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:
rampratapsinha39@gmail.com, rp1153@rediffmail.com
EIGHTH NATIONAL CONFERENCE ON INDIAN
CAPITAL MARKETS-2014
1

Introduction
• The 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-2014
2

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
3

The 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 µF≥µG and VarF ≤VarG (not both
equalities holding simultaneously).
4

The M-V Utility Function
• Markowitz 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+ 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 will
choose his portfolio solely on the basis of
the mean and variance of R.
5

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 σp
2 = ωTσr
2 ω
• Where r is a column vector of returns corresponding to the n
securities and ω is a row vector of weights relating to the n
securities included in the portfolio.
6

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 µT. Then the optimization
program of the investor is:
Min ½ ωTσr
2 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 ωTσr
2 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).
7

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, σp
2= σT
2 (the target level of variance) and
eT ω=1
• The problem can be solved as before by forming a Lagrangean
function.
8

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 (½ ωTσr
2 ω-λ rω)
Subject to eT ω=1
• For λ>0, the term -λ rω seeks to push rω upwards to
counterbalance the downward pull in respect of ½ ωTσr
2 ω.
9

Extension to Non-normal Cases
• Hadar 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.
10

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 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) Δxi ≤ ∑rF(xi) Δxi for all r<n where xn is the largest
value taken by the random variable and Δxi=xi+1- xi . Under
SSD, distributions may be ordered for any utility function which
exhibits 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.
11

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
12

Portfolio Evaluation –A Distance
Function Approach
• In the context of multi-criteria portfolio evaluation, Shephard’s
(1953,1970) distance function approach provides a sound
conceptual basis for the derivation of evaluation criteria. The idea
is invoked from a multi-input multi-output production system
where distance function provide a functional characterisation of
the structure of production technology.
• The input set of the production technology is characterised by the
input distance function while the output set is characterised by
the output distance function.
• We consider a technology T using a nonnegative vector of inputs
X=(x1,x2,......,xn) Rn
+ to produce a nonnegative vector of outputs
Y=(y1,y2,......,ym) Rm
+ . In functional terms, they can be related as:
Y=P(X) and X=L(Y)
13

Input Distance Function
• Given this, an input distance function can be defined as
Dinput= Max*λ:X/λ L(Y)].Intuitively speaking, an input distance
function gives the maximum amount by which the producer’s
input vector can be radially contracted and yet remain
feasible for the output vector it produces. The reciprocal of
the input distance function can be considered as the radial
measure of input oriented technical efficiency. Using DEA we
can compute input oriented technical efficiency as:
Minimise µ
Subject to: µx0-Xλ≥0, y0≤Yλ,   j=1,λ≥0
14

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 the
minimum amount by which the producer’s output vector can be
deflated and yet remain feasible for the input vector it uses. The
output distance function can be considered as the radial measure
of output oriented technical efficiency.
• The output oriented technical efficiency is calculated from:
Max vrs
• subject to  yo   Y, Xo  X,   j=1,j 0
15

Graph Hyperbolic Approach
• This implies maximisation of return and
minimisation of risk at the same time.
Consequently, the optimization problem for
the observed mutual fund is:
• Min G
• Subject to: Gx0≥ Xλ, 1/G y0≤Yλ, λ≥0
• In the VRS case we add the additional
convexity condition   j=1. Technical
efficiency= G
16

Introduction to Bootstrap
• Efron (1979) introduced the concept of bootstrap.
• Bootstrap involves resampling from an original sample of data
through computer-based simulations to obtain the sampling
properties of random variables.
• The starting point of any bootstrap procedure is a sample of
observed data X = {x1, x2, . . . , xn} drawn randomly from some
population with an unknown probability distribution f .
• The premise of the bootstrap method is that the random sample
actually drawn “mimics” its parent population.
• The bootstrap method suggested by Efron (1979) involves drawing
of sample (with replacement) directly from the observed data and
is known as naive bootstrap.
17

Naïve vs Smoothed Bootstrap
• The bootstrap method suggested by Efron (1979)
involves drawing of sample (with replacement) directly
from the observed data and is known as naive
bootstrap.
• In this case the bootstrap sample is effectively drawn
from a discrete population which fails to recognise the
fact that the underlying population density function f is
continuous.
• Simar and Wilson (1998) suggested that the problem
could be overcome by resorting to smoothed bootstrap
which involves resampling via a fitted model.
18

Smoothed Bootstrap
• The smoothed bootstrap methodology involves the use of Kernel
estimators 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 the
following manner: xi**=xi*+hϵ ~f, where h is the smoothing
parameter for the density function while xi* and xi** represent
the ith elements of the naive and smoothed bootstrap samples.
• In case of bootstrapping, every time when we replicate the
bootstrap sample, we get a different sample X**, we will also get
a different estimate of θ* = θ(X**). Thus, we need to select a
large number of bootstrap samples, B, in order to extract as many
combinations of xj ( j = 1, 2, . . . , n) as possible.
19

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).
20

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 technical
efficiency is :θbc=2 -θe
21

Confidence Interval
• By calculating the standard deviation of
technical efficiency scores, we can also
calculate the upper and lower bounds of
technical efficiency with lower and upper
bounds being 2.5% and 97.5% .
22

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 fund
schemes have been collected from AMFI website and the calculations
regarding mean return, mean upside potential and standard deviation
have been calculated by the author.
23

Descriptive Statistics of Technical
Efficiency Scores
Particulars
Input 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
24

Confidence Interval of Technical Efficiency
Confidence Interval
Input 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
25

Summing Up
• In the present study, 16 sectoral mutual fund
schemes have been evaluated for the second
half of 2010 using the concepts of input
oriented, output oriented and graph hyperbolic
measure.
• For the purpose of performance
benchmarking, the present study makes both
point and bootstrap estimates of performance.
The bootstrap measures have been used to
correct bias in pointed technical efficiency
scores.
26

THANK YOU
27

Ibs gurgaon-8 th ncm

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