امور بانکی خرده فروشی و مهندسی مالی رفتاری : مورد محصولات ساختارمند

We apply cumulative prospect theory and hedonic framing to evaluate discount reverse convertibles (DRCs) and reverse convertible bonds (RCBs) as important examples of structured products from a boundedly rational investor’s point of view. While common expected utility theory would also conclude that DRCs and RCBs are of interest to investors with moderate return expectations and underestimated stock return volatility, that theory would overestimate the market success of DRCs and underestimate that of RCBs in comparison to a situation with bounded rationality. Hedonic framing and relatively low subjectively felt competence levels of investors are decisive for the demand for RCBs.

One of the core tasks of investment banking is the constant search for opportunities to create new financial instruments. Typically, this task is fulfilled by innovatively combining already existing components to form new financial instruments. By combining a set of elementary components, it is possible to cater to the special needs of individual groups of customers. This process is called “financial engineering”, as investment bankers act similarly to engineers or natural scientists when planning and creating complex financial innovations, on the basis of some elementary building blocks, in order to meet their customers’ needs. One of the most prominent groups of newly introduced financial instruments resulting from such financial engineering is termed “structured products”. These are combinations of derivatives and underlying financial instruments which exhibit structures with special risk/return profiles that may not be otherwise attainable on the capital market without significant transaction costs being incurred – at least for private investors (see, e.g., Das, 2000). Discount reverse convertibles (DRCs, henceforth) and reverse convertible bonds (RCBs, henceforth) are important examples of structured products. DRCs and RCBs can be interpreted as a combination of a zero bond or a coupon bond plus a short position in put options on stocks. While the creation of new financial instruments of this kind is not too difficult, their market success depends on the costs of their reconstruction for the issuer and the benefits they offer to potential buyers. Certainly, for complex financial instruments like structured products, there is a special need for a quantification of these costs and benefits. Financial engineering thus comprises two quantitative subroutines. First, investment banks have to calculate the costs of creating a certain structured product as the outcome of the combination of several single modules. This is typically done with the help of arbitrage-theoretical tools for perfect capital markets, as investment banks can be considered as acting on capital markets that are near to perfection. Second, however, one has to evaluate a customer’s possible utility gains when he or she buys a certain financial product, whereby it is necessary to abstract from a perfect capital market – at least in the case of retail customers – because these customers do not have the same unhampered market access as investment banks do. Moreover, there would otherwise indeed be no need for any financial innovation at all, as we are told by the celebrated irrelevance theorem introduced by Modigliani and Miller (1958). One straightforward idea would be to apply expected utility theory based on the axioms of rational decision making, as introduced by von Neumann and Morgenstern (1944) in order to assess the utility effects of new financial products. However, ever since Allais published his seminal work in 1953, there have been practically innumerable contributions, all pointing out that real-life human decision behavior is not governed by such rational axioms. In fact, even the evaluation of simple stock holdings by expected utility theory is troubled by a problem known as the “equity premium puzzle”, which describes the fact that real-life risk premia are far higher than expected utility theory would suggest (see, e.g., Mehra and Prescott, 1985). Recent work on the equity premium puzzle, like that of Bernartzi and Thaler, 1995, Barberis et al., 2001 and Barberis and Huang, 2005, therefore tries to exploit the findings of the (cumulative) prospect theory suggested by Kahneman and Tversky, 1979 and Tversky and Kahneman, 1992. This alternative behavioral decision theory seems to be one of the most promising attempts to realistically describe many aspects of actual human decision making. With individuals’ value functions being defined in wealth changes, instead of absolute wealth levels, and exhibiting loss aversion as well as the assumption of overweighting extremely low and underweighting extremely high probabilities, the (cumulative) prospect theory was originally designed for the subjective evaluation of lotteries as the typical form of uncertain prospects. However, this theory can also be used to enhance our understanding of the equity premium puzzle, since uncertain returns of financial instruments are themselves essentially nothing more than a – possibly rather complex – lottery (see Barberis and Huang, 2005). We will follow this path by applying the cumulative prospect theory, in combination with arbitrage theory, to price and to evaluate DRCs and RCBs as examples of structured products. According to Glaser (2001), we will use the term “behavioral financial engineering” to designate this specific approach in the more general field of behavioral finance. In order to contribute to the theory of behavioral finance by examining in detail aspects of optimal pricing and product design, with regard to DRCs and RCBs as important examples of structured products, we build on Shefrin and Statman, 1993 and Glaser, 2001. Shefrin and Statman (1993) seem to have been the first to apply the idea of combining concepts of arbitrage-theory and behavioral finance when examining a covered-call position which consists of a combination of a stock and a short position in corresponding call options and thus is – in principle – equivalent to a DRC. However, Shefrin and Statman compared this position only with a direct stock holding. We extend their analysis by accounting for the possibility of buying a riskless asset and RCBs as further alternatives. Unfortunately, for the typical case of power value functions, in a one-period binomial framework, as utilized by Shefrin and Statman, DRCs could never be preferred by individuals to both the sole purchase of stocks and riskless lending.2 We therefore have to extend the analysis to a multi-period binomial framework. Moreover, we must introduce a new kind of hedonic framing rule in place of that initially suggested by Thaler (1985) in connection with the handling of different mental accounts – as described by Thaler, 1980 and Tversky and Kahneman, 1981. According to the original hedonic framing rule, two distinct certain payments will only be viewed as one single payment (integration) if this leads to a higher subjective value than the separate evaluation of both payments (segregation) would. Our extension of this basic concept explicitly accounts for the possible integration of uncertain prospects with certain ones and makes it possible to evaluate particularly RCBs in a concise way. Finally, allowing for the variation of central parameters of the cumulative prospect theory, we can take into account recent findings of Kilka and Weber (2001) regarding the subjectively felt competence of individuals on their decision making. Our main results are the following: first, expected utility theory and cumulative prospect theory come to the same qualitative conclusion that DRCs and RCBs benefit from low volatility estimates and medium-level return expectation estimates by investors. Actually, this is not too surprising, as investors who purchase DRCs or RCBs are effectively taking a short position in stock options and DRCs and RCBs are competing with the direct holding of stocks. In that respect, such qualitative findings should be expected for any reasonable theory of human decision making. In the same way, analyses of the welfare effects of direct stock holdings show the same qualitative results for both expected utility theory and cumulative prospect theory: they both find positive risk premia. However, according to the literature on the equity premium puzzle, quantitative outcomes differ considerably, as resulting risk premia are significantly higher for boundedly than for fully rational investors. The same holds true in an analogous way for our analysis of DRCs. These are generally much more attractive for fully rational individuals than they are for boundedly rational ones. We show that this result is mainly a consequence of the overweighting of small probabilities, which implies skewness preferences and thus makes the holding of short option positions less attractive. Loss aversion, in contrast, only plays a minor role in the explanation of this finding. Second, the demand for RCBs by individual investors can only be understood against the background of hedonic framing. Without hedonic framing, there is hardly any need for RCBs, as they are simply a combination of a DRC and riskless lending, so that, for fully rational investors, either of these alternatives will generally be better than their holding an RCB. Third, the demand for structured products depends on the subjectively felt competence level of private investors, and RCBs in particular seem to become, ceteris paribus, more attractive for individuals with smaller competence levels. This is because lower competence levels reduce the desirability of uncertain prospects, so that differences in subjective evaluation of certain and uncertain prospects become more relevant, thus leading to a greater importance of the hedonic framing rule. Fourthly, for our numerical analysis, the subjectively felt competence level of possible customers turns out to be more relevant for the market success of DRCs and RCBs than even the optimization of the issuance price and, especially, than the redemption value or periodical interest payments connected with these structured products. Obviously, our approach can be used to derive recommendations for the adequate product design regarding DRCs and RCBs. Our findings imply that issuers of DRCs and RCBs should prefer blue-chip stocks as underlyings, since the volatility estimate of these tends to be relatively moderate. Moreover, in particular with respect to the sale of RCBs, banks should aim at those customers who do not feel too competent, as this renders RCBs, ceteris paribus, more attractive. In addition, issuers of DRCs and RCBs may overestimate the market potential of these structured products when not accounting for individuals’ bounded rationality. Our approach can easily be applied to other financial products and, thus, might generally help investment banks to better understand, design, and even price financial innovations. The rest of our paper is organized as follows: Section 2 presents the basics of the cumulative prospect theory and the hedonic framing rule. In Section 3, we introduce DRCs and RCBs as examples of structured products and we present their arbitrage-theoretical valuation in the basic capital market setting introduced by Black and Scholes (1973). Section 4 presents the main results of our paper described above by contrasting the subjective evaluation of DRCs and RCBs for both fully rational and boundedly rational individuals. Section 5 concludes.

Our paper was mainly motivated by our intention to analyze discount reverse convertibles and reverse convertible bonds as typical examples of structured products against the background of the cumulative prospect theory of Tversky and Kahneman, 1992 and Thaler, 1985 hedonic framing rule for mental accounts, with the latter being extended to the case of uncertain prospects. Moreover, we allowed for the consequences of different individuals’ competence levels as expressions of varying attitudes towards ambiguity. We found that DRCs and RCBs are of interest to investors who moderately estimate the expected return of the underlying stock and who underestimate the corresponding return volatility. While this result holds true for both fully rational individuals and boundedly rational ones, the possible demand for DRCs seems to be significantly overestimated, if full rationality is used as an approximation of boundedly rational investors. Moreover, without hedonic framing, the demand for RCBs cannot be understood. Varying competence levels turn out to be more relevant for the evaluation of DRCs and RCBs than are the optimization of redemption values, interest payments and absolute premia under explicit consideration of aspects of bounded rationality. In particular, RCBs become more interesting for, ceteris paribus, reduced competence. This finding might give rise to some practical recommendations, such as that of addressing less experienced investors. Though we have only considered just one specific DRC and a corresponding RCB, our qualitative findings carry over to other DRCs and RCBs on stock indexes, as long as we do not alter our distributional and preference-related assumptions, since then, only parameters of minor importance – like the time to maturity and the riskless interest rate – could be changed, while almost all the more interesting parameters have already been the object of our analyses. Nevertheless, there are several important possibilities of broadening our examination. As a first interesting aim for further research, our approach should be extended to the analysis of more complex portfolio selection problems. This would give rise to the relevance of some kind of systematic risk from the private investor’s point of view. Moreover, we have only analyzed situations with perfect inter-bank competition, i.e., no positive profit at all for all banks, as in the scenarios underlying Fig. 1 and Fig. 2, and with no competition at all among banks as in the situations examined in Table 1 and Fig. 3. However, since financial innovations can be replicated quite easily by competitors, possible innovators’ monopoly gains will erode in the long run. As an extreme example, Bertrand competition among banks without capacity constraints would eventually force absolute margins to reduce to zero and, thus, imply indeed perfect competition. As known from Kreps and Scheinkman (1983), Bertrand competition with endogenous capacity constraints leads to situations which resemble pricing behavior in Cournot oligopolies. Such situations could also be handled by our approach, because this modification would only reduce accessible private investors’ wealth W0,tot from the issuer’s point of view without changing substantially the relevance of our analysis. Furthermore, in any case, competition will also induce banks to steadily create new financial products and to take advantage of temporary monopoly positions, thus giving a further justification for the analysis corresponding to Table 1 and Fig. 3. Finally, we have only considered the most straightforward scenario for the valuation of DRCs and RCBs, as we did not allow for aspects such as stochastic interest rates and stochastic volatility of stock prices as well as issuers’ credit risk. In the first instance, such additional risk problems will influence the arbitrage-free valuation of these financial derivatives. While such variations should not affect our qualitative findings, optimal absolute und relative margins might be influenced. Nevertheless, because of space constraints, such issues shall be postponed to future research as well.