Revising future valuation: Updating Capacities and Upstating Discount Factors

1
Revising future valuation Updating Capacities
and Upstating Discount Factors

• Robert Kast
• CNRS,Laboratoire Montpelliérain dEconomie
  Théorique et Appliquée
• et Institut dEconomie Publique
• André Lapied
• Professor University of Aix-Marseille
• Groupement de Recherche en Economie Quantitative
  dAix-Marseille
• et Institut dEconomie 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
• axiomsBayes (B), Dempster-Shafer (D), or the
  Full Bayesian Uptdate (FUBU)

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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.

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Notations

• The future is S?T, S 1, , S, T
   1,  , T.
• S is endowed with a filtration F 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

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Preferences valuation and measures

• 1-Pref. on certain 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. on uncertain payoffs are 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 on uncertain cash flows are
  represented
• whether by DE Discounted Expected uncertain
  payoffs
• or by ED Expected Discounted payoff
  trajectories.

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Expectations and Discounting
Expectations and additive expectations (certainty
equivalents)
Dn(s,t) n s? S / X(s,t) ? X(s,t) - ns? S
/ X(s,t) gt X(s,t)
Discounting and time-separable discounting
(present equivalents)

Dr(s,t) r t? T / X(s,t)? X(s,t) - rt? T /
X(s,t) gt X(s,t)
Discounted Expected payoffs
Expected Discounted payoffs
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Information and conditoning
Information is yielded by an adpted process
(Yt)t1T-1 in (I, 2I). At date t, information
is Yti in Ft.
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
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Hierarchy 1 States then Dates
Discounting Expected payoffs, formulas (3) and
(3) DE
Dynamic consistency DE(X) DE (X1, ... ,
Xt-1,
Xt DtEYt i(X) , 0, , 0) .
In the case X (0, ,0, XT) DE(XT)DE
DtEYt i(XT) ,
Separable discounting D(x) DDt(x) r(T) r(t)
rt(t1, ,T) And simplifying away, Dynamic
consistency E(XT)E EYt i(XT) .
i.e. It yields an implicit definition of
conditional Choquet expectation, from which
updating rules are obtained.
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Updtating rules for a capacity
n(A/B) E1B1 (1A),
If information is comonotonic with payoffs,
Bayes' rule prevails
If information is antimonotonic (1B comonotonic
with 1A) with payoffs, Dempster-Shafers rule
obtains
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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).

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Hierarchy 2 Dates then States
Formulas (4) and (4)
Dynamic consistency ED(X) ED(X1(s), ,
Xt-1(s),
Xt(s) EYt iDtX(s), 0, , 0)s?Yt
i.
In the case of a cash flow of the form 1E with E
a subset of T, and let t - 1, , t-1 and t
t1, ... , T and
ED(1E) D(1E) r(E), EDt(1E) rt(E?t),
Then, Dynamic Consistency becomes ED(1E)
r(E) ED(1tE).
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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 xts are strictly positive
payoffs
Let Et tgtt / xt gt xt, then
Drt(t) rt( Et?t/xt xt? t) - rt(Et? t)

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Upstating rules for time non-separable discount
factors
Because of rank dependency
If t ? E
If t ? E and
If t ? E and
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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, doesnt change much the
  valuation.
• An information that arrives when a lot of cash is
  expected does affect valuation and decisions
  flexibilities, options, etc.

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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 EYti (X) ? Dt EYti (X') ? Dt EYti
(Y) ? Dt EYti (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
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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 rt is really rYt i what is
  DE(rYt i EYt 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)?