SLIDE 1
Longevity and pension funds Paris 3-4 February 2011 Ragnar Norberg - - PowerPoint PPT Presentation
Longevity and pension funds Paris 3-4 February 2011 Ragnar Norberg - - PowerPoint PPT Presentation
Longevity and pension funds Paris 3-4 February 2011 Ragnar Norberg ISFA Universit e Lyon 1 Email: ragnar.norberg@univ-lyon1.fr Homepage: http://isfa.univ-lyon1.fr/ norberg/ MANAGEMENT OF MORTALITY AND LONGEVITY RISK IN LIFE INSURANCE
SLIDE 2
SLIDE 3
Thus, (푌, 푍) is a Markov chain with state space 풴 × 풵 and intensities 휅푒푗,푓푘(푡) =
⎧ ⎨ ⎩
휆푒푓 , 푒 ∕= 푓, 푗 = 푘 , 휇푒,푗푘(푡) , 푒 = 푓, 푗 ∕= 푘 , 0 , 푒 ∕= 푓, 푗 ∕= 푘 . Indicator processes 퐼푌
푒 (푡) = 1[푌 (푡) = 푒]
Counting processes 푁푌
푒푓(푡) = ♯{휏; 푌 (휏−) = 푒, 푌 (휏) = 푓, 휏 ∈ (0, 푡]}
Observe that 푑퐼푌
푒 (푡) =
∑
푓;푓∕=푒
푑푁푌
푓푒(푡) −
∑
푓;푓∕=푒
푑푁푌
푒푓(푡) ,
퐼푌
푒 (0) = 훿0푒
Similar for policy process 퐼푍
푗 (푡) and 푁푍 푗푘(푡)
(푌, 푍) generates filtration F = (ℱ푡)푡∈[0,푇], ℱ푡 = ℱ푌
푡 ∨ ℱ푍 푡 .
3
SLIDE 4
FINANCIAL MARKET: Assume there exists arbitrage-free market for environmental risk: Equivalent martingale measure ˜
P under which 푌 is a Markov chain
with transition rate matrix ˜
Λ = (˜
휆푒푓). The compensated environment-related counting processes 푑푀푌
푒푓(푡) = 푑푁푌 푒푓(푡) − 퐼푌 푒 (푡) ˜
휆푒푓 푑푡 are the driving (F푌 , ˜
P)-martingales.
Locally risk free asset: Market money account with price process 푆0(푡) = 푒
∫ 푡
0 푟(푠) 푑푠
yields interest at fixed rate 푟푒 in state 푒: 푟(푡) = 푟푌 (푡) =
∑
푒
퐼푌
푒 (푡) 푟푒
4
SLIDE 5
Risky assets, fundamental or derivatives, with unpredictable price jumps. Generic 푈-claim with a single pay-off 퐻 ∈ ℱ푌
푈 at fixed time 푈. Price at
time 푡 ≤ 푈 is 푆(푡) = ˜
E
[
푒− ∫ 푈
푡 푟(푠) 푑푠 퐻
- ℱ푌
푡
]
. Discounted price process ˜ 푆(푡) = 푆−1
0 (푡) 푆(푡) = ˜
E
[
푆−1
0 (푈) 퐻
- ℱ푌
푡
]
is (F푌 , ˜
P)-martingale:
푑˜ 푆(푡) =
∑
푒∕=푓
˜ 휉푒푓(푡) 푑푀푌
푒푓(푡) ,
where the ˜ 휉푒푓 are F푌 -predictable. Work with “inflated” coefficients, 휉푒푓(푡) := 푆0(푡) ˜ 휉푒푓(푡) : 푑˜ 푆(푡) = 푆−1
0 (푡)
∑
푒∕=푓
휉푒푓(푡) 푑푀푌
푒푓(푡) .
5
SLIDE 6
Analysis for identifying the coefficients 휉푒푓. Consider fairly general 푈-claim 퐻 = 푒− ∫ 푈
0 푔푌 (푠)(푠) 푑푠 ℎ푌 (푈) ,
(푔푒(푡))푡∈[0,푈] and ℎ푒, 푒 = 0, . . . , 퐽푌 are non-random scalars. ˜ 푆(푡) = ˜
E
[
푒
− ∫ 푈
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ℎ푌 (푈)
- ℱ푌
푡
]
= 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ˜
E
[
푒
− ∫ 푈
푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ℎ푌 (푈)
- ℱ푌
푡
]
= 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ∑ 푒
퐼푌
푒 (푡) 푣푒(푡)
푣푒(푡) = ˜
E
[
푒
− ∫ 푈
푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ℎ푌 (푈)
- 푌 (푡) = 푒
]
6
SLIDE 7
Ito: 푑˜ 푆(푡) = − 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 (
푟(푡) + 푔푌 (푡)(푡)
)
푑푡
∑
푒
퐼푌
푒 (푡) 푣푒(푡)
+ 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ∑ 푒
퐼푌
푒 (푡) 푑푣푒(푡)
+ 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ∑ 푒∕=푓
(푣푓(푡) − 푣푒(푡)) 푑푁푌
푒푓(푡)
= 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ∑ 푒
퐼푌
푒 (푡) ×
⎛ ⎝− (푟푒 + 푔푒(푡)) 푣푒(푡)푑푡 + 푑푣푒(푡) + ∑
푓; 푓∕=푒
˜ 휆푒푓
(
푣푓(푡) − 푣푒(푡)
)
푑푡
⎞ ⎠
+ 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ∑ 푒∕=푓
(
푣푓(푡) − 푣푒(푡)
)
푑푀푌
푒푓(푡)
7
SLIDE 8
Drift term vanishes, which gives constructive PDE-s: 푑푣푒(푡) =
⎡ ⎣(푟푒 + 푔푒(푡)) 푣푒(푡) − ∑
푓; 푓∕=푒
˜ 휆푒푓
(
푣푓(푡) − 푣푒(푡)
) ⎤ ⎦ 푑푡 ,
푒 ∈ 풴, with side conditions 푣푒(푈) = ℎ푒 , 푒 ∈ 풴. Martingale dynamics reduces to 푑˜ 푆(푡) = 푒
− ∫ 푡
(
푟(푠) + 푔푌 (푠)(푠)
)
푑푠 ∑ 푒∕=푓
(
푣푓(푡) − 푣푒(푡)
)
푑푀푌
푒푓(푡)
= 푆−1
0 (푡)
∑
푒∕=푓
휉푒푓(푡) 푑푀푌
푒푓(푡)
휉푒푓(푡) = 푒− ∫ 푡
0 푔푌 (푠)(푠) 푑푠 (
푣푓(푡) − 푣푒(푡)
)
8
SLIDE 9
INSURANCE PAYMENTS: Benefits less premiums generate payment function 퐵: 푑퐵(푡) =
∑
푗∈풵
퐼푍
푗 (푡) 푑퐵푗(푡) +
∑
푗,푘∈풵; 푗∕=푘
푏푗푘(푡) 푑푁푍
푗푘(푡)
Policy terminates at a finite time 푇. Market reserve at time 푡 ≤ 푇 is 푉 (푡) = ˜
E
[∫ 푇
푡
푒− ∫ 휏
푡 푟(푠) 푑푠 푑퐵(휏)
- ℱ푡
]
=
∑
푒
∑
푗
퐼푌
푒 (푡) 퐼푍 푗 (푡) 푉푒푗(푡) ,
where (Markov assumptions) state-wise reserves are 푉푒푗(푡) = ˜
E
[∫ 푇
푡
푒− ∫ 휏
푡 푟(푠) 푑푠 푑퐵(휏)
- 푌 (푡) = 푒, 푍(푡) = 푗
]
9
SLIDE 10
Sums at risk: 휂푒,푗푘(푡) = 푏푗푘(푡) + 푉푒푘(푡) − 푉푒푗(푡) , 푒 ∈ 풴, 푗 ∕= 푘 ∈ 풵 , and 휂푒푓,푗(푡) = 푉푓푗(푡) − 푉푒푗(푡) , 푒 ∕= 푓 ∈ 풴, 푗 ∈ 풵 . The 푉푒푗(푡) are solutions to the ODE-s 푑푉푒푗(푡) = 푉푒푗(푡)푟푒푑푡 − 푑퐵푗(푡) −
∑
푓∕=푒
휂푒푓,푗(푡)˜ 휆푒푓푑푡 −
∑
푘∕=푗
휂푒,푗푘(푡)휇푒,푗푘(푡)푑푡 푉푒푗(푇−) = ∆퐵푗(푇) Market value of total cash-flow at time 푡, discounted at time 0, is 푀(푡) = ˜
E
[∫ 푇
0− 푆−1 0 (휏) 푑퐵(휏)
- ℱ푡
]
=
∫ 푡
0− 푆−1 0 (휏) 푑퐵(휏) + 푆−1 0 (푡) 푉 (푡)
10
SLIDE 11
푀 is (F, ˜
P)-martingale, therefore stochastic integral with respect to
the underlying (F, ˜
P)-martingales, 푀푌
푒푓 and 푀푍 푗푘 given by
푑푀푍
푗푘(푡) = 푑푁푍 푗푘(푡) − 퐼푍 푗 (푡)
∑
푒
퐼푌
푒 (푡)휇푒,푗푘(푡) 푑푡
11
SLIDE 12
To determine form of 푀, apply Itˆ
- :
푑푀(푡) = 푆−1
0 (푡)
⎛ ⎝∑
푗
퐼푍
푗 (푡) 푑퐵푗(푡) +
∑
푘; 푘∕=푗
푏푗푘(푡) 푑푁푍
푗푘(푡)
⎞ ⎠
− 푆−1
0 (푡) 푟(푡) 푑푡
∑
푒
∑
푗
퐼푌
푒 (푡) 퐼푍 푗 (푡) 푉푒푗(푡)
+ 푆−1
0 (푡)
∑
푒
∑
푗
퐼푌
푒 (푡) 퐼푍 푗 (푡) 푑푉푒푗(푡)
+ 푆−1
0 (푡)
∑
푗
퐼푍
푗 (푡)
∑
푒∕=푓
(푉푓푗(푡) − 푉푒푗(푡)) 푑푁푌
푒푓(푡)
+ 푆−1
0 (푡)
∑
푒
퐼푌
푒 (푡)
∑
푗∕=푘
(푉푒푘(푡) − 푉푒푗(푡)) 푑푁푍
푗푘(푡) .
(Left limits in the indicators 퐼푍
푗 (푡) and 퐼푌 푒 (푡) have been dropped.)
12
SLIDE 13
Insert 푑푁푌
푒푓(푡) = 푑푀푌 푒푓(푡) + 퐼푌 푒 (푡)˜
휆푒푓 푑푡, 푑푁푍
푗푘(푡) = 푑푀푍 푗푘(푡) + 퐼푍 푗 (푡) ∑ 푒 퐼푌 푒 (푡)휇푒,푗푘(푡) 푑푡
and identify drift terms and martingale terms, to arrive at: The martingale 푀 is given by 푀(0) = ∆퐵0(0) + 푉0,0(0) 푑푀(푡) = 푆−1
0 (푡)
⎛ ⎝ ∑
푗∕=푘
∑
푒
퐼푌
푒 (푡)휂푒,푗푘(푡) 푑푀푍 푗푘(푡)
+
∑
푒∕=푓
∑
푗
퐼푍
푗 (푡)휂푒푓,푗(푡) 푑푀푌 푒푓(푡)
⎞ ⎠
The differential equations for the 푉푒푗 are obtained by setting drift term equal to 0.
13
SLIDE 14
OPTIMAL HEDGING: Financial market consists of money account and 푚 risky securities based on indices related to the vital rates. Discounted price processes are martingales 푑˜ 푆푖(푡) = 푆−1
0 (푡)
∑
푒∕=푓
휉푖,푒푓(푡) 푑푀푌
푒푓(푡) ,
푖 = 1, . . . , 푚. Self-financing portfolio consisting of 휃푖(푡) units of asset No. 푖 at time 푡 has discounted value process ˜ 푉 휽(푡) =
푚
∑
푖=0
휃푖(푡) ˜ 푆푖(푡) Is martingale with dynamics 푑˜ 푉 휽(푡) =
푚
∑
푖=1
휃푖(푡) 푑˜ 푆푖(푡) = 푆−1
0 (푡)
∑
푒∕=푓 푚
∑
푖=1
휃푖(푡) 휉푖,푒푓(푡) 푑푀푌
푒푓(푡)
14
SLIDE 15
Objective is to minimize the hedging error or risk 휌휃 = ˜
E
(
푀(푇) − ˜ 푉 휃(푇)
)2 .
Inserting 푀(푇) = 푀(0) +
∫ 푇
0 푑푀(푡) ,
˜ 푉 휽(푇) = ˜ 푉 휽(0) +
∫ 푇
0 푑˜
푉 휽(푡) , and their dynamics, we get 휌휃 = ˜
E
⎡ ⎣푀(0) − ˜
푉 휽(0) +
∫ 푇
0 푆−1 0 (푡)
∑
푗∕=푘
∑
푒
퐼푌
푒 (푡) 휂푒,푗푘(푡) 푑푀푍 푗푘(푡)
+
∫ 푇
0 푆−1 0 (푡)
∑
푒∕=푓
⎛ ⎝∑
푗
퐼푍
푗 (푡) 휂푒푓,푗(푡) − 푚
∑
푖=1
휉푖,푒푓(푡) 휃푖(푡)
⎞ ⎠ 푑푀푌
푒푓(푡)
⎤ ⎦
2
15
SLIDE 16
RECALLING SOME MARTINGALE RESULTS: The martingales 푀푌
푒푓
and 푀푍
푗푘 are square integrable and mutually orthogonal, i.e the pre-
dictable covariance process of any two distinct martingales is zero. Heuristic proof: ˜
E
[
푑푁푌
푒푓(푡)
- ℱ푌
푡−
]
= 퐼푌
푒 (푡) ˜
휆푒푓 푑푡 + 표(푑푡) , ˜
E
[
푑푁푌
푒푓(푡) 푑푁푌 푔ℎ(푡)
- ℱ푌
푡−
]
= 표(푑푡) , if (푒, 푓) ∕= (푔, ℎ) , and ˜
E
[(
푑푁푌
푒푓(푡)
)2
- ℱ푌
푡−
]
= ˜
E
[
푁푌
푒푓(푡)
- ℱ푌
푡−
]
+ 표(푑푡) = 퐼푌
푒 (푡) ˜
휆푒푓 푑푡 + 표(푑푡) . From this we obtain the orthogonality property and the predictable variance processes:
16
SLIDE 17
푑⟨ ˜ 푀푌
푒푓 , ˜
푀푌
푔ℎ⟩(푡)
= ˜
E
[
푑 ˜ 푀푌
푒푓(푡) 푑 ˜
푀푌
푔ℎ(푡)
- ℱ푌
푡−
]
+ 표(푑푡) = ˜
E
[(
푑푁푌
푒푓(푡) − 퐼푌 푒 (푡)˜
휆푒푓 푑푡
) (
푑푁푌
푔ℎ(푡) − 퐼푌 푔 (푡) ˜
휆푔ℎ 푑푡
)
- ℱ푌
푡−
]
+ 표(푑푡) = ˜
E
[
푑푁푌
푒푓(푡) 푑푁푌 푔ℎ(푡)
- ℱ푌
푡−
]
+ 표(푑푡) = 훿푒푓,푔ℎ ˜
E
[
푑푁푌
푒푓(푡)
- ℱ푌
푡−
]
+ 표(푑푡) = 훿푒푓,푔ℎ 퐼푌
푒 (푡) ˜
휆푒푓 푑푡 + 표(푑푡) 훿푒푓,푔ℎ is Kroenecker delta. Martingale increments over disjoint time intervals are uncorrelated. ˜
E
[
푑 ˜ 푀푌
푒푓(푡) 푑 ˜
푀푌
푔ℎ(푢)
- ℱ푌
푡−
]
= ˜
E
[
푑 ˜ 푀푌
푒푓(푡) ˜
E
[
푑 ˜ 푀푌
푔ℎ(푢)
- ℱ푌
푢−
]
- ℱ푌
푡−
]
= 0 , 푡 < 푢.
17
SLIDE 18
For 퐺 and 퐻 predictable processes (such that expected values exist), ˜
E
[∫ 푇
0 퐺(푡) 푑 ˜
푀푌
푒푓(푡)
∫ 푇
0 퐻(푢) 푑 ˜
푀푌
푔ℎ(푢)
]
= ˜
E
[∫ 푇 ∫ 푇
0 퐺(푡) 퐻(푢) 푑 ˜
푀푌
푒푓(푡) 푑 ˜
푀푌
푔ℎ(푢)
]
= ˜
E
[∫ 푇
0 퐺(푡) 퐻(푡) ˜
E
[
푑 ˜ 푀푌
푒푓(푡) 푑 ˜
푀푌
푔ℎ(푡)
- ℱ푌
푡−
]]
= 훿푒푓,푔ℎ ˜
E
[∫ 푇
0 퐺(푡) 퐻(푡) 퐼푌 푒 (푡) ˜
휆푒푓 푑푡
]
18
SLIDE 19
The risk decomposes into 휌휃 = 휌휃
0 + 휌휃 퐼 + 휌휃 퐸,
where 휌휃
0 :=
(
푀(0) − ˜ 푉 휽(0)
)2
is squared price bias; 휌휃
퐼
:= ˜
E
⎡ ⎣ ∑
푗∕=푘
∫ 푇
0 푆−1 0 (푡)
∑
푒
퐼푌
푒 (푡)휂푒,푗푘(푡)푑푀푍 푗푘(푡)
⎤ ⎦
2
= ˜
E
⎡ ⎣ ∑
푗∕=푘
∫ 푇
0 푆−2 0 (푡)퐼푍 푗 (푡)
(∑
푒
퐼푌
푒 (푡)휂2 푒,푗푘(푡)
)
휇푍
푒,푗푘(푡)푑푡
⎤ ⎦
is the non-systematic individual risk;
19
SLIDE 20
휌휃
퐸 =
˜
E
⎡ ⎣ ∑
푒∕=푓
∫ 푇
0 푆−1 0 (푡)
⎛ ⎝∑
푗
퐼푍
푗 (푡)휂푒푓,푗(푡) − 푚
∑
푖=1
휉푖,푒푓(푡)휃푖(푡)
⎞ ⎠ 푑푀푌
푒푓(푡)
⎤ ⎦
2
= ˜
E
⎡ ⎢ ⎣ ∫ 푇
0 푆−2 0 (푡)
∑
푒
퐼푌
푒 (푡)
∑
푗
퐼푍
푗 (푡)
∑
푓∕=푒
⎛ ⎝휂푒푓,푗(푡) −
푚
∑
푖=1
휉푖,푒푓(푡)휃푖(푡)
⎞ ⎠
2
˜ 휆푒푓푑푡
⎤ ⎥ ⎦
is the systematic environment risk or hedging error.
20
SLIDE 21
Basis risk 휌휃
0 is minimized by setting
˜ 푉 휽(0) = 푀(0) . The individual risk 휌휃
퐼 does not depend on the portfolio strategy. Thus,
we are left with the problem of minimizing the environment risk.
21
SLIDE 22
To this end set 휽 ← (휃1, . . . , 휃푚)′ and introduce
휽(푡) =
∑
푒
∑
푗
퐼푌
푒 (푡−) 퐼푍 푗 (푡−) 휽푒푗(푡) ,
휽푒푗(푡) = (휃푒푗,1(푡), . . . , 휃푒푗,푚(푡))′
풴푒 = {푓; 휆푒푓 > 0} 푛푒 the number of elements 푓 in 풴푒
휼푒푗(푡) = (휂푒푓,푗(푡))푓∈풴푒 , 푗 ∈ 풵,
(푛푒 × 1) ˜
Λ푒 = Diag푓∈풴푒(˜
휆푒푓) (푛푒 × 푛푒)
Ξ푒(푡) =
(
휉푖,푒푓(푡)
)푖=1,...,푚
푓∈풴푒
(푛푒 × 푚)
22
SLIDE 23
휌휃
퐸(푡) = ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (푡)
∑
푒
∑
푗
퐼푌
푒 (푡) 퐼푍 푗 (푡) 푄푒푗,푡(휽) 푑푡
⎤ ⎦ ,
푄푒푗,푡(휽) =
(
휼푒푗(푡) − Ξ푒(푡) 휽
)′ ˜
Λ푒
(
휼푒푗(푡) − Ξ푒(푡) 휽
)
, the distance from 휼푒푗(푡) to a point Ξ푒(푡) 휽 in the linear space spanned by the columns of Ξ푒(푡) under the inner product ⟨x, y⟩ = x′ ˜
Λ푒 y.
The best hedging portfolio, ˜
휽(푡) =
∑
푒
∑
푗
퐼푌
푒 (푡−)퐼푍 푗 (푡−) ˜
휽푒푗(푡) ,
is obtained by, for each state (푒, 푗) and each time 푡, minimizing the quadratic form 푄푒푗,푡(휽). A minimizer ˜
휽푒푗(푡) is obtained by projecting 휼푒푗(푡) onto the column space of Ξ푒(푡):
23
SLIDE 24
˜
휼푒푗(푡) = P푒(푡) 휼푒푗(푡) P푒(푡) = Ξ푒(푡)
(
Ξ′
푒(푡) ˜
Λ푒 Ξ푒(푡)
)−1 Ξ′
푒(푡) ˜
Λ푒
is 푛푒 × 푛푒 projection matrix of column space of Ξ푒(푡). By Pythagoras, min 푄푒푗,푡 = 휼′
푒푗(푡) ˜
Λ푒 휼푒푗(푡) − ˜ 휼′
푒푗(푡) ˜
Λ푒 ˜ 휼푒푗(푡)
= 휼′
푒푗(푡)
(
˜
Λ푒 − P′
푒(푡) ˜
Λ푒 P푒(푡)
)
휼푒푗(푡)
= 휼′
푒푗(푡) ˜
Λ푒 (I − P푒(푡)) 휼푒푗(푡)
24
SLIDE 25
ANALYSIS OF HEDGING ERROR minimized systematic hedging error, ˜ 휌 := 휌˜
휃
퐸, is
˜ 휌 = ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (푡)
∑
푒
∑
푗
퐼푌
푒 (푡)퐼푍 푗 (푡) 휼′ 푒푗(푡) ˜
Λ푒 (I − P푒(푡)) 휼푒푗(푡) 푑푡
⎤ ⎦
= ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (휏)
∑
푒
∑
푗
퐼푌
푒 (푡) 퐼푍 푗 (푡) 휼′ 푒푗(푡) ˜
Λ푒 휼푒푗(푡) 푑푡
⎤ ⎦
− ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (휏)
∑
푒
∑
푗
퐼푌
푒 (푡) 퐼푍 푗 (푡) 휼′ 푒푗(푡) ˜
Λ푒 P푒(푡) 휼푒푗(푡) 푑푡
⎤ ⎦ .
For the standard insurance policy the 휼푒푗(푡) are deterministic. There- fore, the issue of computing ˜ 휌 depends entirely on properties of P푒(푡).
25
SLIDE 26
Simplest situation is when the P푒(푡) are deterministic functions. Then, due to the Markov property, we need only consider the state-wise compounded remaining risks, ˜ 휌푒푗(푡) = ˜
E
⎡ ⎢ ⎣ ∫ 푇
푡
푒−2 ∫ 휏
푡 푟 ∑
푒′
퐼푌
푒′ (휏)
∑
푗′
퐼푍
푗′ (휏)푤푒′푗′(휏)푑휏
- 푌 (푡) = 푒, 푍(푡) = 푗
⎤ ⎥ ⎦
where the 푤푒푗 are the deterministic functions 푤푒푗(푡) = 휼′
푒푗(푡) ˜
Λ푒 (I − P푒(푡)) 휼푒푗(푡)
The integrals are solutions to the Thiele type of backward equations: 푑 푑푡˜ 휌푒푗(푡) = 2푟푒˜ 휌푒푗(푡) − 푤푒푗(푡) −
∑
푓∈풴푒
˜ 휆푒푓
(
˜ 휌푓푗(푡) − ˜ 휌푒푗(푡)
)
−
∑
푘;푘∕=푗
휇푒,푗푘(푡)
(
˜ 휌푒푘(푡) − ˜ 휌푒푗(푡)
)
,
26
SLIDE 27
Fairly general sufficient condition for the P푒(푡) to be deterministic is that, for each 푒 and 푡, the coefficient matrix Ξ푒(푡) is of the form
Ξ푒(푡) = Ξ0,푒(푡) Ψ푒(푡) ,
where Ξ0,푒(푡) is a deterministic 푛푒×푚 matrix and Ψ푒(푡) is some 푚×푚 matrix of full rank. Then projector reduces to
P푒(푡)
= Ξ0,푒(푡)Ψ푒(푡)
(
Ψ′
푒(푡)Ξ′ 0,푒(푡)˜
Λ푒Ξ0,푒(푡)Ψ푒(푡)
)−1 Ψ′
푒(푡)Ξ′ 0,푒(푡)˜
Λ푒
= Ξ0,푒(푡)
(
Ξ′
0,푒(푡) ˜
Λ푒 Ξ0,푒(푡)
)−1 Ξ′
0,푒(푡) ˜
Λ푒 ,
which is deterministic.
27
SLIDE 28
If the 푚 risky securities are of the fairly general type considered before, then
Ξ0,푒(푡) =
(
푣푖,푓(푡) − 푣푖,푒(푡)
)푖=1,...,푚
푓∈풴푒
and
Ψ푒(푡) = Diag
(
exp(−
∫ 푡
0 푔푖,푌 (푠)(푠) 푑푠)
)
An interpretation is that the 푚 risky securities are linearly independent self-financing portfolios based on 푚 fundamental risky securities, and it says that the two sets of derivatives will span the same space of martingale dynamics and are therefore equivalent for the purpose of
- hedging. There is no need to invent complicated derivatives since all
that matters are the martingales appearing in their dynamics.
28
SLIDE 29
Best buy-and-hold strategy: With 휽 constant, 휌휃
퐸(푡)
= ˜
E
⎡ ⎣ ∫ 푇 ∑
푒
퐼푌
푒 (푡)
∑
푗
퐼푍
푗 (푡) 휼′ 푒푗(푡) ˜
Λ푒 휼푒푗(푡) 푑푡
⎤ ⎦
− 2 휽′ ˜
E
⎡ ⎣ ∫ 푇 ∑
푒
퐼푌
푒 (푡) Ξ′ 푒(푡) ˜
Λ푒
∑
푗
퐼푍
푗 (푡) 휼푒푗(푡) 푑푡
⎤ ⎦
+ 휽′ ˜
E
[∫ 푇 ∑
푒
퐼푌
푒 (푡) Ξ′ 푒(푡) ˜
Λ푒 Ξ푒(푡) 푑푡
]
휽.
29
SLIDE 30
The problem boils down to minimizing a quadratic form, and the
- ptimal portfolio is
˜
휽
=
(
˜
E
[∫ 푇 ∑
푒
퐼푌
푒 (푡) Ξ′ 푒(푡) ˜
Λ푒 Ξ푒(푡) 푑푡
])−1
× ˜
E
⎡ ⎣ ∫ 푇 ∑
푒
퐼푌
푒 (푡) Ξ′ 푒(푡) ˜
Λ푒
∑
푗
퐼푍
푗 (푡) 휼푒푗(푡) 푑푡
⎤ ⎦ .
30
SLIDE 31
WORKED EXAMPLE. Multiple decrement model with three causes
- f death, e.g. Cause 1 is cardiovascular diseases, Cause 2 is cancer,
and Cause 3 comprises all other causes of death. At the outset the mortality rate at age 푥 of Cause 1 is 휇1(푥) = 0.0003 and it remains so until it becomes 0 at all ages at a random time which is exponentially distributed with parameter ˜ 휆1 = 0.05 (medi- cal science eliminates cardiovascular diseases, expected to happen in twenty years). At the outset the mortality rate at age 푥 of Cause 2 is 휇2(푥) = 0.00002586 100.038 푥 , and it remains so until it becomes 0 at all ages at a random time which is exponentially distributed with parameter ˜ 휆2 = 0.05.
31
SLIDE 32
Mortality rate at age 푦 of Cause 3 alternates between high value, 휇3,ℎ(푥) = 1.1 휇3(푥) , and low value, 휇3,ℓ(푥) = 0.9 휇3(푥) , where 휇3(푥) = 0.0002 + 0.00005 100.038 푥, transition from high to low being at rate ˜ 휆3 = 0.1 and transition from low to high being at rate ˜ 휆4 = 0.1 (on the average there is a transition every 10 years).
32
SLIDE 33
ENVIRONMENT PROCESS Y: 휇0 = 휇1 + 휇2 + 휇3,ℎ
- ˜
휆1
- ˜
휆2
- ˜
휆3 휇1 = 휇1 + 휇2 + 휇3,ℓ 1
- ˜
휆1
- ˜
휆2
˜
휆4 휇2 = 휇2 + 휇3,ℎ 2
- ˜
휆2
- ˜
휆3 휇3 = 휇2 + 휇3,ℓ 3
- ˜
휆2
˜
휆4 휇4 = 휇1 + 휇3,ℎ 4
- ˜
휆1
- ˜
휆3 휇5 = 휇1 + 휇3,ℓ 5
- ˜
휆1
˜
휆4 휇6 = 휇3,ℎ 6
- ˜
휆3 휇7 = 휇3,ℓ 7
˜
휆4 POLICY PROCESS Z: Alive Dead 1
- 휇푌
1
SLIDE 34
There are 8 푌 -states: 0 = “Cause 1 active, Cause 2 active, Cause 3 high”, 1 = “Cause 1 active, Cause 2 active, Cause 3 low”, 2 = “Cause 1 inactive, Cause 2 active, Cause 3 high”, 3 = “Cause 1 inactive, Cause 2 active, Cause 3 low”, 4 = “Cause 1 active, Cause 2 inactive, Cause 3 high”, 5 = “Cause 1 active, Cause 2 inactive, Cause 3 low”, 6 = “Cause 1 inactive, Cause 2 inactive, Cause 3 high”, 7 = “Cause 1 inactive, Cause 2 inactive, Cause 3 low”. For an individual aged 푥 at time 0, the stochastic mortality rate at time 푡 is 휇푌 (푡)(푡) =
(
퐼푌
0 (푇) + 퐼푌 1 (푇) + 퐼푌 4 (푇) + 퐼푌 5 (푇)
)
휇1(푥 + 푡) +
(
퐼푌
0 (푇) + 퐼푌 1 (푇) + 퐼푌 2 (푇) + 퐼푌 3 (푇)
)
휇2(푥 + 푡) +
(
0.9 + 0.2 [퐼푌
0 (푇) + 퐼푌 2 (푇) + 퐼푌 4 (푇) + 퐼푌 6 (푇)]
)
휇3(푥 + 푡) .
33
SLIDE 35
Market for mortality risk: Consists of bank account with constant interest rate 푟 = 0.03 and two mortality derivatives of digital type: Derivative No. 1 pays the holder a unit at term 푈 = 20 if mortality
- f Cause 3 is then low. Pay-off is
푆1(푈) = 퐼푌
1 (푇) + 퐼푌 3 (푇) + 퐼푌 5 (푇) + 퐼푌 7 (푇) .
Derivative No. 2 pays the holder a unit at term 푈 = 20 if Cause 1 of mortality has been eliminated by then. The pay-off is 푆2(푈) = 퐼푌
2 (푇) + 퐼푌 3 (푇) + 퐼푌 6 (푇) + 퐼푌 7 (푇) .
34
SLIDE 36
Derivative No 1 is effective in all 푌 -states, and its dynamics involves
- ne martingale at any time: in ’even’ states 푒 it involves 푀푌
푒,푒+1 and
in ’odd’ states 푒 it involves 푀푌
푒,푒−1. Derivative No 2 is effective in the
푌 -states 푒 = 0, 1, 4, 5, where Cause 1 is at work, and in those states its involves the single martingale 푀푌
푒,푒+2. In the remaining states it
reduces to the bank account and can be disregarded. The capability of the derivatives to span the random environment sources: In each of the states 0 and 1 there are three sources of randomness (directly accessible states) and two effective derivatives: the projec- tion is from three dimensions onto two dimensions, resulting in a positive hedging error.
35
SLIDE 37
I each of the states 2 and 3 there are two sources of randomness and
- ne effective derivative: the projection is from two dimensions onto
- ne dimension, resulting in a positive hedging error.
I each of the states 4 and 5 there are two sources of randomness and two effective derivatives: the projection is from two dimensions onto two dimensions, resulting in no hedging error. I each of the states 6 and 7 there is one source of randomness and
- ne effective derivative: the projection is from one dimension onto
- ne dimension, resulting in no hedging error.
36
SLIDE 38
Insurance payments dependent only on survival and death. Policy process 푍 has only two states. Policy is a 푇 = 10 year life endowment with sum assured 1 against a single premium paid up front, purchased by a 50 year old at time 0. Payment function is 푑퐵(푡) = 푏 퐼푍
0 (푡) 푑1[푇,∞)(푡) − 푐 푑1[0,∞)(푡) .
37
SLIDE 39
Numerical results. In states 2, 3, 6, and 7, Cause 1 is not effective, leaving Derivative
- No. 2 trivial and with portfolio weight 0. The amounts held in the
derivatives decrease as time progresses and become null at term when no uncertainty remains. The hedging error is null in all states where the number of derivatives equals the number of random sources. Portfolio at times 푡 = 0, 5, 10 in all 푌 -states.
38
SLIDE 40
푡 = 0.0 : 휃0,0 = 0.58142 휃1,0 = 0.33192 휃2,0 = 0.00788 휃0,1 = 0.58141 휃1,1 = 0.33192 휃2,1 = 0.00792 휃0,2 = 0.58219 휃1,2 = 0.33270 휃2,2 = 0.00000 휃0,3 = 0.58553 휃1,3 = 0.33270 휃2,3 = 0.00000 휃0,4 = 0.59605 휃1,4 = 0.34027 휃2,4 = 0.00808 휃0,5 = 0.59604 휃1,5 = 0.34027 휃2,5 = 0.00812 휃0,6 = 0.60027 휃1,6 = 0.34107 휃2,6 = 0.00000 휃0,7 = 0.60027 휃1,7 = 0.34107 휃2,7 = 0.00000 푡 = 5.0 : 휃0,0 = 0.77130 휃1,0 = 0.12355 휃2,0 = 0.00357 휃0,1 = 0.77130 휃1,1 = 0.12355 휃2,1 = 0.00359 휃0,2 = 0.76960 휃1,2 = 0.12371 휃2,2 = 0.00000 휃0,3 = 0.77353 휃1,3 = 0.12371 휃2,3 = 0.00000 휃0,4 = 0.78490 휃1,4 = 0.12573 휃2,4 = 0.00363 휃0,5 = 0.78490 휃1,5 = 0.12573 휃2,5 = 0.00365 휃0,6 = 0.78717 휃1,6 = 0.12590 휃2,6 = 0.00000 휃0,7 = 0.78717 휃1,7 = 0.12590 휃2,7 = 0.00000
39
SLIDE 41
푡 = 10.0 : 휃0,0 = 1.00000 휃1,0 = 0.00000 휃2,0 = 0.00000 휃0,1 = 1.00000 휃1,1 = 0.00000 휃2,1 = 0.00000 휃0,2 = 1.00000 휃1,2 = 0.00000 휃2,2 = 0.00000 휃0,3 = 1.00000 휃1,3 = 0.00000 휃2,3 = 0.00000 휃0,4 = 1.00000 휃1,4 = 0.00000 휃2,4 = 0.00000 휃0,5 = 1.00000 휃1,5 = 0.00000 휃2,5 = 0.00000 휃0,6 = 1.00000 휃1,6 = 0.00000 휃2,6 = 0.00000 휃0,7 = 1.00000 휃1,7 = 0.00000 휃2,7 = 0.00000
40
SLIDE 42
Reserve 푉푒0(0), asset prices 푣1,푒(0) and 푣2,푒(0), and minimized hedg- ing error ˜ 휌푒,0 at time 0. 푒 푉푒0(푡) 푣1,푒 푣2,푒 ˜ 휌푒,0 0.67356 0.26938 0.34692 0.00006257 1 0.67690 0.27943 0.34692 0.00006295 2 0.67516 0.26938 0.54881 0.00006277 3 0.67850 0.27943 0.54881 0.00006315 4 0.69052 0.26938 0.34692 0.00000000 5 0.69394 0.27943 0.34692 0.00000000 6 0.69215 0.26938 0.54881 0.00000000 7 0.69558 0.27943 0.54881 0.00000000
41
SLIDE 43
OPTIMAL DESIGN OF DERIVATIVES Market with 푚 (risky) mortality derivatives and 푞 agents who take
- ptimal hedging positions.
Problem: how to design the very derivatives so as to optimally serve the objectives of the agents? Overall objective is to maximize the weighted average of the agents’ minimized hedging errors, that is, maximize 푄 =
∑
푝
푤푝 ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (휏)
∑
푒
∑
푗
퐼푌
푒 (푡) 퐼푍푝 푗 (푡) 휼′ 푝,푒푗(푡) ˜
Λ푒 P푒(푡) 휼푝,푒푗(푡) 푑푡
⎤ ⎦ ,
for some weights 푤푝, subscript 푝 labels the agents. Maximization is with respect to the projector P푒(푡) or, rather, the coefficients Ξ푒(푡).
42
SLIDE 44
Assuming that 휼푝,푒푗(푡), ˜
Λ푒, and P푒(푡) are deterministic, 푄 reduces to
푄 =
∑
푝
푤푝 ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (휏)
∑
푒
∑
푗
˜ 푝푌 푍푝
00,푒푗(0, 푡) 휼′ 푝,푒푗(푡) ˜
Λ푒 P푒(푡) 휼푝,푒푗(푡) 푑푡
⎤ ⎦ ,
where the ˜ 푝푌 푍푝
00,푒푗(푠, 푡) are the ˜
P counterparts of the simple transition
- probabilities. Introducing
˜
휼푝,푒푗(푡) = ˜ Λ1/2
푒
휼푝,푒푗(푡), ˜ Ξ푒(푡) = ˜ Λ1/2
푒
Ξ푒(푡), ˜ P푒(푡) = ˜ Ξ푒(푡)
(
˜
Ξ′
푒(푡)˜
Ξ푒(푡)
)−1 ˜
Ξ′
푒(푡),
푄 = ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (휏)
∑
푒
∑
푝
푤푝
∑
푗
˜ 푝푌 푍푝
00,푒푗(0, 푡) ˜
휼′
푝;푒푗(푡) ˜
P푒(푡) ˜ 휼푝;푒푗(푡) 푑푡
⎤ ⎦ .
Use trace operator, tr, invariant under cyclical permutations, and that ˜
P푒(푡) is symmetric and idempotent:
43
SLIDE 45
˜
휼′
푝;푒푗(푡) ˜
P푒(푡) ˜ 휼푝;푒푗(푡)
= tr
(
˜
휼′
푝;푒푗(푡) ˜
P′
푒(푡) ˜
P푒(푡) ˜ 휼푝;푒푗(푡)
)
= tr
(
˜
P푒(푡) ˜ 휼푝;푒푗(푡) ˜ 휼′
푝;푒푗(푡) ˜
P′
푒(푡)
)
, 푄 = ˜
E
⎡ ⎣ ∫ 푇
0 푆−2 0 (휏)
∑
푒
tr
⎛ ⎝ ˜
P푒(푡)
∑
푝
푤푝
∑
푗
˜ 푝푌 푍푝
00,푒푗(0, 푡)˜
휼푝,푒푗(푡)˜ 휼′
푝,푒푗(푡)˜
P′
푒(푡)
⎞ ⎠ 푑푡 ⎤ ⎦
Problem now reduced to minimizing, for each 푒 and 푡, tr
⎛ ⎝˜
P푒(푡)
∑
푝
푤푝
∑
푗
˜ 푝푌 푍푝
00,푒푗(0, 푡)˜
휼푝,푒푗(푡) ˜ 휼′
푝,푒푗(푡) ˜
P′
푒(푡)
⎞ ⎠
with respect to the projector ˜
P푒(푡) or, rather, the columns of ˜ Ξ푒(푡).
44
SLIDE 46
Use the spectral representation for symmetric positive definite matri- ces to write
∑
푝
푤푝
∑
푗
˜ 푝푌 푍푝
00,푒푗(0, 푡) ˜
휼푝,푒푗(푡) ˜ 휼′
푝,푒푗(푡) =
∑
푓∈풴푒
푑푒,푓(푡) c푒,푓(푡) c′
푒,푓(푡) ,
where the 푑푒,푓(푡) are the eigenvalues of the matrix on the left, and the
c푒,푓(푡) (c′
푒,푓(푡)) are the corresponding right (left) eigenvectors, which
are orthonormal. The best choice of ˜
Ξ푒(푡) of rank ≤ 푚 is the matrix
with columns equal to the c푒,푓(푡) corresponding to the max(푚, 푛푒) largest eigenvalues. Since tr(c푒,푓(푡) c′
푒,푓(푡)) = c′ 푒,푓(푡) c푒,푓(푡) = 1, the
maximum is (in self-explaining notation) max 푄 = ˜
E
⎡ ⎢ ⎣ ∫ 푇
0 푆−2 0 (휏)
∑
푒
∑
푓;maximal
푑푒,푓(푡) 푑푡
⎤ ⎥ ⎦ .
45
SLIDE 47
The device prescribes a rule for building a market for demographic risk in steps by supplying derivatives in their order of hedging efficiency. The first derivative has dynamics coefficients c푒,푓(푡) corresponding to the largest 푑푒,푓(푡) (in state 푒 at time 푡), the second derivative has dy- namics coefficients c푒,푓(푡) corresponding to the second largest 푑푒,푓(푡), and so on. Derivatives constructed this way may appear less trans- parent than e.g. survivor bonds of digital bonds and, therefore, be deemed unpractical. We reiterate that, what matters from a theo- retical point of view, are the random sources involved in their price dynamics.
46
SLIDE 48
BACK TO BASICS: TRADITIONAL (INTERNAL) ACTUARIAL RISK MANAGEMENT
47
SLIDE 49
PRINCIPLE OF EQUIVALENCE: Future interest and mortality etc are unknown at time 0. Equivalence must now mean 피
[∫ 푇
0− 푒− ∫ 휏
0 푟푑퐵(휏)
- ℱ푌
푇
]
= 0
∫ 푇
0− 푒− ∫ 휏
0 푟 ∑
푗
푝0푗(0, 휏)
⎛ ⎝푑퐵푗(휏) + ∑
푘;푘∕=푗
푏푗푘(휏) 휇푌 (휏); 푗푘(휏) 푑휏
⎞ ⎠ = 0
with probability one. Thus, 퐵 must be adapted to F푌 ∨ F푍.
48
SLIDE 50
INDEX-LINKED INSURANCE: Clear-cut unit linked policy: 퐵푗(푡) = 푒
∫ 푡
0 푟 푝∗
0푗(0, 푡)
푝0푗(0, 푡) 푑퐵∗
푗 (푡)
푏푗푘(푡) = 푒
∫ 푡
0 푟
푝∗
0푗(0, 푡) 휇∗ 푗푘(푡)
푝0푗(0, 푡) 휇푌 (푡); 푗푘(푡) 푏∗
푗푘(푡)
Equivalence requirement reduces to
∫
[0,푇]
⎛ ⎝∑
푗
푝∗
0푗(0, 휏) 푑퐵∗ 푗 (휏) +
∑
푗∕=푘
푝∗
0푗(0, 휏) 휇∗ 푗푘(휏) 푏∗ 푗푘(휏) 푑휏
⎞ ⎠ = 0
Arranged by choice of baseline payments 퐵∗
푗 and 푏∗ 푗푘 at time 0. Envi-
ronment risk managed perfectly from a solvency point of view. NO ASSUMPTIONS ABOUT ENVIRONMENT PROCESSES!
49
SLIDE 51
WITH PROFIT: Payments 퐵∗
푗 and 푏∗ 푗푘 guaranteed at time 0. Designed
by equivalence principle using prudent technical basis with elements 푟∗ and 휇∗
푗푘.
Surpluses that emerge are paid back as bonuses, cash dividends or additional benefits: 퐷(푡) total of dividends paid out cash by time 푡 푄(푡) total additional units of additional benefits 퐵∗+ guaranteed by time 푡. At time 푡 the company promises to pay 푄(푡) (퐵∗+(휏) − 퐵∗+(푡)) for 휏 ∈ (푡, 푇]. 퐷 and 푄 must be non-decreasing, and 퐷(0) = 푄(0) = 0. Payments from the company to the insured are 푑퐵(푡) = 푑퐵∗(푡) + 푄(푡−)푑퐵∗+(푡) + 푑퐷(푡)
50
SLIDE 52
퐵∗, 퐵∗+ are of the standard form with 퐵∗
푗 , 푏∗ 푗푘 deterministic.
푄 and 퐷 are adapted to F푌 ∨ F푍. They are not stipulated in the contract, but controlled by the company in view of the past experience and with a view to customer needs and solvency. Liability in respect of future payments at time 푡 is 푉 ∗
푍(푡)(푡) + 푄(푡) 푉 ∗+ 푍(푡)(푡)
Discounted surplus at time 푡 is ˜ 푊(푡) = −
∫ 푡
0− 푒− ∫ 휏
0 푟 (
푑퐵∗(휏) + 푄(휏−) 푑퐵∗+(휏) + 푑퐷(휏)
)
− 푒− ∫ 푡
0 푟
(
푉 ∗
푍(푡)(푡) + 푄(푡) 푉 ∗+ 푍(푡)(푡)
)
˜ 푊(0) = −∆퐵∗(0) − 푉 ∗
0 (0) = 0
51
SLIDE 53
˜ 푊(푇) = −
∫ 푇
0− 푒− ∫ 휏
0 푟 (
푑퐵∗(휏) + 푄(휏−) 푑퐵∗+(휏) + 푑퐷(휏)
)
If first order basis is chosen on the entirely safe side and bonuses are allotted with sufficient prudence, then one can arrange that ˜ 푊(푡) ≥ 0 for all 푡, and there is no solvency problem. Solvency requirement:
E
[
˜ 푊(푡)
- ℱ푌
푡
]
≥ 0 , 푡 ∈ [0, 푇] (1) Equivalence:
E
[
˜ 푊(푇)
- ℱ푌
푇
]
= 0 (2)
52
SLIDE 54
Applying Itˆ
- to ˜
푊(푡): 푑 ˜ 푊(푡) = 푒− ∫ 휏
0 푟(푑퐶(푡) − 푑퐷(푡) − 푑푄(푡)푉 ∗+
푍(푡)(푡) + 푑푀∗(푡))
퐶(푡) is drift term representing “technical surplus”: 푑퐶(푡) = (푟(푡) − 푟∗)
(
푉 ∗
푍(푡)(푡) + 푄(푡) 푉 ∗+ 푍(푡)(푡)
)
푑푡 +
∑
푘;푘∕=푍(푡)
(
휇∗
푍(푡) 푘(푡) − 휇푌 (푡); 푍(푡) 푘(푡)
) (
푅∗
푍(푡) 푘(푡) + 푄(푡)푅∗+ 푍(푡) 푘(푡)
)
푑푡 푅∗
푗푘(푡) = 푏∗ 푗푘(푡) + 푉 ∗ 푘 (푡) − 푉 ∗ 푗 (푡) ,
푅∗+
푗푘 (푡) = 푏∗+ 푗푘 (푡) + 푉 ∗+ 푘
(푡) − 푉 ∗+
푗
(푡) 푀∗(푡) is martingale representing pure life history randomness: 푑푀∗(푡) =
∑
푔∕=ℎ
(
푅∗
푗푘(푡) + 푄(푡−)푅∗+ 푗푘 (푡)
) (
푑푁푍
푗푘(푡) − 퐼푗(푡)휇∗ 푌 (푡); 푗푘(푡) 푑푡
)
53
SLIDE 55
Writing ˜ 푊(푇) =
∫
[0,푇] 푑 ˜
푊(휏), and forming conditional expectation, equivalence can be recast as
E
[∫ 푇
0 푒− ∫ 휏
0 푟 (푑퐶(휏) − 푑퐷(휏) − 푑푄(푡)푉 ∗+
푍(푡)(푡))
- ℱ푌
푇
]
= 0 NO ASSUMPTIONS ABOUT ENVIRONMENT PROCESSES!
54
SLIDE 56
REFERENCES: Andersen, Borgan, Gill, and Keiding (1993). Statistical Models Based
- n Counting Processes. Springer-Verlag.
Cairns, A.J.G., Blake, D., Dowd, K. (2008): Modelling and manage- ment of mortality risk: an overview. Scandinavian Actuarial Journal 2008, 2/3, 70113. Dahl, M (2004): Stochastic mortality in life insurance: market re- serves and mortality-linked insurance contracts. Insurance: Mathe- matics & Economics 35, 113-136. Dahl, M and Møller, T (2006): Valuation and hedging of life insur- ance liabilities with systematic mortality risk. Insurance: Mathematics & Economics 39, 193-217. Dahl, M., Melchior, M., Møller, T. (2008): On systematic mortality risk and risk-mini mization with survivor swaps. Scandinavian Actu- arial Journal, 2008, 2/3, 114-146.
55
SLIDE 57
Norberg,R (1999): A theory of bonus in life insurance. Finance and Stochastics 3, 373-390. Norberg, R (2003): The Markov chain market. ASTIN Bulletin 33, 265-287. Norberg (2006): The pension crisis: its causes, possible remedies, and the role of the regulator. In Erfaringer og utfordringer, 20 years Ju- bilee Volume of Kredittilsynet, the Financial Supervisory Authority of
- Norway. Preliminary version at http://stats.lse.ac.uk/norberg/links/papers/KredTils20.do