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Revising future valuation: Updating Capacities and Upstating Discount Factors

Revising future valuation: Updating Capacities and Upstating Discount Factors. Robert Kast CNRS, Laboratoire Montpelliérain d’Economie Théorique et Appliquée et Institut d’Economie Publique André Lapied Professor University of Aix-Marseille

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Revising future valuation: Updating Capacities and Upstating Discount Factors

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  1. Revising future valuation: Updating Capacities and Upstating Discount Factors Robert Kast CNRS,Laboratoire Montpelliérain d’Economie Théorique et Appliquée et Institut d’Economie Publique André Lapied Professor University of Aix-Marseille Groupement de Recherche en Economie Quantitative d’Aix-Marseille et Institut d’Economie Publique

  2. Motivation: revising valuation in a present certainty equivalent value (NPV) • Integrate option values and flexibilities • In a dynamic setting, the future is described by, at least: • Uncertain states: S • Dates in future Time: T • Information arrivals: measurable functions that take value at some dates • We concentrate on cash flows X: SXT  R+

  3. The role of information on preferences and valuation • Information is not neutral: the way it is integrated in preferences depends on its type. • Good news, i.e. information that confirms forecasts may induce finance analysts to be too optimistic • Bad news are not integrated in the same way • An information that comes too late may induce a change in the preference for present consumption • An information that comes too early may the DM leave unchanged the discount factors • None of these effects are taken care of by usual NPV calculations

  4. Classical NPV and the introduction of non-additive measures • Classical NPV is done with additive measures on uncertain states (probabilities) and on dates (time separable discount factors). • Bayes updating rule and accountants’ time consistency apply: r(T) = r(t) rt(T) • Other rules are obtained for capacities, depending on axioms:Bayes (B), Dempster-Shafer (D), or the Full Bayesian Uptdate (FUBU)

  5. Problems with Dynamic Consistency • Horie (2006) proposes a general updating rule that obtains all others as particular cases. • Most models weaken or drop altogether DC (Eicheberger et al. 2005), they concentrate on some subset of information sets Denneberg 1994, Chateauneuf et al. 2003) • . Sarin Wakker (1998) : under model consistency (MC), dynamic consistency (DC) and consequentialism (C), only the multiprior model fits, that leaves B and FUBU. • Ghirardato (2004) shows that axioms on the whole sigma-algebra yields B only. • In this paper consequentialism is violated by the rules.

  6. X(s, t), w. l. o. g. And let’s add: X(0, 0) = 0 Notations • The future is: ST,S = {1,… , S}, T = {1, … , T}. • Sis endowed with a filtrationF indexed by time: F1 …  FT = 2S. • T is measurable (T,2T). • A cash flow is a non-negative measurable real function: • X: (S T, 2S 2T) R+, and X = [X(1,1), … , X(s,t), … X(S,T)]. • It can be seen as a stochastic process adapted to F: • X = (X1, … , XT), with tT,Xt= X(.,t), Ft-measurable. • Or as a list of trajectories indexed by states in S: X = [X(s, .)]sS with  s  S,X(s, .):(T, 2T)  R+

  7. Preferences valuation and measures 1-Pref. oncertain cash flows are represented by D: RT R, Axioms by Koopmans (1972), or Gilboa (1989) increasing non negative bounded measures on T: p(additive) or r s. t. D is the Choquet integral of payoffs w.r.t. the measure. 2-Pref. onuncertain payoffsare represented by: E :RS R, Axioms by de Finetti (1932) or Chateauneuf (1991),  probability or capacity measures on S: m (additive) or n s.t. E is the Choquet expectation of payoffs w.r.t. the measure. 3- Preferences onuncertain cash flows are represented whether by DE: Discounted Expected uncertain payoffs or by ED: Expected Discounted payoff trajectories.

  8. E(Xt) = Xt(s) Dn(s,t) (1) ,or:E(X)= . Xt(s) m(s)(1') D[X(s,.)] = X(s,t) Dr(s,t)(2) ,or:D(X(s,.)]= X(s,t) p(t)(2’) X(s,t) Dn(s,t)] Dr(t) (3) DE(X) = X(s,t) Dm(s,t)] Dp(t) DE(X) = X(s,t) Dr(s,t)] Dn(s) (4) ED(X) = X(s,t) Dp(s,t)] Dm(s) = =ED(X) Expectations and Discounting Expectations and additive expectations (certainty equivalents): Dn(s,t) = n{ s’ S / X(s’,t)  X(s,t)} - n{s’ S / X(s’,t) > X(s,t)} Discounting and time-separable discounting (present equivalents): Dr(s,t) = r{ t’ T / X(s,t’) X(s,t)} - r{t’ T / X(s,t’) > X(s,t)} Discounted Expected payoffs: Expected Discounted payoffs:

  9. X(s,t) Dn [Yt= i](s,t) pt(t) (3’) Here: DtE[Yt= i](X) = E[Yt= i] Dt(X) = X(s,t) Drt(s,t)m [Yt= i](s)(4’) Information and conditoning Information is yielded by an adpted process(Yt)t=1…T-1in(I, 2I). At date t, information is [Yt=i]inFt. Model Consistency: for any date t and information [Yt =i], conditional preferences satisfy the same axioms and expectations and discounting are computed according to the same models as those at the initial date. Two assumptions: # Only one measure is non-additive: n or r # Conditional discounting is non-random: r [Yt= i] = rt

  10. Hierarchy 1: States then Dates Discounting Expected payoffs, formulas (3) and (3’): DE Dynamic consistency: DE(X) = DE [ (X1, ... , Xt-1, Xt + DtE[Yt= i](X) , 0, … , 0) ]. In the case X = (0, …,0, XT): DE(XT)=DE[DtE[Yt=i](XT)], Separable discountingD(x) = DDt(x) :r(T) = r(t) rt(t+1, …,T) And simplifying away, Dynamic consistency: E(XT)=E [ E[Yt= i](XT) ]. i.e.: It yields an implicit definition of conditional Choquet expectation, from which updating rules are obtained.

  11. 1B 1A 1 1 1 0 1 1 0 1 X 0 0 or 1 0 1 1 or X' 0 1 Graph 2 Updtating rules for a capacity n(A/B)= E[1B=1](1A), If information is comonotonic with payoffs, Bayes' rule prevails: If information is antimonotonic (1Bcomonotonic with –1A) with payoffs, Dempster-Shafer’s rule obtains:

  12. interpretations • The formulas show the importance of the relation between information and future payoffs, even those that are not possible anymore. • If information is in accordance with future payoffs, it is taken as it is: B conditions the future payoffs (Bayes rule confirms forecasts). • If information goes the other way than future payoffs, its complementary is given more importance (Dempster-Shafer rule infirms forecasts). • But, in both cases, this behaviour shows a lack of confidence in information (Chateauneuf et al. 2003).

  13. Hierarchy 2: Dates then States Formulas (4) and (4’) Dynamic consistency:ED(X) = ED[(X1(s), … , Xt-1(s), Xt(s) + E[Yt= i]Dt[X(s)], 0, … , 0)s[Yt= i]]. In the case of a cash flow of the form: 1E with E a subset of T, and lett- = {1, … , t-1} and t+ = {t+1, ... , T} and ED(1E) = D(1E) = r(E), EDt(1E) = rt(Et+), Then, Dynamic Consistency becomes: ED(1E) = r(E) = ED[(1tE)].

  14. D(X) = xt Dr(t) + r(t) [xt + xt Drt(t) ] Upstating: a special case E = {1, ... ,T} say, a constant coupon on a bond: Which yields, for a riskless cash flow X = (x1, … , xT) where all the xt’s are strictly positive payoffs: Let Et = { t’>t / xt’> xt}, then: Drt(t) = rt( [Et{t’/xt’ = xt}] t+) - rt(Et t+)

  15. Upstating rules for time non-separable discount factors Because of rank dependency: If tE: If tE and If tE and

  16. interpretations • The formulas show the importance of the timing of information arrivals with respect to the payoffs, even the past ones. • It depends also on the preference for present consumption and of aversion to time variability. • An information that arrives after too much payoffs have been cashed, doesn’t change much the valuation. • An information that arrives when a lot of cash is expected does affect valuation and decisions: flexibilities, options, etc.

  17. Updating and Upstating rules violate consequentialism Consequentialism: pref. depend only on future possible consequences Conseqentialism for uncertain payoffs sB, X(s)=X’(s) and Y(s)=Y’(s); and sB, X(s)=Y(s) and X’(s)=Y’(s). Dt E[Yt=i](X)  Dt E[Yt=i](X')  Dt E[Yt=i](Y)  Dt E[Yt=i](Y') But updating depends on payoffs that are NOT POSSIBLE after information Conseqentialism for trajectories Dt(X)  Dt(X')  Dt(Y)  Dt(Y') But upstating depends not only on FUTURE, but also on PAST payoffs

  18. To be continued … • Applications 1: Managing insurance portfolios when static vs dynamic matters: abondon « consequentialism »? • Application 2: The observed behaviours of Finance analysts (forecasts): sensitive on the « comonotonicity » of information and previous predictions?, i.e.: (too) optimistic when information is « good » and (too) pessimistic when information is « bad ». • At the theoretical level: 1. What if rtis really r[Yt= i]: what is DE(r[Yt= i] E[Yt= i])? 2. How to formalise preferences if they are contingent on other parameters than dates and states? 3. Or on the future as a whole (not a product space)?

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