Chapter 3: Model and propositions
3.3 Solving the model
3.3.1 The project evaluation decision
The project evaluation decision constitutes the third and ultimate step of the investment decision process. Project evaluation describes the choice problem a manager faces when reviewing a given ongoing investment project. Following negative feedback on the performance of the investment, meaning that the project is underperforming, the manager may either terminate the investment, resulting in a termination value [L], or continue it. If continued, the outcome of the investment
31 Even though backward induction is typically used in games with several strategic players, it is also plausible to assume that a single decision maker will make choices with consideration of potential outcomes.
Chapter 3: Model and propositions
depends on the realisation of an external binary variable. With probability [q], the NPV of the investment under continuation will be small but non-negative [CVh with CVh ≥0], or negative [CV withl CVl <0 and L>CVl] with probability [(1-q)]. Figure 3.2 shows the relevant section of the decision tree (Figure 3.1).
Figure 3.2: Project evaluation decision problem
Overconfidence and regret aversion are integrated into this decision problem following the general modelling definitions of these two psychological biases that were outlined in the preceding section. Overconfidence leads the manager to make estimates of the probability of the good state of nature with which a value of [CV ] is achievable that h are biased by the degree of his overconfidence in the state of nature, optimistic overconfidence, denoted by [b]. Following the general definition of regret (Eq-3.11), regret in the project evaluation decision depends on whether it is expected (ex-ante) that the outcome of the alternative choice option will be observable ex-post. It is evident that the model thus will lead to different predictions regarding the effect of regret aversion depending on which assumption regarding ex-post outcome observability is made. Theoretically, it is possible to distinguish between four different cases regarding expected outcome observability:
• Only the outcome of the chosen strategy will be known, so that the outcome of the alternative choice option will definitely not be observable (no observability): In the absence of counterfactual information, there should be no regret anticipated for either choice option. This case is thus effectively identical to assuming a decision maker who does not feel regret at all.
• Whichever strategy is chosen, both outcomes (continuation and abandonment) will be observable to the manager ex-post (total outcome observability): Interestingly, for this case, the effects of regret aversion associated with either choice option exactly offset and cancel out in my model (see Appendix A).
M3
N
q
1-q continue
abandon
CVh
CVl
L
Capital investment decisions with managerial overconfidence and regret aversion
• If continued, the abandonment value will not be observable, whilst the continuation value will be observable even if the investment is abandoned (type 1 partial outcome observability): For example, this situation may occur when the outcome of continuing the investment depends on an external and observable variable such as the price of crude oil, whilst at the same time, the abandonment value depends on a number of factors that can only be ascertained if the project were indeed terminated.
• The value of the investment if abandoned is certain and will be known even if the investment is continued, whilst the continuation value is uncertain and will not be known if the investment is abandoned (type 2 partial outcome observability): This is the case where the scrap value of an investment is known, and will be known even if the investment is continued, but where the continuation value would not be observable ex-post subsequent to termination of the investment.
As the decision tree diagram in Figure 3.2 shows, the formal analysis here will be limited32 to the second type of partial observability.
The manager will hence feel regret if he chose to continue the investment but only obtains [CV ] since he would have been better off had he abandoned the investment l [L>CVl]. Consistent with the definition provided in the preceding section (Eq-3.11), the amount of regret in this case – having continued when abandonment would have been better – is given by
(Eq-3.13) Rct =CVl −L with [Rct <0].
Assuming a risk-neutral decision maker, the investment's expected continuation value (general form) is
(Eq-3.14) CV =q(1−b)CVh +(1−q(1−b))⋅(CVl +rRct).
32 Focussing on this particular case limits the complexity of the model. Moreover, of the different assumptions on outcome observability, this seems to be the most reasonable one with respect to the problem of knowing the outcome of a foregone choice option: Ignoring the possibility of alternative investment opportunities that might be taken up following abandonment, the cash flow a discontinued activity is arguably at least less uncertain than that for continuation, so that assuming type 2 partial outcome observability for this analysis would appear to be plausible.
Chapter 3: Model and propositions
Note that in this equation an asymmetry with a bias towards the good outcome emerges. However, this is intentional and induced by the specific modelling of optimistic overconfidence as an unjustified upward bias in probability estimates (in contrast to overconfidence in ability), which is consistent with the phenomenon of overconfidence as outlined previously in the review of the literature.
o For an unbiased decision maker, the bias parameters take neutral values [b=0;r=0] to yield the unbiased continuation value [CV ]: u
(Eq-3.15) CVu =qCVh +(1−q)CVl.
Because of the assumption of type 2 partial outcome observability, the termination value of the investment is known with certainty and unaffected by the cognitive biases.
Consequently, as demonstrated in the chapter on optimal investment decision making, shareholder value maximization requires an investment to be abandoned if the value expected for continuation is less than what would be achievable with termination.
(Cond-3.1) L>CVu.
The optimality of this decision rule is based on [CV ] representing the rationally u expected continuation value.
o The overconfident manager believes that achieving the high continuation value [CV ] is more likely than it really is. Given this intuition, and by considering (Eq-h 3.14) it can be seen that the subjectively expected continuation value is increasing in the level of overconfidence, or >0
∂
∂ b
CV , and henceCV
(
b>0;r =0)
>CVu; that is, an overconfident manager who is otherwise unbiased (no regret aversion) overestimates the continuation value.o In contrast, the regret-averse manager, who is well calibrated (zero overconfidence), will underestimate the continuation value since as the individual regret aversion parameter [ r ] becomes larger, the subjectively perceived continuation value decreases, or <0
∂
∂ r
CV , and, therefore,CV(b=0;r >0)>CVu. Intuitively, the anticipated disutility of regret from having continued the investment and finding out
Capital investment decisions with managerial overconfidence and regret aversion
subsequently that abandonment would have yielded a higher value reduces the expected continuation value with weight [ r ].
For the evaluation decision, the effect of overconfidence is thus directly opposite to that of regret aversion: An overconfident manager will overestimate the continuation value of an investment, whereas a regret-averse manager will underestimate it.
Therefore, it is interesting to consider the interaction effects resulting from a combination of these biases in more detail. To do this, one may thus take any level of managerial regret aversion [ r ] and define a critical level of overconfidence [b′] such thatCV(b=b′;r)=CVu. Solving33 this condition using (Eq-3.14) and (Eq-3.15) for different levels of regret aversion is illustrated for assumptions of [CVh,CVl,L] in the diagram in figure 3.3.
33 For the formal proof please refer to Appendix B.
Figure 3.3: Interaction of overconfidence and regret aversion in project evaluation
0
Assumptions: CVh=20; CVl=(-30); L=(-10); q=0.3;
effect of overconfidence dominates
effect of regret aversion dominates
Chapter 3: Model and propositions
The concave line illustrates combinations of overconfidence and regret aversion that yield the optimal decision behaviour. Any point above this line represents a situation where [b>b′] such that the effect of overconfidence dominates that of regret aversion, and hence that the manager overestimates the continuation value; conversely, at any point below the line, the regret aversion drives the net effect by more than off-setting the impact of overconfidence, and the manager consequently underestimates the continuation value. Based on the preceding analysis, it is possible to state a first result.
Proposition 1: The effect of managerial overconfidence regarding the state of nature and regret aversion on the abandonment / continuation decision given that negative information on the investment's performance was received;
(a) IfCV =CV(b>b′;r),CV >CVu follows, and hence the manager overestimates the continuation value. Therefore,
i) If CV >CVu >L, the biased manager makes the efficient decision to continue the investment.
ii) If L>CV >CVu, the biased manager makes the efficient decision to abandon the investment.
iii) If CV >L>CVu, the biased manager makes the inefficient decision to continue the investment.
(b) If CV =CV(b<b′;r),CV <CVu follows, and hence the manager underestimates the continuation value. Therefore,
i) If CVu >CV >L, the biased manager makes the efficient decision to continue the investment.
ii) If L>CVu >CV the biased manager makes the efficient decision to abandon the investment.
iii) If CVu >L>CV the biased manager makes the inefficient decision to abandon the investment.
(c) If CV =CV(b=b′;r), CV =CVu follows, and hence the manager makes efficient continuation or abandonment decisions.
As Proposition 1(c) demonstrates, optimistic overconfidence can lead to the correct abandonment decision, but it can also lead to sub-optimal decisions that may destroy
Capital investment decisions with managerial overconfidence and regret aversion
shareholder value. Similarly, although regret aversion does not necessarily have to affect a manager's investment evaluation, Proposition 1(b) demonstrates that there are conditions under which regret aversion leads to the sub-optimal (premature) termination34 of an investment. Given these individual effects, it was shown that a manager who is both overconfident and regret averse may not only over- or underestimate the continuation value, but may also arrive at the 'correct' unbiased forecast based on rational expectations. Similar to Besharov (2002), my model thus allows for the optimal behaviour despite the existence35 of psychological biases.