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Working Paper

2014-303

Constant Proportion Portfolio

Insurance under Tolerance and

Transaction Costs

Farid MKAOUAR

Jean-luc PRIGENT

http://www.ipag.fr/fr/accueil/la-recherche/publications-WP.html

IPAG Business School

184, Boulevard Saint-Germain

75006 Paris

France

Constant Proportion Portfolio Insuranceunder Tolerance and Transaction Costs

Farid MKAOUAR�y Jean-luc PRIGENTz

Abstract

Portfolio insurance allows investors to recover at maturity a givenpercentage of their initial investment, whatever �nancial market evolu-tions. This portfolio insurance strategy limits downside risk in fallingmarkets, while it allows potential bene�ts in rising markets. We analyzethis method in the presence of jumps in asset price dynamics, in partic-ular for Lévy processes. First we examine the continuous-time rebalanc-ing case, second we introduce a stochastic-time rebalancing according toinvestor�s tolerance. This latter case is more in accordance with usualpractice and allows to take account of transaction costs. The target mul-tiple may be either deterministic or stochastic. We study in particularthe impact of tolerance and transaction costs. We provide general resultsthat we illustrate when the risky asset price follows a double exponentialLévy process.

AMS 2010 classi�cation: 91G10, 91G80, 60G51.Key words : Portfolio Insurance, CPPI, Lévy processes, risk tolerance,

transaction costs.

1 Introduction

The goal of portfolio insurance is to limit downside risk while allowing someparticipation in upside markets. Such methods allow investors to recover, atmaturity, a given percentage of their initial capital, in particular in falling mar-kets. The �rst main portfolio insurance method has been introduced by Lelandand Rubinstein (1976). It is the Option Based Portfolio Insurance (OBPI),which consists of a portfolio invested in a risky asset S (usually a �nancial in-dex such as the S&P) covered by a listed put written on it. Whatever the valueof S at maturity T , the portfolio value will be always greater than the strike Kof the put. The purpose of the OBPI method is to guarantee a �xed amountonly at maturity. The second important insurance portfolio strategy is the Con-stant Proportion Portfolio Insurance (CPPI) considered by Perold (1986) and

�IPAG Business School, Paris.yLIRSA, ENAss/CNAM, Paris.zTHEMA, University of Cergy-Pontoise, 33 Bd du Port 95011, FRANCE, e-mail:jean-

luc.prigent@u-cergy.fr, Tel: 331 34 25 61 72

1

further studied by Perold and Sharpe (1988) for �xed-income instruments andBlack and Jones (1987) for equity instruments. This strategy is based on adynamic asset allocation over time. The investor starts by setting a �oor equalto the lowest acceptable value of the portfolio. Then, he determines the cushionas the excess of the portfolio value over the �oor. The amount allocated to therisky asset is equal to the cushion multiplied by a predetermined multiple. Theremaining funds are invested in the reserve asset, usually T-bills.

The comparison of both strategies have been examined by Bookstaber andLangsam (2000), who have focused on path dependence and dealed with theproblem of the time horizon and in particular time-invariant or perpetual strate-gies (studied also in Black and Perold, 1992). Black and Rouhani (1989) havecompared CPPI and OBPI when put option must be synthesized. They haveshown that OBPI provides better performance when �nancial market increasesmoderately, while CPPI payo¤ dominates OBPI payo¤ if market drops or in-creases by a small or large amount. Bertrand and Prigent (2003, 2005) haveexamined more systematically probability distributions of both portfolio valuesand compared them by means of various criteria, in particular some of theirquantiles. This allows to emphasize the role of the insured amount : as theinsured percentage of the initial investment increases, the CPPI becomes moredesirable than the OBPI. Cesari and Cremonini (2003) have used Monte Carlosimulations to compare such dynamic strategies of asset allocation. Stochasticdominance criteria can also be introduced, as illustrated by Kraus and Zagst(2009) who provide parameter conditions implying the second- and third-orderstochastic dominance of the CPPI strategy. Empirical stochastic dominancetests have also be conducted by Annaert et al. (2009). Bertrand and Prigent(2008) have also proved that CPPI method performs better than OBPI accord-ing to Kappa performance measures (which include the Sortino and Omegaratios).

In this paper, we focus on CPPI method analysis when the price of therisky asset may have jumps. Such problem have been previously analyzed inPrigent (2001, 2007) and Cont and Tankov (2009), when the investor tradesin continuous-time and when there is no transaction cost. To take accountof usual �nancial practice, we introduce stochastic-time rebalancing accordingto investor�s tolerance and in the presence of transaction costs. The investorrebalances his portfolio as soon as the ratio �exposure/cushion�reaches a loweror an upper bound. These bounds can be chosen equal to percentages of a�xed multiple ("the target multiple"). This paper is organized as follows. InSection 2, �rst basic properties about CPPI method are recalled and some newresults are provided. Section 3 deals with the case of stochastic time rebalancingwith a target multiple. We provide explicit (or quasi-explicit) formulas for theportfolio values and probability distributions of rebalancing times, when assetprice dynamics are driven by Lévy processes. Simulations also allow to illustratetheoretical results. Some of the proofs are gathered in the Appendix.

2

2 The CPPI method with continuous-time re-balancing

2.1 Asset dynamics

The investor is assumed to trade on two basic assets: a money market account,denoted byB, and a portfolio of traded assets such as a composite index, denotedby S. The period of time considered is [0; T ]. Strategies are self-�nancing.The value of the riskless asset B evolves according to :

dBt = Btrdt;

where r is the deterministic interest rate.Dynamics of the risky asset price S are given by a di¤usion process with

jumps1 :dSt = St�[�(t; St)dt+ �(t; St)dWt + �(t; St)dN ];

where (Wt)t is a standard Brownian motion, independent of the Poisson processhaving the jump measure N .These assumptions imply that the sequence of random times of jumps (�n)n

is such that the random variables (�n+1 ��n) are independent and have thesame exponential probability distribution with parameter denoted by �. Therelative jumps of the risky asset �S�n

S�nare equal to �(�n; S�n): These ones are

assumed to be strictly greater than (�1) (this implies the positivity of the assetS). The integral

R +1�1

R t0�(u; Su)dN is equal to the sum

P�n�t �(�n; S�n) of

all of the relative jumps before time t2 .

2.2 Constant Proportion Portfolio Insurance

This strategy consists in managing a dynamic portfolio so that its value isabove a �oor P at any time t of the management period. The value of the �oorindicates the dynamic insured amount. It is assumed to evolve according to:

dPt = Ptrdt:

Obviously, the initial �oor P0 is smaller than the initial portfolio value V CPPI0 .The di¤erence

�V CPPI0 � P0

�is called the cushion. It is denoted by C0. Its

value Ct at any time t in [0; T ] is given by:

Ct = VCPPIt � Pt:

Denote by et the exposure. It is the total amount invested in the risky asset.The standard CPPI method consists of letting et = mCt where m is a constant

1The functions �(:), �(:) and �(:) satisfy the usual conditions to garantee the existence,uniqueness and positivity of the solution of this stochastic di¤erential equation (see Jacod andShiryaev, 2003).

2For detailed explanations of dynamics with jumps, see Karr (1991), Last and Brandt(1995), Shiryaev (1999), Cont and Tankov (2004).

3

called the multiple. The interesting case is when m > 1, that is, when theportfolio pro�le is convex. Thus, the CPPI method is parametrized by P0 andm3 .Both the �oor and the multiple depend on the investor�s risk tolerance. The

total amount allocated to the risky asset is known as the exposure. The higherthe multiple, the more the investor will participate in a sustained increase instock prices. Nevertheless, the higher the multiple, the faster the portfolio willapproach the �oor when there is a sustained decrease in stock prices. As thecushion approaches zero, exposure approaches zero too. In continuous time andwithout asset price jumps, this keeps portfolio value from falling below the �oor.Portfolio value will fall below the �oor only when there is a very sharp drop inthe market before the investor has a chance to trade.The cushion value at any time is given by:

Ct = C0 exp

�(1�m)rt+m

�Z t

0

��� 1

2m�2(s; Ss)

�ds+

Z t

0

�(s; Ss)dWs

���

Y0��n�t

(1 +m�(�n; S�n)) : (1)

Consequently, the guarantee is satis�ed as soon as the relative jumps of assetS satisfy:

�(�n; S�n) � �1=m: (2)

Thus, when jumps are always higher than a negative constant d; condition

0 � m � �1=d (3)

implies the positivity of the cushion. For example, if d is equal to �10%, thenm � 10. Note that this condition does not depend on the probability distribu-tion of jump times �n.Assume that the market value of the risky asset S is given by

dSt = St [�dt+ �dWt] ;

where Wt is a standard Brownian motion. Then, for the standard case, thecushion value is given by:

Ct = C0em�Wt+[r+m(��r)�m2�2

2 ]t with C0 = V0 � P0:

In that case, the cushion value and the portfolio value are path independent.The insurance is perfect. Their probability distributions are lognormal (upto a translation for the portfolio value) with a volatility equal to m�. Theinstantaneous mean rate of return is equal to r+m(�� r). The multiple m canbe viewed as a weight in the volatility and in the excess of return (�� r).

3Note that the multiple must not be too high as proved for example in Prigent (2001) orin Bertrand and Prigent (2002).

4

The value V CPPIt of the portfolio is given by:V CPPIt (m;St) = P0:e

rt + �t:Smt ;

where �t =�C0Sm0

�exp [�t] ; and � =

�r �m

�r � 1

2�2��m2 �2

2

�:

Assume now that there exists a compound Poisson component. Denote byK(dx) the probability distribution of the relative jumps of asset S, and by band c respectively the expectations E[�(�1; S�1

)] and E[�2(�1; S�1)]. Then, the

portfolio value has mean and variance, respectively given by:�E[Vt] = (V0 � P0)e[r+m(�+b��r)]t + P0ert;V ar[Vt] = (V0 � P0)2e2[r+m(�+b��r)]t[em

2(�2+c�)t � 1]:

Denote by g0;t; the cushion pdf at time t when there is no jump. Function g0;tis given by:

g0;t(x) =Ix>0

xp2��2m2t

e� 12�2m2t

�lnh

xC0

i�t(r+m(��r)�m2�2

2 )�2:

In the presence of jumps, the cushion pdf is given by:

gCt(x) =1Xn=0

e��t(�t)n

n!

ZIRn

g0;t(xQ

i�n(1 +myi))Kn(dy1; :::; dyn)Q

i�n(1 +myi);

where Kn designates the n-convolution product of K : Kn(dy1; :::; dyn) =K(dyi). Finally, the pdf of the portfolio value is equal to:

fV t(x) = gCt(x� P0ert):If Condition (2) is not satis�ed, portfolio value can go below the �oor, which

is usually called the �gap risk�. Within the Lévy process framework, Prigent(2001) introduces a quantile condition on the multiple to control gap risk:

P[Ct � 0;8t � T ] � 1� �; (4)

where � is "small". This is equivalent to:

P[8t � T; �StSt

� �1m] � 1� �:

Assume that functionK has an inverseK(�1). Then, Condition (4) is equivalentto:

m � �1K(�1)

�1�T ln(

11�� )

� : (5)

This condition determines a new upper bound on the multiple, which is ofcourse less stringent than Condition (3). This upper bound is decreasing withrespect to the jump intensity � since, if more jumps can occur, the multiplemust be reduced. Cont and Tankov (2009) provides also explicit values of theprobability of loss during a given management period. They compute both theunconditional expectation of loss and the expectation of loss conditional on lossoccurence. Such results can be applied to quantify gap risk, for instance whenthe risky asset price follows a double exponential Lévy process as described inKou and Wang (2003).

5

2.3 Double Exponential Lévy Process Case

This model assumes that there exists a Brownian type component and a jumppart modeled by a compound Lévy process such that jump sizes have a doubleexponential distribution. In that case, the risky asset prices follows the followingstochastic di¤erential equation:

dStSt

= �dt+ �dWt + d

NtXn=1

��STnSTn

�!; (6)

where W is a standard Brownian motion, N is a standard Poisson process withintensity � > 0 and the relative jump sizes �STn

STn(with values in (�1;1)) is a

sequence of i.i.d. variables such that Zn = ln�1 +

�STnSTn

�has an asymmetrical

double exponential distribution with pdf given by:

fZ(z) = pz:�1e��1zIfz�0g + qz:�2e�2zIfz<0g; �1 > 1; �2 > 0;

where pz; qz � 0 are respectively equal to the probability of an upside jump anda downside jump (pz + qz = 1).Figure 1 provides a comparison between the Gaussian distribution and the

double exponential law for null expectation (E =0), for variance equal to 1(V = 1) and for p = 0:5; �1 = �2 =

p2: Therefore, we have:

Figure 1: Gaussian and Double exponential pdfs.

ln

�1 +

�STnSTn

�= Z

n=

��+; with probability pz���; with probability qz

;

where �+ and�� are exponential random variables respectively with means 1�1

and 1�2.Note that the processes (Nt)t; (Wt)t and the sequence (Zn)n are assumed

to be independent. The solution of the SDE (6) is:

St = S0:exp

"��� 1

2�2�t+ �Wt +

NtXn=1

ln

�1 +

�STnSTn

�#:

6

Denote

E [Z] =pz�1� qz�2;V [Z] = pzqz

�1

�1� 1

�2

�2+

�pz�21� qz�22

�and:

E��STnSTn

�= E

�eZ�� 1 = qz

�2�2 + 1

+ pz�1

�1 � 1� 1; �1 > 1; �2 > 0:

Condition �1 > 1 is necessary to insure that Eh�STnSTn

i<1 and E [St] <1:

In particular, this means that the mean of the relativeupside jumps is less than100% (which is rather �reasonable�).4

Denote by Xt the discounted process of the log returns of St. We have:

Xt = ln

�StS0

�� rt =

��� r � 1

2�2�t+ �Wt +

NtXn=1

ZTn :

For the double-exponential distribution, the cushion may be negative sincethe negative jumps are not lower bounded. We can control the probability ofsuch event by using a quantile condition:

P [8t 2 [0; T ]; Ct > 0] � 1� "; for a small " (" = 1%), (7)

and/or by controlling the level of the following expected shortfall:5

8t 2 [0; T ]; E [�t � Ct jCt < �t; Ct� > 0;Ft� ] � �Ct�; (8)

where �t denotes a reference threshold. Usually, we take �t = Ct� where isa �xed parameter satisfying: < 0.

Proposition 1 For the double-exponential distribution, Condition (7) leads tothe following upper bound on the multiple:

m � 1

1��Log[ 1

1�" ]qZ�T

� 1�2

: (9)

For the expected shortfall condition, we deduce:

m � �(�2 + 1) + (1� ): (10)

Proof. See Appendix A.These conditions are not so stringent. For " = 1%, the upper bound deter-

mined from the quantile condition is approximately equal to 46, and the upperbound associated to the expected condition, with = 0 (i.e. the cushion Ctbecomes negative) and � = 10%, is approximately equal to 19. For this latterupper bound corresponding to the failure of protection, we recover standardvalues of the multiple for level � smaller than 5%.

4 If Z has a Gaussian distribution, then the Kou and Wang model is the jump-di¤usionmodel proposed by Merton (1976).

5Ft� denotes the left hand limit at time t of the �ltration generated by the observation of

the processes W , N and the sequence of relative jumps��STnSTn

�Tn�t

.

7

3 CPPI with stochastic-time rebalancing and adeterministic target multiple

In previous section, the investor is assumed to continuously rebalance his portfo-lio. In practice, this rebalancing cannot be made at any time of the managementperiod and the impact of the market timing has to be analyzed, in particularwhen there are transaction costs. One of the standard method is to �x a targetmultiple m and to rebalance the portfolio as soon as the value of the ratio �ex-posure/cushion�is smaller than m(1� �) or higher than m(1+ �): This methodimplies to rebalance the portfolio along a sequence of increasing random times(Tn)n. In what follows, we examine the problem whent the target multiple is�xed.

3.1 The model

When the cushion rises, the exposure can reach the maximal level that theinvestor want to invest or the minimal level that he requires. While the exposurelies between these two bounds, he does not trade. Otherwise, for example whenmarket �uctuations are signi�cant, he may rebalance his portfolio in order tokeep the ratio exposure/cushion within a given set of values. For this purpose, hecan de�ne a tolerance to market �uctuations which determines the two boundson percentages of variations.

3.1.1 Portfolio values

Introduce the lower bound m and the upper bound �m on the multiple m. Theinvestor begins by investing a total amount V0 and by setting a given initial�oor P0. The share �

S0 invested on the underlying S and the share �

B0 invested

on the riskless asset B are given by:

�S0 =m(V0 � P0)

S0and �B0 =

V0 �m(V0 � P0)B0

:

Notation 2 In what follows, we use the following notations for a given �nancialvariable X:

� X�Tn: the value of X at rebalancing time Tn without transaction cost (be-

fore rebalancing).

� X+Tn: the value of X at rebalancing time Tn with transaction cost (after

rebalancing).

We assume that the cushion value is equal to the di¤erence between theportfolio value V +Tn when the transaction cost is deduced and the �oor, sinceportfolio managers usually prefer to propose to their clients a cushion based onportfolio value when transacrion costs are deduced.Note that, since the investor can only trade at random times Tn; the cushion

CTn may be negative. If there exists a time Tn before the maturity T , at which

8

the cushion is negative, we assume that the investor allocates the whole portfoliovalue on the riskless asset: for any t � Tn; �St = 0:The portfolio value V �Tn+1(before rebalancing) at each time Tn+1 is equal to

V �Tn+1 = �B+TnBTn+1 + �

S+TnSTn+1 :

Note that �B+Tn = �B�Tn+1 and �S+Tn= �S�Tn+1 . Thus, we have also:

V �Tn+1 = �B�Tn+1

BTn+1 + �S�Tn+1

STn+1 :

However, the goal of the CPPI strategy is to keep an amount eTn+1

of risk

exposure that is proportional to the cushion: e+Tn+1

= mC+Tn+1with C+Tn+1

=�V +Tn+1 � PTn+1

�: This latter condition allows the determination of the quanti-

ties �S+Tn+1and �B+Tn+1

to invest during the period ]Tn+1; Tn+2[.The portfolio value V +Tn+1 at time Tn+1 after rebalancing is equal to:

V +Tn+1 = �B+Tn+1

BTn+1 + �S+Tn+1

STn+1 :

We suppose that there exist transaction costs which are proportional to therisky amount variation (rate denoted by ). We assume that these costs are nullat time T0: At each rebalancing time Tn+1, the portfolio value V

�Tn+1

is reduced

by the amount of transaction costs ����S+Tn+1 � �S�Tn+1���STn+1 :

Therefore, the portfolio value V +Tn+1 (after rebalancing) is given by:

V +Tn+1 = V�Tn+1

� ����S+Tn+1 � �S�Tn+1���STn+1 : (11)

Proposition 3 If C+Tn � 0 then the whole portfolio value is invested on theriskless asset. If C+Tn > 0; the cushion value C+Tn+1 is determined from the

cushion value C+Tn at time Tn according to the buy/sell condition. We get:

�S+Tn+1 > �S�Tn+1

) C+Tn+1 = C+Tn

0@m (1 + ) STn+1STn� (m� 1)BTn+1

BTn

(1 + m)

1A ; (12)

�S+Tn+1 � �S�Tn+1

) C+Tn+1 = C+Tn

0@m (1� ) STn+1STn� (m� 1)BTn+1

BTn

(1� m)

1A : (13)

Proof. See Appendix B.

The previous results allow us to establish the following proposition.

9

Proposition 4 (Characterization of the buy/sell condition)Assume that, at time Tn, we have: m > 1; 0 < < 1

m(usual assumptions)and the cushion value C+Tn > 0. We deduce the following equivalences:

�S+Tn+1 > �S�Tn+1 ,�STn+1STn

>�BTn+1BTn

;

�S+Tn+1 < �S�Tn+1 ,�STn+1STn

<�BTn+1BTn

;

�S+Tn+1 = �S�Tn+1 ,�STn+1STn

=�BTn+1BTn

:

Proof. See Appendix C.

From previous result, we can determine the portfolio value and strategy attime Tn+1 after rebalancing according to �nancial market variations.

Corollary 5 Assume that, at time Tn, we have: m > 1; 0 < < 1m and the

cushion value satis�es: C+Tn > 0.

If�STn+1STn

>�BTn+1

BTn; then we have:

V +Tn+1 =

�V �Tn+1 + mPTn+1 + �

S+TnSTn+1

�(1 + m)

;

C+Tn+1 = C+Tn

0@m (1 + ) STn+1STn� (m� 1)BTn+1

BTn

(1 + m)

1A ;�S+Tn+1 =

m�V �Tn+1 + �

S+TnSTn+1 � PTn+1

�(1 + m)STn+1

:

If�STn+1STn

<�BTn+1

BTn; then we have:

V +Tn+1 =

�V �Tn+1 � mPTn+1 � �

S+TnSTn+1

�(1� m) ;

C+Tn+1 = C+Tn

0@m (1� ) STn+1STn� (m� 1)BTn+1

BTn

(1� m)

1A ;�S+Tn+1 =

m�V �Tn+1 � �

S+TnSTn+1 � PTn+1

�(1� m)STn+1

:

If�STn+1STn

=�BTn+1

BTn; then we have: �S+Tn+1 = �

S+Tn:

10

Proposition 6 Assume that there exists a �rst time Tn < T at which the cush-ion value is negative: C+Tn < 0: From previous assumption, the whole portfoliovalue is invested on the riskless asset. Therefore, we have:

8k; Tn � Tk � T; �S+Tk = 0;

V +Tk = V +Tn

Yn+1�l�k

�1 +

�BTlBTl�1

�;

C+Tk = V +Tk � PTk :

Proposition 7 (Positivity condition of the cushion values C+Tn in the presenceof transaction costs) Assume m > 1, and 0 < < 1

m : Then, the cushion valueis always positive if and only if we have:

m � Inffn;Tn�TgBTn+1=BTn

BTn+1=BTn � (1� )STn+1=STn;

for any time Tn such that STn+1=STn � BTn+1=BTn :

Proof. We only have to consider the cases�STn+1STn

<�BTn+1

BTn: For these cases,

we have:

C+Tn+1 = C+Tn

0@m (1� ) STn+1STn� (m� 1)BTn+1

BTn

(1� m)

1A ;Thus, we have: C+Tn > 0) C+Tn+1 > 0 if and only if m (1� )

STn+1STn

� (m�

1)BTn+1

BTn> 0; from which we deduce the result.

We now examine the special case when the sequence of rebalancing times(Tn)n is de�ned from tolerance conditions.

3.1.2 Portfolio rebalancing times

We examine now the probability distribution of the rebalancing times with re-spect to the transaction cost rate : In what follows, we assume that the logre-turn discounted process X is a Lévy process. The portfolio rebalancing occursas soon as the ratio et

Ctis smaller than m or higher than �m.

General result Let us examine the �rst period (before the �rst rebalancingtime T1). We have:

V0 = �B0 B0 + �S0S0;

e0 = mC0 = m (V0 � P0) ;

and, for any t < T1;Vt = �

B0 Bt + �

S0St:

11

and,

V +T1 = V�T1�

� ����S+T1 � �S�T1� ���ST1 :

We have also:�S�T1�ST1 = m

�T1�

�V �T1 � PT1

�:

�S+T1 ST1 = m�V +T1 � PT1

�Then:

V +T1 = V�T1�

� ���m �V +T1 � PT1��m�

T1�

�V �T1 � PT1

���� : (14)

Case 1: m�T1=

e�T1

C�T1

� m: In this case we have C�T1 > 0 (otherwise, if

C�T1 < 0 then e�T1� mC�T1 < 0;which is impossible). Therefore, we have:

m�T1� m � m; from C�T1 > 0 and V

�T1� V +T1 ;

from which we deduce:

m�T1

�V �T1 � PT1

�� m

�V �T1 � PT1

�� m

�V +T1 � PT1

�:

From (14) we obtain:

V +T1 = V �T1 � �m�T1

�V �T1 � PT1

��m

�V +T1 � PT1

��;

V +T1 � PT1 = V �T1 � PT1 � �m�T1

�V �T1 � PT1

��m

�V +T1 � PT1

��;

C+T1 (1� m) = C�T1�1� m�

T1

�: (15)

Note that we have C�T1 > 0 sincee�T1

C�T1

� m > 0. We also have 1 � m > 0

(usual assumption: m < 1 ). Therefore, from (15), and since m�

T1> 0 and

< 1m , the condition C

+T1> 0 is equivalent to the condition

�1� m�

T1

�> 0:

Finally, we deduce:

<1

m�T1

: (16)

Case 2: m�T1=

e�T1

C�T1

� m:

i) If m�V +T1 � PT1

�� m�

T1

�V �T1 � PT1

�:

From (14), the portfolio value is given by:

V +T1 = V �T1 � �m�V +T1 � PT1

��m�

T1

�V �T1 � PT1

��;

C+T1 = C�T1 � �mC+T1 �m

�T1C�T1

�:

We deduce:C+T1 (1 + m) = C

�T1

�1 + m�

T1

�:

12

In this case, we have:mC+T1 � m

�T1C�T1 :

If C�T1 < 0; then we deduce m�T1C�T1 � mC+T1 � mC�T1 < 0: Thus, we have

m�T1� m, which is not possible in this case. Therefore, since we must have

C�T1 > 0, and m > 0; m�T1> 0, > 0; we deduce that the condition C+T1 > 0 is

satis�ed.

ii) If m�V +T1 � PT1

�< m�

T1

�V �T1 � PT1

�.

From (14) we obtain:

V +T1 = V �T1 � �m�T1

�V �T1 � PT1

��m

�V +T1 � PT1

��;

V +T1 � PT1 = V �T1 � PT1 � �m�T1

�V �T1 � PT1

��m

�V +T1 � PT1

��;

C+T1 (1� m) = C�T1�1� m�

T1

�: (17)

-1) From (17), if C�T1 > 0, then m�T1> 0 and < 1

m (usual assump-tion) implies < 1

m�T1

: Therefore, C+T1 > 0. But, in this case we have:

V +T1 = V �T1 � �m�T1

�V �T1 � PT1

��m

�V +T1 � PT1

��;

V +T1 � V�T1

= � m�T1

�V �T1 � PT1

�+ m

��V +T1 � V

�T1

�+�V �T1 � PT1

��:

Thus: �V +T1 � V

�T1

�(1� m) =

�V �T1 � PT1

� �m�m�

T1

� :

Let us compare m�T1

�V �T1 � PT1

�with m

�V +T1 � PT1

�: We have:

m�V +T1 � PT1

�= m

��V +T1 � V

�T1

�+�V �T1 � PT1

��;

= m

"�m�m�

T1

�(1� m) + 1

# �V �T1 � PT1

�

Thus, we have to compare m�T1with m

��m�m�

T1

�(1� m) + 1

�. We have:

�m�m�

T1

�(1� m) + 1 =

1� m�T1

1� m > 1:

Therefore,

m�T1< m

"�m�m�

T1

�(1� m) + 1

#;

which is impossible since we suppose thatm�V +T1 � PT1

�is smaller thanm�

T1

�V �T1 � PT1

�.

-2) From (17), if C�T1 < 0, then the whole portfolio value is investedon the riskless asset during the remaining time period and we have C+T1 < 0:

13

Lemma 8 The transaction cost may induce a negative value of C+T1whereas thevalue of C�T1 is positive.

In order to avoid such case, which can only happen whene�T1C�T1

� m; Condition(16) must be satis�ed, which is equivalent to the following conditions:

- Ife�T1C�T1

= m�T1= m, then we must have < 1

m :

- Ife�T1C�T1

= m�T1> m; de�ne the random function em�

T1

(:) by:

em�T1(x) =

�S0ST1� (1 + x)

�S0ST1� (1 + x) + �B0 BT1 � PT1

: (18)

Then, we must have:

<1em�

T1[�Max (��(�n; S�n)]

:

Proof. Note that we have:

e0C0

= m > 1, �S0S0

�B0 B0 + �S0S0 � P0

> 1, �B0 B0 � P0 < 0:

Denotec = �B0 BT1 � PT1 =

��B0 B0 � P0

�exp(rT1) < 0:

The function de�ned by (x) = xx+c is such that

0(x) = c(x+c)2 < 0.

Thus is decreasing. Assume that

�S0ST1�

�S0ST1� + �B0 BT1 � PT1

= m:

We have:

m�T1= em�

T1

��ST1ST1�

�=

�S0ST1� (1 +�ST1ST

1�)

�S0ST1� (1 +�ST1ST

1�) + �B0 BT1 � PT1

> m:

Then: 8<: If �ST1ST1�> 0;m�

T1< m;

If �ST1ST1�< 0;m�

T1> m:

Therefore, from Relation (16), we deduce:

<1

Sup��ST1ST

1�<0

��em�T1

��ST1ST

1�

�� ;

14

which is equivalent to:

<1em�

T1[�Max (��(�n; S�n)]

:

Lemma 9 The value of C�T1 may be negative, due to negative jumps of the riskyasset. To avoid such problem, the following condition must be imposed:

�m � 1

Maxn(��(�n; S�n)):

Proof. Recall that the cushion value CT1� before jump at time T1 satis�es:

CT1� = �S0ST1� + �

B0 BT1 � PT1 > 0:

The cushion value C�T1 after jump at time T1 is given by:

C�T1 = �S0ST1� (1 +

�ST1ST1�

) + �B0 BT1 � PT1 ;

from which, we deduce:

C�T1 � CT1� = �S0ST1�

��ST1ST1�

�:

Therefore, we have:

C�T1 > 0, �S0ST1�

��ST1ST1�

�> �CT1� :

When �ST1ST

1�> 0; the previous condition is immediately satis�ed.

Thus let us examine the case when�ST1ST1�< 0: In that case:

�S0ST1�

��ST1ST1�

�> �CT1� , �S0 <

CT1���ST1

:

We also have:m � et

Ct� �m;8 t < T1:

Thus, in particular:

�S0ST1��m

� CT1� ��S0ST1�m

:

Therefore, as soon as �S0 � 1��ST1

�S0 ST1��m ;we have �S0 <

CT1�

��ST1:

15

But:

�S0 �1

��ST1�S0ST1��m

, �m <1�

��ST1ST

1�

� :Consequently, the value of C�T1 is non negative as soon as:

�m <1

Sup��ST1ST

1�<0

����ST1ST

1�

� = 1

Maxn(��(�n; S�n)):

Finally, from the two previous lemma, we deduce the following result.

Proposition 10 (Positivity condition of the cushion in the presence of trans-action costs) The cushion is always positive if and only if the upper bound onthe multiple �m and the transaction cost rate satisfy:

�m � 1

Maxn(��(�n; S�n))and <

1em�T1[�Max (��(�n; S�n

)]; (19)

which is equivalent to:

�m � 1

Maxn(��(�n; S�n))and <

1�Maxn(��(�n; S�n))1�m �Maxn(��(�n; S�n

)): (20)

Proof. The �rst condition is the results of the two previous lemma. For thesecond condition, note that from relation (18), we have:

em�T1

��ST1ST1�

�=

�1 +

�ST1ST

1�

��1 +

�ST1ST

1�

�+

�B0 BT1�PT1

�S0 ST1�

:

This term is maximal when:-First, at time T1� , the multiple is equal to the upper value m:em�

T1(0) = m;

which implies:1

1 +�B0 BT1

�PT1�S0 ST1�

= m:

- Second, the absolute value of the (negative) jump���ST1ST

1�

�is maximal.

Therefore, the minimal value of 1em�T1[�Max(��(�n;S�n )]

is given by:

1�Maxn(��(�n; S�n))

1�m �Maxn(��(�n; S�n

));

16

which determines the upper bound for the inverse of the transaction cost :Since S is assumed to be an exponential Lévy process, the same result can

be proved for any period [Tn; Tn+1[:

We now determine the probability distribution of the rebalancing times.

Proposition 11 (First rebalancing time)Since (usually) the amount �B0 B0 invested on the riskless asset is smaller

than the initial �oor P0, then the rebalancing condition is given by:

m � etCt� �m; and is equivalent to: A � Xt � B;

where A and B are two constants and (Xt)t is the process de�ned by:

Xt = ln

�StS0

�� rt:

Proof. At time t = 0; we have

S0�S0 = m (V0 � P0) ; �S0 = m

(V0 � P0)S0

and �B0 B0 + �S0S0 = V0:

Denote by T1 the �rst rebalancing time. If t < T1; then the portfolio value,the cushion value, and the exposure are respectively equal to:

Vt = �B0 Bt + �

S0St; Ct = Vt � P0ert; and et = �S0St:

Therefore, the condition m � etCt� �m is equivalent to

m 6 �S0St

�B0 Bt + �S0St � P0ert

6_m;

which also means:

m�P0 � �B0 B0

�(m� 1) �S0

6 Ste�rt �m�P0 � �B0 B0

�� (m� 1) �S0

:

Setting Xt = ln�StS0

�� rt; we deduce that there exist two constants A and

B such thatA � Xt � B:

Corollary 12 The two constants are only functions of the target multiple mand the rebalancing tolerance � : They are given by:8<: A(� ;m) = Ln

��m�m�1

(P0��B0 B0)m�(V0�P0)

�= Ln

��m�m�1

m�1m

�= Ln

�m�1

m� 11+�

�;

B(� ;m) = Ln�

mm�1

(P0��B0 B0)m�(V0�P0)

�= Ln

�mm�1

m�1m

�= Ln

�m�1

m� 11��

�:

17

For any �xed rebalancing rate � ; A(� ;m) is an increasing function with re-spect to m and B(� ;m) is a decreasing function with respect to m.

For any �xed multiple m; A(� ;m) is a decreasing function with respect to�and B(� ;m) is an increasing function with respect to � :

Corollary 13 The corridor fB(� ;m); A(� ;m)g depend only on the target mul-tiple m and on the rebalancing rate � :

B(� ;m)�A(� ;m) = Ln�m

�m

�m� 1m� 1

�= Ln

m� 1

1+�

m� 11��

!:

For any �xed rebalancing rate � ; the corridor is a decreasing function withrespect to m.

For any �xed multiple m; the corridor is an increasing function with respectto � .

Proposition 14 Assuming that the logreturn of the risky asset is a Lévy process,the conditional probability distribution of the hitting time

�T(�)n+1 � T

(�)n

�; given

the information FT(�)n

at time T (�)n ; is de�ned by:

P[T (�)n+1 � T (�)n � t���FT (�)n

] =

1� P[Sup0�s�tXs � B(� ;m); Inf0�s�tXs � A(� ;m)]:

Proof. The proof is the same as for the �rst hitting time T1. Recall that theportfolio rebalancing occurs as soon as the ratio et

Ctis smaller than m or higher

than �m: The condition is now:

A(� ;m) � XT(�)n+1

�XT(�)n� B(� ;m);

with the same values for A(� ;m) and B(� ;m) as previously.

Remark 15 Note that these constants do not depend on the transaction costs.Only the tolerance rate is involved.

18

Examples Consider now some basic examples of risky asset prices driven byLévy processes.

Example 1 (Geometric Brownian case) In this case, the asset price Sis given by:

St = S0 exp

���� 1

2�2�t+ �Wt

�:

Thus, the process (Xt)t is a Brownian motion with drift, de�ned by

Xt = (�� r � 1=2�2)t+ �Wt:

The conditional distribution of time rebalancing is determined from the prop-erty that the Brownian motion with drift goes beyond the corridor fA;Bg: Thisprobability can be computed by using the trivariate distribution trivariate ofthe running maximum, minimum and terminal value of the Brownian motion(See e.g. Revuz and Yor, 1994) after an appropriate change of probability toeliminate the drift (see also Kunitomo and Ikeda, 1992; Geman and Yor, 1994;He, Keirstead and Rebholz, 1998; and Karlin and Taylor, 1975). Recall thatthe pdf of this joint law in the presence of a drift constant � is de�ned for allvalues of x in [A;B] by:

g(x;A;B) = exp[�x

�2� �2t

2�2]�

+1Xn=�1

1

�pt

��(x� 2n(B �A)

�pt

)� �(x� 2n(B �A)� 2A�pt

)

�;

where � is the pdf of the standard Gaussian distribution and N is its cdf:

- If A < 0 and B > 0, then the distribution of the �rst passage time T1 isgiven by:

P[T1 � t] = 1� P[Maxs�tXs � B;Mins�tXs � A]with

P[Maxs�tXs � B;Mins�tXs � A] =+1X

n=�1e2n�(B�A)=�

2

[N(B � �t� 2n(B �A)

�pt

)�N(A� �t� 2n(B �A)�pt

)]

�e2A�=�2

[N(B � �t� 2n(B �A)� 2A

�pt

)�N(A� �t� 2n(B �A)� 2A�pt

)]:

This problem can be also be examined for other Lévy processes. In orderto obtain solutions which are su¢ ciently explicit, some particular models canbe examined. Spectrally negative Lévy process are examined in Dufresne andGerber (1990), in Dozzi and Vallois (1997)), and in Roynette, Vallois and Volpi(2003). Such processes can be introduced in the rebalancing model which isstudied here. Usually, the distribution of the �rst hitting time and of the over-shoot are known through their Laplace transforms.

19

Example 2. Double exponential Lévy process For the double expo-nential Lévy process, the distribution of the sequence of rebalancing times canbe characterized from Laplace transform, by using results provided in Mkaouarand Prigent (2009).Recall that Xt denotes the discounted process of the log returns of St. We

have:

Xt = ln

�StS0

�� rt =

��� r � 1

2�2�t+ �Wt +

NtXn=1

ZTn :

Denote e� = � � r � 12�

2 and de�ne �a;b as the �rst time at which process Xgoes above the lower barrier (a) or below the upper barrier (b):

�a;b = inf ft � 0;Xt � b or Xt � ag ; a < 0 < b:

Then:

P [�a;b � t] = P�max0�s�t

fXs � bg and inf0�s�t

fXs � ag�:

The Laplace transform of �a;b is de�ned by

� �! E [exp [���a;b]] :

An auxiliary function ua;b is introduced. It is related to the in�nitesimalgenerator of X and satis�es:

E [exp [���a;b]] = ua;b(0):

The moment generator of process X is given by:

E(e�Xt) = e�e�tE(e��Wt)E(e�PNt

n=1 ZTn ) = eG(�)t;

with

G(�) = �e�+ 12�2�2 + �

�pz�1�1 � �

+qz�2�2 + �

� 1�: (21)

- EquationG(x) = �; 8 � > 0; (22)

has exactly four roots �1;�; �2;�;��3;� et ��4;�; such that:

0 < �1;� < �1 < �2;� <1; and 0 < �3;� < �2 < �4;� <1:

Then, the Laplace transform of the �rst passage of the double barrier isgiven by:

E�e���a;b

�= u�(0) = Ae

��1;�b +Be��2;�b + Ce�3;�(a�x) +De�4;�a;

with:G(�i;�) = �; 8� > 0 and 8i 2 f1; 2g ;

andG(��i;�) = �; 8� > 0 and 8i 2 f3; 4g :

20

3.2 Empirical Illustrations

We focus on daily returns (dividend adjusted) for S&P500 for the period 1=01=1990through 30=05=2009. We use the maximum-likelihood method to obtain para-meter estimation of double exponential Lévy process. For this time period, we�nd that (pz�)�1 = 1:8357 and (qz�)�1 = 1:79). The expectations of jump sizesare given by: ��11 = 0:65% and ��12 = 0:72% per day. The daily expected return� is equal to 0:0616991% and the daily volatility � is equal to 0:463107%.

3.2.1 Rebalancing times

In what follows, we set T = 1 year. Figure (2) illustrates the pdf and cdf ofthe three �rst rebalancing times and durations, and also the 10th; 20th and30th. Obviously (see Figure (2) (b)), time Ti dominates Ti�1: As proved inprevious Proposition (14), the durations have the same probability distribution(see �gure (2) (c) and (d)).Note, for example, that the median of durations is equal to 0:0086. It cor-

responds to rebalance every three days. The probability that duration betweentwo rebalancing times is smaller than one week is equal to 85%:

0 0.1 0.2 0.3 0.4 0.5 0.60

10

20

30

40

50

60

70

(Rebalancing Time)

f(Reb

alan

cing

 Tim

e)

(a) µ = 10% σ  = 20% γ = 0.5% τ = 10%   m=6

RT1RT2RT5RT10RT20RT30

0 0.1 0.2 0.3 0.4 0.5 0.60

0.2

0.4

0.6

0.8

1

(Rebalancing Time)

F(R

ebal

anci

ng T

ime)

(b) µ = 10% σ  = 20% γ = 0.5% τ = 10%   m=6

RT1RT2RT5RT10RT20RT30

0 0.02 0.04 0.06 0.08 0.1 0.120

10

20

30

40

50

60

70

(Duration)

f(Dur

atio

n)

(c) µ = 10% σ  = 20% γ = 0.5% τ = 10%   m=6

D1D2D5D10D20D30

0 0.02 0.04 0.06 0.08 0.1 0.120

0.2

0.4

0.6

0.8

1

(Duration)

F(D

urat

ion)

(d) µ = 10% σ  = 20% γ = 0.5% τ = 10%   m=6

D1D2D5D10D20D30

Figure 2: Rebalancing times and Durations

21

3.2.2 Portfolio value and cumulated transaction costs

Consider the continuous-time rebalancing portfolio value V ctrT which correspondsto � = 0%. The corresponding cumulated transaction cost is denoted by TCctrT .Denote respectively by V strT the stochastic-time rebalancing portfolio value atmaturity T (� 6= 0%) and by TCstrT the cumulated transaction cost. We examinethe distributions of V strT and TCstrT , jointly with the distributions of the ratiosV ctrT

V strT

and TCctrT

TCstrT. Note that their cdfs depend on and m. The cdf F of V

ctrT

V strT

isgiven by:

F (x) = P�V ctrT (� = 0)

V strT (� 6= 0) � x�( ;m):

We search in particular for the quantile at level 12 and the value of F (1):Figure (3) illustrates how the transaction cost rate and tolerance both de-

termine portfolio value and cumulated transaction costs.

0.88 0.9 0.92 0.94 0.96 0.98 10

0.2

0.4

0.6

0.8

1

(VTctr / V

Tstr)

F(V

Tctr  / 

VTst

r )

(a) µ = 0.0617% σ  = 0.4631% γ = 0.5% τ = 10% m=6

1.5 2 2.5 3 3.5 4 4.50

0.2

0.4

0.6

0.8

1

(TCTctr / TC

Tstr)

F(TC

Tctr  / 

TCTst

r )

(b) µ = 0.0617% σ  = 0.4631% γ = 0.5% τ = 10% m=6

100 110 120 130 140 1500

0.2

0.4

0.6

0.8

1

(Portfolio Value)

F(Po

rtfol

io V

alue

)

(c) µ = 0.0617% σ  = 0.4631% γ = 0.5% τ = 10% m=6

VTctr

VTstr

0 1 2 3 4 5 6 7 80

0.2

0.4

0.6

0.8

1

(Transaction costs)

F(Tr

ansa

ctio

n co

sts)

(d) µ = 0.0617% σ  = 0.4631% γ = 0.5% τ = 10% m=6

TCTctr

TCTstr

Figure 3: Cdf of V ctrT ; V strT ; TCctrT and TCstrT ; and ratios V ctrT

V strT

and TCctrT

TCstrT:

From Figure (3) (a), we note that the probability that V strT is higher than

V ctrT is about 100% and the range of ratio V ctrT

V strT

lies in [0:94; 1:01]:This proves theneed to introduce portfolio rebalancing according to tolerance level. In addition,from Figure (3) (b), the ratio TCctr

T

TCstrTof the cumulative amount of transaction costs

is always higher than 149%: Figure (3) (c) shows that the probability that theportfolio�s value V ctrT is smaller than the initial investment V0 is around 40%

22

and 34% for the stochastic case (V strT ). Looking at Figure (3) (d), we note thatcumulative transaction costs TCctrT is smaller than 2% of the initial investmentV0 with a probability equal to 58%, whereas, for TCstrT , this probability is equalto 93%. Note also that TCctrT can reach 8% of the initial investment V0:

4 Conclusion

In this paper, we have examined the CPPI method in the presence of jumpsin the risky asset prices. When the investor trades in continuous-time, themultiple must be upper bounded according to jump characteristics. This allowsto control the gap risk. However, usually in practice, portfolio rebalancing takesplace in stochastic-time, according to investor�s tolerance and transaction costs.The investor modi�es his portfolio as soon as the ratio �exposure/cushion�reaches a lower or an upper bound. In this framework, we have determinedgeneral formulas for portfolio values and probability distributions of rebalancingtimes, when the risky asset price is an exponential Lévy process. We have alsoillustrated how transaction costs in�uence the portfolio performance. Therefore,the tolerance to the target multiple must be carefully chosen according to thetransaction cost level.

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25

Appendix

Appendix A

Proof of Proposition 1. By conditioning with respect to the number of jumps,the quantile condition

P [8t 2 [0; T ]; Ct > 0] � 1� " (23)

is equivalent to:

1Xn=0

P [8t 2 [0; T ]; Ct > 0 \NT = n] � 1� ";

where Nt denotes the random number of jumps before time t, and also to:

1Xn=0

P [8t 2 [0; T ]; Ct > 0 jNT = n ]P [NT = n] � 1� ":

Using the assumptions on the risky logreturn and on the random variables

Zn =Logh1 +

�STnSTn

i, and since m > 1, we deduce the inequality:

1Xn=0

e��T(�T )

n

n!P�8i � n; Zi > Log

�1� 1

m

��� 1� ":

But, since the random variables Zi are i.i.d., we deduce that each probabilityP�8i � n; Zi > Log

�1� 1

m

��is equal to

�P�Z > Log

�1� 1

m

���n.

Additionally, we have:

P�Z > Log

�1� 1

m

��

= P�Z > Log

�1� 1

m

�\ Z � 0

�+ P

�Z > Log

�1� 1

m

�\ Z < 0

�;

= pZ + qZ

�1�

�1� 1

m

��2�:

Consequently, the condition (23) is equivalent to:

exp

��T

�pZ + qZ

�1�

�1� 1

m

��2�� 1��

� 1� ":

Finally, the upper bound on the multiple, de�ned from the quantile condi-tion, is equal to

m � 1

1��Log[ 1

1�" ]qZ�T

� 1�2

:

26

The upper bound on the multiple associated to the expected shortfall con-dition (8) is determined as follows. We have:

E [ Ct� � Ct jCt < Ct�; Ct� > 0;Ft� ] =

Ct�

� � E

��1 +m

�StSt�

� �����1 +m�StSt�

�< ;Ct� > 0;Ft�

��:

Since S follows a geometric Lévy process, we have:

E��1 +m

�StSt�

� �����1 +m�StSt�

�< ;Ct� > 0;Ft�

�=

E��1 +m

�StSt�

� �����1 +m�StSt�

�<

�:

Standard calculus shows that this later term is equal to �(m� 1)=(�2 + 1).Therefore, we deduce that the expected shortfall condition (8) is equivalent to:

m � �(�2 + 1) + (1� ):

Appendix B

Proof of Proposition (3). At time Tn+1; the portfolio value V �Tn+1 (beforerebalancing) satis�es:

V �Tn+1 = mC+Tn

�STn+1STn

�BTn+1BTn

�+ C+Tn

BTn+1BTn

+ PTn+1 :

We also have:

�S+Tn STn+1 = mC+Tn

STn+1STn

: (24)

Case 1: �S+Tn+1 > �S�Tn+1

. We deduce the value of the cushion at time Tn+1after rebalancing:

C+Tn+1 = V+Tn+1

�PTn+1 =

�V �Tn+1 + mPTn+1 + �

S+TnSTn+1

�� (1 + m)PTn+1

(1 + m);

From (24) we get:

C+Tn+1 =mC+Tn

�STn+1STn

� BTn+1

BTn

�+ C+Tn

BTn+1

BTn+ mC+Tn

STn+1STn

(1 + m);

C+Tn+1 = C+Tn

0@m (1 + ) STn+1STn� (m� 1)BTn+1

BTn

(1 + m)

1A :

27

Case 2: �S+Tn+1 � �S�Tn+1

. We deduce the value of the cushion at time Tn+1after rebalancing:

C+Tn+1 = V+Tn+1

�PTn+1 =

�V �Tn+1 � mPTn+1 � �

S+TnSTn+1

�� (1� m)PTn+1

(1� m) ;

From (24); we have:

C+Tn+1 =mC+Tn

�STn+1STn

� BTn+1

BTn

�+ C+Tn

BTn+1

BTn� mC+Tn

STn+1STn

(1� m) ;

C+Tn+1 = C+Tn

0@m (1� ) STn+1STn� (m� 1)BTn+1

BTn

(1� m)

1A :Appendix C

Proof of Proposition (4).At time Tn+1; the variation of the quantity invested on the risky asset (dif-

ference between the value before and after the rebalancing) is given by:

�S+Tn+1 � �S�Tn+1

= �S+Tn+1 � �S+Tn=mC+Tn+1STn+1

�mC+TnSTn

: (25)

Case 1: �S+Tn+1 � �S�Tn+1

:

From (25) and (12), we have:

�S+Tn+1 � �S�Tn+1

=

mC+Tn

m(1+ )

STn+1STn

�(m�1)BTn+1BTn

(1+ m)

!STn+1

�mC+TnSTn

;

= mC+Tn

m (1 + )STn+1STn

� (m� 1)BTn+1

BTn� (1+ m)STn+1

STn

(1 + m)STn+1;

= mC+Tn

0@ (m� 1)STn+1STn� (m� 1)BTn+1

BTn

(1 + m)STn+1

1A ;= mC+Tn

0@ (m� 1)hSTn+1STn

� BTn+1

BTn

i(1 + m)STn+1

1A ;

28

From m > 1; > 0 and CTn+ > 0; we have:

�S+Tn+1 > �S�Tn+1 ) (m� 1)�STn+1STn

�BTn+1BTn

�> 0;

�S+Tn+1 > �S�Tn+1 )STn+1STn

>BTn+1BTn

;

�S+Tn+1 > �S�Tn+1 )�STn+1STn

>�BTn+1BTn

: (26)

Case 2: �S+Tn+1 � �S�Tn+1

.From (25) and (13); we have:

�S+Tn+1 � �S�Tn+1

=

mC+Tn

m(1� )

STn+1STn

�(m�1)BTn+1BTn

(1� m)

!STn+1

�mC+TnSTn

;

= mC+Tn

m (1� ) STn+1STn� (m� 1)BTn+1

BTn� (1� m)STn+1

STn

(1� m)STn+1;

= mC+Tn

0@ (m� 1)STn+1STn� (m� 1)BTn+1

BTn

(1� m)STn+1

1A ;= mC+Tn

0@ (m� 1)hSTn+1STn

� BTn+1

BTn

i(1� m)STn+1

1A ;From m > 1; < 1

m and C+Tn > 0; we deduce:

�S+Tn+1 < �S�Tn+1 ) (m� 1)�STn+1STn

�BTn+1BTn

�< 0;

�S+Tn+1 < �S�Tn+1 )STn+1STn

<BTn+1BTn

;

�S+Tn+1 < �S�Tn+1 )�STn+1STn

<�BTn+1BTn

: (27)

Finally, from Relations (26) and (27), we deduce the following equivalence:

�S+Tn+1 > �S�Tn+1 ,�STn+1STn

>�BTn+1BTn

:

�S+Tn+1 < �S�Tn+1 ,�STn+1STn

<�BTn+1BTn

:

�S+Tn+1 = �S�Tn+1 ,�STn+1STn

=�BTn+1BTn

:

29