XXIII CNSF International Seminar

Workshop on long dated insurance and pension contracts

14-15 November 2013, Mexico City

RISK SHARING IN LIFE INSURANCE AND PENSIONS

Ragnar Norberg

ISFA, Universite Lyon 1, France

E-mail: ragnar.norberg@univ-lyon1.fr

Homepage: http://isfa.univ-lyon1.fr/∼norberg

1

BACKGROUND IN KEYWORDS

Traditional insurance mathematics: Perfect internal management of

demographic and financial risk through with-profit schemes adapting

payments to indices for interest, mortality, etc.

Deregulation in the 80-es lead to:

In finance: Financial innovation and emergence of huge derivatives

markets. Growth in volume of trades, increasingly research- and

technology-driven. Investment banks growing and hedge funds emerg-

ing (“fonds de speculation”). Financial instability culminating in the

ongoing crisis.

2

In life insurance and pensions: with profit gave way to a plethora ofproducts equipped with various forms of guarantees (risk creation).Risk management increasingly based on market operations: reinsur-ance, swaps, securitization. Overreliance in the efficient market hy-pothesis (on which modern financial mathematics builds).

Parallel development in actuarial science seen as an area of applicationof financial maths – pricing and hedging of guarantees.

Pension crisis triggered by faltering markets and improved longevityand by the guarantee culture.

Reintroduction of regulation. Solvency II, IFRS.

Renewed interest in with profit? Or seek new ways of adapting pre-miums and benefits to indices for interest and mortality?

3

SAVINGS CONTRACT

Sj price of one unit of asset at time j = 1,2, ...

cj invested at time j = 1,2, ...,m

Contribution at time j purchases uj = 1Sjcj units (cj = uj Sj)

Value of portfolio at time t = 1,2, ...,m is

Vt = St

t∑j=1

uj = St

t∑j=1

1

Sjcj

bj benefit withdrawn at time j = m+ 1, ..., T

Value of portfolio at time t = m+ 1, ..., T is

Vt = St

m∑j=1

1

Sjcj −

t∑j=m+1

1

Sjbj

4

Contract terminates at time T with balance VT = 0:

m∑j=1

1

Sjcj =

T∑j=m+1

1

Sjbj

Level contributions cj = c and benefits bj = b:

c

b=

∑Tj=m+1

1Sj∑m

j=11Sj

If Sj increasing then cb <

T−mm .

The “steeper” Sj, the smaller c/b.

5

MORTALITYLarge number `0 of new-born`x survivors at age x = 1,2, . . .dx = `x − `x+1 deaths at age xqx = dx/`x mortality rate at age x

x `x dx qx

0 100 000 58 0.00057925 98 083 119 0.00120650 91 119 617 0.00677460 82 339 1 275 0.01548470 65 024 2 345 0.03606980 37 167 3 111 0.08371190 9 783 1 845 0.188617

100 401 158 0.394000

6

PENSIONS CONTRACT

VT = ST

m∑j=1

1

Sj`x+j cj −

T∑j=m+1

1

Sj`x+j bj

Vt ≥ 0 for all t and VT = 0:

m∑j=1

1

Sj`x+j cj =

T∑j=m+1

1

Sj`x+j bj

Level contributions cj = c and benefits bj = b:

c

b=

∑Tj=m+1

1Sj`x+j∑m

j=11Sj`x+j

7

NUMERICAL ILLUSTRATIONS: x = 30, m = 35, T = 70, values of

c/b:

No interest, no mortality: 1.0000

4% interest, no mortality: 0.2538

4% interest, mortality as in table (mean life length 73): 0.1149

4% interest, mortality half of table (mean life length 81): 0.1592

2% interest, mortality as in table: 0.2035

2% interest, mortality half of table: 0.2918

m = 39, T = 70

2% interest, mortality as in table: 0.2101

8

RISK MANAGEMENT. Balance requirement (the principle of equiv-

alence):

m∑j=1

1

Sj`x+j cj =

T∑j=m+1

1

Sj`x+j bj (1)

If future indices Sj and `x+j are uncertain, equivalence can only be

attained if the payments are allowed to depend on the indices.

We shall consider two ways of doing this.

9

PERFECT UNIT LINKED INSURANCE (index-regulated payments):

At time 0 specify a technical basis S∗j and `∗j representing “best esti-

mates” of future indices, and baseline payments c∗j and b∗j representing

intended profile of the payments, designed to satisfy equivalence un-

der the technical scenario:

m∑j=1

1

S∗j`∗x+jc

∗j =

T∑j=m+1

1

S∗j`∗x+jb

∗j

Actual payments are index-regulated:

cj =Sj

S∗j

`∗x+j

`x+jc∗j bj =

Sj

S∗j

`∗x+j

`x+jb∗j

Equivalence (1) is then attained under all scenarios.

10

Solvency for sure.

No assumptions about future interest and mortality needed

Unit linked in its pure form is not used in practice. Usually premiums

are not index linked and benefits are equipped with a guarantee that

they cannot go below a certain nominal level.

Such guarantees reintroduce risk on the part of the insurer.

11

WITH PROFIT INSURANCE

Contractual contributions c∗j and benefits b∗j specified at time 0 such

that reserve is non-negative and balance is attained under prudent

technical assumptions (S∗0 = 1), S∗j , `∗x+j, j = 1, . . . , T :

V ∗t ≥ 0, t = 1, . . . , T − 1

V ∗T = 0 (2)

Technical reserve at time t (retrospective = prospective due to (2)):

V ∗t = S∗tt∑

j=1

1

S∗j`∗x+j (c∗j − b

∗j) = S∗t

T∑j=t+1

1

S∗j`∗x+j (b∗j − c

∗j) (3)

(Notation: b∗j = 0, j = 1, . . . ,m, c∗j = 0, j = m+ 1, . . . , T .)

12

Factual interest and mortality are given by (S0 = 1), Sj, `x+j, j =1, . . . , T . Non-negative dividends (bonuses) dj, j = 1, . . . , T , are paidif technical assumptions generate surplus. Define surplus at time t asaccumulated value of past payments less discounted value of futurepayments under technical assumptions:

Wt =t∑

j=1

St

Sj`x+j(c

∗j − b

∗j − dj) − `x+t

T∑j=t+1

S∗tS∗j

`∗x+j

`∗x+t

(b∗j − c∗j)

Use (3) and rearrange to write Wt as technical surplus less dividends:

Wt =t∑

j=1

(1 − δt,j)St

Sj`x+j (b∗j − c

∗j) −

t∑j=1

St

Sj`x+j dj (4)

δt,j =S∗t /S

∗j

St/Sj

`x+t/`x+j

`∗x+t/`∗x+j

≤ 1 if S∗t /S∗j ≤ St/Sj and `∗x+t/`

∗x+j ≥ `x+t/`x+j

13

Pay back surplus in arrears as bonus (not a contractual obligation) in

a manner that balance is attained at time T .

Solvency ensured if technical assumptions are sufficiently prudent.

No assumptions about future interest and mortality needed

The contractual payments are binding to the insurer throughout the

term of the contract. Thus, the technical basis represents a guarantee

to the policy holder that benefits will not be less than stipulated

in the contract (and premiums cannot be raised) no matter what

future mortality and investment experience will be. Such guarantees

represent risk on the part of the insurer.

14

MARKET METHODS

MORTALITY SWAPS:

Essentially a form of reinsurance. OTC - not liquidly traded. No price

record.

MORTALITY DERIVATIVES:

Can work for catastrophic mortality risk in the short term. Risk

is limited in time and space and can be absorbed by capital mar-

kets. Catastrophes can be predicted by mathematical models fitted

to historical data.

15

Longevity risk is all different:

Longevity derivatives need to be exceedingly long term. Longevity

is the resultant of complex demographic, societal, and technological

processes. Cannot be reliably predicted by mathematical models fit-

ted to historical data. (Mathematical hedging theories are out.)

In the shorter term longevity developments can be anticipated from

current collateral information to which investors will not have equal

access. Arbitrages to be fetched by big investors/speculators.

The scale is too large. Pensions are a major accumulator of savings

(bigger than the mortgage market), and longevity risk is at macroe-

conomic scale.

The very idea of longevity guarantees is questionable.

16

FINANCIAL MATHEMATICS

Traded assets:Bank account with price process Bt = ert

Stock with price process St = eαt+σ∑ti=1Xi

(Physical) Probability measure P: the Xi i.i.d. Bin(0, p), 0 < r−α < σ

Portfolio at time t is (ηt, ξt) chosen at time t− 1Its value at time t is Vt = ηtBt + ξtStIt is self-financing (SF) ifVt = ηt+1Bt + ξt+1St or (ηt+1 − ηt)Bt + (ξt+1 − ξt)St = 0Since Vt+1 = ηt+1Bt+1 + ξt+1St+1, SF means∆Vt = Vt+1 − Vt = ηt+1∆Bt + ξt+1∆St

An SF portfolio is an arbitrage if V0 < 0 and VT ≥ 0There should be no arbitrage in an efficient market (model!).

17

Discounted values B = DtBt, S = DtSt, V = DtVtSF property is preserved under discounting:

Vt = ηt+1Bt + ξt+1St

Convenient to take Dt = B−1t . Then

Bt = 1, St = e(α−r)t+σ∑ti=1Xi,

∆Bt = 0

∆St = St[Xt+1(eα−r+σ − 1) + (1−Xt+1)(eα−r − 1)

]= Ste

α−r(eσ − 1)(Xt+1 − q)∆Vt = ξt+1 ∆St for SF portfolio

q = (er−α − 1)/(eσ − 1) ∈ (0,1)

18

Equivalent Martingale Measure (EMM) Q: the Xi i.i.d. Bin(0, q).

S is Q-martingale: EQ[∆St|Ft] = 0.

V is Q-martingale for SF portfolio: EQ[∆Vt|Ft] = ξt+1EQ[∆St|Ft] = 0

Therefore, EQVT = V0, so there exists no arbitrage.

Suppose a traded derivative of the stock pays h(ST ) at time T .

It is attainable if there exists SF portfolio with P[VT = h(ST )] = 1.

Then the price of the derivative at time t ≤ T must be Vt.

The present market is complete: every financial derivative is at-

tainable. Prove this by constructing SF portfolio such that VT =

e−rTh(ST ) a.s.

19

Start from martingale Vt = EQ[e−rTh(ST )|Ft

]= EQ [H(X1, . . . , XT )|Ft]

Using independence, and writing E∗ for EQ w.r.t. Xt+2, . . . , XT only,

∆Vt = EQ[H(X1, . . . , XT )|Ft+1

]− EQ [H(X1, . . . , XT )|Ft]

= Xt+1 E∗H(X1, . . . , Xt,1, Xt+2, . . . , XT )

+(1−Xt+1)E∗H(X1, . . . , Xt,0, Xt+2, . . . , XT )

−q E∗H(X1, . . . , Xt,1, Xt+2, . . . , XT )

−(1− q)E∗H(X1, . . . , Xt,0, Xt+2, . . . , XT )

=[E∗H(X1, . . . , Xt,1, Xt+2, . . . , XT )

− E∗H(X1, . . . , Xt,0, Xt+2, . . . , XT )]

(Xt+1 − q)= ξt+1 ∆St

Thus, (ηt, ξt)t=0,...,T with ηt = Vt − ξt St is SF portfolio attaining the

claim.20

LIFE INSURANCE AND PENSIONS

A T -year endowment with sum b is sold to an x year old against single

premium π.

Remaining life length Tx, P[Tx > t] = exp[−∫ t0 µx+u du]

Ulpianus (about 40 AD): Replace the uncertain life length with ex-

pected.

De Witt (17th century): Principle of equivalence:

π = E[b e−

∫ T0 ru du 1[Tx > T ]

]= b e−

∫ T0 (ru+µx+u) du

21

The contract is long term and the development of r and µ is uncertain.

The principle of equivalence must be recast as

π = E[b e−

∫ T0 ru du 1[Tx > T ]

∣∣∣∣ Ft] = b e−∫ T

0 (ru+µx+u) du

A resolution is perfect index-linked insurance:

b = π e∫ T

0 (ru+µx+u) du

Another possibility is with profit insurance.

NB! Solvency with probability 1 (if implemented properly).

No model assumptions about r and µ.

A different approach is Alternative Risk Transfer.

Securitization merits discussion.22

Market methods through securitization. Model:

r constant

µx+u = mx+u + σXi, i− 1 ≤ u < i, i = 1, . . . , T

(Physical) Probability measure P: the Xi i.i.d. Bin(0, p),

Introduce mortality derivative with payment G(X1, . . . , XT ) at time T .

Hedge the claim

H(X1, . . . , XT ) = e−∫ T

0 mx+u du−σ∑Ti=1Xi

perfectly by trading in the bank account and the derivative (the same

maths as for the financial claim seen before).

And we are done!

23

Second thoughts:

How to determine Q? Is the model right?

If H = G things are simple, no model needed. If not, there is super-

imposed model risk.

Dangers are looming: Speculation, Asymmetry of information, Bub-

bles and panic, ...

Does the product make sense to customers?

Does the very idea make economic sense?

24

Nothing wrong with with profit. The problem was that it was not im-

plemented properly. Presumably, under the pressure of competition,

the insurers weren’t sufficiently prudent in their assumptions about

future interest and mortality developments (technical basis), the sur-

pluses were squandered on premature bonuses, and bonuses were even

guaranteed before the surpluses were gained. The blame should not

be put on the with profit concept, it should be placed at the door of

the actuaries. (If you get a ticket for speeding, you should not blame

your Mercedes.)

Analysis of interest, mortality, expenses and other collective risk fac-

tors is an essential part of actuarial work, but an even more important

part - that only the actuary can do - is the design of the scheme and

the insurance contracts.25

THE MATHEMATICS OF RISK SHARING

26

Consider multi-state life insurance contract with states {0, . . . , n},starting in state 0 at time 0 and terminating at time T . State of

policy at time t is Z(t).

Total payments (benefits less premiums) in time interval [0, t] are B(t)

given by

dB(t) =∑j

Ij(t−)dBj(t) +∑j 6=k

bjk(t) dNjk(t) t ∈ [0, T ] ,

Ij(t) = 1[Z(t) = j] indicator processes,

Njk(t) = ]{s ∈ (0, t]; Z(s−) = j, Z(s) = k} counting processes,

Bj is payment stream (general annuity) running in state j,

bjk is sum assured payable upon transition from state j to state k.

Filtration generated by the policy is FN = (FNt )t∈[0,T ].

27

Payments are deposited/withdrawn from investment portfolio with

interest rate r(t) at time t.

Transition intensities for the policy are µjk(t), j 6= k.

Y (t) = (r(t), µjk(t), j 6= k) is stochastic process generating filtration

FY = (FYt )t∈[0,T ].

Thus, N is Markov given Y .

28

PRINCIPLE OF EQUIVALENCE requires balance on the average in

an infinitely large portfolio.

Future interest and mortality are unknown at time 0.

Therefore, equivalence must mean

E[∫ T

0−e−∫ τ

0 rdB(τ)

∣∣∣∣∣FYT]

= 0

or ∫ T0−

e−∫ t

0 r∑j

p0j(0, t)

dBj(t) +∑

k;k 6=j

bjk(t)µjk(t) dt

= 0 (5)

with probability one.

Therefore, B must be adapted to FY ∨ FN .

29

INDEX-LINKED INSURANCE

Let r∗ and µ∗jk, j 6= k, be base-line interest and transition rates repre-

senting best estimates at time 0.

Stipulate base-line payments B∗j and b∗jk that would be suitable under

this scenario and satisfying equivalence:∫[0,T ]

e−∫ t

0 r∗∑

j

p∗0j(0, t) dB∗j (t) +∑j 6=k

p∗0j(0, t)µ∗jk(t) b∗jk(t) dt

= 0 .(6)

30

Clear-cut index-linked policy has state-wise annuities Bj and assur-ances bjk given by

dBj(t) =e−∫ t

0 r∗p∗0j(0, t)

e−∫ t

0 r p0j(0, t)dB∗j (t)

bjk(t) =e−∫ t

0 r∗p∗0j(0, t)µ∗jk(t)

e−∫ t

0 r p0j(0, t)µjk(t)b∗jk(t)

Inserting this into (5) gives (6), hence equivalence with probability 1.

NO ASSUMPTIONS ABOUT ENVIRONMENT PROCESSES. Byconditioning on the factual outcome of the environment process,nothing depends on its distribution, which cannot be reliably mod-elled for the long term anyway.

31

MORAL: Do not let vital decisions that determines the survival of an

insurance scheme depend on speculative assumptions about interest,

mortality, etc over the next century.

32

WITH PROFIT: Payments B∗j and b∗jk guaranteed at time 0. Designed

by equivalence principle using prudent technical basis with elements

r∗ and µ∗jk.

Surpluses that emerge are paid back as bonuses, cash dividends or

additional benefits:

D(t) total of dividends paid out cash by time t

Q(t) total additional units of additional benefits B∗+ guaranteed by

time t.

At time t the company promises to pay Q(t) (B∗+(τ) − B∗+(t)) for

τ ∈ (t, T ].

D and Q must be non-decreasing, and D(0) = Q(0) = 0.

Payments from the company to the insured are

dB(t) = dB∗(t) +Q(t−)dB∗+(t) + dD(t)

33

B∗, B∗+ are of the standard form with B∗j , b∗jk deterministic.

Q and D are adapted to FN ∨ FY . 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 t is

V ∗Z(t)

(t) +Q(t)V ∗+Z(t)

(t)

Discounted surplus at time t is

W (t) = −∫ t

0−e−∫ τ

0 r(dB∗(τ) +Q(τ−) dB∗+(τ) + dD(τ)

)− e−

∫ t0 r

(V ∗Z(t)

(t) +Q(t)V ∗+Z(t)

(t))

(7)

W (0) = −∆B∗(0)− V ∗0 (0) = 0

34

W (T ) = −∫ T

0−e−∫ τ

0 r(dB∗(τ) +Q(τ−) dB∗+(τ) + dD(τ)

)

If first order basis is chosen on the entirely safe side and bonuses are

allotted with sufficient prudence, then one can arrange that W (t) ≥ 0

for all t, and there is no solvency problem.

Solvency requirement:

E[W (t)

∣∣∣ FYt ] ≥ 0 , t ∈ [0, T ] (8)

Equivalence:

E[W (T )

∣∣∣ FYT ] = 0 (9)

35

Applying Ito to W (t):

dW (t) = e−∫ τ

0 r(dC(t)− dD(t)− dQ(t)V ∗+Z(t)(t) + dM∗(t))

C(t) is drift term representing “technical surplus”:

dC(t) = (r(t)− r∗(t))(V ∗Z(t)

(t) +Q(t)V ∗+Z(t)

(t))dt

+∑

k;k 6=Z(t)

(µ∗Z(t) k(t)− µZ(t) k(t)

) (R∗Z(t) k(t) +Q(t)R∗+

Z(t) k(t))dt

R∗jk(t) = b∗jk(t) + V ∗k (t)− V ∗j (t) , R∗+jk (t) = b∗+jk (t) + V ∗+k (t)− V ∗+j (t)

M∗(t) is martingale representing purely individual life history random-ness:

dM∗(t) =∑g 6=h

(R∗jk(t) +Q(t−)R∗+jk (t)

) (dNjk(t)− Ij(t)µjk(t) dt

)

36

Writing W (T ) =∫[0,T ] dW (τ), and forming conditional expectation,

equivalence can be recast as

E[∫ T

0e−∫ τ

0 r (dC(τ)− dD(τ)− dQ(t)V ∗+Z(t)(t))

∣∣∣∣∣ FYT]

= 0

NO ASSUMPTIONS ABOUT ENVIRONMENT PROCESSES!

37

REFERENCES:

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.

Norberg, R. (1999): A theory of bonus in life insurance. Fi-

nance and Stochastics 3, 373-390.

Norberg, R. (2013): Optimal hedging of demographic risk in life in-

surance. Finance and Stochastics 17, 197-222.

38

Resume: A sustained trend of improving longevity prospects and

repeated down-falls of the financial markets have been identified as

the principal causes of the pension crisis, ongoing since more than

a decade now. This perception needs to be completed by a broader

historical view: already about a century ago insurance mathematics

provided principles for perfect management of demographic and fi-

nancial risk through so-called risk sharing in the form of with-profit

schemes designed as follows: Upon the inception of the contract the

premiums and the benefits are guaranteed in nominal amounts for

the entire contract period; Premiums are determined by the principle

of equivalence (expected balance between discounted premiums and

discounted benefits) based on a worst-case economic-demographic

scenario called the first order (technical) basis; The systematic sur-

pluses generated by the conservative premiums are redistributed to

39

the insured in arrears as bonus. However simple the idea, its technical

implementation is non-trivial since the scheme needs to be, not only

equitable, but also fair in the following precise actuarial sense: with

probability 1 the principle of equivalence should ultimately be attained

conditionally, given the course of economic-demographic events over

the contract period.

The purpose of the present paper is to pursue the idea of risk sharing

in life insurance and develop it further in two specific directions: (1)

Revisit the with-profit concept and reshape it in terms of the cash flow

dynamics in a manner that, firstly, comprises in a unified manner the

various forms of bonuses (cash dividends, terminal bonus, and guar-

anteed added benefits) and, secondly, allows the first order reserve to

be based on technical elements that are dynamically adapted

40

to experience (“prudent predictions”). (2) Work out in full generality

what can be called perfectly index-linked insurance, whereby payments

(premiums as well as benefits) are linked to financial and demographic

indices in a manner that automatically establishes solvency and fair-

ness as defined above.

41