Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Convex order approximations in case of
cash flows of mixed signs
J. Dhaene, M.Goovaerts, Michèle Vanmaele, K. Van Weert
KULeuven and UGent
11th Scientific Day DGVFM
April 27, 2012
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
1/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Outline
1 Motivation
2 Problem description
3 Saving & Terminal Wealth
4 Reserves for Future Obligations
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
2/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
J. Dhaene, M. Goovaerts, M. Vanmaele, and K. Van Weert.
Convex order approximations in case of cash flows of mixed
signs.
Insurance: Mathematics and Economics, 2012, accepted.
K. Van Weert, J. Dhaene, and M. Goovaerts.
Optimal portfolio selection for general provisioning and
terminal wealth problems.
Insurance: Mathematics and Economics 47(1):90–97, 2010.
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
3/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Outline
1 Motivation
2 Problem description
3 Saving & Terminal Wealth
4 Reserves for Future Obligations
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
4/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Motivation
distribution of sum of r.v.
series of future payments
stochastic returns
provisioning or savings context
maximize target capital for given probability level
maximize probability level s.t. terminal wealth > target
wealth
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
5/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Motivation
all payments same sign
Dhaene, Vanduffel, Goovaerts, Kaas, Vyncke (2005)
saving-consumption (+ followed by −)
Vanduffel, Dhaene, Goovaerts (2005)
more general cash flow patterns?
e.g., periodic savings and periodic liabilities in pension
fund
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
6/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Motivation
all payments same sign
Dhaene, Vanduffel, Goovaerts, Kaas, Vyncke (2005)
saving-consumption (+ followed by −)
Vanduffel, Dhaene, Goovaerts (2005)
more general cash flow patterns?
e.g., periodic savings and periodic liabilities in pension
fund
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
6/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Motivation
all payments same sign
Dhaene, Vanduffel, Goovaerts, Kaas, Vyncke (2005)
saving-consumption (+ followed by −)
Vanduffel, Dhaene, Goovaerts (2005)
more general cash flow patterns?
e.g., periodic savings and periodic liabilities in pension
fund
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
6/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Outline
1 Motivation
2 Problem description
3 Saving & Terminal Wealth
4 Reserves for Future Obligations
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
7/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Problem description
deterministic amounts αl
time period and investment horizon: long
Gaussian model for stochastic returns
k-1
1
k
Yk
e
i.i.d. normally distributed with mean µ − 12 σ 2 and
variance σ 2
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
8/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Problem description
S=
n
X
Zi
αi e , with Zi =
i=0
n
X
λij Yj
j=1
(Y1 , Y2 , . . . , Yn ) multivariate normal
Problem: impossible to determine distribution of S
analytically in closed form
Convex order bounds: Kaas et al. (2000)
αi ≥ 0
E[S | Λ] = S ` ≤cx S ≤cx S c , with
S` =
Sc =
n
X
i=0
n
X
αi e
E[Zi ]+ 21 (1−ri2 )σZ2 +ri σZi Φ−1 (U)
i
−1 (U)
αi e E[Zi ]+σZi Φ
i=0
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
9/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Problem description
S=
n
X
Zi
αi e , with Zi =
i=0
n
X
λij Yj
j=1
(Y1 , Y2 , . . . , Yn ) multivariate normal
Problem: impossible to determine distribution of S
analytically in closed form
Convex order bounds: Kaas et al. (2000)
αi ≥ 0
E[S | Λ] = S ` ≤cx S ≤cx S c , with
S` =
Sc =
n
X
i=0
n
X
αi e
E[Zi ]+ 21 (1−ri2 )σZ2 +ri σZi Φ−1 (U)
i
−1 (U)
αi e E[Zi ]+σZi Φ
i=0
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
9/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Problem description
E[S | Λ] = S ` ≤cx S ≤cx S c
αi ≥ 0, ∀i ⇒ S c and S ` are comonotonic sum
⇒ straightforward to compute Qp [S c ] and Qp [S ` ]
αi ’s with changing signs:
Sc =
n
X
−1 (U)
αi e E[Zi ]+sign(αi )σZi Φ
i=0
Problem: suitable Λ such that S ` is comonotonic sum?
Vanduffel, Dhaene, Goovaerts (2005): solution for
Saving-Consumption schemes
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
10/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Problem description
E[S | Λ] = S ` ≤cx S ≤cx S c
αi ≥ 0, ∀i ⇒ S c and S ` are comonotonic sum
⇒ straightforward to compute Qp [S c ] and Qp [S ` ]
αi ’s with changing signs:
Sc =
n
X
−1 (U)
αi e E[Zi ]+sign(αi )σZi Φ
i=0
Problem: suitable Λ such that S ` is comonotonic sum?
Vanduffel, Dhaene, Goovaerts (2005): solution for
Saving-Consumption schemes
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
10/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Problem description
E[S | Λ] = S ` ≤cx S ≤cx S c
αi ≥ 0, ∀i ⇒ S c and S ` are comonotonic sum
⇒ straightforward to compute Qp [S c ] and Qp [S ` ]
αi ’s with changing signs:
Sc =
n
X
−1 (U)
αi e E[Zi ]+sign(αi )σZi Φ
i=0
Problem: suitable Λ such that S ` is comonotonic sum?
Vanduffel, Dhaene, Goovaerts (2005): solution for
Saving-Consumption schemes
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
10/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Outline
1 Motivation
2 Problem description
3 Saving & Terminal Wealth
4 Reserves for Future Obligations
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
11/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Deterministic cash flows α0 , α1 , . . . , αn−1 with n ≥ 1:
α0 > 0
αi , 0 < i < n can be postive or negative
Generalization of Saving-Consumption scheme
Available surplus at time k:
Vk =
k
X
l=0
αl e
Zl,k
=
k
X
αl e
Pk
j=l+1
Yj
l=0
Terminal wealth:
Wn = max[Vn , 0]
Goal: determine distribution of Wn
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
12/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Approximation:
Vn ≥cx Vn` = E[Vn | Λ]
for some appropriate r.v. Λ
P
Maximizing Var[Vn` ] leads to Λ = nj=1 βj Yj with
βj =
j−1
X
l=0
αl e
E[Zl,n ]+ 12 σZ2
l,n
=
j−1
X
αl e (n−l)µ
l=0
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
13/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Distribution of Vn` :
d
Vn` =
n−1
X
1 2
αl e (n−l)µ− 2 rl (n−l)σ
2 +r σ
l
√
n−lΦ−1 (U)
l=0
≡ f (U)
Pn
Cov (Zl,n , Λ)
j=l+1 βj
qP
with rl =
=√
n
σZl,n σΛ
2
n−l
j=1 βj
Distribution of Wn` :
d
Wn` = max[Vn` , 0] = max[f (U), 0]
If all terms are non-decreasing functions of U, Vn` is
comonotonic sum
n−1
X
√
1 2 2
−1
`
⇒ Qp [Vn ] = f (p) =
αl e (n−l)(µ− 2 rl σ )+rl σ n−lΦ (p)
l=0
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
14/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
If not all terms are non-decreasing functions of U, Vn` is
not comonotonic sum.
If total sum f is non-decreasing function in interval
(p ? , 1), Qp [Vn` ] can still be used for p ∈ (p ? , 1)
conditions on amounts αl ?
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
15/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Theorem
P
If the conditioning random variable Λ equals nj=1 βj Yj with
P
(n−l)µ
βj = j−1
, and if the surplus Vl satisfies
l=0 αl e
E[Vl ] > 0,
l = 0, . . . , n − 1,
then the quantiles of Wn` are given by
Qp [Wn` ] = max[f (p), 0] = f (p),
p ? < p < 1.
The distribution function of Wn` follows from
f (FWn` (x)) = x,
x ≥ Qp? [Wn` ].
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
16/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Equivalent conditions of E[Vl ] > 0, l = 0, . . . , n − 1
E[Vl ] = e µ E[Vl−1 ] + αl , l = 1, . . . , n − 1, with
E[V0 ] = α0 :
E[Vl ] > 0,
for all l s.t. αl < 0
E[Vl ] = e −(n−l)µ βl+1 :
βl+1 > 0,
l = 0, . . . , n − 1
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
17/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Sufficient conditions for E[Vl ] > 0, l = 0, . . . , n − 1
E[Vl ] =
Pl
k=0
αk e (l−k)µ =
µ > µ? ⇒ E[Vl ] > 0,
(
µ? =
max
l=0,...,n−1
Pl
k=0
αk x l−k :
l = 0, . . . , n − 1
max µ | µ > 0 and
E[Vl ] = e −(n−l)µ βl+1 = e −(n−l)µ
l
X
k=0
Pl
l
X
αk e (l−k)µ = 0
k=0
k=0
)!
αk e (n−k)µ
αk ≥ 0 l = 0, . . . , n − 1
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
18/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Example 1 : n = 20
10 10 10 10 −40 10 10 10 10 −40
10 10 10 10 −40 10 10 10 10 −40
Example 2 : n = 20
10 10 10 10 −50 10 10 10 10 −50
10 10 10 10 −50 10 10 10 10 −50
E[Vl ]> 0, l = 4, 9, 14, 19 for µ ≥ 0.088
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
19/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Example 1 : n = 20
10 10 10 10 −40 10 10 10 10 −40
10 10 10 10 −40 10 10 10 10 −40
Example 2 : n = 20
10 10 10 10 −50 10 10 10 10 −50
10 10 10 10 −50 10 10 10 10 −50
E[Vl ]> 0, l = 4, 9, 14, 19 for µ ≥ 0.088
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
19/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
How to determine p ? in
Qp [Wn` ] = max[f (p), 0] = f (p),
p ? < p < 1?
Answer: s.t. function f is positive and non-decreasing in p
f (p) =
n−1
X
1 2 2
)+rl σ
αl e (n−l)(µ− 2 rl σ
√
n−lΦ−1 (p)
l=0
1
split f (p) in saving-consumption cases fi (p) and apply
Vanduffel et al. (2005) then
p ? = maxi∈{1,...,m} pi
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
20/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
40
50
40
20
30
0
20
-20
10
0
-40
-10
-60
-20
-80
-30
f1
1
f
f2
f3
f4
Example 2 : n = 20
10 10 10 10 −50 10 10 10 10 −50
10 10 10 10 −50 10 10 10 10 −50
p ? = maxi=1,...,4 pi = p4 = 0.999999
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
21/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Proof that there exists p ? ≤ maxi∈{1,...,m} pi :
2 use Descartes’ rule of sign for generalized polynomials
f (p) = h(x) =
n−1
X
1
2
αl e (n−l)(µ− 2 σ ) x rl
√
n−l
−1 (p)
, x = e σΦ
l=0
f 0 (p) = h0 (x)x
X
√
√
σ
, h0 (x) =
al rl n − lx rl n−l−1
ϕ(p)
l=0
n−1
1
0
)
p ? = Φ( ln x ? ) with x ? = max(xmax , xmax
σ
Example 2:
(µ, σ) = (0.09, 0.10) ⇒ p ? = 0.437329957844209
(µ, σ) = (0.10, 0.10) ⇒ p ? = 0.326242535523647
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
22/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Proof that there exists p ? ≤ maxi∈{1,...,m} pi :
2 use Descartes’ rule of sign for generalized polynomials
f (p) = h(x) =
n−1
X
1
2
αl e (n−l)(µ− 2 σ ) x rl
√
n−l
−1 (p)
, x = e σΦ
l=0
f 0 (p) = h0 (x)x
X
√
√
σ
, h0 (x) =
al rl n − lx rl n−l−1
ϕ(p)
l=0
n−1
1
0
)
p ? = Φ( ln x ? ) with x ? = max(xmax , xmax
σ
Example 2:
(µ, σ) = (0.09, 0.10) ⇒ p ? = 0.437329957844209
(µ, σ) = (0.10, 0.10) ⇒ p ? = 0.326242535523647
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
22/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Saving & Terminal Wealth
Proof that there exists p ? ≤ maxi∈{1,...,m} pi :
2 use Descartes’ rule of sign for generalized polynomials
f (p) = h(x) =
n−1
X
1
2
αl e (n−l)(µ− 2 σ ) x rl
√
n−l
−1 (p)
, x = e σΦ
l=0
f 0 (p) = h0 (x)x
X
√
√
σ
, h0 (x) =
al rl n − lx rl n−l−1
ϕ(p)
l=0
n−1
1
0
)
p ? = Φ( ln x ? ) with x ? = max(xmax , xmax
σ
Example 2:
(µ, σ) = (0.09, 0.10) ⇒ p ? = 0.437329957844209
(µ, σ) = (0.10, 0.10) ⇒ p ? = 0.326242535523647
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
22/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Outline
1 Motivation
2 Problem description
3 Saving & Terminal Wealth
4 Reserves for Future Obligations
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
23/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Reserves for Future Obligations
Deterministic obligations α1 , α2 , . . . , αn with n ≥ 1:
αn > 0
αi , 0 < i < n can be postive or negative
Future obligations at time l:
Rl =
n
X
k=l+1
αk e
Zl,k
=
n
X
αk e −
Pk
j=l+1
Yj
k=l+1
Initial provision:
S0 = max[R0 , 0]
Goal: determine distribution of S0
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
24/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Reserves for Future Obligations
Approximation:
R0 ≥cx R0` = E[R0 | Λ]
for some appropriate r.v. Λ
P
Maximizing Var[R0` ] leads to Λ = nj=1 βj Yj with
βj = −
n
X
αl e l(−µ+σ
2)
l=j
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
25/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Reserves for Future Obligations
Distribution of R0` :
n
X
√ −1
1 2
2
` d
R0 =
αj e −jµ+(1− 2 rj )jσ +rj σ jΦ (U) ≡ f (U)
j=1
P
− jk=1 βk
with rj = √ qP
j
2
j
k=1 βk
Distribution of S0` :
d
S0` = max[R0` , 0] = max[f (U), 0]
If all terms are non-decreasing functions of U, R0` is
comonotonic sum
n
X
√ −1
1 2
2
⇒ Qp [R0` ] = f (p) =
αj e −jµ+(1− 2 rj )jσ +rj σ jΦ (p)
j=1
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
26/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Reserves for Future Obligations
Theorem
P
If the conditioning random variable Λ equals nj=1 βj Yj with
P
2
βj = − nk=j αk e k(−µ+σ ) , and if the obligations Rl satisfies
E[Rl ] > 0,
l = 0, . . . , n − 1,
then the quantiles of S0` are given by
Qp [S0` ] = max[f (p), 0] = f (p),
p ? < p < 1.
The distribution function of S0` follows from
f (FS0` (x)) = x,
x ≥ Qp? [S0` ].
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
27/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Reserves for Future Obligations
invert time axis ⇒ terminal wealth setting
0
1
α1
n
n−1
α
e1
···
l
αl
··· n − l
α
en−l
···
n−1
n
···
1
0
α
e1
α
e0
-
αn−1 αn
el = −Yn−l+1 ∼ N (µ̃ − 1 σ 2 , σ 2 ) with µ̃ = −µ + σ 2
Y
2
µ̃ in general negative
en = R0
V
el ] = αn−l + E[Rn−l ]
E[V
d e`
` d
f˜(U) = V
n = R0 = f (U)
P
E[Rl ] > 0, l = 0, . . . , n − 1 ⇒ nk=j αk ≥ 0 (µ − σ 2 > 0)
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
28/29
Motivation Problem description Saving & Terminal Wealth Reserves for Future Obligations
Thank you for your attention
Michèle Vanmaele — Convex order approximations in case of cash flows of mixed signs
29/29