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Theory and Decision

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13-08-2016

The valuation “by-tranche” of composite investment instruments

Authors: Doron Sonsino, Mosi Rosenboim, Tal Shavit

Published in: Theory and Decision | Issue 3/2017

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Abstract

The return on composite investment instruments takes the form of weighted-average, derived from two economic indicators or more. Three experiments illustrate that prospective investors tend to valuate composites “by-tranche”, consistently violating the premise of reduction. Valuation-by-tranche shows for uncertain and risky composites and reflects in allocation problems and binary choice. The willingness to invest still strongly increases when one tranche hedges against the other, suggesting that reduced-form considerations may interfere with the inclination to value by part. A hybrid model where investors weight the values of tranches, but also respond to the reduced-form, approximates the data most accurately.

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The magnitude and diversity of the market may be glimpsed at designated commercial Web portals such as SRP (http://www.StructuredRetailProducts.com), SVSP (http://www.sspa-association.ch), and SPA (http://structuredproducts.org).

In the context of mortgage securitization, the term “tranche” is used to address the array of securities issued against a pool of loans. The current figurative use is different. Another definitional issue deals with structures that track the performance of large indices such as SP100. The suggestion that investors valuate such investments by weighting the values of 100 stocks seems far-fetched, although intuitively it is plausible that specific securities occasionally tint the willingness to invest in SP100 in the spirit of valuation-by-tranche. For the current use, we treat indices as a “primitive” so that the deposit must refer to 2 indices (or an index and some other asset) to classify as composite.

The tendency to avoid risk with gains while taking risk with losses has been evoked to explain the disposition effect (Shefrin and Statman 1985); Benartzi and Thaler (1995) build on loss aversion and the short sightedness of investors to explain the equity premium puzzle; the overweighting of small win-chances has been utilized to explain the IPO anomalies (Green and Hwang 2012). Barberis (2013) provides a general survey and additional examples.

Formally, if the probability assigned to conditions A (B) are P1 (P2), it is easy verified that the shift of 2 % return from tranche 1 to tranche 2 decreases the absolute value of the difference between the expected return on tranche 1 and the expected return on tranche 2, independently of P1 and P2.

The symbol (x, p) denotes a lottery paying x with probability p and 0 with probability 1-p. The term “50–50 lottery” is applied for binary lotteries paying each prize with probability p $=$ 0.5.

Following studies of financial decision (e.g., Benartzi and Thaler 1995), we assume that the primitive of choice is the return prospect rather than gains and losses (see Sect. 6 for further discussion).

See block 4 of Web supplement E for an example. More generally, Eyster and Weizsäcker (2011) methodology of testing for correlation neglect could be adapted to construct equivalent composites; e.g., a 0.6–0.4 composite with X and Y as tranches vs. a 0.2–0.8 version with X and Y’ $=$ 0.5*X $+$ 0.5*Y as components, but the parallel change in weights and returns complicates the analysis of such shifts.

The binary model is insufficient for the reduced-form prospect that generally pays four distinct returns (Table 1), but since reduction simply stipulates that the willingness to invest is unaffected by shifts in the composite representation, general assumptions on the valuation function are unnecessary.

As explained at the introduction, the restriction is motivated by our results. The limited role of losses in the experimental composites also motivated two other modeling choices: (1) The basic structure ignores the possibility of distinct weighting of loss and gain events (Tversky and Kahneman 1992) (2) The tranche weighting scheme precludes dissimilar weighting of negative-valued tranches. Background estimations suggested that more general frameworks that relax (1) or (2) do not provide additional insights.

The US$ was traded for about 3.5 NIS around the experiments. The prime rate in 12/2014 was $2.5~\,\%$.

The average check was close to 100 NIS (about 25 US$ at the time of payment: early February 2015). All web supplements are available at http://www2.colman.ac.il/business/doron/papers.

We intended to run Q1 on a larger sample, but had to stop recruitment with 120 subjects as the test year (2014) arrived. Thirteen outliers were removed. The mean confidence score (1–100 scale; 1 for the most confident) of the 13 filtered subjects was 66, compared to 36 for the 107 selected questionnaires ($\hbox {\textit{p}}<0.01$). In general, the results strengthen when the least confident, or the questionnaires with close to zero correlation between expected returns and levels of composite investments are removed, but the effective filter is problem dependent. The results are statistically stronger for Q2–Q3 where reduction is rejected for VBT at $\hbox {\textit{p}}<0.001$ by likelihood ratio tests.

To quickly verify that version R may appear more attractive than L under VBT when $-U(-5)<U(15)-U(10)$, assume W(E1) $=$ 0 and W(E2) $=$ 1. With this respect note that Q1 builds on specific properties of V to make directional predictions for VBT, but the joint hypothesis concern is alleviated in Q2–Q3, where we use individual-level estimations to contrast reduction and VBT without making a priori assumptions on risk and loss attitudes.

Since the results of the experiments provide very limited support to tranche weighing, we assume $ \varphi (0.5)=0.5$ in the discussion of $w=0.5$ composites. Web supplement C.1 shows that the preference for version L of “FTSE + TA-Finance” extends to VBT+ when $\varphi (0.5)\le 0.5$. Supplement C.2 derives general bounds on the intensity of tranche weighting so that VBT strict preference for given version extends to VBT+ if tranche weighting meets these restrictions.

In the case of “Finance Sector”, 4 subjects allocated $100~\,\%$ and 1 invested $0~\,\%$ and Table 2 results are robust to removal of these 5 observations. The slope of allocations with respect to E(R) was not affected by version. The mean allocations of subjects with expected return $<6~\,\%$ were 61 vs. 49, compared to 69 vs. 59 for the subjects with expected return $>8~\,\%$. The results discussed in Sects. 3.3–3.6 show similar robustness.

The “ %-preference for the composite” was similar, around $60\,\%$, in versions $\hbox {L}$ and $\hbox {R}$, but more than fifth of the version R subjects that preferred the $60~\,\%$ investment in the binary choice chose to decrease their allocation, below $60~\,\%$, in the open allocation problem.

Blue-Tech groups the technology and biomed stocks traded at the Tel-Aviv exchange.

If P1 $=$ 0 and P2 $=$ 1, version $\hbox {L}$ pays $-3\,\%$ on one tranche and 12 % on the second, while version $\hbox {R}$ yields −6 % and 15 %, respectively. If $U(-3)-U(-6)\ge U(15)-U(12)$, L is preferred to $\hbox {R}$ under VBT.

Intuitively, version $\hbox {R}$ subjects could turn suspicious because of the extreme returns on tranche 2.

To summarize Q1, we ran tentative estimations on the $107\cdot 6=642$ allocation decisions, taking individual heterogeneity (106 intercepts), problem-specific adjustments (6 indicators), and subjective expected returns into account. The VBT indicator ($=$ 1 for the versions that attracted significantly larger demand) was significant at $\hbox {\textit{p}}<0.01$ by Tobit regressions on the allocations $(\hbox {\textit{T}}=2.8)$. Logistic regressions on “preference for the composite” gave p $=$ 0.03 (Chi-square $=$ 4.5).

Except for the three problems of block 14 (supplement E) where one alternatively dominated the other.

Following the literture conventions, we let $U(x)=\ln (x)$ for $x\ge 0$ when $\alpha _G\rightarrow 0$ and $U(x)=-\lambda *\ln (-x)$ for $x<0$ when $\alpha _L\rightarrow 0$.

In example (a) of Fig. 7 the composite version noise is the average of U(26)-U(22) and U(8)-U(4), while the reduced-form noise is U(3.5)-U(3.1). When the heteroskedastic adjustment is removed, assuming constant noise in all problems, the $-2LL$ (by individual-level estimations) increase from 2191 to 2285 for Q2 and from 2144 to 2284 for Q3. The estimation results are generally robust except for strong increase in the estimated $k_{i}$’s.

The random choice model can be considered a special case of the hybrid model with large enough noise $k_{i}$. The random choice $-2LL$ is easily derived: $66\cdot 36\cdot -2\cdot \hbox { ln}(0.5)=3294$ for Q2 and $81 \cdot 34 \cdot -2 \cdot \hbox {ln}(0.5)=3818$ for Q3.

Intuitively, tranche weighting could play stronger role with framed-field composites such as “Blue-Tech (+5 % DJIA)” compared to the more stylized risky composites of Q2-Q3.

For example, take problems 2.2–2.3 of Q3: 2.2 involves choice between a lottery paying 15 % or −1 % and fixed return of 5.55 %. In 2.3, the lottery pays 15.6 % or −0.6 %, while the fixed return is 5.4 %. Consistency is violated if the lottery is selected in 2.2 and the safe alternative is selected in 2.3.

The Q2 (Q3) consistency scores ranged between 2/14 and 12/14 (1/9 and 9/9) with medians 0.54 (0.55), respectively. $\mu $ also shows negative correlation −0.21 with $\alpha _{G}$. A median split of the samples by $\alpha _{G}$ reveals median $\mu $ 0.1 (in both samples) for the relatively risk seeking, compared to 0.4 (Q2) and 0.9 (Q3) for the relatively risk averse. The risk averse thus appear to process the composites more rigorously than others. The correlations between $\mu $ and other estimates were close to zero.

Again, the 8 Q2 subjects with estimated $\mu =1$ are ignored for the $\alpha _L$ statistics.

The remaining subjects (about 13 % of the pool) exhibit the reversed type of behavior with loss aversion replaced by gain-seeking when returns exceed some threshold level $x_{{0}}< 20\,\%$. The prediction ${\vert }\hbox {Error}{\vert }$ for these subjects is 1/3 larger than others (median 2.4 % vs. 1.8 %; p $=$ 0.06) which suggests that the hybrid model shows relatively small success in explaining their choices.

Take, for example, the case where a composite consists of one tranche paying 8 % with probability 0.8 or 2 % with probability 0.2 and a second tranche paying 10 % or 0 % with equal 0.5 probabilities. Assume w $=$ 0.5 and independence of tranche returns. While the concavity of U plays for the reduced-form version (since, for example, U(1 %)>0.5$\cdot $U(2 %) $+$ 0.5$\cdot $U(0 %)), strong overweighting of the 0.1 worst case scenario (2 % and 0 % returns) may oppositely decrease the appeal of the reduced-form lottery.

Formally, concavity of U implies that $U(6300)-U(4500)<W(0.5)\cdot [U(3000)-U(1200)]+ (1-W(0.5))\cdot [U(2600)-U(800)]$, so that $U(6300)+W(0.5)\cdot U(1200)+(1-W(0.5))\cdot U(800)<U(4500)+W(0.5)\cdot U(3000)+(1-W(0.5))\cdot U(2600)$, and R is preferred to L under VBT with payoffs as the primitive of valuation. The point could be made with example (a) of Fig. 7, but the Fig. 9 example is more convincing.

The exact terms of deposit A (Appendix A) are provided at https://www.cibcnotes.com (product code CBL221). The weight of GOLD and SILVER in ETN B changes with the underlying index composition; see http://etfdb.com/factsheets/BLNG.The past performance table in C was replicated from http://tih.co.il/docs/heb/tih-1129527.pdf.

Loss aversion is used for 12–15 % losses; for compatibility with assumption (II) of the basic structure, assume that $x_{0}\le 10\,\%$

To illustrate that $\varphi (0.05)\ge l$ is insufficient for preference reversal unless $\varphi (0.95)\le 1-l$, let $W(E1)=1$ and $W(E2)=0$. Assume that preferences take the Vendrik and Woltjes form with $x_{\mathrm{G}}=0.8$. It then follows that $l\approx 0.057989$, so that L and R obtain similar values under VBT+ if $\varphi (0.05)=0.057989$ and $\varphi (0.95)=0.942011$. Version R becomes more attractive if, for example, $\varphi (0.05)=0.06$ and $\varphi (0.95)=0.94$. However, if $\varphi (0.05)=0.057989$ (sufficient overweighting of 0.05) but $\varphi (0.95)=0.95$ (insufficient), L is still preferred to R.

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Title: The valuation “by-tranche” of composite investment instruments
Authors: Doron Sonsino
Mosi Rosenboim
Tal Shavit
Publication date: 13-08-2016
Publisher: Springer US
Published in: Theory and Decision / Issue 3/2017
Print ISSN: 0040-5833
Electronic ISSN: 1573-7187
DOI: https://doi.org/10.1007/s11238-016-9567-7

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