Philippe Bertrand
email: philippe.bertrand@univ-amu.fr

Jean-Luc Prigent
email: jean-luc.prigent@u-cergy.fr

On the Optimality of Path-Dependent Structured Funds: the Cost of Standardization

Keywords: JEL classi…cation: G11, G13, G21, D03 Finance, Structured …nancial products, portfolio insurance, Asian options, average of calls, stochastic volatility, compensating variation

This paper examines the suitability of an important class of standard …nancial structured products, namely those whose performances are based on smoothing the return of a given risky underlying asset while providing a guarantee at maturity. Using various assumptions about the customers attitudes towards risk, we show that such standardized products are not optimal, even if the …nancial market volatility is constant. As a by-product, we provide in particular the optimal portfolio value in the regret/rejoice framework to go further with the notion of aversion of getting a return smaller than the risk-free one. Using the notion of compensating variation, we determine for the …rst time, the monetary losses of providing these standardized products instead of the optimal ones to the customers. We show that these monetary losses can be very signi…cant when the volatility of the risky asset is stochastic. From the operational point of view, such results highly suggest to trade on the Volatility Index (VIX) and/or to introduce derivatives written on it, when selling standardized funds in order to better meet investors needs and preferences.

Introduction

Structured …nancial products issued by …nancial institutions have been introduced in order to allow investors to invest on complex …nancial instruments such as derivatives while satisfying speci…c needs. Generally, the proposed portfolios provide a capital protection against market ‡uctuations while allowing to bene…t from market rises. They are based on more or less complex options the underlying assets of which can be a single security, a basket of securities or …nancial indices.

A …rst problem that arises with structured products is their fair pricing. It has been examined by several authors using comparisons of prices in the primary or secondary market to the theoretical fair values: Chen and Kensinger (1990) and Chen and Sears (1990) deal with the pricing of convex instruments on the S&P 500; [START_REF] Wasserfallen | Portfolio insurance for the small investor in Switzerland[END_REF] and Burth et al. (2001) examine the Swiss market; Stoimenov and Wilkens (2005) investigate the fair pricing of equity-linked structured products in the German private retail banking sector; Bertrand and Prigent (2015a) study the fair pricing of French …nancial structured products. All these results show that almost all types of equity-linked structured products are priced above their theoretical values, depending on their complexity degree.

The second important problem, which has been less examined in the …nancial literature, is the suitability of a given structured product to the investor's risk pro…le (see [START_REF] Shefrin | Behavioral aspects on the design and marketing of …nancial products[END_REF][START_REF] Driessen | An empirical portfolio perspective on option pricing anomalies[END_REF][START_REF] Das | Options and structured products in behavioral portfolios[END_REF]. Both the U.S. Dodd-Frank Act 1 and the European MiFID directive 2 have emphasized that …nancial institutions must check the adequacy of structured products to their retail investors, in light of investor …nancial protection (see [START_REF] Chang | Suitability checks and household investments in structured products[END_REF]. Most of these structured products provide a capital guarantee 3 . Therefore, they are related to portfolio insurance (PI), which has been extensively used by the …nancial management industry in equities, bonds and hedge funds. One family of such products is devoted to the smoothing of the performance of a given underlying asset over time (typically a …nancial index or a basket of indices). Two main methods are introduced to reach such goal: one is based on the introduction of an Asian call option which yields part of the positive performance of the average of the benchmark asset values along the management period; the other one is based on an average of past positive performances of the benchmark asset. 4 Both types of funds allow the portfolio to be less sensitive to the terminal value of the underlying asset. Indeed, due to their averaging property, they exhibit lower volatilities than their underlying assets and take better account of the whole path of the benchmark asset. However, [START_REF] Bertrand | On path-dependent structured funds: Complexity does not always pay (Asian versus Average performance funds)[END_REF] show that complexity of such products does not always gain advantage over simpler products, when using performance measures such as Kappa ratios (that include for instance both Omega and Sortino ratios). Therefore a natural question arises: Are these funds optimal or not according to main decision criteria ? And, if not, how to measure their inadequacy to customers needs ? For this latter purpose, de Palma and [START_REF] De Palma | Utilitarianism and fairness in portfolio positioning[END_REF][START_REF] De Palma | Standardized versus customized portfolio: a compensating variation approach[END_REF] introduce the compensating variation approach as a quantitative methodology to provide investment recommendations relying both on the performances of the …nancial products and on the investor preferences. For the standard allocation problem on simple assets such as equities, bonds and money market accounts, they show that the customers can su¤er from substantial losses when …nancial institutions o¤er a limited number of standardized portfolios which imperfectly match investor preferences.

The objective of the paper is twofold. First, we prove that standardized path dependent structured products issued by …nancial institutions are not optimal when considering main decision criteria, such as the expected utility maximization and one of its extensions in the portfolio insurance framework, namely the regret/rejoice criterion introduced by [START_REF] Bell | Regret in decision making under uncertainty[END_REF][START_REF] Bell | Risk premiums for decision regret[END_REF] and Loomes andSugden (1982, 1987). For this purpose, we begin by recalling optimization results within the standard concave utility framework (with or without kink in the utility function). Then we investigate also the regret/rejoice criterion for which we provide the optimal portfolio solution. This latter criterion allows to introduce for example the regret of not having chosen a whole investment on the risk free asset if the structured product …nally provides only the guarantee, which is an important issue with portfolio insurance. The non optimality of the standardized path dependent structured products is true even if the …nancial market is driven by a geometric Brownian motion or a usual Markov di¤usion process. Second, using the notion of compensating variations, our numerical results show that the monetary losses resulting from not having access to the optimal structured portfolio can be substantial. This is in particular signi…cant for investors who regret not having invested on the risk free asset when they get a return smaller than the risk-free one (the socalled "cash-lock risk"). This is also an important element when taking account of volatility randomness, what these standardized products do not do.

Our results are in line with previous literature about the standard portfolio allocation problem which also emphasizes the added value when introducing volatility indices exposures in portfolios, including for example VIX call options. [START_REF] Szado | VIX futures and options: A case study of portfolio diversi…cation during the 2008 …nancial crisis[END_REF] emphasizes for instance the diversi…cation bene…ts of such strategy based on VIX for institutional investors (but illustrated only for a static allocation and during the 2008 crisis). Using the mean-variance analysis, Chen et al. (2010) show that introducing short VIX futures exposures into portfolios enhances the e¢cient frontier. Looking at pension funds, Warren (2012) also proves that VIX futures improve the in-sample performance of equity portfolios. Going beyond the standard mean-variance framework which can be unsuitable given the statistical characteristics of the VIX, [START_REF] Bahaji | How rational could VIX investing be[END_REF] illustrates the attractiveness of VIX derivatives for risk-averse investors. In this paper, we show the high interest of VIX instruments for the optimal management of …nancial structured portfolios by measuring for the …rst time the monetary loss of buying standardized portfolios (which do not take account of the volatility randomness) instead of getting the optimal ones. Indeed, our numerical results emphasize that such kind of monetary loss can be higher than the corresponding risk free return on the given portfolio management period. It implies in particular that options written on volatility should be introduced in structured portfolios issued by …nancial institutions. From an operational point of view, they should trade on volatility indices, for example on the volatility index VIX for the S&P 500 index and the VSTOXX indices based on EURO STOXX 50, and/or should introduce options written on them in the structured portfolios.

The paper is organized as follows. Section 2 is devoted to the modelling of the …nancial market when taking account of the stochastic volatility of the risky asset. It provides also the analytical expressions of optimal portfolios corresponding to various assumptions about the expected utility maximization criterion. In Section 3, we examine the optimality of main path-dependent structured products issued by the …nancial institutions (i.e. the standardized funds). We prove that they do not correspond to the optimal solutions. Then, we introduce the notion of compensating variation to compute the monetary losses of not providing the optimal portfolios to the customers. We illustrate numerically our theoretical results by considering various values for both the risk aversion and …nancial parameters. Several technical details are relegated in the Appendix (see also "Supplementary Materials" for additional …gures and tables).

The …nancial market model and portfolio optimization 2.1 The …nancial market

In what follows, we consider a …nancial market modelling as in [START_REF] Pham | Optimal portfolio in partially observed stochastic volatility models[END_REF]. It encompasses main stochastic volatility models, such as the [START_REF] Hull | The pricing of options on assets with stochastic volatilities[END_REF], [START_REF] Heston | A closed-form solution for options with stochastic volatility with applications to bond and currency options[END_REF] and Black, Derman and Toy (1990) models together with the more simple standard Markovian di¤usion case.

Let ( F P) be a complete probability space equipped with a …ltration F = fF   0 •  •  g with  a …xed maturity date and F  = F. Recall that F corresponds to the dynamic available information from the …nancial market. Consider a …nancial market with a risk free asset  with instantaneous rate  and with a risky asset the dynamics of which is de…ned as follows:

  =   h    + (      ) (1)  i  ( 1
)
  =    + (      ) (1)  + (      ) (2)   ( 2
)
where  (1) and  (2) are two independent standard Brownian motions under the historical probability P.

We denote by  0 and  0 the initial values of previous processes. Both processes   and   are supposed to be adapted to the …ltration F. The three deterministic functions (  ), (  ) and (  ) are measurable mappings from

[0  ] £ R £ R into R.
They are assumed to satisfy the standard conditions to ensure the existence and the uniqueness of the solution to the system (1) of stochastic di¤erential equations, namely: 51) H1: The functions

(  ) (  ) and (  ) are Lipschitz in ( ) 2 R + £ R uniformly in  2 [0  ];
2) H2: The functions (  ) and (  ) are non negative on

[0  ] £ R + £ R.
We deduce that the dynamics of the volatility  has the following form:

(      ) = (      ) h  ()   +  () (      ) (1)  +  () (      ) (2)  i  (3)
We assume that the function

 () (  ) is non negative on [0  ] £ R + £ R.

Portfolio optimization

In what follows, we consider that both processes  and  are observable, which is equivalent to the observation of both Brownian motions  (1) and  (2) under the previous assumptions. This yields to a complete …nancial market, meaning that any contingent claim is perfectly hedged from trading on the two basic assets  and . Such assumption is obviously satis…ed when the process  has not to be introduced (i.e. for the standard Markovian case). From the practical point of view, such assumption assumes that the portfolio manager can invest in …nancial products such as the volatility indices. 6We investigate three kinds of decision criterion: The …rst one corresponds to the standard expected utility maximization with a regular and concave utility function; the second one introduces a kink in the utility function to better take account of aversion to the loss relatively to the risk-free investment; …nally, the third one introduces explicitly the regret/rejoice of not having chosen the risk-free investment. 7

Computation of the optimal portfolio for the standard decision criterion

To represent the investor's preferences, consider a utility function  () : R + ! R, which is assumed to be continuous, monotonically increasing and twice di¤erentiable. For the standard decision criterion, the utility  is also supposed to be concave, meaning that the investor is risk averse. Let denote by  the inverse function of the marginal utility, i.e.  = ( 0 ) ¡1 . In what follows, we …rst determine the (unique) risk neutral probability measure Q. We denote by  the riskless interest rate. The probability measure Q is characterized by means of its Radon-Nikodym derivative process (  )  . This latter one is de…ned by:

  = E P • Q P jF  ¸= exp • ¡ 1 2 Z  0 ³  (1)  ´2  ¡ Z  0  (1)   (1)  ¡ 1 2 Z  0 ³  (2)  ´2  ¡ Z  0  (2)   (2)  ¸ (4)
where the premium processes 

(1)

 and 

(2)

 are given by:8

 (1)  =   ¡  (      )
and

 (2)  =  ()  ¡  ¡  () (      ) (1)   () (      )  ( 5
)
Proposition 1 The optimal portfolio value is given by:

 ¤  = (  ) (6)
where the parameter  is determined from the budget constraint, namely:

 ¡ E Q [ ¤  ] =  0 
Proof. We apply results of [START_REF] Cox | Optimal consumption and portfolio policies when asset prices follow a di¤usion process[END_REF].

Remark 2 Looking at Formulas ( 6) and ( 4), the optimal portfolio value is function of both the whole paths of the risky asset process and the volatility process.

Computation of the optimal portfolio with kink

As illustrated in Ben-Akiva et al. (2008), the investor may exhibit a kink in her utility function. For example, instead of having a CRRA utility function, the relative risk aversion may take two di¤erent values. In the portfolio insurance framework, such a kink can occur when the investor compares her portfolio return to the riskless rate. It corresponds to an aversion of getting a return smaller than the risk-free one.

Proposition 3 When there is a kink in the utility function, the optimal portfolio is de…ned from the following equality:

 ¤  =  0¡1 + ( ¤   )  (7)
where  0¡1 + denotes the inverse of the right derivative of the utility function.

Proof. This is a particular case of [START_REF] Cox | Optimal consumption and portfolio policies when asset prices follow a di¤usion process[END_REF] result.

To provide an illustration of such a kink, consider the modi…ed CRRA utility de…ned as follows:

( ) = (  (1¡ 1 ) (1¡ 1 ) if  ¸   (1¡ 2 ) (1¡ 2 ) if     with  = (1¡ 2 ) (1¡ 1 )  ( 2 ¡ 1 ) and either 0   2   1  1 or 1   1   2 .
We get:

 0 + ( ) = ½  ¡ 1 if  ¸  ¡ 2 if     and  0¡1 + () = 8 > > < > > :   ¡ 1  1  if    ¡ 1  if  ¡ 1 •  •  ¡ 2    ¡ 1  2  if    ¡ 2  Corollary 4
For the GBM case, where   has the form   ¡  (see Relation 35), the optimal portfolio value when there is a kink in the CRRA type utility function, is given by:

 ¤  =  0¡1 + ¡  ¤  ¡  ¢ = ¡  ¤  ¡  ¢  ¡ 1  1  I f( ¤  ¡  ) ¡ 1 g + I f ¡ 1 •( ¤  ¡  )• ¡ 2 g + µ  ¤  ¡   ¶  ¡ 1  2  I f( ¤  ¡  ) ¡ 2 g = ( ¤ )  ¡ 1  1     1  I f ( ¤   1 ) 1 g +I f( ¤   2 ) 1 •  •( ¤   1 ) 1 g + µ  ¤   ¶  ¡ 1  2     2  I f ( ¤   2 ) 1 g
For example, consider the GBM case with the following parameter values:

 = 006;  = 02;  = 8 ;  0 = 100;  0 = 100;  =  0   ;  1 = 16;  2 = 24
Figure 1 illustrates the optimal portfolio value as function of the risky asset value . As expected, the payo¤ is equal to the constant

 =  0   on the interval [( ¤   2 ) 1  ( ¤   1 ) 1 ].

Computation of the optimal portfolio within the regret/rejoice framework

To go further with the notion of aversion of getting a return smaller than the risk-free one, we can base our analysis on the regret/rejoice theory. The regret/rejoice criterion Among generalizations of the standard maximization of expected concave utility functions, a well-known approach is the cumulative prospect theory with the notion of loss aversion (see [START_REF] Tversky | Advances in prospect theory: cumulative representation of uncertainty[END_REF]. However, in the portfolio insurance framework, loss (after deduction of management fees) cannot exist, implying that loss aversion cannot be applied in such a context. But a well-known problem arises: Does the insured portfolio provide higher bene…ts than a straightforward investment on the risk free asset? 9 Similarly, the investor can regret not having su¢ciently investing on the risky asset when the …nancial market rises. To model such attitude towards risk and return, we can involve the regret aversion as suggested by [START_REF] Bell | Regret in decision making under uncertainty[END_REF] and further developed by [START_REF] Loomes | Some implications of a more general form of regret theory[END_REF] who emphasize that "one signi…cant factor is an individual's capacity to anticipate feelings of regret and rejoicing." 10 Regret is also often observed when examining …nancial investment decisions. Using regret theory framework, Michenaud and Solnik (2005) examine usual …nancial problems such as international portfolio optimization. They analyze in particular the home bias equity, which corresponds to weak international diversi…cation for investors having strong preference for domestic …nancial assets. 11 The regret theory supposes that individuals are «rational » but that their decisions are not only based on potential bene…ts (as for EU criterion) but also on expected regret with respect to other possible outcomes. Therefore, it includes two types of aversions: aversion to risk and aversion to regret.

The regret theory involves the regret or rejoice that an individual can feel when she gets outcome  instead of outcome . The di¤erence between these two outcomes is a measure of the rejoice or regret ( ) de…ned by: ( ) = () ¡ () (8) 9 Bertrand and Prigent (2015 b) shows that the probability that the fund becomes monetized (known as "the cash-lock") is null for a structured fund based on the average of the past performances of the risky underlying asset while it can be signi…cant for the OBPI or Asian type funds.

1 0 This theory succeeds in explaining paradoxes such as the fanning out and the preference reversal e¤ects (see e.g. [START_REF] Seidl | Preference reversal[END_REF]. A large literature has been devoted to experimental psychology while a recent literature in neurobiology shows that regret has an impact on decision making under uncertainty and is not su¢ciently taken into account by usual approaches (see [START_REF] Gilovich | The experience of regret: what, when, and why[END_REF][START_REF] Zeelenberg | Anticipated regret, expected feedback and behavioral decision making[END_REF][START_REF] Bleichrodt | Regret theory: A bold alternative to the alternatives[END_REF]. 1 1 As mentioned by Solnik (2007), the regret in investment choices leads to simple rules such as formulated by Harry Markowitz: "I should have computed the historical covariance of the asset classes and drawn an e¢cient frontier. Instead I visualized my grief if the stock market went way up and I wasn't in it-or if it went way down and I was completely in it. My intention was to minimize my future regret, so I split my [pension scheme] contributions 50/50 between bonds and equities."(Harry Markowitz, as quoted in Zweig (1998).).

which is negative if the individual feels regret and positive if she feels rejoice. More generally, Loomes and Sugden (1982) and [START_REF] Bell | Regret in decision making under uncertainty[END_REF] introduce the following modi…ed version of the utility of  in the presence of :

V( ) = () +  [() ¡ ()]  (9)
with  [0] = 0,  increasing, concave and  000  0.

Optimal portfolio within regret/rejoice utility In what follows, the utility  of the investor is supposed to be increasing and piecewise di¤erentiable. The regret function  satis…es:  [0] = 0,  is increasing,  00 concave and  000  0. We assume that the marginal utility  0 is invertible, and we denote its inverse by . When the investor may regret or rejoice of not having received the reference terminal wealth e    the optimal payo¤  ¤  is solution of the following problem:

   E P [(  ) +  ³  (  ) ¡  ³ e   ´´] with  0 =  ¡ E P [    ]
Suppose that the function ©  [] de…ned by:

©  [] =  0 [] ¡ 1 +  0 [ [] ¡  []] ¢ 
has an inverse for any value of . Under previous assumptions on utility  and regret function , the function ©  is strictly decreasing.

Proposition 5

The optimal portfolio in the regret/rejoice framework is given by

 ¤  = ©    [  ]  ( 10
)
where  corresponds to the Lagrange multiplier associated to the budget constraint.

Proof. See Appendix 2 in Supplementary Materials 1.

When the terminal wealth e   is a function of the terminal value of the risky asset price, namely e   =  0 (  ), we introduce the function © 0 which is de…ned by:

©  0 () () =  0 () £ 1 +  0 [ () ¡  ( 0 ())] ¤ 
It is invertible as function of .

Corollary 6 In the GBM framework, we deduce that the optimal portfolio value is equal to the payo¤  ¤  (  ) given by:

 ¤  = © ¡1  0 () (  ) ( 11
)
where   is the scalar Lagrange multiplier such that:

 0   = Z R + © ¡1 0() (  ())()   () Example 7
Consider the following regret function, as suggested in [START_REF] Bell | Risk premiums for decision regret[END_REF]:

() = 1 ¡ exp[¡]   with   0 Then, for a CRRA utility () =  1¡ (1 ¡ ) we have: © 0() () =  ¡ µ 1 + exp µ ¡ •  1¡ 1¡ ¡  0 () 1¡ 1¡ ¸ ¶ ¶ 
For example, consider the GBM case with the following parameter values:

 = 006;  = 02;  = 8 ;  0 = 100;  0 = 100;  = 2;  = 20 (12)
Figure [START_REF] Arrow | The role of securities in the optimal allocation of risk-bearing. Econometrie; as translated and reprinted in 1964[END_REF] illustrates the optimal portfolio value as function of the risky asset value . As expected, since we have considered here a rather high regret aversion, the optimal payo¤ for the regret/rejoice utility is much more close to the payo¤ corresponding to the risk free investment (horizontal line in the …gure). Proposition 8 The optimal insured portfolio corresponds to the maximum of a portfolio value without guarantee constraint and the guarantee itself. When this guarantee is equal to the initial portfolio value, we get:

 ¤¤  =  [(  )  0 ]  ( 13
)
Remark 9 We note that the previous optimal solution can be viewed as a combination of an investment on the risk free asset and on a call option written on a portfolio value without guarantee constraint since we have:

 ¤¤  =  0 +  [(  ) ¡  0  0]  (14)
Relation ( 13) is true for all cases that we investigate in the paper. In Section 3, this optimal solution is compared with the standardized …nancial products.

Optimality of standardized path-dependent structured products

In this section, we examine whether or not the standardized structured products issued by …nancial institutions to smooth risky market ‡uctuations are optimal with respect to usual decision criteria. we introduce the notion of compensating variation to determine the monetary losses of the customers when …nancial institutions do not provide their optimal portfolios. We examine two main types of such products: the …rst one corresponds to a combination of a riskless investment (providing the guarantee) with an Asian call option which yields part of the positive performance of the average of the benchmark asset values along the management period; the second one is based on an average of past positive performances of the benchmark asset. We also consider the most popular insured portfolio, namely the portfolio corresponding to the Option Based Portfolio Insurance (OBPI), introduced by Leland and Rubinstein (1976). 12 The portfolio is invested in a risky benchmark asset  covered by a listed put written on it. The strike  of the put is equal to a predetermined proportion of the initial investment. This amount corresponds to the capital that is insured at maturity whatever the value of  at the terminal date  . 13 3.1 The standardized structured products

The OBPI portfolio

The OBPI method consists basically in purchasing an amount invested on the money market account and  shares of European call options written on asset  with maturity  and exercise price , usually the initial one  0 . The payo¤ is de…ned as follows. At maturity, the payo¤ of this fund is de…ned by: 14

  =  0 +    0 (  ¡  0 ) +  ( 15
)
We denote by  (  0   ) the price of each call option with maturity  , strike , underlying , and riskless interest rate . We deduce that, at time 0, the budget constraint is given by:

 0 =  0  ¡ +    0  (  0   )  ( 16
)
In what follows, we set  =  0 . The parameter   is chosen such that the initial price  0 solves equation [START_REF] Buhlmann | No-arbitrage, change of measure and conditional Esscher transforms[END_REF]:

  = ¡ 1 ¡  ¡ ¢  (  0   0  )  (17)
Therefore, the return of the OBPI strategy over the period [0  ] is given by:

   0 = 1 + ¡ 1 ¡  ¡ ¢ (  ¡  0 ) +  (  0   0  )  3.1.

Guaranteed funds based on average of positive performances

This …nancial structured product is based on the average of past positive performances with respect to a given underlying asset value, usually the initial one  0 . Usually, the performances (monthly, quarterly or half-yearly) are de…ned by averages of calls with strike equal to the initial value  0 of the underlying risky asset. In what follows,  corresponds to the number of dates per year (for example;  = 12 4 or 2),  is the time horizon (number of years) and    denotes the value of the risky asset at time   = . The payo¤ of a guaranteed fund based on average (excess) performances is de…ned as follows. At maturity, the payo¤ of this fund is de…ned by:

  =  0 +    0 1  X  =1 (   ¡  0 ) +  (18)
Therefore, only the    values of the underlying index which are higher than the initial  0 value are taken into account in the formula.

We denote by  (     ) the price of each call option with maturity   , strike , underlying , and riskless interest rate . We deduce that, at time 0, the budget constraint is given by:

 0 =  0  ¡ +    0 1  X  =1  ¡( ¡  )  (     0  )  (19)
Thereafter, we denote:

  (   0  ) = 1  X  =1  ¡( ¡  )  (     0  )
The parameter   is chosen such that the initial price  0 solves equation [START_REF] Chen | An analysis of market-index certi…cates of deposit[END_REF]:

  = ¡ 1 ¡  ¡ ¢   (   0  )  ( 20
)
The return of the average strategy over the period [0  ] is given by:

   0 = 1 + ¡ 1 ¡  ¡ ¢ 1  P  =1 (   ¡  0 ) +   (   0  ) 

Guaranteed funds based on Asian options

Asian options were introduced to take account of the average of the underlying asset values, unlike the European-type options linked to the standard OBPI method which only use value at maturity. It is not surprising, therefore, that they are being used with …nancial structured funds, particularly insured funds.

A guaranteed fund based on discrete-time Asian options delivers a payo¤ de…ned by:

  =  0 +    0 [ ( ) ¡  0 ] +  ( 21
)
with [START_REF] Cox | Optimal consumption and portfolio policies when asset prices follow a di¤usion process[END_REF] represents the payo¤ of a discrete arithmetic Asian Call on the index with a strike price equal to the initial price  0 of the index. We denote its price by   (  ( )   0  ) where  corresponds to the riskless interest rate. Therefore, at time 0, the budget constraint is given by:

 ( ) = 1  P  =1    . The term [ ( ) ¡  0 ] + in expression
 0 =  0  ¡ +    0   (  ( )   0  )  ( 22
)
from which we deduce that the parameter   must be chosen such that:

  = ¡ 1 ¡  ¡ ¢   (  ( )   0  )  (23)
Note that the return of the strategy over the period [0  ] is also de…ned by:

   0 = 1 + ¡ 1 ¡  ¡ ¢ [ ( ) ¡  0 ] +   (  ( )   0  )  (24)

Comparison with optimal portfolios

In what follows, we examine the optimality of the previous standardized structured products. As it can be immediately deduced, these products can only be optimal in the case of a Markovian di¤usion, since they do not involve explicitly the random paths of the volatility. To make the comparison between standardized structured products and optimal portfolios, we …rst detail these ones in what follows.

Optimal portfolio in the Markovian di¤usion framework

This is a particular case of the general model ( 1) where both functions (  ) and (  ) do not depend on  . The stock index price dynamics is given by the following stochastic process:

  =   [(   ) + (   )  ] (25)
where ( ) and ( ) are deterministic functions that satisfy the usual assumptions and  denotes a standard Brownian motion with respect to a given …ltration (F  )  . Then, the risky asset price  is a Markovian di¤usion process. One particular case is known as the CEV model for which the volatility is given by: (  ) =  1¡ .

It implies:

  =  0 exp •Z  0 µ (   ) ¡ 1 2  2 (   ) ¶  + Z  0 (   )  ¸ (26)
Under the (unique) risk neutral probability measure Q, we have:

  =  0 exp •Z  0 µ  ¡ 1 2  2 (   ) ¶  + Z  0 (   ) f   ¸ (27)
where  denotes the riskless interest rate and process f  is a Brownian motion with respect to Q. Note that :

  = E P • Q P jF  ¸= exp " ¡ 1 2 Z  0 µ (   ) ¡  (   ) ¶ 2  ¡ Z  0 (   ) ¡  (   )   # 
Assuming ( )  0, we get:

  =     ¡ (   ) (   )  (28)
Therefore, we have:

Proposition 10 For the Markovian case, the optimal portfolio value is given by:

 ¤  = (  ) (29)
with

  = exp •Z  0 • 1 2 µ (   ) ¡  (   ) ¶ µ (   ) +  (   ) ¶¸  ¡ µZ  0 (   ) ¡     2 (   ) ¶  

¸

As emphasized previously, the optimality of the main standardized path dependent structured products issued by …nancial institutions has a chance to be satis…ed only in the Markovian framework. It is the reason why we detail this case under various assumptions on the drift and volatility functions.

Consider two fundamental examples of utility functions, namely the Constant Relative Risk Aversion (CRRA) case and the Constant Absolute Risk Aversion (CARA) case.

1) Assume that the utility function is a power function

() =  1¡
1¡ with a constant relative risk aversion   0 and  6 = 1. We have:  0 () =  ¡ and () =  ¡1  . Thus the optimal portfolio value is given by:

 ¤  =  ¡1  exp • ¡1  Z  0 • 1 2 µ (   ) ¡  (   ) ¶ µ (   ) +  (   ) ¶¸  + 1  Z  0 µ (   ) ¡     2 (   ) ¶   ¸(30)
2) Assume that the utility function is an exponential () =  ¡ (¡) for  ¸0 with a constant absolute risk aversion   0. We have:

 0 () =  ¡ and () = ¡ 1  ().
Thus the optimal portfolio value is given by:

 ¤  = ¡ 1  • () + Z  0 • 1 2 µ (   ) ¡  (   ) ¶ µ (   ) +  (   ) ¶¸  ¡ Z  0 µ (   ) ¡     2 (   ) ¶   ¸(31)
Optimal portfolio under speci…c assumptions on the modi…ed Sharpe ratio In the Markovian framework, usual discussions are based on the following modi…ed Sharpe ratio ()¡  2 () . We investigate two main cases: the …rst one assumes that this modi…ed Sharpe ratio is constant; the second one supposes that this ratio divided by the risky asset price is constant (i.e. ()¡    2 (  ) ). We examine what happens for the two standard utility functions.

Assume now that the modi…ed Sharpe ratio is equal to a constant e , which is a common assumption in the theoretical literature about portfolio choice. It implies that the Sharpe ratio ()¡ () is increasing with the volatility.

Corollary 11 When the modi…ed Sharpe ratio is constant, the optimal portfolio value for the CRRA case is given by:

 ¤  =  ¡1  exp • ¡1  Z  0 • 1 2 e  ((   ) + ) ¸ ¸exp • e  2 Z  0  2 (   ) ¸£      
For the CARA case, it is given by:

 ¤  = ¡ 1  • () + Z  0 • 1 2 e  ((   ) + ) ¸ ¸+ e   µ ln •    0 ¸+ 1 2 Z  0  2 (   ) ¶  Proof.
For the CRRA case, applying Relation (30), we get:

 ¤  =  ¡1  exp • ¡1  Z  0 • 1 2 e  ((   ) + ) ¸ + e   Z  0    
¸ from which we deduce the result. For the CARA case, using Relation (31), we get:

 ¤  = (  ) = ¡ 1  • () + Z  0 • 1 2 e  ((   ) + ) ¸ ¸+ e   Z  0    
Note that we have:

Z  0     = Z  0 (   ) + (   )  = Z  0 1 2  2 (   ) £ •Z  0 µ (   ) ¡ 1 2  2 (   ) ¶  + Z  0 (   )  = exp •Z  0 1 2  2 (   ) ¸£ ln µ    0 ¶ 
Therefore we deduce the result. We note that, for the CRRA case, the portfolio value is a power of the risky asset value, weighted by exponentials of stochastic integrals involving respectively the drift and volatility of the risky asset. For the CARA case, the portfolio value is a linear combination of the risky asset logreturn and of temporal means of respectively the drift and volatility of the risky asset.

Assume now that the ratio ()¡  2 () = b  is constant with  6 = . We get:

Corollary 12 When the ratio ()¡  2 () is constant, the optimal portfolio value for the CRRA case is given by:

 ¤  =  ¡1  exp • ¡1  Z  0 • 1 2 b   ((   ) + ) ¸ + b   (  ¡  0 )
¸ For the CARA case, it is given by:

 ¤  = ¡ 1  • () + Z  0 • 1 2 b   ((   ) + ) ¸ + b  0 ¸+ b      Additionally, if (   ) ´ constant (thus (   ) = q ¡ 
), then we deduce:

 ¤  = ¡ 1  () ¡ b    0 ¡ b  ( + ) 2 Z  0    + b     
We note that, for the CRRA case, the portfolio log return is a linear combination of the risky asset value at maturity and of the temporal mean of its current values weighted by the sum of its drift with the risk free asset. For the CARA case, the portfolio value is a linear combination of the risky asset terminal value and of the temporal mean of the risky asset values.

Finally, note that the Markovian di¤usion case with constant Sharpe ratio encompasses the usual geometric Brownian motion (GBM) case. 15 Due to its fundamental applications, we detail this case in what follows.

Optimal portfolio for the basic example (GBM case) Consider the risky asset dynamics de…ned by:

    =  +    (32)
with constant parameters  and  (  0). Denote:

 =  ¡ 1 2  2 ;  =  ¡   (Sharpe ratio) ( 33
)
 = ¡ 1 2  2  +    ;  =   ;  =   ( 0 )  
We consider the risk-neutral probability measure Q to price options. The ¡algebra F is generated by the Brownian motion  . We deduce that the probability density function (pdf) of Q  is a function (  ) of the terminal value of the risky asset price with respect to the -algebra generated by  . This function is given by:

() =  ¡ 
Therefore, from Relation (6), we deduce that the optimal portfolio is equal to:

 ¤ (  ) =  £    ¡  ¤  ( 34
)
where  is the Lagrange parameter associated to the budget constraint.

The CRRA case Corollary 13 For the GBM case, the optimal solution for the CRRA case is given by:

 ¤ (  ) =  ¡ 1       ( 35
)
where the power   of   is equal to the Sharpe type ratio 16  = ¡  2 times the inverse of the relative risk aversion . The ratio   corresponds to the Merton ratio. Applying budget constraint, the coe¢cient  is equal to:

 =  0    • ¡  ¡  ¢ ¡1  ¸with  • ¡  ¡  ¢ ¡1  ¸= exp µ 1 2  2  1 ¡   2 ¶  (36)
Therefore, the optimal portfolio is a power of the terminal risky asset value. Note that  ¤ (  ) =  ¤ (  ) is increasing. This property is satis…ed for all concave utilities, as soon as the density  is decreasing, for instance within the Black-Scholes asset pricing framework. The concavity/convexity of the optimal payo¤ is determined by the comparison between the relative risk-aversion  and the ratio  = ¡  2 , which is the Sharpe ratio divided by the volatility  17 :

i)  ¤ is concave if   ; ii)  ¤ is linear if  = ; iii)  ¤ is convex if   .
To illustrate the previous result, consider the following numerical base case:

 = 3%  = 20%  0 = 1  0 = 100
The initial investment is  0 = 100 and the time horizon  is equal to 1 (one year). Figure 3 illustrates how the optimal portfolio pro…le depends on the expected instantaneous rate of return  of the risky asset, which has a straightforward impact on the Sharpe type ratio . 18 When  is smaller than , the optimal payo¤ is a decreasing function of the risky asset value   , which leads in practice to buy put options on ; for  higher than , the optimal is an increasing function of the risky asset value   which is concave for moderate values of  then convex for higher values. Indeed, when the expected instantaneous rate of return  of the risky asset is su¢ciently high, the investor searches to better bene…t from potential market rises. The CARA case Corollary 14 For the GBM case, the optimal solution for the CARA case is given by:

 ¤  = ¡ 1  () ¡    0 ¡  ( + ) 2 Z  0    +     
Therefore, the optimal portfolio is a combination of the terminal risky asset value and of the average of its values over the time period.

Optimality of the OBPI

The standard OBPI fund with no cap can be optimal but under very strong assumptions (GBM case and speci…c parameter values). Using previous results (see Corollary 13 and Proposition 8), we get:

  =  0 + (   ¡  0 ) +  with  constant. ( 37
)
For  = 1, we get the OBPI fund. 19 But recall that this fund is not used to smooth the volatility along the management period.

Optimality of the guaranteed fund based on average of positive performances

To examine the optimality of the guaranteed fund based on average of positive performances, we can introduce its continuous time version, namely:

    0 = 1 + ¡ 1 ¡  ¡ ¢ R  0 (  ¡  0 ) +    (   0  ) 
Let us examine the optimality of such a product. 20 Due to terms (  ¡  0 ) + or equivalently (  ¡  0 ) + in continuous time, this portfolio cannot be optimal when compared with previous solutions for the Markovian di¤usion case. First, the volatility would be constant; second it would be assumed that the investor searches a portfolio optimization for each time subperiod. For example, assume that the investor maximizes her expected utility on each subperiod [ ¡1    ]. Assume also that she wants a guarantee on each subperiod (i.e.    ¸0 ).

Then, the optimization problem is the following:

 E [  (  )] under    ¸0  with   () =  (1¡) (1 ¡ ).
Using previous results (see Corollary 13 and Proposition 8), we get:

   =  0 +  0 (  ¡      ¡1 ¢  ¡ 1) + 
If the investor splits her initial investment into  equal amounts  0 for all the subperiods with condition (i.e.  0 =  0 ), then we have:

  =  0  +  0 (  ¡      ¡1 ¢  ¡ 1) +  1 9
Consider the HARA utility case:

() = ( ¡  ¤ ) (1¡)  (1 ¡ ) 
where  ¤ corresponds to the initial invested wealth 0. The optimal portfolio has the following form:

 = 0 +      with   constant.
It corresponds to another portfolio insurance method, namely the Constant Proportion Portfolio Insurance (CPPI) method. As opposed to the Option Based Portfolio Insurance (OBPI) strategy, the CPPI portfolio payo¤ does not truly include an option whereas, for the OBPI method, the strike corresponds to the initially invested amount which insures the capital at maturity). In the paper, we only consider structured products which provide capital protection while allowing investors to bene…t from market rises through more or less sophisticated derivatives written on a risky asset. The payo¤ at maturity is characterized by a given formula that de…nes the optional component.

2 0 In the GBM framework, Shackleton and Wojakowski (2007) provides the explicit pricing formula of   (  0 ).

Thus, the global portfolio value at maturity is given by:

  =  X =1   =  0 " 1 + 1   X =1 (  ¡      ¡1 ¢  ¡ 1) + # 
Thus, the portfolio value at maturity is a sum of call options written on the return of the underlying asset . However there are several shortcomings in order this fund to be truly optimal: …rst, this fund does not correspond to returns     0 ; second, we must have  = 1, which is a very speci…c case; …nally, the decision criterion is only "local" (i.e. optimization only on each subperiod).

Optimality of the Asian product

To examine the optimality of the Asian product, we can also introduce its continuous-time version, namely:

    0 = 1 + ¡ 1 ¡  ¡ ¢ h 1  R  0    ¡  0 i +   (   0  ) 
Therefore, this structured fund has no component de…ned on the volatility process itself. Again, for this Asian fund to be optimal, the …nancial market must evolve according to the Markov di¤usion process described previously.

Note that, for the computation of the current value of

A  = 1  R  0   ,
we can use the standard no-arbitrage valuation. Therefore, the current value of A  is equal to:

 ¡( ¡) E Q [A  jF  ] =  ¡( ¡) E Q • 1  Z  0    jF  ¸ Since we have: for  ¸   =  ¡(¡) E Q [  jF  ],
we deduce:

 ¡( ¡) E Q [A  jF  ] = 1   ¡( ¡) µZ  0    +   Z    (¡)  ¶  = 1   ¡( ¡) µZ  0    +   h³  ( ¡) ¡ 1 ´ i ¶  Let () =  ± ().
We search a function  such that

 ¤  =  ± (  ) = A   This is equivalent to: (ln  + ln   ) = A  
which gives:

 µ ln  + Z  0 • 1 2 µ (   ) ¡  (   ) ¶ µ (   ) +  (   ) ¶¸  ¡ µZ  0 (   ) ¡     2 (   ) ¶   ¶ = 1  Z  0    Denote ()¡  2 () = (   ).  µ ln  + Z  0 • 1 2 (   )  ((   ) + ) ¸ ¡ µZ  0 (   ) ¶   ¶ = 1  Z  0   
The only possible case corresponds to the CARA case (i.e.  is an exponential function). Looking at Corollary [START_REF] Bernard | Explicit representation of cost-e¢cient strategies[END_REF], the temporal mean 1  R  0    appears in the fund's formula. However, the terminal payo¤   is also involved.

To avoid such problem, we can also rely on the intertemporal consumption framework since the Asian product is based on temporal mean of the risky asset price. In the GBM framework, in order to get such products in the optimal portfolio, we can also introduce a consumption based model. This leads to the following expected utility:

E •Z  0 b (   ) + (  ) ¸ ( 38
)
Proposition 15 Using the Hamilton-Jacobi-Bellman approach (see Karatzas et al., 1985), we deduce the optimal consumption and portfolio values.

 ¤  =  (  ) and  ¤  = b  (  ) 
Remark 16 Consider a particular case (see Prigent, 2007, pages 198-202) for which we assume that

 = 0 b () =  1¡
1 ¡  with   0 and  6 = 1 and …nally the Merton ratio

1  (   ) ¡  (   ) 2 = 1
Then, there exists only a consumption process of type  ¤  = ()  where () is the deterministic function of time  given by: 21

() =  ¡   with   = ¡ 1 2  2  +    ( 39
)

Thus the cumulated consumption process is based on the temporal mean of the risky asset values weighted by a deterministic function, namely

R  0  ¤   = R  0 ()  .
Looking at the previous optimal solution, for the Asian option to be optimal, …rst we must require a Merton ratio 1  ()¡ () 2 equal to 1. In that case, the cumulated consumption process is based on the temporal mean of the risky asset values weighted by a deterministic function, namely

R  0  ¤   = R  0 ()  .
Second, to get exactly the Asian payo¤, the deterministic function () must be constant which is equivalent to (   ) ¡  = 0. Obviously, the two previous conditions cannot be simultaneously satis…ed.

Compensating variation (monetary loss)

As shown in Bertrand and Prigent (2015b), when using Kappa performance measures (including both Omega and Sortino ratios), Asian funds are preferred to Average fund for risk-neutral investors, since the expectation of the Asian fund return is higher than the expectation of the Average fund return. But, as soon as potential risk aversion or implicitly loss aversion are taken into account, the ranking is reversed.

In what follows, our goal is to go further into this comparison by …rst introducing various risk aversions and second to compare standardized funds with optimal ones. For this purpose, our approach relies on the notion of compensating variation.

Compensating variation

Several …nancial institutions have understood the importance to evaluate the investor's risk aversion although there are di¢culties to rigorously link those measures to investment recommendations. To the best of our knowledge, one of the reasons is that those evaluations often do not provide a quantitative evaluation of investor risk aversion. Moreover, these approaches fail to describe the dynamics of risk aversion over the investor's time horizon. An exception is provided by Ben-Akiva, de Palma and Bolduc (2002) who propose an econometric method to measure quantitatively the investors' risk aversion. Typically …nancial institutions o¤er a limited number of standardized portfolios which imperfectly match investor preferences. In this paper, we quantify the e¢ciency losses of an investor acquiring given standardized structured funds instead of getting the portfolio corresponding to her own attitude towards risk. The compensating variation approach, introduced by Hicks (1939) in economics and by De Palma and Prigent (2008,2009) in …nance, allows to quantitatively measure the monetary loss of not receiving her own optimal portfolio.

Consider an investor with utility function parametrized by  and time horizon  . Denote by  ¤  her optimal portfolio and by   the standardized structured product sold by the …nancial institution. Since this latter one is generally not optimal, we search the initial value b  0 necessary to reach the same utility level. Such condition leads to the following indi¤erence condition:

[  ( ¤  ;  0 )] = [  (  ; b  0 )] (40)
The ratio b  0  0 , called the compensating variation, provides a quantitative (monetary) measure of the lack of adequacy of the standardized structured product. 22

Compensating variation of the standardized path-dependent structured products

In what follows, we provide numerical examples of the compensating variations for various cases. 23 First, we focus on the standard GBM framework, for which we consider several extensions of the standard concave utility case. Second, assuming that the investor has a standard concave utility, we study the impact of the ‡uctuations of the stochastic volatility. For this purpose, we choose the Log Ornstein model, which is one of the main modelling of stochastic volatility.

Compensating variation for the concave utility in the GBM framework For the riskless rate, we choose two main values, namely  = 1% (low rate) and  = 3% (standard rate). Since the compensating variation is increasing with respect to , we get its bounds for  2 [1% 3%], in particular for  = 2%. For both these cases, we vary the volatility level of the risky asset (here, it corresponds to an equity index). For this latter purpose, we choose four main values of the volatility, namely  = 15% (low volatility),  = 20% (moderate volatility),  = 25% (rather high volatility) and …nally  = 30% (high volatility). Finally, we choose two main values for the drift  such that the risk premium ( ¡ ) is equal to its usual value namely 5%. 24 In what follows, our numerical base cases correspond to the following …nancial parameters: -First, we can note that the three standardized products, namely the OBPI, the Average and the Asian funds are rather close from the compensating variation point of view. Indeed, looking at Tables 1 to 4, the maximum compensating variation is equal to 10327 (OBPI versus Average,  = 1  = 6%,  = 15% and  = 1%). Since the fund horizon is equal to 8 years for this example, it corresponds to a monetary loss per year equal to about 04%; -Second, we can notice that, as volatility increases, the OBPI is less dominant compared to the two other structured funds. For example, the Asian strategy is never dominant for  = 15% 20% and becomes dominant only from  = 25% and generally for investors having moderate or high risk aversions ( ¸4). It means that, according to their objective of smoothing the return of the risky asset, both Asian and Average funds are improving with respect to the standard OBPI fund when the volatility increases. Thus, as volatility increases, the performance smoothing feature of the path dependent strategies goes into action. The Average fund is generally dominant when compared to the Asian fund, especially when the volatility increases or/and the risk aversion increases. As expected, it provides a higher protection against volatility ‡uctuations for investors having moderate or high risk aversions ( ¸4).

( = 6%  = 1%) ; ( = 8%  = 3%) ;  = 8;  0 = 100;  0 = 100
-Third, the compensating variation between those standardized structured products and the optimal fund is very signi…cant, compared to the compensating variations between only standardized structured products. For the compensating variation between the OBPI fund and the optimal one, it can reach for example 1176 for  = 10,  = 25% and  = 3% meaning a monetary loss per year equal to about 224% (see Tables 5 to 8). This would induce an implicit managing cost higher than 2% per year! However, for both interest rate levels, namely  = 1% and  = 3%, the maximum compensating variation is always smaller than the return of the risk free asset on the same time period. Such feature illustrates the lack of adequacy to customers needs, especially for both "aggressive investors" ( = 1) in a moderate ( = 20%) or low ( = 15%) volatility environment and for "conservative" investors ( = 10); -Fourth, the behavior of the compensating variation as a function of risk aversion and volatility levels is rather complex. It depends on all the …nancial parameter values: For …xed …nancial parameter values, the compensating variation is a function of the risk aversion  which can be either …rst decreasing then increasing or always increasing. Indeed, as it can be seen in Relation [START_REF] Loomes | Regret theory: an alternative theory of rational choice under uncertainty[END_REF], the compensating variation is null when the risk aversion  is such that the OBPI fund is optimal ( =    ). Thus, as soon as the distance between  and    increases, the compensating variation does also. As it can be seen in Appendices 3 and 4 (see Supplementary Materials 1), for a simpli…ed case with no insurance constraint, for …xed risk aversion parameter , the compensating variation between the optimal fund and an optimal fund related to the OBPI fund is increasing with the volatility if  is su¢ciently high. Such property is rather robust when dealing with portfolio insurance. Indeed, Relation 13 characterizes the optimal insured portfolio as the maximum between an optimal portfolio without insurance constraint and, for example, a …xed guaranteed amount. Looking at Figures [START_REF] Bell | Regret in decision making under uncertainty[END_REF] and ( 5), (see also Figures [START_REF] Bertrand | Portfolio insurance strategies: OBPI versus CPPI[END_REF] to [START_REF] Chen | Pricing the SPIN[END_REF] in Supplementary Materials 2), we can notice such feature. Indeed, when  = 15%, the compensating variation between the optimal fund and the three standardized funds is higher for small values of the risk aversion  than for the three other values of the volatility, namely  = 20% 25% 30%. For moderate or high risk aversion , it is the converse: for …xed  ¸5, the higher the volatility, the higher the compensating variation.

-Finally, as function of time horizon, the compensating variation is increasing, as illustrated by Figures [START_REF] Cox | Optimal consumption and portfolio policies when asset prices follow a di¤usion process[END_REF], [START_REF] Das | Options and structured products in behavioral portfolios[END_REF] and [START_REF] De Palma | Utilitarianism and fairness in portfolio positioning[END_REF] (see Supplementary Materials 2). Finally, note that the choice of the utility function does not change very signi…cantly the numerical values of the compensating variation (as explained in Supplementary Materials 2).

Compensating variation for the concave utility with kink case in the GBM framework In this case, the dynamic of the risky asset process is given by a geometric Brownian motion (  =   [ +   ]). The utility with kink is given by:

( ) = (  (1¡ 1 ) (1¡ 1 ) if  ¸   (1¡ 2 ) (1¡ 2 ) if     with  = (1 ¡  2 ) (1 ¡  1 )  ( 2 ¡ 1 ) 
Looking at Tables [START_REF] Bertrand | French retail …nancial structured products: A typology and assessment of their fair pricing[END_REF] to [START_REF] Chen | The diversi…cation e¤ects of volatility-related assets[END_REF] in Supplementary Materials 3, we deduce similar results as previously when comparing the three standardized products between them and with the optimal fund. We still observe that higher volatility generally implies higher compensating variations for moderate or high risk aversion . These latter ones are more sensitive to the volatility than to the threshold determining the kink around the riskless return. Most of the time, the compensating variation is decreasing with respect to the drift of the risky asset. We can also examine the compensating variation as function of the kink level. Recall that we choose  = 075 exp[ ],  = 075 exp[ ] and …nally  = 125 exp [ ].

For the case  = 1%, we compare results for the standard concave case with  = 6% and  = 20% (see Table 2) to those in Tables [START_REF] Bertrand | French retail …nancial structured products: A typology and assessment of their fair pricing[END_REF] to [START_REF] Bernard | Explicit representation of cost-e¢cient strategies[END_REF] corresponding to the kink case. We use the following approximate correspondence between the standard concave and the kink cases: For  = 2,  1 = 16 and  2 = 24 for the utility with kink; for  = 6,  1 = 52 and  2 = 68 for the utility with kink; for  = 10,  1 = 85 and  2 = 115 for the utility with kink. For  = 2, the compensating variations for the standard concave case are higher than for the kink case, while for both  = 6 and  = 10, it is generally the converse. Most of the time, the compensating variation is slightly increasing with respect to level of the kink  but the di¤erence is relatively weak.

For the case  = 3%, we compare results for the standard concave case with  = 6% and  = 20% (see Table 6) to those in Tables [START_REF] Bleichrodt | Regret theory: A bold alternative to the alternatives[END_REF] to [START_REF] Chen | The diversi…cation e¤ects of volatility-related assets[END_REF] corresponding to the kink case. Most of the time, the compensating variations for the kink case is very close to that for the standard utility case. We note also that, for  = 6 and  = 10, the compensating variation is not sensitive to the level of the kink (at least for the three considered values, namely  = 075 exp[ ],  = 075 exp[ ] and  = 125 exp[ ]).

Therefore, to better investigate the impact of the aversion of getting a return smaller than the risk-free one, we analyze numerically the regret/rejoice criterion in what follows.

Compensating variation for the regret/rejoice case in the GBM framework To illustrate the compensating variation for the regret/rejoice case, we choose the regret function , as suggested in [START_REF] Bell | Risk premiums for decision regret[END_REF], namely:

 () = 1 ¡ exp[¡]   with   0
Then, we choose a CRRA utility () =  1¡ (1 ¡ ). Thus, we get:

© 0() () =  ¡ µ 1 + exp µ ¡ •  1¡ 1¡ ¡  0 () 1¡ 1¡ ¸ ¶ ¶ 
Applying results of Proposition 5, we can numerically analyze the impact of both the relative risk aversion  and the parameter  de…ning the regret/rejoice function. For this purpose, we set the following …nancial parameters: The drift  of the risky asset is equal to 6%; the volatility  of the risky asset is equal to 20%; the interest rate is equal to either to 1% or to 3%. We consider also a time horizon  equal to 8 years. To calibrate the parameter  associated to the regret function, we use results in [START_REF] Michenaud | Applying regret theory to investment choices: currency hedging decisions[END_REF] based on regret aversion. This latter parameter has been de…ned by [START_REF] Bell | Risk premiums for decision regret[END_REF] who has introduced the ratio  = ¡ 0 00  0 1+ 0 . This leads us to consider the following values for the parameter :  = 1 10 20 and …nally 30. Note that these values yield to regret aversion levels  lying mainly between 2 and 30. Note also that, when letting  converges to 0, the regret function converges to the linear function (namely lim !0 + () = ), which ultimately corresponds to the standard utility case.

Cv Opti Average (r=1%)

Cv Opti Average (r=3%)

Cv Opti Asian (r=1%) Cv Opti Asian (r=3%)

Cv Opti OBPI (r=1%) Cv Opti OBPI (r=3%)

Figure 6: Compensating variations for utility with regret/rejoice Figure [START_REF] Ben-Akiva | Risk, uncertainty and discrete choice models[END_REF] illustrates the compensating variation for utility with regret/rejoice (see also Tables [START_REF] Bertrand | Portfolio insurance strategies: OBPI versus CPPI[END_REF] and [START_REF] Bertrand | Omega performance measure and portfolio insurance[END_REF]. For the case  = 1%, we compare results in Table [START_REF] Arrow | The role of securities in the optimal allocation of risk-bearing. Econometrie; as translated and reprinted in 1964[END_REF] for the standard concave case with  = 6% and  = 20%. We note …rst that the compensating variations for the regret/rejoice case are much higher than for the standard utility case. Indeed, they can reach values equal to about 65% corresponding approximately to 08% per year, which is close to the risk-free rate itself. We note also that, except for the case  = 2, the compensating variation for the regret/rejoice criterion is an increasing function of the parameter  associated to the regret/rejoice function. Table [START_REF] Bertrand | Portfolio insurance strategies: OBPI versus CPPI[END_REF] displays the compensating variations when comparing the optimal portfolio to the three standardized structured products for the case  = 3%. For the case  = 3%, we compare results in Table [START_REF] Bertrand | Omega performance measure and portfolio insurance[END_REF] to those in Table [START_REF] Ben-Akiva | Risk, uncertainty and discrete choice models[END_REF] for the standard concave case with  = 6% and  = 20%. As previously, we note also that the compensating variations for the regret/rejoice case are much higher than for the standard utility case. Indeed, they can reach very high values for weak relative risk aversion such as  = 2. Even for stronger relative risk aversions, for example  = 10, the compensating variation for the regret/rejoice case can be equal to about 258% corresponding approximately to 287% per year, which is still close to the risk-free rate itself. We note also that here the compensating variation for the regret/rejoice criterion is always an increasing function of the parameter  associated to the regret/rejoice function.

Compensating variation for the standard concave utility case with the Log Ornstein Uhlenbeck model for stochastic volatility This is the model introduced by Black, Derman and Toy (1990). Let   ´ constant and (      ) ´(  ) with () =   . Here the process  denotes the logarithm of the volatility. It is supposed to follow an Ornstein-Uhlenbeck model. In this case, the dynamics of the risky asset process is given by: (In Relation 1, it corresponds to the choice

  = ( 1 ¡  ); (      ) = ; (      ) = p 1 ¡  2 )   =   h  + (  ) (1)  i  with   = ( 1 ¡  ) +  (1)  + p 1 ¡  2  (2)  
with constant parameters  (the drift of the risky asset),   0,   0 (standard deviation of  ),  (correlation of the two Brownian motions). Thus  is mean reverting and converges to its long term value  1 with a speed equal to . Applying Ito's lemma to ( ), we deduce that the parameters of volatility  de…ned in Relation (3) are given by: 

()  = (ln  1 ¡ ln ) + 05 2 ;  () (      ) = ;  () (      ) = p 1 ¡  2 . The two premium processes  (1)
 and 

(2)

 de…ned in Relation 5 are given by:

 (1)  =  ¡  exp[  ]   (2)  = ( 1 ¡   ) ¡  ¡  ¡ exp[] p 1 ¡  2  
For the numerical illustration, we consider the following …nancial parameters:

( = 006;  = 001); ( = 008;  = 003)  = 02 We set  1 =  0 = [02],
which allows a better comparison to the GBM case with constant volatility  equal to 20%. 25 Indeed, by setting both the initial volatility  0 =  [ 0 ] and the long term volatility  1 =  [ 1 ] to the same level equal to 20%, we only take account of the stochastic feature of the volatility. For this purpose, we consider four choices for the pair of parameters ( ), namely  = ¡05 or 0, and  = 005 or  = 010. Recall that parameter  allows to calibrate the dependency between the return of the risky asset  and the logarithm of its volatility (usually  is negative) while parameter  allows to consider various levels of the "volatility of the volatility."

For the case  = 1%. Looking at Figure [START_REF] Bertrand | Gestion de portefeuille avec garantie: l'allocation optimale en actifs dérivés[END_REF], we note that the three standardized products, namely the OBPI, the Average and the Asian funds, are rather close from the compensating variation point of view. Indeed, the maximum compensating variation is equal to 10245. Since the fund horizon is equal to 8 years for this example, it corresponds to a monetary loss per year about equal to 03% These weak compensating variations between the three standardized structured products are also illustrated in Tables [START_REF] Chen | An analysis of market-index certi…cates of deposit[END_REF] to [START_REF] Cox | Optimal consumption and portfolio policies when asset prices follow a di¤usion process[END_REF] (see Supplementary Materials 4), from which we can also deduce that parameters  and  do not have a signi…cant impact on these compensating variations.

On the contrary, the compensating variations between those standardized structured products and the optimal fund are very signi…cant. Compared to results in Table [START_REF] Ben-Akiva | Risk, uncertainty and discrete choice models[END_REF] corresponding to the standard GBM case, they are much higher. For example, they can reach 12113 meaning a monetary loss per year equal to about 243%. This induces an implicit managing cost higher than 2% per year ! We note that most of the compensating variations are close to 107 which is only slightly smaller than the risk fee return 1083 on 8 years. For the GBM case, the maximum value is equal to 10436 corresponding "only" to 053% per year. Such feature illustrates the lack of adequacy of those three standardized products to customers needs when the volatility is stochastic, especially for "aggressive" investors ( = 1 or 2). Figure [START_REF] Bertrand | Gestion de portefeuille avec garantie: l'allocation optimale en actifs dérivés[END_REF] illustrates also the compensating variations between standardized structured products and the optimal fund. We note that we get the maximal compensating variations for the cases  = 0,  = 005,  = ¡05,  = 010 and  = ¡05,  = 005, while the minimal compensating variations is reached for the case  = 0,  = 010. Indeed, for instance when the correlation  is null, it is interesting to be able to use a …nancial instrument de…ned on the volatility itself but the higher its volatility the smaller the interest to diversify on it since it becomes riskier. When the correlation is negative (which is for example usually the case between a stock index and its own volatility), the higher the risk aversion from the level 4, the higher the expected utility since the investor can bene…t from the diversi…cation (see Appendix 4 in Supplementary Materials 1 for a simpli…ed example explaining most of these features).

For the case  = 3%, we choose  = 8% so that we get the same risk premium as in previous case, namely the standard value  ¡  = 5%. We set the portfolio horizon to 5 years, which implies that the initial proportion of wealth invested on the risky asset is about 14% (it is equal to about 77% for the case with  = 1%). Note that, from the operational point of view, the horizon equal to 5 years is the standard value used by practitioners as soon as the interest rate is about 3%. We recover the same main features, namely the compensating variations between the standardized structured products and the optimal fund are very signi…cant, compared to those between the standardized structured products themselves. We note in particular that the compensating variations can reach levels signi…cantly higher than the risk free return on the given time period (i.e.  [ ] ' 1162), for the cases  = 0,  = 005 and  = ¡05,  = 005, while the minimal compensating variations is reached for the cases  = 0,  = 010 and  = ¡05,  = 010. (see Figure 8). As previously explained, for  = 005, the Sharpe type ratio of the Log volatility is higher than for  = 010, implying higher compensating variations. Compared to the case  = 1% and  = 6%, when the risk free rate is equal to 3%, the factor loading 

(2)  (see Relation 5) can have a signi…cant smaller absolute value for  = ¡05,  = 010, explaining why, for this second case (i.e.  = 3% and  = 8%) , it is less interesting to be able to trade on the volatility (see remark about condition 0   ¡  (2) in Appendix 4). Obviously, when  is high, this condition has much more chance to be met). On the contrary, for the case  = 3%,  = 8% and portfolio horizon  equal to 5 years, we can better …t from trading on the volatility when its Sharpe ratio is relatively high since the initial proportion of wealth invested on the risky asset is equal to about twice that for the case  = 1%,  = 6% and  = 8. Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma Risk Aversion, Gamma

Conclusion

The suitability of standard …nancial structured products is a very important topic for the costumer protection. We look at an important class of such products, namely those whose performances are based on smoothing the return of a given risky underlying asset while providing a guarantee at maturity. We illustrate the potential inadequacy of such standardized …nancial funds to customers, depending on their attitudes towards risk. From the theoretical point of view, we prove that such standardized products are not optimal, even if the …nancial market volatility is constant. By using the notion of compensating variation, we compute the monetary losses of providing these standardized products instead of the optimal ones to the customers. We show that these (theoretical) losses can be severe, since they can be higher than the risk-free return on the given portfolio management period. It implies in particular that trading on volatility indices and/or options written on them should be introduced in structured portfolios issued by …nancial institutions (for example on the volatility index VIX for the S&P 500 index and the VSTOXX indices based on EURO STOXX 50). Another possible study would consist of calculating also the lowest cost strategies to achieve the given payo¤ distribution of each standardized fund (see e.g. the distributional analysis of portfolio choice and the theory of cost-e¢ciency of Dybvig (1988a[START_REF] Dybvig | Ine¢cient dynamic portfolio strategies or how to throw away a million dollars in the stock market[END_REF] and [START_REF] Bernard | Explicit representation of cost-e¢cient strategies[END_REF].

Figure 1 :

 1 Figure 1: Optimal portfolio payo¤ (utility with kink)

Figure 2 :

 2 Figure 2: Optimal portfolio payo¤ (regret utility)

Figure 3 :

 3 Figure 3: The optimal payo¤ for various values of 

Figure ( 4 )

 4 Figure (4) is devoted to the case  = 1% while Figure (5) corresponds to the case  = 3%.

Figure 4 :

 4 Figure 4: The CV as a function of gamma (CRRA) with GBM (mu=6% and r=1%)

Figure 5 :

 5 Figure 5: The CV as a function of gamma (CRRA) with GBM (mu=8% and r=3%)

Figure 7 :

 7 Figure 7: Compensating variations for the Ornstein-Uhlenbeck case case (mu=0.06, r=0.01)

Figure 8 :

 8 Figure 8: Compensating variations for the Ornstein-Uhlenbeck case (mu=0.08,r=0.03)

Enacted into law on July

[START_REF] Cox | Optimal consumption and portfolio policies when asset prices follow a di¤usion process[END_REF], 2010.[START_REF] Arrow | The role of securities in the optimal allocation of risk-bearing. Econometrie; as translated and reprinted in 1964[END_REF] Initiated in 2004 and implemented in November 2007.[START_REF] Bahaji | How rational could VIX investing be[END_REF] For example, in France, 91% of the total amount of structured funds provide a capital guarantee at maturity. Assets under management on the French retail market for structured products are equal to about 50 billion euros.[START_REF] Bell | Regret in decision making under uncertainty[END_REF] As an illustration, about one third of French capital-guaranteed structured products provide a guaranteed return with a bonus based on smoothing the price variations of a reference portfolio. Products based on Asian call options can reach 14% of the guaranteed funds, while the second based on an average of past positive performances of the benchmark asset can reach 18% of the guaranteed funds.

See e.g.[START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF] for details about these conditions.

If only the risky price process  is observed and tradable, then the …nancial market is incomplete. In such a case, we can determine optimal portfolios using results as in[START_REF] Pham | Optimal portfolio in partially observed stochastic volatility models[END_REF]. Roughly speaking, the portfolio optimization problem is solved in a complete market but for a modi…ed dynamics of the risky asset. To simplify the whole presentation of the present paper, we omit this case. Note that the introduction and development of volatility indices (VIX) instruments allow to drastically reduce this market incompleteness. Additionally, our approach allows to better illustrate the interest of introducing options written on the stochastic volatility in standardized portfolios to better …t customers needs.

We could consider as well the Kahneman and Tversky's framework based on loss aversions. However, since we deal with portfolio insurance, the investor does not su¤er from losses. Instead, she is more sensitive to comparisons with the risk free strategy.

See Appendix 1 in Supplementary Materials 1.

2 See[START_REF] Bertrand | Portfolio insurance strategies: OBPI versus CPPI[END_REF]

2011) for more details about portfolio insurance methods and their comparison.1 3 Equivalently, a call option can be bought on  with strike  and cash held equal to the discounted value of .1 4 Notation:  + = ( 0).

[START_REF] Bell | Risk premiums for decision regret[END_REF] Such an assumption is commonly introduced when dealing with the pricing of structured funds.

[START_REF] Ben-Akiva | Risk, uncertainty and discrete choice models[END_REF] We call it "Sharpe type ratio" since it is equal to the Sharpe ratio when we consider the standard deviation instead the variance.1 7 See e.g.[START_REF] Prigent | Portfolio Optimization and Performance Analysis[END_REF].

[START_REF] Bertrand | Portfolio insurance strategies: a comparison of standard methods when the volatility of the stock is stochastic[END_REF] For this example, the relative risk aversion is …xed at the level  = 09.

[START_REF] Anderson | A Discrete Choice Theory of Product Di¤erentiation[END_REF] Using Relation[START_REF] Jacod | Limit Theorems for Stochastic Processes[END_REF].

[START_REF] Arrow | The role of securities in the optimal allocation of risk-bearing. Econometrie; as translated and reprinted in 1964[END_REF] In De Palma and[START_REF] De Palma | Utilitarianism and fairness in portfolio positioning[END_REF], it is shown that the compensating variation can be also related to the certainty equivalent notion.2

We use Monte Carlo method when there is no explicit formula as in[START_REF] Detemple | A Monte Carlo method for optimal portfolios[END_REF]. 2

Note also that, in Supplementary Materials 2, we examine how the compensating variation varies with respect to interest rate  (see Figures1 to 16) and drift  (see Figures17 to 20). The main conclusions about the compensating variation are not signi…cantly modi…ed.

[START_REF] Bell | Risk premiums for decision regret[END_REF] In Supplementary Materials 4, we have conducted the same analysis for two other levels, namely 1 = 0 = [015] and 1 = 0 = [025]. As shown by Figures[START_REF] De Palma | Standardized versus customized portfolio: a compensating variation approach[END_REF] to[START_REF] Dybvig | Distributional analysis of portfolio choice[END_REF], we get the same features of the compensating variation.