITAL MARKETS-2014

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Introduction

• The performance of mutual funds is generally evaluated in thecontext of a risk-return framework and the conceptual basiswas provided by Markowitz (1952,1959) and Sharpe-Lintner(1964).

• The M-V framework provided by Markowitz permitted to findout a set of minimum variance portfolios corresponding to agiven/target rate of return. The CAPM framework , on theother hand, linked the excess return from a portfolio to theexcess return available from the market portfolio therebypermitting exogenous benchmarking.


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Objective of the Study

The 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 performance


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The Mean-Variance Criteria

One of the earliest attempt towards portfolio benchmarkingwas by Markowitz (1952) and Tobin(1958) in the form of themean-variance criterion.

The basic idea behind the mean-variance approach is that theoptimal portfolio for an investor is not simply any collection ofsecurities but a balanced portfolio which provides the investorwith the best combination of return and risk where return ismeasured by the expected value and risk is measured by thevariance of the probability distribution of portfolio return.

Given two discrete return distributions f(x) and g(x) , investorswill prefer F(x) over F(G) if µF≥µG and VarF ≤VarG (not bothequalities holding simultaneously).


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The M-V Utility Function

• Markowitz pointed out that in the context of risk aversion, aquadratic of the form a+bR+cR2 provides a closeapproximation of a smooth and concave utility function. Inthis case, maximization of expected utility implies:

Max E[U(R)]= Max [a+ bµ +c E(R2)] =Max [a +b µ +c (µ2+σ2)]

Where µ =expected value of R and σ2 =variance of R. Therefore, this investor willchoose his portfolio solely on the basis ofthe mean and variance of R.


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The 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 σr

2.the expected return from the portfolio is

µ= r1ω1+r2 ω 2+…..+rn ωn

= ∑riωi

= rω and σp2 = ωTσr

2 ω

• Where r is a column vector of returns corresponding to the nsecurities and ω is a row vector of weights relating to the nsecurities included in the portfolio.


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Minimising Risk Relative to a Target Rate of Return

• Suppose the portfolio manager/investor wants to minimizerisk relative to a target rate of return µT. Then the optimizationprogram of the investor is:

Min ½ ωTσr2 ω

• Subject to rω =µT (the target rate of return) and eT ω=1

Where e is a row vector of whose all elements are unity.

• For solving the problem, we form the Lagrangean

L=½ ωTσr2 ω +λ1(rω- µT)+ λ2(eT ω-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- ω1, ω2,-----, ωn).


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Maximising Return Relating to a Target Level Risk

• The optimization problem of the investor now becomes

Max µ, where µ= r1ω1+r2 ω 2+…..+rn ωn

= rω

Subject to, σp2= σT

2 (the target level of variance) and

eT ω=1

• The problem can be solved as before by forming a Lagrangeanfunction.


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Minimizing Risk and Maximising Return

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

Min (½ ωTσr2 ω-λ rω)

Subject to eT ω=1

• For λ>0, the term -λ rω seeks to push rω upwards tocounterbalance the downward pull in respect of ½ ωTσr

2 ω.


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Extension to Non-normal Cases• Hadar and Russell (1969) pointed out that excepting some

special cases (like the quadratic utility function), thespecification of distributions in terms of their moments is notlikely to yield strong results as information about themoments can not be used efficiently for the purpose ofordering uncertain prospects in a situation where the utilityfunction is unknown.

• In this context, Hadar and Russell proposed two decision rulesbased on stochastic dominance(ordering) which are strongerthan the moment method.

• In order to provide a very brief introduction to the concept ofstochastic dominance, let us consider a random variable xtaking the values xi. Let f and g denote the probabilityfunctions of x and F(xi) and G(xi) be the respective cumulativedistributions.


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Concept of Stochastic Dominance• First Order Stochastic Dominance (FSD):

In our example elaborated above, f(x) dominates g(x) if F(x)≤G(x)for all xiϵX. Hadar and Russell proved that under this ruledistributions may be ordered according to preference under anyutility functions.

• Second Order Stochastic Dominance (SSD):The second rule is weaker than the first rule. In the discrete casesecond order stochastic dominance implies that f(x) dominatesg(x) if ∑rG(xi) Δxi ≤ ∑rF(xi) Δxi for all r<n where xn is the largestvalue taken by the random variable and Δxi=xi+1- xi . UnderSSD, distributions may be ordered for any utility function whichexhibits non-increasing marginal utility everywhere.

• Third Order Stochastic Dominance(TSD):Whitmore(1970) introduced the concept of third degree

stochastic dominance as follows: f(x) dominates g(x) if ∑rG(xi)( Δxi)

2 ≤ ∑rF(xi)( Δxi)2 for all r<n where xn is the largest value taken

by the random variable and Δxi=xi+1- xi.


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Stochastic Dominance, Downside Risk & Finance Literature

• The concept of downside risk in the context of portfolio evaluation could be

found in Roy (1952).

• However, path-breaking development in the field of downside risk measures

occurred with the development of the Lower Partial Moment (LPM) risk

measure by Bawa (1975) and Fishburn (1977). Bawa (1975) was the first to

define lower partial moment (LPM) as a general family of below-target risk

measures provided a proof that the LPM measure is mathematically related

to stochastic dominance for risk tolerance values of 0, 1, and 2.This model

was later further generalised by Fishburn who formulated the conditions for

identifying optimal and dominated choice sets i.e. Conditional Stochastic

Dominance which enables the decomposition of the choice set in to optimal

and dominated sets.EIGHTH NATIONAL CONFERENCE ON

INDIAN CAPITAL MARKETS-2014

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Portfolio Evaluation –A Distance Function Approach

• In the context of multi-criteria portfolio evaluation, Shephard’s(1953,1970) distance function approach provides a soundconceptual basis for the derivation of evaluation criteria. The ideais invoked from a multi-input multi-output production systemwhere distance function provide a functional characterisation ofthe structure of production technology.

• The input set of the production technology is characterised by theinput distance function while the output set is characterised bythe output distance function.

• We consider a technology T using a nonnegative vector of inputsX=(x1,x2,......,xn) Rn

+ to produce a nonnegative vector of outputsY=(y1,y2,......,ym) Rm

+ . In functional terms, they can be related as:Y=P(X) and X=L(Y)


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Input Distance Function

• Given this, an input distance function can be defined asDinput= Max*λ:X/λ L(Y)].Intuitively speaking, an input distancefunction gives the maximum amount by which the producer’sinput vector can be radially contracted and yet remainfeasible for the output vector it produces. The reciprocal ofthe input distance function can be considered as the radialmeasure of input oriented technical efficiency. Using DEA wecan compute input oriented technical efficiency as:

Minimise µ

Subject to: µx0-Xλ≥0, y0≤Yλ, j=1,λ≥0


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Output Distance Function

• An output distance function can be defined as Doutput=Min[μ:Y/λ,f(X)].

• Intuitively speaking, an output distance function gives theminimum amount by which the producer’s output vector can bedeflated and yet remain feasible for the input vector it uses. Theoutput distance function can be considered as the radial measureof output oriented technical efficiency.

• The output oriented technical efficiency is calculated from:

Max vrs

• subject to yo Y, Xo X, j=1,j 0


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Graph Hyperbolic Approach

• This implies maximisation of return andminimisation of risk at the same time.Consequently, the optimization problem forthe observed mutual fund is:

• Min G

• Subject to: Gx0≥ Xλ, 1/G y0≤Yλ, λ≥0

• In the VRS case we add the additionalconvexity condition j=1. Technicalefficiency= G


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Introduction to Bootstrap

• Efron (1979) introduced the concept of bootstrap.

• Bootstrap involves resampling from an original sample of datathrough computer-based simulations to obtain the samplingproperties of random variables.

• The starting point of any bootstrap procedure is a sample ofobserved data X = {x1, x2, . . . , xn} drawn randomly from somepopulation with an unknown probability distribution f .

• The premise of the bootstrap method is that the random sampleactually drawn “mimics” its parent population.

• The bootstrap method suggested by Efron (1979) involves drawingof sample (with replacement) directly from the observed data andis known as naive bootstrap.


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Naïve vs Smoothed Bootstrap

• The bootstrap method suggested by Efron (1979)involves drawing of sample (with replacement) directlyfrom the observed data and is known as naivebootstrap.

• In this case the bootstrap sample is effectively drawnfrom a discrete population which fails to recognise thefact that the underlying population density function f iscontinuous.

• Simar and Wilson (1998) suggested that the problemcould be overcome by resorting to smoothed bootstrapwhich involves resampling via a fitted model.


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Smoothed Bootstrap

• The smoothed bootstrap methodology involves the use of Kernelestimators as weight functions.

• If we write the naive bootstrap sample as Xnbs ={x1*, x2*,......, xn*}and the smoothed bootstrap sample as Xsbs ={x1**, x2**,......, xn**}then the elements of the two are related to each other in thefollowing manner: xi**=xi*+hϵ ~f, where h is the smoothingparameter for the density function while xi* and xi** representthe ith elements of the naive and smoothed bootstrap samples.

• In case of bootstrapping, every time when we replicate thebootstrap sample, we get a different sample X**, we will also geta different estimate of θ* = θ(X**). Thus, we need to select alarge number of bootstrap samples, B, in order to extract as manycombinations of xj ( j = 1, 2, . . . , n) as possible.


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Steps in Bootstrapping

• The steps followed in bootstrapping are briefly as follows:• (a)Compute the technical efficiency θ from the observed sample

X.• (b)Select rth (r = 1, 2, . . . ,B) independent bootstrap sample X∗

r, which consists of n data values drawn with replacement from the observed sample X. From this, compute the naïve bootstrap.

• (c) Compute the statistic θsb = θ(X**sb ) from the rth bootstrap

sample X**b

• (d) Construct pseudo-data from the smoothed bootstrap efficiency scores and compute technical efficiency

• (e) Repeat steps (b),(c) and (d) a large number of times (say, N times).

• (f) Calculate the average of the bootstrap estimate as the arithmetic mean (θe).


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Bias Correction

• A measure of the accuracy of an estimator θe

of the parameter θ is the bias measure E( )-θ. The bias-corrected estimator is: θbc = -bias.In our case, we compute bias =θe- θ .

• Thus the bias corrected estimated technicalefficiency is :θbc=2 -θe


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Confidence Interval

• By calculating the standard deviation oftechnical efficiency scores, we can alsocalculate the upper and lower bounds oftechnical efficiency with lower and upperbounds being 2.5% and 97.5% .


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Inputs And Outputs And Period of Study

• In the present study we make use of the distance function approach to benchmark select sectoral mutual fund schemes on the basis of information collected for a half-year.

• The distinguishing feature of the study is that it uses stochastic dominance output indicator.

• Thus mean daily return and mean upside potential (an indicator of second order stochastic dominance) as the two outputs whereas variance of return is taken as the input indicator.

• Thus the input-output correspondence in the present study is:• Output [Mean daily Return, Mean Upside Potential] = f(Standard

Deviation)

• The requisite information about daily NAV for the in-sample mutual fundschemes have been collected from AMFI website and the calculationsregarding mean return, mean upside potential and standard deviationhave been calculated by the author.


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Descriptive Statistics of TechnicalEfficiency Scores


ParticularsInput Oriented

Model

Output Oriented

Model

Graph Hyperbolic

Model

Mean Technical

efficiency

(Uncorrected

Estimation)

0.963 0.980 0.987

Standard Deviation 0.0299 0.0161 0.0105

Mean Technical

efficiency (Bias

Corrected

Estimation)

0.942 0.969 0.897

Standard Deviation 0.0256 0.0129 0.1972

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Confidence Interval of Technical Efficiency


Confidence IntervalInput Oriented

Model

Output Oriented

Model

Graph Hyperbolic

Model

Lower Bound

(2.5%)0.9148 0.9529 0.9255

Upper Bound

(97.5%)0.9618 0.9799 0.9312

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Summing Up

• In the present study, 16 sectoral mutual fundschemes have been evaluated for the secondhalf of 2010 using the concepts of inputoriented, output oriented and graph hyperbolicmeasure.

• For the purpose of performancebenchmarking, the present study makes bothpoint and bootstrap estimates of performance.The bootstrap measures have been used tocorrect bias in pointed technical efficiencyscores.


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THANK YOU

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Ibs gurgaon-8 th ncm

Economy & Finance

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