# Value-at-Risk-Based Portfolio Insurance: Performance Evaluation and
Benchmarking Against CPPI in a Markov-Modulated Regime-Switching Market

Peyman Alipour<EMAIL_ADDRESS>Ali Foroush Bastani<EMAIL_ADDRESS>School of Business, Stevens Institute of Technology,
P.O. Box 07030, Hoboken, NJ, USA  Department of Mathematics, Institute for
Advanced Studies in Basic Sciences,
P.O. Box 45195-1159, Zanjan, Iran

###### Abstract

Designing dynamic portfolio insurance strategies under market conditions
switching between two or more regimes is a challenging task in financial
economics. Recently, a promising approach employing the value-at-risk (VaR)
measure to assign weights to risky and riskless assets has been proposed in
[Jiang C., Ma Y. and An Y. “The effectiveness of the VaR-based portfolio
insurance strategy: An empirical analysis”, International Review of Financial
Analysis 18(4) (2009): 185-197]. In their study, the risky asset follows a
geometric Brownian motion with constant drift and diffusion coefficients. In
this paper, we first extend their idea to a regime-switching framework in
which the expected return of the risky asset and its volatility depend on an
unobservable Markovian term which describes the cyclical nature of asset
returns in modern financial markets. We then analyze and compare the resulting
VaR-based portfolio insurance (VBPI) strategy with the well-known constant
proportion portfolio insurance (CPPI) strategy. In this respect, we employ a
variety of performance evaluation criteria such as Sharpe, Omega and Kappa
ratios to compare the two methods. Our results indicate that the CPPI strategy
has a better risk-return tradeoff in most of the scenarios analyzed and
maintains a relatively stable return profile for the resulting portfolio at
the maturity.

###### keywords:

Finance, Constant Proportion Portfolio Insurance, Value at Risk, Regime-
Switching, Omega Performance Measure.

##  1 Introduction

Portfolio insurance products are popular structured investment tools widely
used by private and institutional investors, providing their holders with
capital protection in down turning markets, while allowing benefits from
upside market potentials (see e.g. [38, 41, 50] and the many references
therein). The first milestone research in this direction is due to Leland and
Rubinstein [39] in 1976 who proposed a synthetically constructed put option by
trading on the underlying risky portfolio and a risk-free asset to dynamically
hedge the risk of issuer/guarantor liability. At the same time, Brennan and
Schwartz [15], studying equity-linked life insurance policies guaranteeing a
minimum return, reached a similar result by using the then new arbitrage-free
pricing methodology of Black and Scholes [13] and Merton [44]. These ideas
were culminated in option-based portfolio insurance (OBPI) strategies which
invest the initial endowment in a risky reference portfolio covered by a put
option with a strike chosen to be proportional to the guaranteed amount [6].

In a parallel line of research, Perold [46] (see also Perold and Sharpe [47]
and Black and Jones [11]) introduced an alternative dynamic trading strategy,
called constant proportion portfolio insurance (CPPI), based on continuous
rebalancing of a portfolio containing a safe asset (e.g. treasury bills) and a
risky one (e.g. a financial index) in response to fluctuations in market
conditions. By choosing a floor value, $F_{T}$, as the minimum acceptable
portfolio level at the maturity, $T$, at each rebalancing time, $t$, the
difference between the portfolio value, $V_{t}$ and the (discounted) floor is
computed as $C_{t}=V_{t}-F_{T}e^{-r(T-t)}$ (called the cushion) and the
exposure to the risky asset is determined by $E_{t}=mC_{t}$ in which $m$ is
called the multiple. The excess will naturally be invested in the riskless
asset.

It is shown (see e.g. Black and Perold [12]) that in the absence of borrowing
constraints and transaction costs, CPPI is a special case of the HARA utility-
maximizing rules that have appeared in the literature of continuous-time asset
allocation (see e.g. Merton [43]). For more details on the basic CPPI model
and its extensions, see e.g. Hirsa [32], Boulier and Kanniganiti [14],
Temocin, et al. [54], Bertrand and Prigent [10], Ben Ameur and Prigent [1, 2]
among many other references.

When the basic CPPI strategy is exploited with a constant multiple during the
holding-period, some inevitable shortcomings will arise due mainly to ignoring
the investor’s beliefs and risk preferences (see e.g. Hakanoglu et al. [28]).
This drawback has led the researchers to the family of dynamic proportion
portfolio insurance (DPPI) strategies which basically adopt the choice of the
multiple (and so the risk exposure) to the volatility of market prices (see
e.g. Chen and Chang [19], Chen et al. [20], Hamidi et al. [30]). The basic
idea is to vary the multiple in such a way that some “guarantee condition” at
the maturity is satisfied with a given probability for specified market
conditions. This condition is closely related to the probability of violating
the floor protection widely known in the literature as the gap risk (see e.g.
Jessen [1, 33]).

Value-at-Risk (VaR) based portfolio insurance (VBPI) is a new approach to
dynamic asset allocation which is based on the portfolio’s VaR concept (see
e.g. [7] for a general overview of VaR-based risk management strategies). For
a given time horizon, $t$ and confidence level, $p$, the value at risk is the
loss in market value over the time horizon $t$ that is exceeded with
probability $1-p$. In this strategy, the insured portfolio is constructed and
rebalanced frequently in such a way that the portfolio level at each time step
exceeds the floor at a given confidence level [34]. Based on the fact that
this approach targets the gap risk faced by the insurer, it leads naturally to
a path-dependent portfolio strategy consistent with the performance measure
used to evaluate the insurance strategy. It could also be viewed as a
generalized CPPI strategy which provides the buyer with additional flexibility
to benefit more from upward market movements while limiting the potential loss
from downward moves.

While the basic CPPI strategy has been proposed and analyzed mainly for
diffusion-based dynamics of the underlying risky portfolio, there have been
some efforts in the literature to extend this methodology to more complex
situations in which the underlying assete follows a dynamic process such as
jump-diffusion (see e.g. [23, 18]), regime-switching diffusion (see e.g. [27])
and regime-switching exponential Lévy model (see e.g. [55]). However, a
systematic study in the literature is still lacking for the extension of the
VaR-based DPPI strategies to such families of asset return distributions. As a
first step to fill this gap, we consider here a continuous-time regime-
switching market in which the model parameters switch from one regime to
another according to an unobservable Markov process (see e.g. [24] and the
many references therein). These models provide a natural way to capture
discrete shifts in market behavior in an efficient and flexible way.

We compare the performance of constrained CPPI and VBPI strategies in this
case to demonstrate the effect of regime shifts. In this respect, we perform a
Monte-Carlo simulation study by generating sample paths from the underlying
asset’s risky dynamics and show that the CPPI strategy has a better risk-
return tradeoff in most of the scenarios examined and maintains a relatively
stable return profile for the resulting portfolio at the maturity. This
complements the available results in the literature of portfolio insurance
which claim the effectiveness of CPPI-based strategies in the presence of
realistic circumstances and incompleteness assumptions on the market (see,
e.g. [48]).

The structure of this paper is as follows: In Section 2, we provide the
necessary background material about regime-switching diffusion processes.
Section 3 in concerned with the details of stylized and constrained CPPI
strategies when the underlying risky asset follows a regime-switchning
geometric Brownian motion process. In Section 4, we first present the details
of our proposed VaR-based strategy for the regime-switching case and then
demonstrate the required computational procedure to estimate the VaR quantity
needed in our proposed algorithm. Section 5 is devoted to introduce the risk-
adjusted performance measures used in this paper and also the downside risk
measures employed to evaluate the possible pitfalls of the proposed strategy.
In the remainder and Section 6, we present the details of our comprehensive
numerical experiments to validate and benchmark the VaR-based strategy against
the well-known CPPI method. We conclude the paper by commenting on some
possible research directions in this field.

##  2 The Financial Market Model

In the sequel, we consider an investment horizon of $[0,T]$ and assume that
the value of the riskless asset, denoted by $B$, grows with a constant risk-
free rate, $r$ as

$\displaystyle{dB_{t}=rB_{t}dt.}$ (2.1)

On the other hand, the market value of the risky asset, denoted by $S$, is
given by a regime-switching geometric Brownian motion (GBM) of the form

$\displaystyle{dS_{t}=\mu_{\alpha_{t}}S_{t}dt+\sigma_{\alpha_{t}}S_{t}dW_{t},}$
(2.2)

with a positive initial value, $S_{0}$. In the above equation,
$W={(W_{t})}_{0\leqslant t\leqslant T}$ is a standard Wiener process defined
on a complete filtered probability space,
$(\Omega,\mathcal{F},\\{\mathcal{F}_{t}\\}_{0\leqslant t\leqslant
T},\mathbb{P})$ and $\mu_{\alpha_{t}}$ and $\sigma_{\alpha_{t}}$ are the drift
and diffusion terms depending on a continuous-time stationary Markov process,
${(\alpha_{t})}_{0\leqslant t\leqslant T}$, independent of $W$.

The process, ${(\alpha_{t})}_{0\leqslant t\leqslant T}$, takes values in the
set $\mathcal{H}=\\{1,2,....H\\}$ where each element represents a possible
economic or financial regime or state of the world. The generator of this
Markov chain is given by the matrix $Q=(q_{ij})_{H\times H}$ in which
$q_{ij}\geqslant 0$ for all $i\neq j$ and
$\sum_{j=1}^{H}{q_{ij}}=0,~{}i\in\mathcal{H}$. The transition probability
matrix could be obtained as

$\displaystyle{P(t,s)=\exp(Q(s-t)),}$ (2.3)

for each $s,t~{}(0<s\leq t)$ with the elements

$\displaystyle{p_{ij}(t,s)={\mathbb{P}}(\alpha_{s}=j|\alpha_{t}=i),\quad
i,j\in\mathcal{H}.}$ (2.4)

The probability of being in a specific state at time $t$ will be denoted by
$p_{i}(t)$ and is given by

$\displaystyle{p_{i}(t)={\mathbb{P}}(\alpha_{t}=i)=\sum_{k=1}^{H}p_{k}(0)p_{ki}(0,t),\quad\forall
i\in\mathcal{H}.}$ (2.5)

Also the stationary transition probabilities corresponding to the Markov
process are given by

$p_{i}=\lim_{t\rightarrow\infty}p_{i}(t),\quad\forall i\in\mathcal{H}.$ (2.6)

In the special case where the continuous-time Markov chain, $\alpha_{t}$,
contains only two states (e.g. state 1 denoting a stable low-volatility regime
and state 2 denoting a more unstable high volatility regime), the matrix $Q$
could be written as

$Q=\left(\begin{array}[]{cc}-q_{11}&q_{11}\\\ q_{22}&-q_{22}\\\
\end{array}\right),$ (2.7)

with positive $q_{ij}$s and the stationary probabilities are obtained as

$\begin{array}[]{cc}p_{1}=\frac{q_{11}}{q_{11}+q_{22}},&\quad
p_{2}=\frac{q_{22}}{q_{11}+q_{22}}.\end{array}$ (2.8)

In the remainder, we describe the details of CPPI and VBPI strategies analysed
in this paper.

##  3 Stylized Constant Proportion Portfolio Insurance

CPPI is a dynamic self-financing strategy in which positions in risky and risk
free assets are rebalanced dynamically so that the terminal value of the
portfolio lies above a guaranteed level, $F_{T}$ (the floor), which is usually
given as a percentage, $\pi$ ($0\leq\pi\leq 1$), of the initial investment

$\displaystyle{F_{T}=\pi V_{0},}$ (3.1)

and the value of the floor at any given time, $t\in[0,T]$, is obtained as

$\displaystyle{F_{t}=e^{-r(T-t)}F_{T}.}$ (3.2)

The difference between the market value of the portfolio, ${V_{t}}^{\rm CPPI}$
and the floor is called the cushion and is denoted by $C_{t}$.

In the standard CPPI strategy, there is no restriction on the risky part of
the portfolio111The standard CPPI strategy allows the exposure to be leveraged
at any level i.e., there is no constraint on the borrowing., but to apply more
realistic conditions, constraints are usually imposed on the cushion to
prohibit short-selling of the risky asset (see e.g. [22]). Under these
conditions, the cushion value at any time $t\in[0,T]$ is given by

$\displaystyle{C_{t}=({V_{t}}^{\rm CPPI}-F_{t})^{+},}$ (3.3)

and the total amount invested in the risky asset (which is called the
exposure) is obtained by multiplying a constant coefficient, $m$, in the
cushion value

$\displaystyle{E_{t}=mC_{t}.}$ (3.4)

It is obvious that higher multiples will result in more profits from stock
price increases. Nevertheless, this will also cause faster convergence of the
portfolio value to the floor in the case of a dramatic decrease in the stock
prices. Note also that different values of the parameter, $m$, change the
behavior of the payoff function and $m>1$ provides a convex payoff structure.
From (3.3) and (3.4), it is deduced that if ${V_{t}}^{\rm CPPI}\leqslant
F_{t}$, then the exposure will be zero and so the entire portfolio value will
be invested in the risk-free asset (see e.g. [23] for more details). It could
also easily be seen that the value of the portfolio at any time is equal to
the current floor plus the cushion value. As cushion is non-negative, the
value of the CPPI portfolio is always above the current floor.

###  3.1 Constrained CPPI

In realistic situations, we usually impose a constraint on the portfolio by
limiting the exposure to the bounds

$0<E_{t}<p{V_{t}}^{\rm CPPI},$

in which $p>0$ is a given constant. So the value of the exposure in the
constrained CPPI strategy will be obtained as

$\displaystyle{E_{t}=\min\\{mC_{t},p{V_{t}}^{\rm CPPI}\\}.}$ (3.5)

Note that in the constrained CPPI case, the portfolio composition is path-
dependent (see [14]) and the fraction of the portfolio invested in the risk-
free asset is given by

$\displaystyle{B_{t}={V_{t}}^{\rm CPPI}-E_{t},}$ (3.6)

and the evolution of the portfolio value is described by the following
stochastic differential equation (SDE) (see e.g. [14])

$\displaystyle{d{V_{t}}^{\rm CPPI}=E_{t}\frac{dS_{t}}{S_{t}}+({V_{t}}^{\rm
CPPI}-E_{t})\frac{dB_{t}}{B_{t}}}.$ (3.7)

As mentioned above, we assume that the dynamics of the risky part of the
portfolio is given by a regime-switching GBM given by (2.2) and so
substituting it in (3.7), we obtain the equation

$\ \displaystyle{d{V_{t}}^{\rm
CPPI}=E_{t}\Big{(}(\mu_{\alpha_{t}}-r)d_{t}+\sigma_{\alpha_{t}}dW_{t}\Big{)}+r{V_{t}}^{\rm
CPPI}d_{t}.}$ (3.8)

Based on restrictions on the exposure given by (3.3) and (3.5), the dynamics
of the CPPI portfolio could be re-written as

$d{V_{t}}^{\rm CPPI}=\left\\{\begin{array}[]{ll}\displaystyle{r{V_{t}}^{\rm
CPPI}d_{t}},&\hbox{$C_{t}\leqslant 0$,}\\\
\displaystyle{mC_{t}((\mu_{\alpha_{t}}-r)d_{t}+\sigma_{\alpha_{t}}dW_{t})+{V_{t}}^{\rm
CPPI}rd_{t}},&\hbox{$0<mC_{t}<p{V_{t}}^{\rm CPPI}$,}\\\
\displaystyle{p{V_{t}}^{\rm
CPPI}((\mu_{\alpha_{t}}-r)d_{t}+\sigma_{\alpha_{t}}dW_{t})+{V_{t}}^{\rm
CPPI}rd_{t}},&\hbox{$p{V_{t}}^{\rm CPPI}\leqslant mC_{t}$.}\\\
\end{array}\right.$ (3.9)

By applying Itô’s lemma (see e.g. [45]) and discretizing the dynamics (3.9) at
a set of discrete nodes $t_{n}=n\Delta t$, the solution of this SDE could be
approximated by a discrete process of the form

${V^{\rm CPPI}_{t_{n+1}}}={V^{\rm
CPPI}_{t_{n}}}+\left\\{\begin{array}[]{ll}\displaystyle{{rV^{\rm
CPPI}_{t_{n}}}\Delta t},&\hbox{$C_{t_{n}}\leqslant 0$,}\\\
\displaystyle{C_{t_{n}}(m(\mu_{\alpha_{t_{n}}}-r)+r)\Delta
t+m\sigma_{\alpha_{t_{n}}}\Delta W_{t_{n}}},&\hbox{$0<mC_{t_{n}}<p{V^{\rm
CPPI}_{t_{n}}}$,}\\\
\displaystyle{C_{t_{n}}(p(\mu_{\alpha_{t_{n}}}-r)+r)\Delta
t+p\sigma_{\alpha_{t_{n}}}\Delta W_{t_{n}}},&\hbox{$p{V^{\rm
CPPI}_{t_{n}}}\leqslant mC_{t_{n}}$.}\\\ \end{array}\right.$ (3.10)

##  4 VaR-Based Portfolio Insurance

Similar to the CPPI method, VBPI is a dynamic trading strategy which
rebalances the portfolio composition according to the Value-at-Risk (VaR)
concept. As the value at risk measure concentrates on the downward tail of the
return distribution, the VBPI strategy could address the gap risk by
allocating the funds between risky and risk-less assets in such a way as the
maximum loss is equated to the VaR at a specified confidence level. This
strategy gives a specific discipline for rebalancing portfolios such that the
gap risk is controlled.

In the remainder, we assume that the risky part of the portfolio follows a
regime-switching geometric Brownian motion as given by (2.2). Let $0\leq
w_{t}\leq 1$ be the fractional allocation of funds to the risk-free asset
which will result in $\beta_{t}=w_{t}\frac{V_{t}}{B_{t}}$ as the number of
riskless assets and $\eta_{t}=(1-w_{t})\frac{V_{t}}{S_{t}}$ as the number of
risky assets in the portfolio. The value of riskless assets grows at a
constant risk-free rate according to

$\displaystyle{B_{t}=B_{0}\exp(rt),}$ (4.1)

and the value of the risky portfolio will follow a regime-switching dynamics
given by

$\displaystyle{S_{t}=S_{0}\exp\bigg{[}\sum_{i=1}^{H}\int_{0}^{t}\bigg{(}\mu_{i}-\frac{\sigma_{i}^{2}}{2}\bigg{)}\delta(i,\alpha_{s})ds+\sum_{i=1}^{H}\int_{0}^{t}\sigma_{i}\delta(i,\alpha_{s})dW_{s}\bigg{]}.}$
(4.2)

In the above expression, $\delta(i,\alpha_{s})$ is an indicator function being
equal to 1 if we are in the state $\alpha_{s}=i$ and 0, otherwise. So, the
value of the portfolio at time, $t$, is given by

$\displaystyle{{V_{t}}^{\rm
VBPI}=\beta_{t}B_{t}+\eta_{t}S_{t}=w_{t}V_{0}\exp(r{t})+(1-w_{t})V_{0}\Big{(}\frac{S_{t}}{S_{0}}\Big{)}.}$
(4.3)

According to the dynamic VaR-based approach of Jiang et al. [34], our goal is
to adjust the weights in such a way that the constraint

$P({V_{t}}^{\rm VBPI}\leq F_{t})=\alpha,$ (4.4)

will hold in all rebalancing times. By substituting (4.3) in (4.4) and
denoting the log-return process by
$R_{t}=\ln{\big{(}\frac{S_{t}}{S_{0}}\big{)}}$, we could show that (4.4) is
equivalent to

$P\bigg{(}\frac{S_{t}}{S_{0}}\leq\frac{F_{t}-w_{t}V_{0}\exp(rt)}{(1-w_{t})V_{0}}\bigg{)}=P\bigg{(}R_{t}\leq\ln\bigg{(}\frac{F_{t}-w_{t}V_{0}\exp(rt)}{(1-w_{t})V_{0}}\bigg{)}\bigg{)}=\alpha.$
(4.5)

Equation (4.5) could be re-written as

$\int^{\ln\big{(}\frac{F_{t}-w_{t}V_{0}exp(rt)}{(1-w_{t})V_{0}}\big{)}}_{-\infty}sf_{R_{t}}(s)ds=\alpha,$
(4.6)

in which $f_{R_{t}}(\cdot)$ is the probability density function (pdf) of the
log-return process, $R_{t}$.

According to (4.6), the upper limit in the integral term is the VaR of the
random variable $R_{t}$ at confidence level, $\alpha$, which is defined as the
possible maximum loss of a portfolio over a given time horizon within a fixed
confidence level and is given by

$\ln\Big{(}\frac{F_{t}-w_{t}V_{0}\exp(rt)}{(1-w_{t})V_{0}}\Big{)}=q_{t}.$
(4.7)

So the weight of the risky asset will be obtained as

$w_{t}=\frac{F_{t}-V_{0}\exp(q_{t})}{V_{0}\Big{(}\exp(rt)-\exp(q_{t})\Big{)}}.$
(4.8)

In the remainder of this section, we describe an efficient way to calculate
the value of $q_{t}$ at each rebalancing time.

###  4.1 Details of Calculating Value at Risk

In order to calculate the VaR in (4.8) above, we need to know the distribution
of $R_{t}$. As described in Hainaut [27], the characteristic function of
$R_{t}$, denoted as $\varphi_{t}(\cdot)$ and defined by

$\varphi_{t}(\vartheta)=\mathbb{E}\big{(}e^{i\vartheta
R_{t}}\big{)}=\int^{+\infty}_{-\infty}e^{i\vartheta s}f_{R_{t}}(s)ds,$ (4.9)

could be obtained analytically and so using the characteristic function, we
could determine the probability density function of $f_{R_{t}}$ in a simple
and efficient way. Substituting $R_{t}=\ln\Big{(}\frac{S_{t}}{S_{0}}\Big{)}$
in (4.9), we could write

$\varphi_{t}(\vartheta)=\mathbb{E}\bigg{(}\Big{(}{\frac{S_{t}}{S_{0}}}\Big{)}^{i\vartheta}\bigg{)}.$
(4.10)

In order to obtain an analytic expression for $\varphi_{t}(\vartheta)$, we
need the following result.

######  Proposition 4.1 (Hainaut [27]).

Let the matrix, $B_{\gamma}$, be defined by

$\displaystyle{B_{\gamma}=Q^{\prime}+diag\left(\begin{array}[]{c}\gamma\bigg{(}\mu_{1}-\frac{\sigma_{1}^{2}}{2}\bigg{)}+\frac{1}{2}\gamma^{2}\sigma_{1}^{2}\\\
.\\\ .\\\ .\\\
\gamma\bigg{(}\mu_{H}-\frac{\sigma_{H}^{2}}{2}\bigg{)}+\frac{1}{2}\gamma^{2}\sigma_{H}^{2}\\\
\end{array}\right),\quad\forall\gamma\in\mathbb{R}.}$ (4.11)

Then we have

$\mathbb{E}\bigg{(}\bigg{(}\frac{S_{t}}{S_{0}}\bigg{)}^{\gamma}|\mathcal{F}_{0}\bigg{)}=\mathbb{E}(\langle\exp(B_{\gamma}t)\delta(0);\textbf{1}\rangle|\mathcal{F}_{0})=\sum_{i=1}^{M}p_{i}(0)(\langle\exp(B_{\gamma}t)e_{i};\textbf{1}\rangle),$
(4.12)

in which $\delta(t)=(\delta(i,\alpha_{t}):i\in\mathcal{H})^{\prime}$ is a
vector taking its values in the set of unit vectors
$\\{e_{1},e_{2},\cdots,e_{H}\\}$ and 1 is a vector of $H$ ones.

According to equation (4.12), the characteristic function of $R_{t}$ could be
calculated as

$\varphi_{t}(\vartheta)=\sum_{i=1}^{H}p_{i}(0)(\langle\exp(B_{i\vartheta}T)e_{i};\textbf{1}\rangle).$
(4.13)

By inverting the Fourier transform, the probability density function of
$R_{t}$ could now be obtained as

$f_{R_{t}}(s)=\frac{1}{2\pi}\int^{+\infty}_{-\infty}\varphi(\vartheta)e^{-i\vartheta
s}d\vartheta=\frac{1}{\pi}\int^{+\infty}_{0}\varphi(\vartheta)e^{-i\vartheta
s}d\vartheta.$ (4.14)

As described in Hainaut [27], the integral in (4.14) could be calculated by
the fast Fourier transform (FFT) method. The output of the FFT algorithm is
the distribution function of $R_{t}$. Using the obtained distribution, the
$\alpha$-quantile of $R_{t}$ which is the desired VaR level could be
calculated by

$q_{t}=w_{t}V_{t}\exp(rT)+(1-w_{t})V_{t}\exp(r_{\alpha}).$ (4.15)

##  5 Risk-Adjusted Performance Measures

Evaluating portfolio performance is a key activity in financial economics [40,
51]. During the years, a wide variety of performance measures have been
introduced into the field of finance to guide the investors (both private and
institutional) in comparing and ranking of investment portfolios and
evaluating the added value of portfolio managers (see, e.g. [31, 49, 26, 25,
16]).

There exist a number of performance measures in the literature which focus
mainly on the mean and variance of the return distribution (see e.g. [21] and
references therein). A commonly used measures is the Sharpe ratio (see e.g.
[52]) calculated as the ratio of the expected excess return of an investment
to its return volatility. Originally motivated by mean-variance analysis and
the Capital Asset Pricing Model (CAPM), the Sharpe ratio is routinely used in
many different contexts, from performance attribution to tests of market
efficiency and risk management. This measure has been used in various
researches in order to compare different strategies ([3], [30], [29]).

On the other hand, Keating and Shadwick [36] and Cascon et al. [17] have
introduced a new performance measure, called “Omega” which considers the
returns both below and above a given loss threshold, $L$, selected by the
investor.

###  5.1 Omega Measure

By dividing the returns into two classes according to the loss threshold, $L$,
the returns below it are considered as losses and above it as gains. More
precisely, let $F_{X}(x)$ denote the cumulative distribution function (CDF) of
the return distribution defined on the interval $(a,b)$. The Omega measure is
defined by the expression

$\displaystyle{\Omega_{X}(L)=\frac{\int^{b}_{L}(1-F(x))dx}{\int^{L}_{a}F(x)dx}},$
(5.1)

which could equivalently be written in terms of the final portfolio value,
$V_{T}$, as (see e.g. [35])

$\displaystyle{\Omega_{V}(L)=\frac{{\mathbb{E}}(V_{T}-L)^{+}}{{\mathbb{E}}(L-V_{T})^{+}}}.$
(5.2)

It is obvious that the Omega measure takes account of the entire return
distribution while requiring no parametric assumptions on the distribution. It
provides an appropriate performance measure to compare different strategies.
At any threshold level, $L$, investors prefer the strategies with a higher
Omega value [8].

###  5.2 Kappa Measure

The other performance measure used to compare the performance of two or more
strategies is the “Kappa” measure defined by (see e.g. [8])

$\kappa_{n}(L)=\frac{{\mathbb{E}}(V_{T})-L}{\bigg{[}{\mathbb{E}}\big{[}(L-V_{T})^{+}\big{]}^{n}\bigg{]}^{\frac{1}{n}}}.$
(5.3)

It is shown that the Sortino ratio (see e.g. [53]) could be recovered by
considering the case $n=2$ in the above definition.

In the context of portfolio insurance, Bertrand and Prigent [8] have employed
both the Omega and Kappa measures to compare the performance of OBPI and CPPI
strategies. They show that the CPPI method performs better than the OBPI
strategy for jump-diffusion dynamics of the underlying risky asset. We could
also mention the work of Zagst and Kraus [56] in which they analyze and
compare the performance of OBPI and CPPI strategies by means of stochastic
dominance criteria. They derive parameter conditions implying the second and
third order stochastic dominance of the CPPI strategy.

###  5.3 Downside Risk Measures

A well-studied risk in portfolio insurance, called the “gap-risk”, is the
probability of the portfolio value to fall below the floor and failing to
guarantee the desired final amount. This risk is measured by a quantity called
the expected shortfall given default (ES) which describes the amount which is
lost if a shortfall occurs (see e.g. [54, 5]). In order to make it precise, we
first define a loss variable, $L_{T}$, taken to be the amount the portfolio
value is below the guarantee level at maturity and expressed as

$L_{T}=[F_{T}-V_{T}|F_{T}<V_{T}].$ (5.4)

Expected value (expected shortfall) of the loss is a measure that estimates
the average amount of the loss when the value of the portfolio at the maturity
is below the guarantee amount (see, e.g. [37])

${\mathbb{E}}[L_{T}]={\mathbb{E}}[G-V_{T}|V_{T}<F_{T}].$ (5.5)

A portfolio insurance strategy incurs a shortfall (breaks through the floor),
if $V_{t}<F_{t}$ occurs during the investment horizon. Percentage of times a
loss occurs, when measured over a large number of simulations, $M$, could be
estimated by

${\mathbb{P}}[L_{T}]\approx\frac{1}{M}\sum_{j=1}^{M}\mathbbm{1}_{\\{V_{T}(\omega_{j})<F_{T}\\}},$
(5.6)

and could be interpreted as the probability that the PI strategy to experience
a loss. Note that ${\mathbb{P}}[L_{T}]$ could be interpreted from the buyers
perspective as the probability that they will receive only the guarantee at
maturity.

##  6 Numerical Experiments

In this section, we study both the VBPI and CPPI strategies with daily, weekly
and monthly rebalancing frequencies of the portfolio between a risky and risk-
free asset for a planning horizon of $T=1$ years222We assume $260$ trading
days and $52$ weeks in a typical year.. We take the portfolio initial level to
be $V_{0}=100$ and the guaranteed level at the maturity to be $F_{T}=100$
(i.e. $\pi=1$). We also let the yield of the bond to be constant at an annual
rate of $r=4\%$. We assume that the risky asset is driven by a time-
inhomogeneous Markov-modulated diffusion process with two distinct regimes
(considered here as bullish and bearish markets) with parameter values for
each regime as

$\mu_{1}=0.14,\quad\sigma_{1}=0.16\quad\ \mu_{2}=-0.01\quad\sigma_{2}=0.2.$

The generator matrix of the underlying Markov process is assumed to be given
by

$Q=\left(\begin{array}[]{cc}-0.25&0.25\\\ 0.25&-0.25\\\ \end{array}\right).$

In order to demonstrate the effect of regime shifts on the performance of the
CPPI and VBPI strategies, we have conducted a Monte-Carlo simulation study by
generating sample paths from the underlying asset’s risky dynamics.

To make the two strategies comparable in different scenarios, multiples in the
CPPI strategy are chosen such that the strategy’s initial allocation to the
risky asset is the same as that under the corresponding VBPI strategy (see
e.g. [34]). Using (4.8) and the relation

$mC_{0}=(1-w_{0})V_{0},$ (6.1)

the value of $m$ could be calculated. In the above equation, the weight of the
risky asset in the CPPI strategy is assumed the same as the weight of risky
asset in the VBPI strategy at the initial time.

In Figures 1-3, we have demonstrated the Omega performance measure as a
function of different threshold levels. As pointed out by Bacmann and Scholz
[4], Omega involves all the moments of the return distribution including
skewness and kurtosis and so, it is an appropriate indicator of the
effectiveness of insurance strategies (see also [34]). In all figures and for
high threshold levels, the VBPI strategy has a better performance than the
CPPI method while in low thresholds, CPPI has a better Omega measure. Also by
increasing the confidence level (CL), the Omega measure increases, showing
that the performance of the portfolios in higher confidence levels are
improved.

The histogram (frequency distribution) of the terminal values of CPPI and VBPI
portfolios for daily, weakly and monthly rebalancing periods is depicted in
Figures 4-6. They show that the left tail of the frequency distribution for
the CPPI strategy is shorter than VBPI. Also by increasing the confidence
level, the left tail becomes shorter in both cases. They collectively show
that the performance of CPPI strategy is better than VBPI.

In Figure 7, we have plotted the expected final portfolio levels under the
VBPI and CPPI strategies (denoted respectively by EVVBPI and EVCPPI) versus
the rebalancing period and confidence level. As it is evident from these
figures, the expected value increases in both cases as we increase the
rebalancing period and decreases as we increase the confidence level. Figure 8
which is concerned with the standard deviation of the final portfolio levels
versus the rebalancing period and confidence level, shows a similar behavior
but here the CPPI method is indifferent to changing the rebalancing period. In
Figure 9, we have plotted the normalized number of portfolios which fail to
give the floor value (shortfall probability) versus the rebalancing period and
confidence level. In this case, increasing the confidence level will result in
decreasing the number of portfolios which fail to give the floor value. Here
again we conclude that the CPPI performance is better than VBPI for low
confidence levels.

In Figure 10, we have compared the expected terminal value of the two
strategies. It shows that the initial behavior of the two strategies are the
same, however by increasing the rebalancing period, the value of VBPI
portfolio attains higher levels.

In Table 1 we have compared the two strategies in terms of Omega and Kappa
performance measures for different threshold levels. We observe that for all
threshold values, daily rebalancing leads to better results. Among different
confidence levels and for daily rebalancing, the 99 percent confidence level
will lead to better results while at weekly and monthly rebalancing periods,
lower confidence levels will lead to better results. Table 2 also compares the
two strategies in terms of the Sharpe ratio. For daily rebalancing, we obtain
better results in comparison to the weekly and monthly rebalancing periods.

It is evident from the above experiments that by increasing the confidence
level, the return and the standard deviation of the corresponding portfolios
decreases, as expected. Also by increasing the rebalancing period, the
performance of both strategies degrades. In all the presented results, the
CPPI strategy has a better performance than the VBPI method except for the
Omega measure in which for high threshold levels, the VBPI strategy has a
slightly better performance.

(a) Daily

(b) Weekly

(c) Monthly

Figure 1: Omega measure of CPPI and VBPI portfolio at $90\%$ CL.

(a) Daily

(b) Weekly

(c) Monthly

Figure 2: Omega measure of CPPI and VBPI portfolio at $95\%$ CL.

(a) Daily

(b) Weekly

(c) Monthly

Figure 3: Omega measure of CPPI and VBPI portfolio at $99\%$ CL.

(a) CPPI-Daily-CL 90%  (b) VBPI-Daily-CL 90%

(c) CPPI-Weekly-CL 90%  (d) VBPI-Weekly-CL 90%

(e) CPPI-Monthly-CL 90%  (f) VBPI-Monthly-CL 90%

Figure 4: Frequency of the terminal value of the CPPI and VBPI portfolios at
$90\%$ CL.

(a) CPPI-Daily-CL 95%  (b) VBPI-Daily-CL 95%

(c) CPPI-Weekly-CL 95%  (d) VBPI-Weekly-CL 95%

(e) CPPI-Monthly-CL 95%  (f) VBPI-Monthly-CL 95%

Figure 5: Frequency of the terminal value of the CPPI and VBPI portfolios at
$95\%$ CL

(a) CPPI-Daily-CL 99%  (b) VBPI-Daily-CL 99%

(c) CPPI-Weekly-CL 99%  (d) VBPI-Weekly-CL 99%

(e) CPPI-Monthly-CL 99%  (f) VBPI-Monthly-CL 99%

Figure 6: Frequency of the terminal value of the CPPI and VBPI portfolios at
$99\%$ CL

(a) Expected value of CPPI portfolio.

(b) Expected value of VBPI portfolio.

Figure 7: Comparing terminal expected value of the CPPI and VBPI portfolios.

(a) Standard deviation of the CPPI portfolio.

(b) Standard deviation of the VBPI portfolio.

Figure 8: Comparing Standard deviation of CPPI and VBPI portfolios.

(a) Shortfall probability of the CPPI portfolio.

(b) Shortfall probability of the VBPI portfolio.

Figure 9: Comparing shortfall probability of the CPPI and VBPI portfolios.

(a) Expected shortfall of the CPPI portfolio.

(b) Expected shortfall of the VBPI portfolio.

Figure 10: Comparing expected shortfall of the CPPI and VBPI portfolios.

|  | Omega CPPI |  | Omega VBPI |  |  | Kappa CPPI |  | Kappa VBPI
---|---|---|---|---|---|---|---|---|---
CL |  | Daily | Weekly | Monthly |  | Daily | Weekly | Monthly |  |  | Daily | Weekly | Monthly |  | Daily | Weekly | Monthly
| Thershold 1% |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
90 |  | 46.48 | 0.71 | 0.49 |  | 4.08 | 0.72 | 0.49 |  |  | 15.61 | -0.22 | -0.41 |  | 1.81 | -0.22 | -0.41
95 |  | 629.76 | 0.66 | 0.37 |  | 7.66 | 0.67 | 0.37 |  |  | 85.65 | -0.25 | -0.49 |  | 2.97 | -0.24 | -0.49
99 |  | 947.24 | 0.61 | 0.27 |  | 99.87 | 0.62 | 0.27 |  |  | 425.12 | -0.28 | -0.57 |  | 25.62 | -0.27 | -0.56
| Thershold 2% |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
90 |  | 4.74 | 0.37 | 0.28 |  | 1.91 | 0.38 | 0.28 |  |  | 2.14 | -0.52 | -0.61 |  | 0.60 | -0.51 | -0.60
95 |  | 12.63 | 0.25 | 0.16 |  | 2.65 | 0.29 | 0.16 |  |  | 4.87 | -0.6 | -0.71 |  | 0.99 | -0.58 | -0.71
99 |  | 50.45 | 0.16 | 0.08 |  | 6.75 | 0.19 | 0.09 |  |  | 13.04 | -0.68 | -0.79 |  | 2.96 | -0.66 | -0.78
| Thershold 3% |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
90 |  | 1.36 | 0.22 | 0.17 |  | 1.04 | 0.22 | 0.17 |  |  | 0.26 | -0.67 | -0.72 |  | 0.03 | -0.67 | -0.72
95 |  | 2.02 | 0.12 | 0.08 |  | 1.21 | 0.14 | 0.08 |  |  | 0.62 | -0.76 | -0.81 |  | 0.15 | -0.74 | -0.81
99 |  | 3.37 | 0.05 | 0.03 |  | 1.65 | 0.07 | 0.03 |  |  | 1.25 | -0.83 | -0.88 |  | 0.44 | -0.82 | -0.87
| Thershold 4% |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
90 |  | 0.58 | 0.14 | 0.11 |  | 0.63 | 0.14 | 0.11 |  |  | -0.33 | -0.76 | -0.79 |  | -0.29 | -0.76 | -0.79
95 |  | 0.57 | 0.06 | 0.04 |  | 0.66 | 0.08 | 0.04 |  |  | -0.32 | -0.84 | -0.87 |  | -0.26 | -0.82 | -0.87
99 |  | 0.58 | 0.02 | 0.01 |  | 0.62 | 0.03 | 0.02 |  |  | -0.29 | -0.90 | -0.92 |  | -0.29 | -0.89 | -0.92

Table 1: Comparing the performance of CPPI and VBPI strategies via the Omega and Kappa performance measures. |  | CPPI |  | VBPI
---|---|---|---|---
CL |  | Daily | Weekly | Monthly |  | Daily | Weekly | Monthly
90 |  | 0.36 | -0.19 | -0.31 |  | 0.36 | -0.18 | -0.31
95 |  | 0.41 | -0.64 | -0.86 |  | 0.37 | -0.47 | -0.86
99 |  | 0.45 | -1.24 | -1.59 |  | 0.41 | -1.06 | -1.53
Table 2: Comparing the performance of CPPI and VBPI strategies via the Sharpe
performance measure.

##  7 Conclusion

The main contribution of this paper is the development of a constrained CPPI
as well as a VBPI strategy under a regime-switching diffusion model. In this
respect and for the CPPI strategy, we derive a stochastic differential
equation for the portfolio value consisting of a risky and a riskless asset
where the exposure is constrained both from below and above. In the VBPI case,
we derive expressions for the weights of the risky and riskless assets in the
portfolio based on the value at risk of the portfolio return process at each
discrete rebalancing time. We employ a Fourier-based method to approximate the
VaR by inverting the characteristic function of the underlying asset. We
compare the two approaches based on some performance measures and show that
the constrained CPPI method performs well in most of the scenarios examined
and provides us with a better control on the gap risk of the investment
strategy. For future research, we propose to add a jump term into the model to
better capture the real effects present in the market environment. We could
also consider a flexible floor value which potentially improves the efficiency
of the proposed portfolio insurance strategies.

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##  Appendix A Monte-Carlo Simulation Method

In the Monte-Carlo approach, we first subdivide the time interval $[0,T]$ into
$N=\frac{T}{\triangle t}$ grid points and then simulate a number, $M$, of
risky asset sample paths at these discrete points denoted as
$\\{S(t_{k}),k=1,...,N\\}$. To do so, it suffices to specify the values
${\alpha(t_{k}),k=1,...,N}$ and then simulate $V(t_{k})$. We use the following
method from Yuan and Mao [42] to obtain a sample realization of
${\alpha(t),0\leq t\leq T}$:

  * 1

Let $\alpha(t_{0})=\alpha_{0}$ as the initial value of $\alpha(t)$;

  * 2

Generate $u\thicksim U([0,1])$ from a uniform distribution in the interval
(0,1). Let $\alpha(t_{k}-1)=i$ for some $i\epsilon\mathcal{H}$ be the governed
regime at time $t_{k-1}$ . Now define

$\alpha(t_{n})=\left\\{\begin{array}[]{ll}\alpha_{n},\ \ \ where\ \
\alpha_{n}\varepsilon\mathcal{H}\backslash\\{H\\}\ \ \ if\ \ \ \
\Sigma^{\alpha_{n-1}}_{j=1}P^{\Delta t}_{i,j}\leq
u<\Sigma^{\alpha_{n}}_{j=1}P^{\Delta t}_{i,j}\\\ H,\ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if\ \ \ \Sigma^{H-1}_{j=1}P^{\Delta
t}_{i,j}\leq u\\\ \end{array}\right.$ (A.1)

in which $\Sigma^{0}_{j=1}P^{\Delta t}_{i,j}=0$.

Iterating this algorithm, we obtain the values
$\\{\alpha(1),\cdots,\alpha(t_{N})\\}$ and then we could simulate $V_{t}^{\rm
CPPI}$ (and also $V_{t}^{\rm VBPI}$) using the following expression:

${V_{t_{n+1}}}={V_{t_{n}}}+\left\\{\begin{array}[]{ll}\displaystyle{{rV_{t_{n}}}\Delta
t},&\hbox{$C_{t_{n}}\leqslant 0$,}\\\
\displaystyle{C_{t_{n}}(m(\mu_{\alpha_{t_{n}}}-r)+r)\Delta
t+m\sigma_{\alpha_{t_{n}}}\Delta
W_{t_{n}}},&\hbox{$0<mC_{t_{n}}<p{V_{t_{n}}}$,}\\\
\displaystyle{C_{t_{n}}(p(\mu_{\alpha_{t_{n}}}-r)+r)\Delta
t+p\sigma_{\alpha_{t_{n}}}\Delta W_{t_{n}}},&\hbox{$p{V_{t_{n}}}\leqslant
mC_{t_{n}}$.}\\\ \end{array}\right.$ (A.2)


# Note on clique polynomials and independent set polynomial of graphs

Hany Ibrahim<EMAIL_ADDRESS>Applied university Mittweida, Depertment
of Mathematics

###### Abstract

In 1994, Cornelis Hoede and Xueliang Li introduced the clique polynomial of a
graph. Also, a theorem for the edge subgraph expansion for clique polynomials.
In this note we present a counter example for it and explain which case it
could be a valid theorem.

## Introduction

For notions, definitions and other reference, consult [1]. We copy here the
theorem under consideration as it is and we index the formulas in its proof.

######  Theorem 3.4.

[1] (Edge-subgraph expansion). Let $M$ be a nonempty subset of the edge set of
a graph $G$. Then we have

$C(G;x)=C(G-M;x)+\sum_{r=2}^{|M|}(-1)^{r}\sum_{\begin{subarray}{c}S\subset
M\\\ |S|=(1/2)r(r-1)\end{subarray}}(r-1)x^{r}C^{*}(G_{S};x)$

where

$C^{*}(G_{S};x)=\begin{cases}C(G_{S};x)&\text{if}\left<S\right>\text{is a
clique of }G\text{,}\\\ 0&\text{otherwise.}\end{cases}$

###### Proof.

Let $H_{k}$ be the set of $k$-cliques of $G$. Then $|H_{k}|=a_{G}$. Let
$M={e_{1},e_{2},\cdots,e_{h}}$. Condition $Ci(i=1,2,\dots,h)$, reads ‘contains
edge $e_{i}$’. Then the number of $k$-cliques in $H_{k}$ that satisfy
conditions $C_{i_{1}},\cdots,C_{i_{t}}$, is given by

$\displaystyle N(C_{i_{1}},C_{i_{2}},\cdots,C_{i_{t}})=$ $\displaystyle
a^{*}_{k-\left|V\left(\left<\left<e_{i_{1}},e_{i_{2}},\cdots,e_{i_{t}}\right>\right>\right)\right|}\left(G_{\\{e_{i_{1}},e_{i_{2}},\cdots,e_{i_{t}}\\}}\right)$
$\displaystyle=$
$\displaystyle\begin{cases}a_{k-t^{*}}\left(G_{\\{e_{i_{1}},e_{i_{2}},\cdots,e_{i_{t}}\\}}\right)&\parbox[t]{130.08731pt}{
if $\left<\left<e_{i_{1}},e_{i_{2}},\cdots,e_{i_{t}}\right>\right>$ is a
$t^{*}$-clique of $G$, }\\\ 0&\text{otherwise.}\\\ \end{cases}$ (1)

By the principle of inclusion and exclusion again we have the same formula as
in the proof of Theorem $3.1$ ($M$ replaces $U$ and $e$’s replace $u$’s):

$\displaystyle a_{k}(G-M)=$ $\displaystyle a_{k}(G)-\sum_{1\leq i\leq
h}a^{*}_{k-\left|V\left(\left<\left<e_{i}\right>\right>\right)\right|}\left(G_{\\{e_{i}\\}}\right)$
$\displaystyle+$ $\displaystyle\sum_{1\leq i\leq j\leq
h}a^{*}_{k-\left|V\left(\left<\left<e_{i},e_{j}\right>\right>\right)\right|}\left(G_{\\{e_{i},e_{i}\\}}\right)$
$\displaystyle-$ $\displaystyle\sum_{1\leq i\leq j\leq l\leq
h}a^{*}_{k-\left|V\left(\left<\left<e_{i},e_{j},e_{l}\right>\right>\right)\right|}\left(G_{\\{e_{i},e_{i},e_{l}\\}}\right)+\cdots.$
(2)

The difficulty in this formula stems from the fact of the following sort. A
$K_{6}$ can be induced by the vertices of $3$ independent edges, while
$4,5,6,\cdots,15$ edges may also induce a $K_{6}$. We want to rearrange the
sums according to the orders of the graphs
$\left<\left<e_{i_{1}},\cdots,e_{i_{t}}\right>\right>$. We observe:

  1. 1.

If $\left<e_{i_{1}},\cdots,e_{i_{t}}\right>$ is a clique, then
$t=\binom{P}{2}$ for some $p$, i.e.
$|V(\left<e_{i_{1}},\cdots,e_{i_{t}}\right>)|=p$.

  2. 2.

A clique $K_{p}$, can have a spanning subgraph with $q$ edges for
$\lceil\frac{p}{2}\rceil\leq q\leq\binom{P}{2}$, where $\lceil x\rceil$ is the
smallest integer not smaller than $x$. Contributions to $a_{p}(G-M)$ for a
specific value $q$ stem from different terms in the expansion with a sign
$(-l)^{q}$.

  3. 3.

By Lemma 1.1 there are $f(p,q)$ spanning subgraphs with $q$ edges in $K_{p}$.
These differ from the spanning subgraphs with $q$ edges in another graph
$K_{p}$, as at least one vertex is different. Summing the contributions of
edge sets that induce a specific $K_{p}$, gives

$\sum_{q=1}^{\binom{P}{2}}(-1)^{q}f(p,q)$ (3)

which, by Lemma 1.2, is $(-1)^{p}p(p-1)$. The numbers of $k$-cliques
containing a specific $p$-clique is

$a^{*}_{k-p}\left(G_{\\{e_{i_{1}},e_{i_{2}},\cdots,e_{i_{t}}\\}}\right)=\begin{cases}a_{k-p}\left(G_{\\{e_{i_{1}},e_{i_{2}},\cdots,e_{i_{\binom{P}{2}}}\\}}\right)&\parbox[t]{121.41306pt}{
if$\left<e_{i_{1}},e_{i_{2}},\cdots,e_{i_{\binom{P}{2}}}\right>$ is a
$p$-clique,}\\\ 0&\text{otherwise.}\\\ \end{cases}$ (4)

So, finally, we have

$\displaystyle a_{k}(G-M)=$ $\displaystyle a_{k}(G)-1\sum_{1\leq i\leq
h}a^{*}_{k-2}\left(G_{\\{e_{i}\\}}\right)+2\sum_{1\leq i<j\leq l\leq
h}a^{*}_{k-3}\left(G_{\\{e_{i},e_{i},e_{l}\\}}\right)$
$\displaystyle-3\sum_{1\leq i_{1}\leq i_{2}<\cdots<i_{6}\leq
h}a^{*}_{k-4}\left(G_{\\{e_{i_{1}},\cdots,e_{i_{6}}\\}}\right)+\cdots.$ (5)

By the same method as in Theorem 3.1, we obtain

$C(G;x)=C(G-M;x)+\sum_{r=2}^{|M|}(-1)^{r}\sum_{\begin{subarray}{c}S\subset
M\\\ |S|=(1/2)r(r-1)\end{subarray}}(r-1)x^{r}C^{*}(G_{S};x)$

Morever $C^{*}(G_{S};x)$ satisfies the condition in Theorem 3.4. ∎

######  The counterexample.

The graph $G$ in figure 1 has $C(G;x)=1+3x+3x^{2}+x^{3}$. If we choose
$M=\\{\\{a,b\\},\\{a,c\\}\\}$, then $C(G-M;x)=1+3x+x^{2}$ and the graph
induced from the neighbors of $\\{a,b\\}$ has $C^{*}(G_{\\{a,b\\}};x)=1+x$. By
applying the formula in Theorem 3.4, then we have

$\displaystyle 1+3x+3x^{2}+x^{3}\neq$ $\displaystyle(1+3x+x^{2})+2x^{2}(1+x)$
$\displaystyle\neq$ $\displaystyle 1+3x+3x^{2}+2x^{3}$

abc Figure 1: $G$ c Figure 2: $G_{\\{a,b\\}}$

But, if we choose $M=\\{\\{a,b\\},\\{a,c\\},\\{b,c\\}\\}$, then
$C(G-M;x)=1+3x$ and the graph induced from the neighbors of $\\{a,b\\}$ has
$C^{*}(G_{\\{a,b\\}};x)=1+x$. $M$ induce and empty graph. By applying the
formula in Theorem 3.4, then we have

$\displaystyle 1+3x+3x^{2}+x^{3}=$
$\displaystyle(1+3x+x^{2})+3x^{2}(1+x)-2x^{3}$ $\displaystyle=$ $\displaystyle
1+3x+3x^{2}+x^{3}$

In general, the theorem is valid only when $M$ induces a clique. This is
because, it was assumed that in equation Proof. there will be enough terms to
be ordered and transformed to equation Proof., but this is true only if $M$
induces a clique.

## References

  * [1] Cornelis Hoede and Xueliang Li.  Clique polynomials and independent set polynomials of graphs.  Discrete Mathematics, 125(1-3):219–228, 1994.