Demographic risk sharing in overlapping generations: the case of - - PDF document
Demographic risk sharing in overlapping generations: the case of DC pension funds Daniel Gabay, CNRS, EHESS-CAMS and ESILV Martino Grasselli, Univ. Padova and ESILV on the occasion of the 65th birthday of Wolfgang Runggaldier Brixen July
Daniel Gabay, CNRS, EHESS-CAMS and ESILV Martino Grasselli, Univ. Padova and ESILV
Wolfgang Runggaldier Brixen July 2007
1) Defined Benefit, Defined Contribution pen- sion funds 2) The classic actuarial approach: no market interaction and stationarity 3) The modern approaches for DB and DC 4) Demographic risk and generational overlap- ping: a delayed optimization problem 5) Optimal design of a DC pension scheme without market interaction 6) Adding the market: how can pension funds beat the market?
1) reduction of the birth rate + 2) longer average life + 3) expanded school period and consequent 4) delay in beginning the working-life ⇓
⇓
(France: ∼ 88% of total pension, about 68% in 2020)
talization (France: ∼ 7,8% of total pension, should be 25% in 2020)
(France: ∼ 0,9% of total pension, should be 3,3% in 2020)
2nd PILLAR: PENSION FUNDS
benefits to workers, when they mature the rights provided for by the regulation.
10,000 BILLIONS OF DOLLARS!!!! (WITH 10% INCREASE/YEAR, see Boulier and Dupr´ e1999)
solution preferred by workers, while the spon- soring employers will face the “contribution rate risk”
solution preferred by the corporate, because the investment’s risk is totally charged to the beneficiary (“solvency risk”) NOWADAYS:
antee
allocate the fund wealth into the financial market.
between present value of contributions and present value of liabilities
1987),
tion, salary..) ↓ stability of the pension fund wealth. Haberman (1993a) and (1993b), Haberman (1994) and Haberman and Zimbidis (1993), Dufresne (1989), Cairns (1995), Cairns (1996) and Cairns and Parker (1996)
bility to invest in a portfolio, fund returns depend on the funding method adopted (Boulier, Florens, Trussant 1995).
crease, then fund wealth decreases and li- ability increase, so that ALM has strong interdependence with the allocation strat- egy (Martellini 2003)
What is the role of the guarantee? 0 ≤ EQ(GT) ≤ interest rate
tributor to the manager of the fund
mal dynamic allocation with deterministic guarantee and Vasicek interest rates (finite horizon and CRRA utility)
timal dynamic allocation with determinis- tic guarantee and CIR interest rates (finite horizon and CRRA utility)
namic allocation with deterministic guar- antee (infinite time span and general util- ity)
El Karoui, Jeanblanc, Lacoste (2003) (also in the American guarantee)
Boyle and Imai (2000), Gerber and Pafumi (2000), El Karoui, Jeanblanc and Lacoste (2001)
Optimal strategy = strategy without guarantee +continuum of American put options
(the Lagrange multiplier is a process: contin- uous time almost sure constraint) Quantile approach is perhaps better: Pr(Ft ≥ target) ≥ 1 − α
Optimal for whom? ACTORS:
Jensen and Sørensen (2000): the presence of the guarantee is an obstacle = ⇒ no guarantee!!
Deelstra, Grasselli and Koehl (2003): necessity to specify how the fund surplus will be shared!! (F π
T − GT)
= Fund wealth - Guarantee = (1 − β) (F π
T − GT) + β(F π T − GT)
Insurance companies face this risk due to huge changes in mortality tables together with low interest rates (then increase in the liabilities)
nuity into lumpsum and vice versa (pension funds typically hedge mortality risk by del- egating insurance companies)
(2005), Menoncin, Scaillet (2005)
contributes during [0,a] (Accumulation Phase) and keeps pension till his death [a,τ] (De- cumulation Phase)
ministic distribution function for τ (Gom- pertz) ⇒ classic Merton trading strategy (see also PhD dissertation of Nicolas Rousseau, 1999)
El Karoui, Martellini (2001), Bouchard and Pham (2004), Zitkovic (2005),Blanchet-Scalliet, El Karoui, Jeanblanc and Martellini (2003) Utility maximization strategies with random hori- zon can be different from Merton’s strategies
lated with the market!!
risk due to stochastic entry process (opti- mal contribution rate in a DB scheme with-
pension funds: a theoretical framework”: Demographic risk for a PAYG pension fund (no generational overlapping!!)
repartition vs. capitalization!! ⇓
move demographic risk typical of PAYG systems!!
pension fund? (overlapping generations..)
mulation Phases for a pension fund?
and ”demographic” (fluctuations of global contribution) risk?
sion fund w.r.t. invest directly into the market? (apart from taxes, legal and ad- ministrative incentives..)
t = −a t = 0 t = T
contributions) enter the fund
described by a stochastic process ct includ- ing inflation and demographic fluctuations
sion (lumpsum!! annuity⇔ longevity risk!!) ⇒ [−a, 0] = FUND ”APh”
clients paying contributions (T could be correlated with the market and fund per- formance!!)
time T) receive the (last lumpsum) pension ⇒ [T, T + a] = FUND ”DPh”
PHASE [0, T]: at each time there are new entries and pensions to be paid out!!
the fund clients allowing for demographic fluctuations ⇓ STOCHASTIC CONTROL PROBLEM WITH DELAY (pensions depends on past contributions!!)
variables by keeping finite the dimension
works in very special cases
sional Markovian setting where state vari- ables belong to a Hilbert space (Gozzi-Marinelli 2004, Federico 2007): difficult to obtain explicit solutions even in simple cases
contribution ct and will receive a lumpsum pension at time t + a (No mortality risk for the participants!! and r = 0)
dFt = (ct − ct−aft) dt F0 = x0, so that FT = x0 +
T
0 csds −
T
0 cs−afsds.
rate greater than zero (not restrictive as- sumption)
ft ≥ g a.s.
the clients receiving ct−aft at (current) time t;
for the fund (i.e. manage the ruin prob- ability)
J0 = sup
fs≥g
T+a
cs−aU (fs − g) ds
cumulation Phase: then ct = 0 for t > T
−a csds =
initial fund reserve);
der the historical (actuarial) measure (cer- taintly equivalent valuation)
benefit/number of clients (easy to extend to cγ
s−aU (fs − g) in order to allow for unions
power)
constant Lagrange multiplier!! (Ruin prob- ability could be > 0!!)
ft ≥ g a.s. at optimality
Jt = inf
λ≥0 sup fs≥g
T+a
t
cs−aU (fs − g) ds +λ
T
t
csds −
T+a
t
cs−afsds − K
λ≥0 sup fs≥g
T
t
csds − K
T
t
cs−aU (fs − g) ds − λ
T
t
cs−afsds
λ≥0 Et
T
t
csds − g
T+a
cs−ads − K
sup
fs−g≥0
T
t
cs−aU (fs − g) ds − λ
T
t
cs−a (fs − g) ds f∗
s = g + I (λ) , s ≥ t
x≥0
(U (x) − λx) , d dλ
Jt = inf
λ≥0 Et
λ
T
t csds − g
T+a
cs−ads − K
U (λ)
T−a
t−a Etcsds
.
First order conditions on λ give = Et
T
t
csds − g
T+a
cs−ads − K
T−a
t−a
⇓ I
λ∗ =
T
0 csds − g
T
−a csds − K
T
0 Ecsds
.
λ∗ ≥ 0 ⇓ 0 ≤ g ≤ x0 + E
T
0 csds − K −a csds +
T
0 Ecsds
= gmax. f∗
s = cs−a
g + I λ∗
= cs−a x0 +
T
0 Ecsds − K −a csds +
T
0 Ecsds
= cs−agmax
ing on the same rule!! (FAIR CONTRACT)
distribute the fund reserve, that is K ≶ x0−
−a csds
−a csds,then
f∗
s = g + I (λ)
= 1, (the fund simply gives back the contribu- tion)
dynamics: dFt = (ct − ft) dt, (1) F0 = x0, (2) and the following criterion: J0 = sup
fs≥gcs−a
T
0 U (fs − gcs−a) ds
leads to the same admissibility conditions, but f∗
s
= gcs−a + I (λ) , UNFAIR!!!
dFt = (µFt + ct − ct−aft) dt F0 = x0, is handled similarly
T+a may become negative!! (no trading
strategy to hedge the ruin probability)
egy can (partially) replicate the contribu- tion process!! dFt = (σθπtFt + ct − ct−aft) dt + FtσπtdWt (3) F0 = x0, (4) where σ = volatility of the risky asset θ = risk premium of the risky asset πt = dynamic trading strategy Wt = B.m. under the historical (actuarial) measure P
J0 = sup
π
sup
fs≥g
T
0 cs−aU (fs − g) ds
s.to : E
T+a
straint!! dHt Ht = −θdWt H0 = 1,
f∗
s = g + I (λHs) , MOVING!!
I(xy) = I(x)I(y) (ex. power or log utility) ⇓ I
λ∗ =
x0 +
T
0 E [Hscs] ds − g
T
−a E
T+a
,
0 ≤ g < x0 +
T
0 E [Hscs] ds − K
T
−a E
.
HtF ∗
t −
t
0 Hs
cs − cs−af∗
s
ds = martingale
⇓ HtF π
t
= Et
HTF π
T
− Et T
t
Hs (cs − cs−afs) ds = Et
T
T−a Hs+acsds + S∗
T
t
Hs (cs − cs− ⇓ ITO + identify the Brownian terms to find π
hedge the terminal fund wealth
EXAMPLE:
sup
π E [ln (Xa)]
s.to : E [HaXa] = 1, ⇓ X∗
a = 1
Ha
f∗
a
= 1 λHa = 1 Hs x0 +
T
0 E [Hscs] ds − K −a csds +
T
0 E [cs] ds
X∗
a
< f∗
a
< x0 +
T
0 E [Hscs] ds − K −a csds +
T
0 E [cs] ds
Suppose x0 =
−a cs Hsds, K = 0
f∗
a > X∗ a
cs Hs ds+
T
0 E [Hscs] ds > −a csds+
T
0 E [cs] ds
1
Hs − 1
T
were good) then individual will choose to enter the fund only if EQ [cs] >> EP [cs]
were not good) then individual will choose to enter the fund even if EQ [cs] < EP [cs], but there will exist a maximal T ∗ for which this choice is possible.
When EQ [cs] > EP [cs] ? Intuition: when market returns are negatively correlated with the contribution process: EQ [cs] =
IDEA the investment problem for the fund is special because its initial wealth is RANDOM, then it can use the correlation between ct and Ht in order to make EQ [cs] > EP [cs]
Vasicek, ..)
by the (incomplete) market
tion when T → +∞
analysis