VIMADES VPPI: a New Approach to Asset

VIMADES
Viabilité, Marchés, Automatique et Décision
Jean-Pierre Aubin, Luxi Chen
Olivier Dordan, Patrick Saint-Pierre
VPPI: a New Approach to Asset-Liability
Management Eradicating Gap Risk
Frontières en Finance
Petits Déjeuners de la Finance
15 Juin 2011
1
1
Introduction
Yet the possibility of going below the floor, known as “gap risk”, is widely
recognized by CPPI ( Constant Proportional Portfolio Insurance) managers:
there is a nonzero probability that, during a sudden downside move, the fund
manager will not have time to readjust the portfolio, which then crashes
through the floor. In this case, the issuer has to refund the difference, at
maturity, between the actual portfolio value and the guaranteed amount. It is
therefore important for the issuer of the CPPI note to quantify and manage
this gap risk.
From Cont R. & Tankov P. (2009) Constant Proportion Portfolio Insurance
in presence of jumps in asset prices, Mathematical Finance, 19, 379-401
Our aim is to design another management rule, the VPPI (Viabilist Portfolio
Performance and Insurance) rule, which eradicates the gap risk.
2
The problem of hedging a liability by a portfolio made of a riskless asset and
an underlying requires a dynamical mechanism ensuring that, at each date, the
value of the portfolio is “always” exceeding liabilities (zero gap risk).
This is a typical tychastic viability problem under uncertainty allowing to
“eradicate” risk instead of quantifying of evaluating it using adequate methods
(VaR, expected loss, probability of loss, etc. ).
“Fixed Mix” and CPPI (Constant Proportional Portfolio Insurance) managing rules are not fit for that insurance purpose since there are cases when the
liability floor is pierced when they are used. The
1. initial investment (cushion);
2. fixed weights or multipliers;
have to be estimated by statistical methods optimizing several criteria.
3
Viability tools are designed and used for computing the
1. “Minimum Guaranteed Investment” (MGI), above which one can (really)
guarantee that the value of the portfolio is always larger than the liabilities
and provides opportunistically profits when, at each date after investment,
the risky return is larger than the forecast lower bound of returns:
“Take advantage of highs while protect against laws”;
2. VPPI management rule which prescribes the number of shares (or the
exposition) at each future date when the return is known;
3. “Impulse Management Mechanism” which computes the minimal amount
for reinsuring the portfolio when prediction errors occur, so that the management continues.
The VPPI management rule is not given as a simple analytical explicit formula, but computed through viability algorithms. The VPPI software computes the insurance, the value, the exposition of the portfolio and the prediction
error penalties at each date.
4
2
Management of Asset Portfolios Hedging Liabilities
The liability flow to insure is described as a constraint floor not to be
“pierced”. The floor is defined by a function
k : t ≥ 0 7→ k(t) ≥ 0
The capital to guarantee (at exercise time) is the final value k(T ) of the floor
to the exercise date T .
In this study, the floor is assumed to depend only on time (the case when
the floor is itself uncertain can be studied as well).
The question is “to hedge” the liability by a portfolio made of a riskless asset
and of an underlying (a “risky asset”, in many examples).
5
6
Assets portfolios and their exposure
Denote by 0 the investment date, by T > 0 the exercise time, t ≥ 0 the
current (or spot) time and T − t ≥ 0 the time to maturity. We set:
 0
S (t) the price of the riskless asset,





S(t) the price of the underlying,


d 0


S (t)

dt
0

the return of the riskless asset,
R
(t)
=

0 (t)

S



d


dt S(t)
R(t) =
the return of the underlying,
(1)
S(t)



P 0 (t) the number of shares of the riskless asset,




P (t) the number of shares of the underlying,




W (t) = P 0 (t)S 0 (t) + P (t)S(t) the value of the portfolio,




E 0 (t) = P 0 (t)S 0 (t) the liquid component of the portfolio,



E(t) = P (t)S(t) the exposure (risky component) of the portfolio,
7
The value of the portfolio and its exposure summarize the information for
deriving relevant properties in the portfolio. It follows that, in the case of
self-financing portfolios, the formula
∀ t ∈ [0, T ], W 0 (t) = R0 (t)W (t) + E(t)(R(t) − R0 (t))
(2)
describes the dynamic evolution of the portfolio controlled by the exposures
E(t) and dependent on the returns R(t) of the underlying (tyches) unknown at
investment date but known afterwards.
8
Portfolio insurance: a viability problem
To insure or guarantee a liability by asset-liability management can be formulated in the following way:
∀ t ∈ [0, T ], W (t) ≥ k(t) and W (T ) = k(T )
(3)
Viability Constraints
It is expressed by stating that
∀ t ∈ [0, T ], (t, W (t)) ∈ Ep(k) and (T, W (T )) ∈ Graph((k)
where the epigraph Ep(k) of the function k is defined by
Ep(k) := {(t, W ) such that W ≥ k(t)}
9
(4)
The hedging problem of a liability amounts to looking for evolutions t 7→
(t, W (t)) “viable in the epigraph” Ep(k) of the function k in the sense that
∀ t ∈ [0, T ], (t, W (t)) ∈ Ep(k)
Instead of handling functions as in classical analysis, viability theory manipulates subsets as in set-valued analysis.
10
Financial constraints on exposures require that each period t
∀ t ∈ [0, T ], ∀ W ≥ 0, E ∈ [E [ (t, W ), E ] (t, W )]
(5)
Example — If
B(t) ≥ 0
is the target allocation
A(t) ≥ B(t) − 1 is the maximum cash −P 0 (t)S 0 (t)
(6)
we set
E [ (t, W ) := B(t)W (t) and E ] (t, W ) := (1 + A(t))W (t)
If A(t) = −|A(t)| < 0, condition −P 0 (t)S 0 (t) ≤ A(t)W can be written P 0 (t)S 0 (t) ≥
|A(t)|W , which means that the portfolio should include a minimum share of
the monetary value of the portfolio.
These bounds A(t) and B(t) describe more or less severe prudential constraints. 11
3
Predictive Approach
At investment date, the investor is supposed to provide the forecasted lower
bounds R[ (t) of the returns.
The point is to hedge the liabilities at each instant whatever the return of the
risky asset larger their forecasted lower bounds, taking into account “downward
jumps” for instance, or using any available “volatilometer” or extrapolation
method.
1 VIMADES Extrapolator. It is a mathematical technique that allows to
extend historical evolution defined on an interval [−H, 0] called extrapolator
depth. The originality of the VIMADES Extrapolator is to take into account
not only past values of the historical evolution, but also those of its derivatives
of order less than or equal to a given order. They thus describe the trends
(velocity, acceleration, jerk, etc.) These past trends are taken into account
by the VIMADES Extrapolator.
12
The value of the portfolio is governed by a (very simple) tychastic controlled
system, where controls are the exposures E(t, W ) ∈ [E [ (t, W ), E ] (t, W )] of the
portfolio and the tyches are the returns R(t) ≥ R[ (t) of the underlying:




∀ t ∈ [0, T ],
(i) W 0 (t) = R0 (t)W (t) + E(t)(R(t) − R0 (t)) (evolutionary engine)
(ii) E(t) ∈ [E [ (t, W (t)), E ] (t, W (t))] (controls)



(iii) R(t) ≥ R[ (t) (tyches)
(7)
subject to the floor constraints
∀ t ∈ [0, T ], W (t) ≥ k(t) and W (T ) = k(T )
13
(8)
Definition 1 Minimum Guaranteed Investments. Suppose that the floor t 7→
k(t) and the bounds [E [ (t, W ), E ] (t, W )] describing the contingent uncertainty
are given.
Assume known the lower bounds R[ (t) of the returns on the underlying describing tychastic uncertainty. The problem is to find each date t
1. the (exposure) management rule E ♥ (t, W ) ∈ [E [ (t, W ), E ] (t, W )];
2. the minimum guaranteed investment (MGI) W ♥ (t);
3. and in particular the initial minimum guaranteed investment (“viability
insurance”) W ♥ (0)
14
such that
1. starting at investment date 0 from W0 ≥ W ♥ (0), then regardless be the
evolution of tyches R(t) ≥ R[ (t), the value W (t) of the portfolio governed
by the management module
W 0 (t) = R0 (t)W (t) + E ♥ (t, W (t))(R(t) − R0 (t)) (VPPI management module) (9)
is always above the floor, and, actually, above the minimum guaranteed
investment ;
2. starting at investment date 0 from W0 < W ♥ (0), regardless the manb W ) ∈ [E [ (t, W ), E ] (t, W )] (including the CPPI manageagement rule E(t,
ment rule and its variants), there exists at least one evolution of returns
R(t) ≥ R[ (t) for which the value of the portfolio managed by
b W (t))(R(t) − R0 (t))
W 0 (t) = R0 (t)W (t) + E(t,
pierces the floor.
15
(10)
4
Replaying the 2008 Subprime Crisis
16
Extrapolating each historical (past) time series of upper bounds (HIGH )
and lower bounds (LOW ) of the underlying prices provided daily by brokerage
firms, the VIMADES Extrapolator calculates lower bounds of underlying returns of the future needed for the computation of the VPPI management rule.
No need of a “volatilometer” for measuring volatile volatilities.
17
4.1
Synopsis
The following table summarizes the principal characteristic features of the
portfolio:
1. At investment date, the insurance
(a) value of the initial Minimal Guaranteed Investment
(b) value of the initial Minimal Guaranteed Cushion
2. At each date, when the return is known, the Management of the portfolio
(a) actualized value of the portfolio at exercise time
RT
− 0 R0 (τ )dτ W (T ) − k(T )
(b) liquidating dividend e
W ♥ (0) − k(0)
(c) net liquidating dividend (including prediction penalties)
(d) actualized value of cumulated prediction penalties
18
Synopsis for the 2008 Crisis
minimum guaranteed investment (MGI)
minimum guaranteed cushion (MGC)
actualized exercise value
liquidating dividend
net liquidating dividend
cumulated prediction penalties
2268.88
2229.25
43099.16
19.29 %
14.01 %
15065.18
Even though the cost of insurance in this extreme situation is very high, the
insurance not only guarantees the values of the portfolio, but actually provide
also a (very) high liquidating dividend, even after prediction penalties whenever
it turns out that the returns are (well) above their forecasted lower bounds.
19
2 Floor and MGI.
20
3 Impulse Management.
21
4 Shares of the Underlying.
22
5 Ratchet Management.
23
4.2
Mobile Horizon Minimum Guaranteed Investment
The Mobile Horizon Minimum Guaranteed Investment (MGI) provides at
each date not only the insurance up to the exercise time, but to all shorter
horizon ranging between this date and the exercise time : the shorter the insurance duration, the least costly the insurance. These insurance costs are
displayed in the table displaying at each date (row) and at each column (insurance duration) the amount of the MGI.
The following graphic displays the first row of the table, indicating the MGI
cost at investment date for all maturation dates (the MGI for the whole exercise
time is the last, and highest, value).
24
Mobile Horizon MGI at Investment Date
2,000
MGI
1,500
1,000
500
0
5
10 15 20 25 30 35 40 45 50 55 60 65 70
time
Mobile Horizon MGI
25
6 Comparison of the MGI’s between the predicted and “retropredicted” lower
bounds of returns.
26
4.3
VPPI and CPPI
exposition
.
cushion
1. VPPI : The MGI and the VPPI management rule are deduced from the
floor and lower bounds of the future risky returns, and
The (cushion) multipliers are ratios
2. CPPI : The floor, the initial investment, a prediction mechanism are given,
as well as the multipliers (constant as the CPPI or not as in other rules).
Both rules compute the value or the portfolio and its exposition, but only the
VPPI is designed to eradicate the risk.
27
Comparisons between VPPI and CPPI
VPPI
CPPI
regulation
computed
given structure
insurance
YES (MGI)
NO (loss evaluations)
prediction
YES (impulse
YES (jump processes)
errors
management)
Forecasting Extrapolator of VIMADES
Stochastic
methods
or any other one
processes
28
7 a posteriori VPPI multipliers :=
exposition
.
cushion
29
8 Comparison between the MGI of the CPPI and VPPI management rule.
30
31
Flow Chart of the VPPI Software
Minimum Guaranteed Investment
Underlying Returns
Inputs
VIMADES
Extrapolator
High and Lows
of Underlying Prices
Riskless Return
and
Lower Bounds
of Forecasted
Risky Asset
Returns
Liabilities/Floor
Bounds on
Exposure
Bounds on
Liquidity
Outputs
Minimum
Guaranteed Investment (MGI)
Coupon
Schedule
Other Prediction Modules
Bounds on
Transaction Values
Management Module of the Portfolio
Inputs
Outputs
Porfolio Value
Actual Returns
of the Underlying
Porfolio Exposure
Number of Shares
32
33
5
Uncertainties
Non-deterministic evolutionary systems:
An evolutionary systems associates with any initial state a subset of evolutions.
34
Stochastic uncertainty
Stochastic uncertainty on the returns is described by a space Ω, filtrations
Ft , the probability P, a Brownian process B(t), a drift γ(R) and a volatility σ(R):
dR(t) = γ(R(t))dt + σ(R(t))dB(t).
1. The random events are not explicitly identified. One can always choose the
space Ω of all evolutions. Only the drift and volatility are known;
2. Stochastic uncertainty does not study the ”package of evolutions” (depending on ω ∈ Ω), but functionals over this package, such as the different
moments and their statistical consequences (averages, variance, etc.) used
as evaluation of risk;
3. Required properties are valid for “almost all” constant ω.
35
Naturally, the geometric model used in the Black and Sholes approach is an
approximation model excluding the advent on jumps. For instance, one can
use
1. standard deviations,
2. Lévy processes followed by the risky asset,
3. stochastic volatility,
4. non standard analysis (where noise is replace by rapid fluctuations deduced
from the 1995 Cartier-Perrin decomposition theorem,
5. etc.
36
Tychastic Uncertainty
Tyches are returns of underlyings on which the investor has no influence.
The uncertainty is described by the tychastic map defined by
n
o
[
R(t) := R ∈ R such that R ≥ R (t)
where R[ (t) are the lower bounds on returns (instead of assuming that dR(t) =
γ(R(t))dt + σ(R(t))dB(t)).
1. Tyche are identified (returns of the underlying, for example) which can then
be used in dynamic management systems when they are actually observed
and known at each date during the evolution;
2. For this reason, the results are computed in the worst case (eradication of
risk instead of its statistical evaluation);
3. required properties are valid for “all” evolutions of tyches t 7→ R(t) ∈ R(t).
37
Western etymology
Stochastic refers to the Greek stokhastikos designating the draw of the rulers
of the Athenian democracy. Random comes from the French “randon”, from
the verb “randir”, sharing the same root than the English “to run” and the
German “rennen”.
We borrow from Charles Peirce the use of tychastic evolution he introduced
in 1893 in an article nicely entitled Evolutionary Love for describing the evolution of a system dependent on tyches arising without any statistical regularity.
The word tychastic comes from the Greek tyche, meaning luck, represented
by the goddess Tyche , daughter of Oceanus and Tethys, whose goal was to disrupt the course of events either for good or for bad. Tyche became “Fortuna”in
“rizq” in Arabic (with a positive conLatin, “rizikon” in Byzantine Greek, †PP
notation in these three cases).
38
Chinese etymology
The four ideograms follow, opportunity, reaction, change
are combined to express in Chinese:
1. by the first half, “follow, opportunity”,
ness or stochasticity,
, the concept of random-
2. while by the second half, “reaction, change”,
concept of tychasticity (according to Shi Shuzhong)
3. and “no, necessary”,
, translates contingent.
39
, translates the
Contingent Uncertainty
In fact, the guaranteed threshold decreases when the “tychastic reservoir”
increases, that is to say, in this study, when the lower bounds of underlying
returns decrease.
The question arises whether contingent uncertainty “offsets” tychastic uncertainty.
The word contingent comes from the Latin verb contingere, to arrive unexpectedly or accidentally. Leibniz: “Contingency is a non-necessity, a characteristic attribute of freedom.”
40
Impulse uncertainty.
Since it may be difficult to determine the lower bounds R[ (t), the question
arises to address the inverse problem. Instead of computing the minimum
guaranteed investment W ♥ (t), we assume instead known authorized minimum
investment h♦ (t) ≥ k(t) and we derive lower bounds R♦ (t) of underlying returns
guaranteeing that the floor will never be pierced.
This possible by using an impulse management rule allowing the investor to
set instantly by an impulse (infinite velocity) a new higher level of investment
by using its right to borrow, and start again from the minimum guaranteed
investment.
41
Probability of Ruin
It is possible to interpret otherwise the impulse management mode, regarding
1. h♦ (t) as the liability;
2. and k(t) ≤ h♦ (t) as a tolerance to ruin.
Instead of trying to compute the probability of ruin tolerance, we seek and
obtain the Guaranteed Minimum Return which forbids to go beyond that tolerance to ruin.
The framework of “Solvency 2”, for example, requires that the difference
between the value of portfolio assets and provisions to hedge liabilities must be
positive at every date, possibly with a “probability of failure” (where equity is
negative) below a given threshold.
In our framework, the probability of ruin is replaced by the Guaranteed
Minimum Returns (GMR).
42
Two “inverse” management modules
In a nutshell:
1. The predictive management module assumes the tychastic reservoir to be
known and computes both the evolution of contributions and the local
guaranteed thresholds;
2. The impulse management module assumes the global and local thresholds
known and provides both the evolution of contributions and the computed
tychastic reservoir.
43
6
Impulse Approach
Instead of computing the minimum guaranteed investment W ♥ (t) from R[ (t),
we assume instead known authorized minimum investment h♦ (t) ≥ k(t) and we
derive upper bounds R♦ (t) of underlying returns guaranteeing that the floor
will never be pierced.
Impulse management rule allows the investor to set instantly by an impulse
(infinite velocity) a higher level of investment by using its right to borrow, and
start again from the minimum guaranteed investment.
The impulse tychastic system operates as a regulated tychastic system
W 0 (t) = R0 (t)W (t) + E ♦ (t, W (t))(R(t) − R0 (t))
but, requiring further that at each time t when the value W (t) = k(t) hits the
floor, then it is reset to W (t) = k (t).
44
Definition 2 Guaranteed Minimum Returns. The floor t 7→ k(t) and the bounds
[E [ (t, W ), E ] (t, W )] are given. Consider the amount h♦ (t) ≥ k(t) of authorized
investments (or authorized loans π ♦ (t) := h♦ (t) − k(t) ≥ 0 for borrowing π(t) :=
h♦ (t) − W (t) ≤ π ♦ (t)).
The goal is to find each date t
1. the management rule of exposures E ♦ (t, W ) ∈ [E [ (t, W ), E ] (t, W )];
2. the Guaranteed Minimum Return R♦ (t)
such that, starting from W0 ≥ k(0) and knowing the return R(t) ≥ R♦ (t) of the
underlying, the impulse management module
0
R (t)W (t) + E ♦ (t, W (t))(R(t) − R0 (t)) if W (t) ≥ h♦ (t)
(11)
W 0 (t) :=
R0 (t)h♦ (t) + E ♦ (t, W (t))(R(t) − R0 (t)) if k(t) ≤ W (t) ≤ h♦ (t)
manages a portfolio the value of which is always above the floor.
45
Remark: Tychastic Reliability — The approach provides an answer to a
problem that could be called ”tychastic reliability” as it provides lower bounds
of returns (describing the boundary of the tychastic map) above which the
guarantee sought (the value of the portfolio must be greater than the floor)
and the means of ensuring it (by paying for a cash flow higher than the floor)
to be reliable reliable at 100 %. This question opens a new chapter in viability
theory. 46
9 Example of authorized investments.
47
10 Guaranteed Minimum Returns.
48
11 Impulse Management.
49
50
Flow Chart of the VIPPI Software
Guaranteed Minimum Returns
Inputs
Outputs
Input of Authorized Investments
Liabilities/Floor
Bounds on
Exposure
Riskless Return
Guaranteed
Minimum
Return (GMR)
Bounds on
Liquidity
Coupon
Schedule
Authorized
Investments
Bounds on
Transaction Values
Impulse Management of the Porfolio
Inputs
Outputs
Porfolio Value
Actual Returns
of the Underlying
Porfolio Exposure
Number of Shares
51
52
The VPPI Software
The software of the demonstration version is available on the internet free
of charge for exercises between 10 and 30 dates.
Tests with longer exercises can be produced by VIMADES when the data
(floor and high, low and last prices are provided)
The operational version is also on internet.
Practical modalities for for adapting the VPPI software to the needs of the
prospective user are welcomed.
This concerns specifically the automated interfaces for reading prices and
sending the value and the exposition to the user.
53
Merci pour votre attention
Thank You for Your Attention
1. Aubin J.-P. (2010) La mort du devin, l’émergence du démiurge. Essai sur
la contingence, la viabilité et l’inertie des systèmes, Éditions Beauchesne;
2. Aubin J.-P., Bayen A. and Saint-Pierre P. (2011) Viability Theory. New
Directions, Springer-Verlag
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