Conventional or Reverse Magnitude Effect for Negative Outcomes: A Matter of Framing
Wolfgang Breuer, Can K. Soypak, and Bertram I. Steininger
Working Paper 2020:16
Division of Real Estate Economics and Finance Division of Real Estate Business and Financial Systems Department of Real Estate and Construction Management
School of Architecture and the Built Environment KTH Royal Institute of Technology
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Conventional or Reverse Magnitude Effect for Negative Outcomes: A Matter of Framing
Wolfgang Breuer
RWTH Aachen University,
Department of Finance, Aachen, Germany Email: wolfgang.breuer@bfw.rwth-aachen.de
Can K. Soypak
RWTH Aachen University,
Department of Finance, Aachen, Germany
Bertram I. Steininger
Division of Real Estate Economics and Finance
Department of Real Estate and Construction Management Royal Institute of Technology, Stockholm, Sweden
Email: bertram.steininger@abe.kth.se
Abstract:
We present and expand existing theories about why individuals may assess positive outcomes differently from negative outcomes in intertemporal choices. All of our theories – based on utility or cost considerations – predict a conventional magnitude effect for positive outcomes, i.e., a negative relation between outcome size and subjective discount rates. For negative outcomes, however, implications are different for utility- and cost-based approaches. We argue that the relevance of utility-based aspects is strengthened in a money frame, leading to a conventional magnitude effect even for negative outcomes, whereas cost-based considerations gain in importance in an interest rate frame, implying, in contrast, a “reverse” magnitude effect, i.e. higher discount rates for (absolutely) higher outcome size. By conducting a web-based experiment with 676 participants, we confirm our theoretical findings and conclude:
the conventional magnitude effect prevails for positive outcomes in the money and the interest rate frame and for negative outcomes in the money frame. However, there is a reverse magnitude effect for negative outcomes in the interest rate frame. Our results might help to better understand prevailing magnitude effects in practical applications and might also be apt to derive suggestions for better designing of intertemporal decision problems.
Keywords: Discounting anomalies, Intertemporal choice, Framing, Magnitude effect, Reverse magnitude effect
JEL-codes: D14, D90, D91, G02, G12
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1 Introduction
According to traditional decision-making theories, subjects are forward-looking, and they maximize their aggregate lifetime utility in intertemporal decision settings using backward induction (Samuelson, 1937; Samuelson, 1969). Time preferences are crucial for solving such problems, since they determine the weight given to future utility. It is therefore essential to define discounting functions and discounting behavior in order to understand how private and managerial decision-makers choose their consumption path (Shefrin and Thaler, 1988).
Accordingly, numerous researchers have attempted to identify the shape and the determinants of discounting functions both with the help of experimental studies (see, e.g., Thaler, 1980;
Kirby and Herrnstein, 1995; Andersen et al., 2006) and field investigations (see, e.g., Hausman, 1979; Viscussi and Moore, 1989; Warner and Pleeter, 2001). Typically, those studies found several violations of the original exponential discounting model suggested by Samuelson (1937), which are generally called “discounting anomalies”. One of the most important discounting anomalies is the so-called “magnitude effect”. In the case of positive outcomes, this effect states that the individuals’ required rates of return are decreasing with increasing magnitude of the outcomes. There are many theoretical (e.g., Benzion et al., 1989, Benhabib et al., 2010, Read et al., 2013) and empirical (e.g., Chapman, 1996, Green et al., 1997) papers addressing this “conventional” magnitude effect.
However, there are far fewer papers which examine magnitude effects for negative outcomes, and they find far less evidence for a “conventional” magnitude effect than in the case of positive outcomes (see, e.g., Thaler, 1981, Baker et al., 2003, and Estle et al., 2006). For instance, Thaler (1981) identifies a conventional magnitude effect with median discount rates of 26 % for $15, 6 % for $100, and 1 % for $250 losses in an experiment conducted with around 20 participants. With larger sample sizes, Baker et al. (2003) and Estle et al. (2006) report that discount rates for choices involving losses amounting to $100 and $100,000 are not
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significantly different from each other. Moreover, this result can be confirmed when comparing the discount rates in intertemporal decisions involving losses of $10, $100, and $1,000, respectively. Therefore, the authors conclude that the magnitude effect may not exist for negative outcomes. Based on Pigou’s (1920) diminished visibility theory of future outcomes or impatience argument, Noor (2011) states that individuals are more patient toward larger outcomes, so that we can generalize this finding to a conventional magnitude effect for positive and negative outcomes. Inspired by Mitchell and Wilson’s (2010) finding of a non-significant trend for a reverse magnitude effect in the domain of losses, Hardisty et al. (2013) try to systematically derive and test the potential of a reverse magnitude effect. According to this paper, for positive and negative outcomes, there is a present bias at work that induces subjects to prefer immediate consequences to future consequences regardless of the sign and magnitude of either. Due to the present bias, the authors expect to observe a conventional magnitude effect for positive outcomes and a reverse magnitude effect for negative outcomes but only when comparing immediate outcomes with delayed outcomes and not when an individual has to decide between two outcomes, where one is farther and one is less far in the future.
Furthermore, according to Hardisty et al. (2013), the reverse magnitude effect should be weaker in an interest rate frame than in a money frame. In a money frame, subjects have to choose between two mutually exclusive (predefined) monetary outcomes (for example: EUR 100 today or EUR 102.47 in 6 months). However, the implicit interest rate between the sooner and the later alternative (for our example: 5 % per year) is not revealed to participants by the experimenters. Contrarily, when disclosing the internal rate of return of alternative payments (instead of the payments themselves), we may speak of an interest rate frame (for a more differentiated characterization of different kinds of frames, see Read et al., 2013). As individuals in a money frame are more directly confronted with the monetary consequences of
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different alternatives, there should be a more pronounced present bias effect than in an interest rate frame.
In addition, there are also some empirical papers, especially on P2P lending, that show that interest rates paid by borrowers are increasing with the amount borrowed; e.g., Berger and Gleisner, 2009, see Tables 8, 9, 10, and 14, although they do not directly refer to the magnitude effect. Breuer et al. (2020), who explicitly analyze the conditional relationship between interest rates and the respective amounts on a P2P platform, conclude that the conventional magnitude effect applies for lending situations, but a reverse magnitude effect for borrowing situations.
Apparently, these findings hint at the possibility of a reverse magnitude effect for negative outcomes in an interest rate frame.
Due to these contradicting experimental and empirical results regarding the magnitude effect, there still seems to be need for additional research. In this regard our main contributions are threefold. First of all, we show that the status quo bias is another alternative to explain for a money frame why we may observe a conventional magnitude effect only for positive outcomes but a reverse magnitude effect for negative ones. The status quo bias refers to (mental) adjustment cost-based considerations of intertemporal decision-making inducing individuals to stick to a given initial consumption pattern and making them reluctant to changes. This bias is relatively more important for small absolute sizes inducing in such a case higher discount rates for investors and smaller discount rates for borrowers than in situations with higher absolute outcome sizes. When simultaneously assuming utility functions which support a conventional magnitude effect for positive outcomes in line with empirical evidence, then the conventional magnitude effect will also prevail for a combined utility-based and cost-based consideration.
For negative outcomes, however, we are able to show the existence of a lower bound so that only for smaller mental adjustment costs a conventional magnitude effect results. The reason is that the utility-based impact supports a conventional magnitude effect even for negative
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outcomes while the cost-based influence is in favor of a reverse magnitude effect. For sufficiently high mental adjustment costs the latter effect prevails.
Secondly, we know of no other study where the difference between decision-making in a money frame and in an interest rate frame is presented in a formal way. While the former refers to absolute monetary outcomes, the latter is concerned with corresponding rates of return.
Similarly to the money frame, an interest rate frame implies a conventional magnitude effect for positive outcomes and a transition from a conventional to a reverse magnitude effect for negative outcomes with increasing mental adjustment costs. However, in an interest rate frame the corresponding critical value for mental adjustment costs is typically smaller than in a money frame, as the relative importance of mental adjustment costs tends to be higher, when decisions are based on relative figures like rates of return leading to lower upper bounds in the interest rate frame for the emergence of a reverse magnitude effect. Against this background, we argue that there are reasons to expect a combination of a conventional magnitude effect in the money frame and of a reverse magnitude effect for the interest frame for negative outcomes (and moderate values of mental adjustment costs).
Thirdly, we perform quite a sizable web-based experiment with 676 participants to test our theoretically derived hypotheses. In doing so, we are not only able to find evidence for a reverse magnitude effect in an interest rate frame with negative outcomes, while all three other possible combinations of frames and outcome signs lead to a conventional magnitude effect, but also identify some limitations on the working of the present bias as propagated by Hardisty et al. (2013). Indeed, we deem our study to be one of the most comprehensive analyses of magnitude effects for different frames and outcome signs. For an excellent review of experimental literature with real incentives in the positive – and mostly money – frame see Andersen et al. (2014), Table 3 and Appendix D.
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The remaining part of the paper is organized as follows. In Section 2, we present several theoretical explanations for different kinds of magnitude effects and examine their relationship with a problem presentation in a money frame or an interest rate frame. Based on these theoretical considerations, we formally derive three hypotheses in Section 3. The experimental setting for testing our hypotheses is described in Section 4, while results are presented in Section 5. Section 6 concludes by highlighting practical applications of our findings.
2 Theoretical explanations for magnitude effects
We follow Benzion et al. (1989) by assuming that an individual’s overall positive or negative monetary outcome at a future point in time t is F (see also Breuer et al., 2020). Here,
1 is a scale parameter introduced in order to investigate potential magnitude effects, whereas F stands for the resulting future outcome for = 1. Moreover, we describe by P the subjective present value implied by the future outcome F, this means, the present amount of money that is considered to be equivalent to F. P is preference-dependent due to the discount function applied and can be understood as a willingness to “pay” or even more precise for the standard choice in an intertemporal decision where subjects must forego smaller sooner (SS) positive outcomes in exchange for larger later (LL) positive outcomes as a willingness to forego money.
Such a willingness is measured in monetary terms, but it is also subjective. Against this background, it is possible to determine a resulting overall rate of return of F/P1 from time 0 to time t with the conventional magnitude effect being characterized by F/P11 > F/P1 for > 1. In contrast, a reverse magnitude effect would be identified by F/P11 < F/P1 for
> 1. These definitions do not only apply to cases with positive outcomes, i.e. for F > 0, but also for those with negative outcomes, i.e. for F < 0, as we can restrict our analysis to situations with sgn(F) = sgn(P1) = sgn(P) (see below for further discussion).
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2.1 Utility-based based approaches
We now assume that v(F) is an individual’s utility (measured in monetary units) resulting from the outcome position F > 0, and that d(t) is a time-dependent subjective discounting function. Then, we have d(t)v(F) = P, implying that F/P11 > F/P1 for v(F)/v(F) > . We call this “the property of more than proportionally increasing utility”. It means that a future outcome being times the amount of another future outcome at the same time t will give more than times the utility of the latter outcome. Therefore, we have P >
P1implying smaller rates of return for higher values of . This property also guarantees a conventional magnitude effect for negative outcomes (F < 0) and describes the “utility-based”
explanation of the conventional magnitude effect for positive and negative outcomes. Its base form was developed by Loewenstein and Prelec (1992) (they called it the “sub- proportionality”), but a similar argument applies for other approaches which rely on utility comparisons like the one above (see, e.g., the attribute-based tradeoff model of Scholten and Read, 2010).
2.2 Cost-based approaches
Another way to explain magnitude effects is via the introduction of (mental) adjustment costs for changing financial positions. According to the status quo bias, people resist adjusting their financial positions. Due to these perceived adjustment costs, later positive outcomes are less valuable in a delay frame, i.e., in a situation where subjects must forego SS positive outcomes in exchange for LL positive outcomes – as is typical for lending decisions. For the same reason, later negative outcomes are going to be costlier, where subjects can avoid sooner negative outcomes in exchange for larger negative outcomes – which is typical for borrowing decisions. Such an approach was introduced by Benzion et al. (1989). To be more specific, they propose the following model for calculating the present value P of a future positive or negative outcome amounting in total to F:
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(𝐹 − 𝐵𝐴𝐶) ∙ 𝑑(𝑡) = 𝑃. (1)
BAC denotes the so-called added compensation premium that is fixed independently of
and expressed in monetary terms. BAC is always positive, since postponing a positive or negative payment requires a financial adjustment, i.e., a financial compensation. Consequently, the LL positive outcomes become less valuable and the LL negative outcomes become costlier according to this assumption. It is noteworthy that for sufficiently high mental adjustment costs in the case of positive outcomes Pα may become negative, although we have F > 0. In such situations with negative present values, the individual will not be willing to switch from the SS to the LL positive outcome. In our later experiment, we fix Pα and then try to determine the corresponding value of F so that there is indifference between the SS and the LL outcome.
Therefore, the problem of different signs cannot become relevant for our experimental setting.
For negative outcomes, there is not a similar problem since the subjective present value of a negative payment F will always be negative, as mental adjustment costs are subtracted as well and otherwise individuals would be willing to pay in the present to pay something (additionally) in the future.
While Benzion et al. (1989) assume BAC to be discounted as well, it would also be pos- sible to assume that discounting is only applied to F. This yields the following equation instead of (1), which corresponds to Equation (6) of Benhabib et al. (2010) and therefore is the base form of their so-called fixed-cost approach (with BFC being the corresponding fixed costs):
𝐹 ∙ 𝑑(𝑡) − 𝐵𝐹𝐶 = 𝑃. (1’)
Apparently, the differentiation between (1) and (1’) can only become relevant for varying values of t, which is not the case in our analysis. As we focus our study on the magnitude effect, we do not delve deeper into details with respect to the relevance of t and simply speak of “(adjustment) cost-based explanations for magnitude effects”. In our Online Appendix, Section 1 we replicate the finding of the literature that mental adjustment costs BAC
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and BFC imply a conventional magnitude effect for positive outcomes. In addition, we extend the literature by showing that these cost-based approaches support a reverse magnitude effect for negative outcomes. The reason for both results is a reduced relative relevance of BAC or BFC
for absolutely higher values of F, which makes an individual willing to accept ceteris paribus lower interest rates for higher positive outcomes and to pay higher interest rates for absolutely higher negative outcomes.
Moreover, as is also shown in our Online Appendix 1, Section 2, both the conventional and the reverse magnitude effect should become less pronounced for ceteris paribus higher values of (or – alternatively – F). This conjecture of a non-constant impact is in accordance with previous literature on positive money frames (e.g., Kirby and Maraković, 1996, or Green et al., 1997), and Hardisty et al. (2013, p. 357) also hint at this possibility.
2.3 A unified approach and a distinction between money and interest rate frame1 The analysis so far has two shortcomings. First of all, the utility- and the cost- based approach are presented as alternatives rather than complements. In what follows, we want to address this issue by assuming a monotone increasing utility function v(F) with the properties v(F)/v(F) > and (for normalization purposes) v(0) = 0 together with an amount BFC of fixed mental adjustment costs for switching from SS to LL outcomes (once again, similar derivations would be possible on the basis of BAC). By postulating v(F)/v(F) > , we acknowledge that there is overwhelming evidence for a conventional magnitude effect in the case of positive outcomes and this condition is necessary to explain this empirical finding on the basis of pure utility considerations. Moreover, until now we have mainly considered choices based on absolute monetary or utility values, but not on rates of return. However, this distinction seems
1 We are very grateful to an anonymous reviewer whose recommendations inspired us to this sub-section.
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to be essential to grasp the potential different consequences of a money and an interest rate frame.
With a utility function v(.) for monetary outcomes, equation (1’) becomes
𝑣(𝐹) ∙ 𝑑(𝑡) − 𝐵𝐹𝐶 = 𝑃. (2)
While d(t) in (1) and (1’) can be regarded as dimensionless, in (2), d(t) describes the monetary equivalent at time 0 of a utility value of 1 at time t. The conventional money frame results for F/P11 > F/P1, which – for positive outcomes and thus v(.) > 0 – means
𝐵𝐹𝐶 > 𝑣(𝛼 ∙ 𝐹) − · 𝑣(𝐹)
1 − · 𝑑(𝑡). (3)
For v(F)/v(F) > and F > 0, the numerator of inequality (3) becomes positive, while the nominator is negative due to > 1. This means that for any nonnegative fixed mental adjustment costs, we arrive at a conventional magnitude effect. Certainly, this is not too surprising, as both the utility- and the cost-based approach support this conclusion when considered separately.
For F < 0, the sign in inequality (3) reverses for the derivation of a conventional magnitude effect. With v(F)/v(F) > and > 1, the fraction of the right-hand side of (3) is positive, implying indeed an upper bound for BFC, up to which we observe a conventional magnitude effect (since the utility-based impact dominates), while for higher values of mental adjustment costs, we end up with a reverse magnitude effect (due to the prevalence of the cost- based argument). This means, for negative outcomes, inequality (3) describes a condition for BFC leading to a reverse magnitude effect. The precise value for this critical lower bound for BFC apparently depends on the utility function under consideration as well as the values of F and . In particular, when considering a class of utility functions v(F) = c·u(F) with u(F)/u(F)
> ( > 1), then v(F)/v(F) > also holds true for all parameter values c > 0 and for given mental adjustment costs BFC, there will always be a critical minimum value for c leading to a
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conventional magnitude effect. To put it another way, as long as the utility effects of negative outcomes are sufficiently pronounced in comparison to the relevance of mental adjustment costs, there will be a conventional magnitude effect for a money frame also in the case of negative outcomes.
The main goal of this paper is to take a closer look at magnitude effects in interest rate frames. Rather surprisingly, until now, there seems to be no attempt to formalize decision- making in an interest rate frame in a similar way, as has been done for the money frame. Though it would in principle be possible to apply the same decision process in an interest rate frame as in a money frame, we conjecture that an interest rate frame may induce people to base their decision on rates of return instead of absolute money amounts. In the present (or near future), the decision-maker is facing a positive SS monetary outcome α·x which can be exchanged against a certain LL monetary outcome α·x·(1+r)t with r being the offered rate of return for this exchange. We now expect the subject to be concerned with a necessary subjectively determined interest rate realized by switching from the SS positive outcome α·x to the LL outcome α·x·(1+r)t so that he/she is indifferent between the SS and the LL outcome. Apparently, compared to a money frame α·x replaces P and F is substituted by α·x·(1+r)t reassuring a generally analogous setting in both frames.
The critical question now is what kind of interest rate will be elicited in an interest rate frame. Here we deem it plausible that the subject will ask him-/herself which rate of return iα he/she earns on his/her SS outcome α·x+BFC after adding the mental adjustment costs as another investment besides foregoing α·x, when v(α·x·(1+r)t) is returned. This gives us the following (implicit) definition of iα with α 1:
(𝛼 · 𝑥 + 𝐵𝐹𝐶) ∙ (1 + 𝑖𝛼)𝑡 = 𝑣(𝛼 · 𝑥 · (1 + 𝑟)𝑡 )
(1 + 𝑖𝛼)𝑡 =𝑣(𝛼·𝑥·(1+𝑟)𝑡 ) 𝛼·𝑥+𝐵𝐹𝐶 .
(4)
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(1+iα)t is thus the utility equivalent at time t of 1 monetary unit at time 0. The individual will demand iα to reach at least a given minimum required rate icrit which we assume to be independent of α, x, and BFC, thus ruling out any kind of ad hoc magnitude effect. This leads to some requirement r = rα on the right-hand side of equation (4). As this is true for all α
1, we can conclude:
𝑣(𝑥 · (1 + 𝑟1)𝑡 )
𝑥 + 𝐵𝐹𝐶 = 𝑣(𝛼 · 𝑥 · (1 + 𝑟𝛼)𝑡 ) 𝛼 · 𝑥 + 𝐵𝐹𝐶
𝑣(𝛼 · 𝑥 · (1 + 𝑟𝛼)𝑡 )
𝑣(𝑥 · (1 + 𝑟1)𝑡 ) =𝛼 · 𝑥 + 𝐵𝐹𝐶 𝑥 + 𝐵𝐹𝐶 .
(5)
For BFC = 0, we immediately get r1 > rα due to our assumption v(α·F)/F > α, as the left-hand side of the last equation of (5) would be greater than α for r1 = rα. Because the right-hand side of the last equation in (5) is a decreasing function of BFC, the conventional magnitude effect will prevail even in the case of positive mental adjustment costs.
In the case of negative outcomes and thus utility values, savings of negative outcomes amounting to an absolute value of α·x are reduced as a consequence of the additional fixed mental adjustment costs BFC. This gives us once again:
−𝑣(𝑥 · (1 + 𝑟1)𝑡 )
−𝑥 − 𝐵𝐹𝐶 = −𝑣(𝛼 · 𝑥 · (1 + 𝑟𝛼)𝑡 )
−𝛼 · 𝑥 − 𝐵𝐹𝐶
𝑣(𝛼·𝑥·(1+𝑟𝛼)𝑡 )
𝑣(𝑥·(1+𝑟1)𝑡 ) =𝛼·𝑥+𝐵𝐹𝐶
𝑥+𝐵𝐹𝐶 .
(6)
Now, for BFC = 0, due to our assumption v(α·F)/F > α, we once again get a conventional magnitude effect, but as the right-hand side of the last equation is an increasing function of BFC, there will be a critical value BFC from which on a reverse magnitude effect prevails. However, in contrast to the situation for a money frame, this critical value is the same for all utility functions of the kind v(F) = c·u(F). In this sense, higher utility effects of negative outcomes will not influence the critical value for BFC beyond which a reverse magnitude effect can be observed. Moreover, for situations with BFC converging to x from below, the right-hand
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side of the last equation in (6) will certainly exceed any upper bounds and therefore imply a reverse magnitude effect, i.e. r1 < rα. The same is not true with respect to the money frame with negative outcomes. Here, a reverse magnitude effect for negative outcomes would require that BFC is satisfying inequality (3). It is by no means clear that BFC in the region of, say, F would imply (3).
In addition, we can use (6) to give an explicit formula for the minimum value of BFC implying a reverse magnitude effect. In order to do so, we simply have to set r1 = rα and determine the corresponding value for BFC. For higher values of BFC, we then will have r1 < rα.
and thus a reverse magnitude effect (note that this critical value is 0 for linear utility).
𝐵𝐹𝐶= 𝑥 · [𝑣(𝛼 · 𝑥 · (1 + 𝑟1)𝑡 ) − 𝛼 · 𝑣(𝑥 · (1 + 𝑟1)𝑡 )]
𝑣(𝑥 · (1 + 𝑟1)𝑡 ) − 𝑣(𝛼 · 𝑥 · (1 + 𝑟1)𝑡 ) . (7) Summarizing, we may expect a reverse magnitude effect for rather small values of BFC in the interest rate frame, while at the same time a conventional magnitude effect should be of more relevance in a money frame with negative outcomes. As a rough intuition, it may be argued that mental adjustment costs of a given amount gain comparatively in importance when relative figures like rates or return are considered and these mental adjustment costs support the occurrence of a reverse magnitude effect in an interest rate frame. Further insights can be obtained when v(.) is specified in detail.
As an example and following other authors (e.g. Chapman & Winquist, 1998), we apply the negative exponential function c·(1–exp(–b·x)) since it exhibits the above stated assumption that v(F)/v(F) > for all values of b. Under this assumption, we are able to determine for which values for BFC we observe which kind of magnitude effect for negative outcomes, when we assume choice situations in line with our experiment of Section 4. More details of our numerical example and calibration are shown in our Online Appendix 1, Section 3. Indeed, we find a region for plausible parameter combinations for which a conventional
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magnitude effect is at work in the money frame and at the same time a reverse magnitude effect in the interest rate frame (see the shaded area in Fig. A1). This numerical analysis supports the theoretical insights according to which moderate values for mental adjustment costs (in our case values of BFC exceeding about EUR 1.07 are necessary) should imply a conventional magnitude effect in the money frame and simultaneously a reverse magnitude effect in the interest rate frame for negative outcomes. Nevertheless, eventually, the relative importance of both kinds of magnitude effects for negative outcomes is an issue that has to be settled empirically, and therefore we now formulate our hypotheses.
3 Hypotheses
All of the theories presented in Section 2 predict a conventional magnitude effect for positive outcomes but may lead to opposing predictions for negative outcomes with a money frame giving more support to a conventional magnitude effect for negative outcomes than the interest rate frame. This gives us the following hypotheses as the basis for our experimental analysis in the following sections.
Hypothesis 1. For positive outcomes, we observe a conventional magnitude effect regardless
of the prevailing frame.
Hypothesis 2. For negative outcomes, we observe a conventional magnitude effect in the money
frame, which is weaker than for positive outcomes due to the possibility that some individuals with rather high mental adjustment costs exhibit a reverse magnitude effect.
Hypothesis 3. For negative outcomes, we observe a reverse magnitude effect in the interest
rate frame, which is weaker in absolute terms than the conventional magnitude effect for positive outcomes due to the possibility that some individuals with rather low mental adjustment costs exhibit a conventional magnitude effect.
It is worth mentioning that our Hypotheses 2 and 3 stand in contrast to Hardisty et al.
(2013) and their theory. Another difference between our mainly status quo-driven explanation
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for a reverse magnitude effect and that of them is that according to Hardisty et al. (2013, p. 359) the reverse magnitude effect should be weakened in a delay frame where negative outcomes at different future points in time are compared, because here there is no possibility of realizing immediate losses. We will not describe this distinction as a hypothesis, since we expect “no influence” from a potential delay of all outcomes
4 Experiment
The main objective of our experimental setting is to find out whether the theoretical considerations are of practical relevance for describing the individual intertemporal choice behavior for different positive or negative outcomes in the money or the interest rate frame. We examine these two conditional aspects – outcome sign and disclosing information – by conducting an experiment with a Multiple Price List (MPL).
4.1 Method and general setting
The MPL is a widely used and simple procedure for eliciting subjective discount rates with an array of ordered prices in a table. In general, participants are asked to choose in each row between an SS and an LL outcome. We determine the corresponding observable subjective discount rates by using the “full” MPL method instead of the iterative MPL method in order to alleviate potential problems of anchoring effects, i.e., decision-makers’ tendency to rely too heavily on the first piece of information that is offered (see Frederick et al., 2002). In contrast, Andersen et al. (2006) raise the question of whether the MPL might bias a participant’s choice towards the middle of the table due to a corresponding general psychological bias towards the middle. However, their findings imply that this framing effect is negligible.
The experimental design as a money frame or an interest rate frame is only alternated for the LL outcome. The outcome sign is swapped against a different textual description in each MPL from the beginning (see Section 4.3 and Table 1 for more details of our MPL). To control for other influencing factors, we also alter the questions in the following dimensions: outcome
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size (EUR 4, EUR 36, and EUR 324 for the SS outcome) and time frame (now vs. in 9 months, now vs. in 18 months, and in 9 vs. in 18 months).
In total, our dimensions of interest (outcome size, time, outcome sign, and disclosing information) result in 3 × 3 × 2 × 2 = 36 different choices neglecting the ordering of individual characteristics in each dimension. Since we also want to control for potential ordering effects in two dimensions (outcome size and time), the number of different choices increases by a permutation of these two factors to 3! × 3! × 36 = 1,296 choices. As this high number would require too many participants for our experiment, we use a mixture of a within-subject and a between-subject design for the different dimensions. Outcome sign and disclosing information only vary among subjects, while outcome size and time vary within subjects, but simultaneously not for the same subject. A list of all used choices is shown in Table A.2 in our Online Appendix 2 as supplementary material.
Our procedure splits our participant pool into two halves. In the former half, outcome size varies within subjects while time is fixed; for the latter, this structure is swapped. The three outcome sizes small (EUR 4), medium (EUR 36), and large (EUR 324) grow with a constant factor of 9: EUR 4 × 9 = EUR 36 and EUR 36 × 9 = EUR 324. For the first 12 groups, time (now vs. 9 months, now vs. 18 months, and 9 vs. 18 months), outcome sign (positive vs.
negative), and disclosing information (money frame vs. interest rate frame) are between-subject arranged, implying 3 × 2 × 2 = 12 different groups, while outcome size is within-subject specified. To control for order effects, the sequence of the presentation of the three outcome sizes (4, 36, and 324) is varied across subjects so that each outcome size ranks first for roughly the same number of individuals. Therefore, we present three different size-ordered variants of the 12 groups and, consequently, 36 subgroups to our subjects.
For the second half, we swap outcome size and time in our within-/between-subject design. Now, the time dimension is within-subject and the outcome size between-subject,
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leading once again to 3 × 2 × 2 = 12 different groups. In turn, a potential order effect is controlled for by varying the sequence of the presentation across subjects – in this case – of the 3 time spans, so that we have 36 variants of groups again. Summing up, we have 72 subgroups with which we can test the magnitude effect within and between subjects. Every participant is randomly forwarded from a fixed URL to one of these 72 subgroups until each subgroup has reached roughly a number of 10 (median) valid subjects.
4.2 Participants
We conducted a web-based experiment in order to achieve a large number of participants. At the beginning, we informed around 2,000 students of a German university enrolled in different programs about the opportunity to participate in an experiment. The survey was accessible for all students, and participation was incentivized by a random lottery. Subjects were truthfully informed that six respondents would be fully paid based on their choices made in one randomly selected question from one of the presented MPLs. In order to assure that each student would participate in the questionnaire only once, participants had to register with their unique university email address. In total, we have 676 valid participants excluding 8 who switched more than once in the MPL and ignoring 69 who stated that they were not interested in the offered remunerations.
4.3 Design of individual parts
The participation rules, payments, and the expected duration were announced on the first screen of the questionnaire (see Online Appendix 3 as supplementary material). Then, all participants had to state basic socio-demographic data (Female and Age) and whether a subject was familiar with the concept of discounting (DCF Exp.). By using Major (students enrolled in Economics or Business programs) as a surrogate for DCF Exp., we find similar results.
In the next step, the MPL section started with an introductory part and three choice tasks out of all 72 subgroups. In general, participants were asked to choose between an SS and an LL
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outcome arranged by different outcome size, time, outcome sign, and disclosing information.
The outcome sign (positive vs. negative) and disclosing information (money frame vs. interest rate frame) are always between-subject arranged, the outcome size (EUR 4, EUR 36, and EUR 324) and the time frame (now vs. in 9 months, now vs. in 18 months, and in 9 vs. in 18 months) vary between-subject and within-subject.
On the first screen of the MPL section, we wanted to make sure that participants had understood the concept of compound interest rate in the interest rate frame in order to avoid the confounding influence of potential misunderstandings on our experimental results. For this purpose, we started with an example with which we explained how the compounding principle works. The illustration contained the same text but differed in numbers for the money amount, interest rate, and time spans from the subsequent MPLs. Afterwards, the participants were confronted with the MPL tasks, which had the following basic form as in Table 1. To counteract the perception that future outcomes are inherently uncertain (Halevy, 2008), we pointed out that payoffs for all experiments are always guaranteed and paid by the corresponding professor. The selected students had to pick up the reward at his office which is centrally located on campus so that we assume that potential transaction costs of the pick-up – even if present and affecting all magnitudes in both frames in the same way – are of minor importance. This assumption is supported as all students picked up their rewards a few days after the informing email in which they were told to do so. We actually followed the desired time of payment (e.g. now vs. in 9 months) as chosen by the students.
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Table 1
Tabular format of an MPL in our questionnaire.
[translated from German by the authors]
Because of your good work, your employer will pay you a bonus of EUR 4.00 right now. Alternatively you can choose to get a later payment. These two options (Options A and B) are now provided in 20 different constellations. Which of the two options do you prefer?
Note: You can assume that there is no uncertainty about payment, since payment is guaranteed by the University’s Department of Finance.
Option A Option B
You change nothing and keep the bonus amounting to EUR
4.00 now.
You waive the original bonus and will receive the bonus plus interest according to one of the following yearly interest rates in
9 months’ time.
Which option do you prefer for each question?
Option A Option B
1 EUR 4.00 2.5 % ○ ○
2 EUR 4.00 5.0 % ○ ○
… … … … …
20 EUR 4.00 50.0 % ○ ○
○ Ideally, I would choose neither Option A nor B; e.g. because I am not interested in a cash payment of this kind.
The outcome for the SS amount is varied by three size alternatives (EUR 4, EUR 36, and EUR 324); the three time dimensions alternate between now vs. 9 months, now vs. 18 months, and 9 months vs. 18 months. Consequently, a complete choice task consists of three MPLs. The LL amount (option B) is increased stepwise by 2.5 percentage points in each step up to 50 % per year in the interest rate frame. In the money frame, the LL amounts are presented in euro by using exactly the same percentage point increases as in the interest rate frame displayed in Table 1. The remainder part – structure and text – is basically the same. The screenshots of the full experiment and their translation from German into English are shown in our Online Appendix 3.
We wanted our participants to put themselves in the shoes of an actual person who is facing this specific decision problem. Therefore, we tried to describe a tangible real-life scenario for our decision-makers in positive and negative outcome settings. This helps us to
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close the so-called empathy gap in such decision tasks (Faro and Rottenstreich, 2006). For the positive outcome, we described it in such a way as if it had already been decided that the subject was going to receive a bonus (the SS amount) from the employer. This payment announcement denotes a status quo for the subject. A participant could change this position by choosing the LL amount.
For negative outcomes, we had to organize a lottery where only the rewarded participants paid the lottery price in the end in order to avoid that all our participants must immediately pay for conducting the experiment. This idea follows the approach that experiments with students have normally no negative final rewards and that such a procedure would be in breach of the rules of our experimental laboratory. Thus, we had to select one of the following concepts: hypothetical loss, loss from an initial endowment, or loss from a voucher. The general idea for negative outcomes is to have an incentive-compatible procedure following the argument that hypothetical choices may be unrealistic resulting in unreliable data (e.g. Harrison, 1994; Cox and Grether, 1996). As a consequence, Andersen et al. (2014) ignored all hypothetical survey studies in their review since they assume systematic “hypothetical biases”. Thus, we deny hypothetical losses to avoid a hypothetical bias even if this procedure is the general standard in other intertemporal choice experiments (e.g. Mitchell and Wilson, 2010; Hardisty et al., 2013). An initial endowment without or with effort tasks could be a solution to be able to pay for real losses. However, we see the following two opposing problems (Thaler and Johnson, 1990): “house money” effect and “prospect theory with memory” effect.
For the former, subjects may not consider the presented loss as real and play with the money.
This may affect their behavior. For the latter, subjects may balance the initial endowment with the latter stated loss resulting in a positive money domain in their mental account. Based on the general finding from prospect theory that individuals act in different ways in loss domains and gain domains, we would actually test more on positive outcomes rather than on negative ones
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as intended. Thus, we decided to use a voucher procedure as an initial endowment. The objective value of the voucher (movie theater, restaurant, smartphone) is always higher than the respective loss. Assuming that the subjective values of our selected vouchers are higher than the losses, we avoid that the participants have a negative net payment in the end but minimize or even prevent the “prospect-theory-with-memory” effect since it is harder for the subjects to mentally combine the voucher value with the loss.
Within our negative setting, a successful subject had actually to buy a lottery ticket in three different voucher categories (outcome size: small (2 tickets for a visit to a movie theater), medium (a meal for 2 persons in a fine restaurant), and large (a smartphone)). Certainly, the perceived value/utility of a voucher category may be different for different individuals, but for each voucher category the corresponding reward and the reception date were fixed and thus not the object of our choice tasks. As is common in experiments on intertemporal choice, we assume some kind of mental accounting, i.e. that individuals choose between intertemporal alternatives regardless of their overall (fixed) wealth and consumption position. In this sense, we also neglect different utility levels for individuals connected with the same voucher category. In any case, even if there were some confounding effects of this kind, we would expect them to be independent of the framing of the intertemporal choice and thus to be only of minor relevance for our main research question. Nevertheless, we will return to this issue later on. Then, the subjects participated in a lottery where they could choose between paying the SS amount (small: EUR 4, medium: EUR 36, or large: EUR 324) or switching to the LL amount in each size category when the lottery prize was delivered. This means that only lottery winners had to pay money according to their choice either in the present or in the future. The remaining manipulation dimensions – time and disclosing information – are the same as in the positive outcome scenarios. At the end of each negative MPL, we asked whether a subject was actually interested in winning a movie theater ticket, a meal, or a smartphone. 46 participants
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who responded with “no” were excluded from the sample. For reasons of symmetry, we also asked subjects in the positive scenario whether they were interested in “a cash payment of this kind” (see Table 1) and also excluded 23 participants who were not willing to accept a cash payment.
To cover further (interval- or delay-related) discounting anomalies, which are well discussed in previous literature (see, e.g., Read et al., 2005, or Andersen et al., 2014 for an excellent overview), we included two interval lengths and a delay variable in our experiment.
The variable Long Interval takes the value 1 for longer intervals (18 months) and 0 for shorter intervals (9 months) between the SS and LL choice. Similarly, Delay is coded with 1 if SS is delayed for 9 months and with 0 if the SS choice is immediately executed. With Position, we control for the sequence of the three MPLs presented to each subject (position 1-3). As already mentioned above, interactions between Delay and the (reverse) magnitude effect for negative outcomes would support Hardisty’s et al. (2013) argument of a present bias at work. Therefore, later on we will take a closer look at this relationship.
Furthermore, we see the possibility that subjects may face difficulties paying their future obligations. This is probably most relevant for the highest size category, where negative payments ranged between EUR 324 and EUR 595, even if a subject’s net payment is always positive (e.g., equivalent market value of a smartphone minus the lottery ticket price). Such participants should have a stronger tendency to think in absolute monetary values even if results are presented in an interest rate frame. Consequently, a potential reverse magnitude effect should be of minor importance in the interest rate frame with negative outcomes. Therefore, we asked our participants in the case of a negative future outcome whether they would be able to repay this amount, and we code a denial as Cannot Afford. For Cannot Afford = 1, the conventional magnitude effect for negative outcomes should gain more relevance in the interest rate frame.
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At the end of the questionnaire, all participants had to calculate two rates of return per annum in order to test their mental arithmetic skills (Comput. Error) in computing effective interest rates, as these are not directly disclosed in the money frame. They were informed about two different money amounts (the second amount is either 10 % or 25 % higher) of different outcome sizes (ranging from EUR 4 to EUR 2,916) for two different investment durations (2 and 3 years, respectively). The complexity of the discounting/compounding principle might lead to calculation errors, since it requires working with exponential functions, and this task might be too much to handle for naïve investors. The cognitive psychology literature related to this “exponential growth bias” began with Wagenaar and Sagaria (1975). Stango and Zinman (2009) give an excellent review of people’s tendency to linearize exponential functions if they estimate them intuitively.
Computational errors for converting money amounts into rates of return (or the other way around) may cause larger required positive outcomes or smaller ones, depending on the sign of the computational error when calculating interest rates and depending on the question of whether subjects convert changes in money amounts into rates of return or vice versa. For participants primarily thinking in rates of return, computational errors should be more influential in the money frame, while subjects relying more on absolute money amounts should be prone to computational errors in the interest rate frame. A lack of relevance of computational errors would hint at the possibility that people think in rates of return in the interest rate frame and in absolute money amounts in the money frame, making it unnecessary to recalculate the LL outcomes. Moreover, computational errors may be smaller in absolute terms for those subjects who relied on electronic devices to perform their calculations. As the subjects’
computational abilities are not the main focus of our analysis, it suffices to include as a control variable Comput. Error being equal to the average relative deviation between the estimated and
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the actual interest rate in order to account for any significant influence of different computational abilities.
Furthermore, to check subjects’ financial literacy (Fin. Literacy) and to assess each individual’s opportunity costs for investing and lending, all participants had to estimate five current market rates (deposit rate for a holding period of one year, 10-year government bond rates for Ireland and Germany, consumer loan rate, and student loan rate). The order of all questions was permutated among all participants. We expect individuals with higher market rates estimates to demand ceteris paribus higher LL positive outcomes due to opportunity cost- based considerations. For the same reason, these subjects should be willing to accept ceteris paribus higher LL negative payments. To measure financial literacy, we calculate the average relative deviation between the estimated and the actual rate across all five current market rates (Fin. Literacy).
Comput. Error and Fin. Literacy describe the relative estimation error for each rate and are averaged for each subject as well as winsorized at the 10th and 90th percentiles in order to control for extreme outliers (e.g. overestimations of Fin. Literacy with more than 1,000 %).
Table 2 presents major descriptive statistics regarding participants’ socio-demographic information, arithmetic skills, and financial literacy. In the experiment, 35 % of our subjects are female and the average age is 23 years; 66 % of the subjects have Economics or Business as their major and 50 % of them are familiar with the concept of discounting. For negative payments, 11 % of the participants state that they cannot afford the price of a lottery ticket.
Furthermore, they underestimate by 4.3 % the two rates of return and overestimate the five current market rates by 296 %. Even after winsorizing, the extreme values are high in absolute terms, e.g., 985 % for Fin. Literacy.
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Table 2
Summary statistics for socio-demographic variables.
Mean Min. Median Max. Std. Dev. N
Female 0.35 0 0 1 0.48 676
Age 22.9 17 22.5 35 3.2 676
Major 0.66 0 1 1 0.47 676
DCF Exp. 0.50 0 1 1 0.50 676
Cannot Afford 0.11 0 0 1 0.31 323
Comput. Error -4.3 -57.9 -1.7 54.9 26.7 676 Fin. Literacy 295.9 -69.0 210.9 985.0 284.4 676 This table shows the descriptive statistics for the socio-demographic variables of our subjects. Std. Dev. stands for standard deviation. Female, Major, Age, DCF Exp., and Cannot Afford are dummy variables. Female indicates a participant’s sex (male = 0, female = 1), Participants’ Age is measured in full years, Major expresses whether students have Economics or Business as their major subject (non-Economics/Business students = 0, Economics/Business students = 1), DCF Exp. whether they had comple- ted a basic finance course which contains the concept of discounting (no = 0, yes = 1), and Cannot Afford whether they state that they cannot afford the price of a lottery ticket for negative outcomes (no = 0, yes = 1). Comput. Error measures the averaged relative estimation error between two estimations and their actual rates of return. Fin.
Literacy is the averaged relative estimation error of five current market rate estimations; both estimation errors are in %.
5 Analysis and results
We start with univariate tests to see whether the theoretical considerations are suitable to describe the magnitude effect in a distinctive way conditionally on their outcome sign and framing. The dependent variable is the midpoint of the switching point of the subjective discount rate between the SS and the LL choice (middle of 2.5 percentage point increments in the MPL), denoted as Switching Point. According to our hypotheses, these switching points may differ for positive and negative outcomes in the money frame and in the interest rate frame.
In the next step, we switch to multivariate analyses in order to account for differences among participants in different groups in our between-subject design. Furthermore, we are able to control for other potential influencing determinants of subjective discount rates and can use interval regressions for our interval-censored dependent variable.
5.1 Univariate tests
According to Hypotheses 1 and 3, we expect a conventional magnitude effect for an interest rate frame with positive outcomes, i.e., decreasing observable subjective discount rates,
