2
Course content
1. Main features of pension systems (see ([1]), ([4]))
(a) Three pension pillars
(b) Classification of pensions
(c) Methods for calculation (determinations) of pension size
(d) Necessity of solidarity between generations
(e) Pension systems in European Union and USA.
2. Demographical and financial models of pension systems (see ([2]), p.p.259-274; 511-520; 533-550 )
(a) Pure and relative probabilities
(b) Lexis diagram
(c) Funding models of pension systems: terminal funding; fundingmethods; individual / aggregate funding methods
3. Pension annuities (see ([3]), p.p. 2-20; 58-63; 137-148; 152-158; 186-195)
(a) Deterministic vs. stochastic approach
(b) Longevity risk: cohort and period mortality tables
(c) Extrapolation methods for construction of projected mortality ta-bles
(d) Single entry mortality table (age adjustment method)
(e) Parametric methods
(f) Lee Carter method and its modifications
4. Longevity risk (see ([3]), p.p. 267-303; 318-333)
(a) Coefficient of variation and other risk measures
(b) Risk and mortality scenario
(c) Pooling and non pooling parts of risk
(d) Methods for management of Longevity risk: hedging, reinsurance,longevity bonds, etc.
Chapter 1
Pension systems
Period when individual is able to work is shorther than the period ofconsumption. During periods when individual is unable to work for remu-neration (is unable to earn for living) he / she must either relay on thefinancial support from others or to ensure his financial situation by savings(investing) and / or participating in some scheme thus acquiring the rightfor pension benefits. To save for retirement means to delay consumption oftoday for the sake of the future.
Typical problems that my arise when saving for retirement:
1. Unknown period of retirement (including longevity risk)
2. Unknown level of prices and living standards
3. Unknown needs (e.g. health status)
4. Inability to save (invest) today (dependents, mortgage, low salary)
1.1 Three pension pillars. Classification of pen-sions
Probably in most modern countries pension benefits are provided viathree main pillars:
Ist pillar - Pension provided via public scheme (social insurance, govern-ment, etc.)
IInd pillar - Pension provided via employer
IIIrd pillar - Individual savings / investments
3
4 CHAPTER 1. PENSION SYSTEMS
Intergenerational solidarity is essential part of almost every system. Sol-idarity between generations may be achieved:
• In informal way, through different generations in one family
• Through labor market (employer)
• Through taxation system in particular state
• Through capital markets
Pension systems may be classified according many peculiarities, e.g..
• Basis for provision of pension benefits
Citizenship - pension benefits are provided for citizens of country
Living conditions - pension benefits are provided only for those un-able to maintain lowest acceptable living standard
Employment - pension benefits are provided for those who were em-ployed in labour market for specified period
Profession - pension benefits are provided for specific professionalsonly
Individual contracts - pension benefits are provided for those whoconcluded individual contracts
• Method for determination of pension size:
Defined benefit (DB)
Defined Contribution (DC)
Flat rate
Means tested
• Administration of scheme
Public
Private
• Method of financing
PAYG
Funded system
1.2. PENSION SYSTEMS IN EU AND USA 5
1.2 Pension systems in EU and USA
1.3 Sample exercises1 Exercise. Prepare short (5-10 min.) presentation about pension systemin specific country. Main accents:
• Combination of (cooperation between) three main pillars
• Methods of financing
• Basis for provision of pension benefit
• Other essential information
Chapter 2
Demographical and financialmodels of pension systems
2.1 Population models
2.1.1 Closed population with one possibility of with-drawal
Lifetime of newborn is considered to be continous random variable X,determined in [0, ω], where 0 < ω <∞ - maximum age of individual.
2.1.1 Definintion. F (x) = P [X ≤ x]; x ≥ 0 is d.f. of X and s(x) =F (x) = 1− F (x) = P [X > x] - X is called survival function.
Obvious that F (0) = 0 and s(0) = 1.
It is easy to see that:
P (x < X ≤ z) = F (z)− F (x) = s(x)− s(z), (2.1.1)
P (x < X ≤ z|X > x) =F (z)− F (x)
1− F (x)=s(x)− s(z)
s(x). (2.1.2)
For a person aged (x), (0 ≤ x ≤ ω) future life time is defined by T (x).We will use notation:
tqx = P (T (x) ≤ t); t ≥ 0
tpx = 1− tqx = P (T (x) > t); t > 0;
t|uqx = P (t < T (x) ≤ t+ u) =t+u qx −t qx =t px −t+u px; (2.1.3)
7
8CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
andxp0 = s(x). (2.1.4)
Alternative expressions:
tpx =x+tp0
xp0
=s(x+ t)
s(x); (2.1.5)
tqx = 1− s(x+ t)
s(x);
t|uqx =s(x+ t)− s(x+ t+ u)
s(x).
and:
t|uqx =s(x+ t)
s(x)· s(x+ t)− s(x+ t+ u)
s(x+ t)= tpx · uqx+t. (2.1.6)
If function F (x) is differentiable then F ′(x) = f(x) and:
P (x < X ≤ x+M x|X > x) =F (x+ M x)− F (x)
1− F (x)≈ f(x)
1− F (x)M x. (2.1.7)
Let’s defineµx =
f(x)
1− F (x)=−s′(x)
s(x), (2.1.8)
µx - is called force of mortality at age x.
Using properties of f(x) and F (x) we get:
µx ≥ 0. (2.1.9)
Moreover:
d(ln(s(y))) =s′(y)
s(y)dy = −µydy, (2.1.10)
Integrating we get:
−∫ x+n
x
µydy = lns(x+ n)
s(x)= ln npx, (2.1.11)
or equivalently:
npx = e−∫ x+n
x µydy (2.1.12)
2.1. POPULATION MODELS 9
and
xp0 = s(x) = e−∫ x0 µsds. (2.1.13)
It is easy to see that:
F (x) = 1− s(x) = 1− e−∫ x0 µsds (2.1.14)
andf(x) = F ′(x) = e−
∫ x0 µsds · µx = xp0µx. (2.1.15)
Analogical formulas for random variable T (x) with d.f. G(t) and p.d.f.g(t) are:
tqx = P (T (x) ≤ t), (2.1.16)
G(t) = tqx, (2.1.17)
g(t) = G′(t) =d
dttqx.
Then
g(t) =d
dttqx =
d
dt(1− s(x+ t)
s(x)) = −s
′(x+ t)
s(x)=s(x+ t)
s(x)· (−s
′(x+ t)
s(x+ t)).
(2.1.18)
g(t) = tpxµx+t; t ≥ 0 (2.1.19)is probability that (x) will die in interval: (t; t+ dt). Then:∫ ∞
0tpxµx+t = 1. (2.1.20)
(since g(t) - is density of random variable).Using
d
dttqx =
d
dt(1− tpx) = − d
dttpx = tpxµx+t. (2.1.21)
we getlimn→∞ npx = 0, (2.1.22)
thenlimn→∞
(− ln(npx)) =∞ (2.1.23)
and finally
limn→∞
∫ x+n
x
µydy =∞. (2.1.24)
10CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
2.1.1.1 Mortality table
Suppose l0 is set of newborns (e.g. l0 = 100000). For a newborn j =1, 2, ..., l0 his / her surival function is the same - s(x). Let, L(x) - set ofindividuals from l0 who attained age x. Then:
L(x) =
l0∑j=1
Ij, (2.1.25)
where
Ij =
{1, if individual attained age x;0, otherwise. (2.1.26)
It is easy to see that:
E(Ij) = s(x) (2.1.27)
and then
E(L(x)) =
l0∑j=1
E(Ij) = l0s(x). (2.1.28)
We will definelx:lx = l0s(x). (2.1.29)
L(x) has binomial distribution with parameters n = l0, p = s(x).
Let random variable nDx - number of individuals from l0 who died be-tween (x, x+ n]. Consider E(nDx) = ndx, then:
ndx = l0[s(x)− s(x+ n)] = lx − lx+n. (2.1.30)
2.1.2 Theorem. Suppose T (x) - continous random variable with d.f. G(t)and G(0) = 0; G′(t) = g(t). Let z(t) - nonnegative, monotonic and differen-tiable function. Assume that E(z(t)) exist. Then
E[z(t)] =
∫ ∞0
z(t)g(t)dt = z(0) +
∫ ∞0
z′(t)(1−G(t))dt. (2.1.31)
Proof. see Bowers, p.p. 62-63.
2.1. POPULATION MODELS 11
Complete expectation of life :
oex= E[T (x)] =
∫ ∞0
ttpxµx+tdt. (2.1.32)
Using Theorem (2.1.2) with z(t) = t ir G(t) = 1− tpx, we get:
oex=
∫ ∞0
tpxdt. (2.1.33)
Let Lx - total expected number of years lived x and x+ 1 by individualsfrom l0:
Lx =
∫ 1
0
tlx+tµx+tdt+ lx+1, (2.1.34)
Easy to see that:
Lx = −∫ 1
0
tdlx+t + lx+1 = −tlx+t|10 +
∫ 1
0
lx+tdt+ lx+1
=
∫ 1
0
lx+tdt. (2.1.35)
Central death rate at age x:
mx =
∫ 1
0lx+tµx+tdt∫ 1
0lx+tdt
=lx − lx+1
Lx. (2.1.36)
2.1.2 Main equations describing development of popu-lation
lx = l0e−
∫ x0 µsds;
lx = l0 −∫ x
0
lsµsds;
lx = ly −∫ x
y
lsµsds; x > y > 0; (2.1.37)
lx = lye−
∫ xy µsds; x > y > 0.
12CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
2.1.3 Closed population with several possibilities of with-drawal (decrement). Pure and relative probabili-ties
Again let T (x) - p.f. of future lifetime of (x) . Let discrete randomvariable J = J(x) describe reason of withdrawal from population (e.g. J = 1- death; J = 2 - lapse, etc.). Define by f(t, j) - joint p.d.f. of variables T (x)and J , by h(j) - marginal p.d.f. of J ; and g(t) - marginal p.d.f. of T (x).
Probability to withdraw due to reason j we will define by:
tq(j)x =
∫ t
0
f(s, j)ds; t ≥ 0; j = 1, 2, ...,m. (2.1.38)
Then h(j) =∫∞
0f(s, j)ds = ∞q
(j)x ; t ≥ 0; j = 1, 2, ...,m.
By analogy:
g(t) =m∑j=1
f(t, j); G(t) =
∫ t
0
g(s)ds. (2.1.39)
Let tq(τ)x - probability to withdraw due to any reason. Then:
tq(τ)x = P (T ≤ t) = G(t) =
∫ t
0
g(s)ds;
tp(τ)x = 1− tq
(τ)x ;
µ(τ)x+t =
g(t)
1−G(t)=
1
tp(τ)x
d
dttq
(τ)x = (2.1.40)
= − 1
tp(τ)x
d
dttp
(τ)x = − d
dtln tp
(τ)x .
Note that formulas (2.1.40) are identical to formulas of closed populationwith one possibility of withdrawal.
Moreover,
f(t, j)dt = P (t < T ≤ t = dt, J = j) (2.1.41)= P ((t < T ≤ t = dt) ∩ J = j|T > t) · P (T > t).
Then in analogy with (2.1.19) we may define the force of decrement dueto cause j:
µjx+t =f(t, j)
1−G(t)=f(t, j)
tp(τ)x
, (2.1.42)
2.1. POPULATION MODELS 13
then:f(t, j)dt = tp
(τ)x µjx+tdt, j = 1, 2, ...,m; t ≥ 0. (2.1.43)
Differentiating (2.1.38) and using (2.1.40) we obtain:
µ(j)x+t =
1
tp(τ)x
d
dttq
(j)x . (2.1.44)
Then from (2.1.38), (2.1.39) and (2.1.40) we get:
tq(τ)x =
∫ t
0
g(s)ds =
∫ t
0
m∑j=1
f(s, j)ds = (2.1.45)
=m∑j=1
∫ t
0
f(s, j)ds =m∑j=1
tq(τ)x .
And:
µ(τ)x+t =
m∑j=1
µ(j)x+t. (2.1.46)
So:
f(t, j) = tp(τ)x µ
(j)x+t
g(t) = tp(τ)x µ
(τ)x+t
h(j) = ∞q(j)x
h(j|T = t) =f(t, j)
g(t)=
tp(τ)x µ
(j)x+t
tp(τ)x µ
(τ)x+t
=µ
(j)x+t
µ(τ)x+t
(2.1.47)
tq(j)x =
∫ t
0sp
(τ)x µ
(j)x+sds
2.1.3.1 Multiple decrement table
Suppose we investigate group of individuals l(τ)a , each of them may with-
draw due to reasons µ(τ)y ; y ≥ a. Then:
l(τ)x = l(τ)
x exp[−∫ x
a
µ(τ)y dy], (2.1.48)
so total number of withdrawals is:
14CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
d(τ)x = l(τ)
x − l(τ)x+1 = l(τ)
x (1−l(τ)x+1
l(τ)x
) =
= l(τ)x (1− exp[−
∫ x+1
x
µ(τ)y ])) = l(τ)
x (1− p(τ)x ) = l(τ)
x q(τ)x ,(2.1.49)
and:
µ(τ)x = − 1
l(τ)x
dl(τ)x
dx. (2.1.50)
Suppose l(j)x , j = 1, 2, ...,m, - subset of individuals aged (x) from l(τ)a who
will withdraw due to reasonj, then:
l(τ)x =
m∑j=1
l(j)x (2.1.51)
and
µ(j)x = lim
h→0
l(j)x − l(j)x+1
hl(τ)x
= − 1
l(τ)x
dl(j)x
dx. (2.1.52)
So:
µ(τ)x = − 1
l(τ)x
d
dx
m∑j=1
l(j)x =m∑j=1
µ(j)x . (2.1.53)
l(j)x − l(j)x+1 = d(j)
x =
∫ x+1
x
l(τ)y µ(j)
y dy. (2.1.54)
Dividing (2.1.54) by l(τ)x we obtain:
d(j)x
l(τ)x
= q(j)x =
∫ x+1
xy−xp
(τ)x µ(j)
y dy, (2.1.55)
where q(j)x - probability to withdraw due to reason j when there are m
possible causes of withdrawal.
2.1.3.2 Associated single decrement table
Now we will introduce so called "pure" (independent, net, uncom-peting) probabilities of withdrawal, that is probabilities of decrement in casethere are no other causes of decrement:
2.1. POPULATION MODELS 15
tp′(j)x = exp[−
∫ t
0
µ(j)x+sds] (2.1.56)
tq′(j)x = 1− tp
′(j)x (2.1.57)
Now we will not require:
limt→∞ tp
′(j)x = 0.
But from∫ x
0µ
(τ)x+t =∞ it is obvious that
∫ x0µ
(j)x+t =∞ for at least one j.
Obviuos that
tp(τ)x = exp[−
∫ t
0
(µ(1)x+s + µ
(2)x+s + ...+ µ
(j)x+s)] =
j∏i=1
tp′(i)x , (2.1.58)
sotp
(τ)x ≤ tp
′(j)x ∀j,
andtp
(τ)x µ
(j)x+t ≤ tp
′(j)x µ
(j)x+t. (2.1.59)
Integrating (2.1.59) we get:
q(j)x =
∫ 1
0tp
(τ)x µ
(j)x+tdt ≤
∫ 1
0tp
′(j)x µ
(j)x+tdt = q
′(j)x . (2.1.60)
Let j = 2 and t = 1, then from (2.1.45) ir (2.1.59) we have:
p(τ)x = p
′(1)x · p′(2)
x = (1− q′(1)x ) · (1− q′(2)
x ); ⇒⇒ 1− q(τ)
x = (1− q′(1)x ) · (1− q′(2)
x ); ⇒⇒ q(τ)
x = q′(1)x + q
′(2)x − q′(1)
x · q′(2)x ; ⇒ (2.1.61)
⇒ q(1)x + q(2)
x = q′(1)x + q
′(2)x − q′(1)
x · q′(2)x .
Assume that deaths are uniformly distributed during the year, e.g.
tq(j)x = t · q(j)
x ; j = 1, 2...,m; 0 ≤ t ≤ 1,
then we have:
tq(τ)x = t · q(τ)
x = 1− p(τ)x .
16CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
From (2.1.44):
µ(j)x+t · tp(τ)
x =d
dttq
(j)x = q(j)
x .
Then
µ(j)x+t =
q(j)x
tp(τ)x
=q
(j)x
1− tq(τ)x
,
so
q′(j)x = 1− exp[−
∫ 1
0
µ(j)x+tdt] = 1− exp[−
∫ 1
0
q(j)x
1− tq(τ)x
dt]
= 1− exp[q
(j)x
q(τ)x
ln(1− tq(τ)x )|10] = 1− exp[
q(j)x
q(τ)x
ln(1− q(τ)x ).
2.2 Demographical models of pension funds
2.2.1 Lexis diagram
See ([2]), p.p. 511-518.
2.2.1.1 Stationary and stable populations
See ([2]), p.p. 518 - 525
2.3 Funding methods in pension funds
Assume that all members enter pension fund at age a, retirement age - - r.Survival function - s(x) with s(a) = 1. For a ≤ x < r, there may be severalcauses of withdrawal, for x ≥ r - the only cause of withdrawal is death.
Let
l(x, u) = n(u)s(x); u = t− x+ a, (2.3.1)
where n(u) - density of new entrants at time u = t−x+a - time of joiningpension fund.
Suppose that at t = 0 person is aged x (a ≤ x < r). Salary rate for suchperson is w(x), let g(t) - inflation related factor, then annual salary rate of
2.3. FUNDING METHODS IN PENSION FUNDS 17
(x) at time t is:w(x)g(t); a ≤ x < r.
Total annual salary at time t:
W(t) =
∫ r
a
l(x, t− x+ a)w(x)g(t)dx.
Suppose pension replacement factor is f (calculated as percentage of lastsalary), then:
fw(r)g(t).
Assume pension is indexed by factor h(x); h(r) = 1, so annual pensionbenefit:
fw(r)g(t− x+ r)h(x); x ≥ r.
2.3.1 Terminal funding
Actuarial present value of continuous pension annuity:
ahr =
∫ ∞r
e−δ(x−r)h(x)s(x)
s(r)dx,
where δ - interest rate.Then Normal cost rate, or Required contribution rate) for individ-
uals attaining age r between t and t+ dt is:
PT(t) = fw(r)g(t)l(r, t− r + a)ahr
2.3.2 Main actuarial functions of pension fund
2.3.2.1 Accrual functions of Actuarial liability
LetM(x) - nondecreasing, right continuous function and 0 ≤M(x) ≤ 1; x ≥a. Assume that M(a) = 0; M(x) = 1; x ≥ r.
For terminal funding:
M(x) =
{0, x < a;1, x ≥ a.
Suppose there exist function m(x):
18CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
M(x) =
∫ x
a
m(y)dy;x ≥ a.
Assume that
m(x) =
continuous, a < x < r;right continuous, x = a;left continuous, x = r;0, x > r.
Then
M ′(x) = m(x), a < x < r.
2.3.2.2 Actuarial present value of future pensions for active mem-bers (aA)(t)
(aA)(t) =
∫ r
a
e−δ(r−x)PT (t+ r − x)dx. (2.3.2)
Notice that:
∂
∂tPT (t+ r − x) = − ∂
∂xPT (t+ r − x).
Then:
d
dt(aA)(t) =
∫ r
a
e−δ(r−x) ∂
∂tPT (t+ r − x)dx
= −∫ r
a
e−δ(r−x) ∂
∂xPT (t+ r − x)dx
= −e−δ(r−x)PT (t+ r − x)|ra + δ
∫ r
a
e−δ(r−x)PT (t+ r − x)dx
= e−δ(r−a)PT (t+ r − a)−PT (t) + δ(aA)(t).
2.3.2.3 Normal cost rate - P(t)
P(t) =
∫ r
a
e−δ(r−x)PT (t+ r − x)m(x)dx, (2.3.3)
It is easy to show that:
2.3. FUNDING METHODS IN PENSION FUNDS 19
∫ u+r−a
u
PT (u+ r − a)m(a+ t− u)dt = PT (u+ r − a)
∫ u+r−a
u
m(a+ t− u)dt =
= PT (u+ r − a)(M(r)−M(a)) = PT (u+ r − a).
2.3.2.4 Accrued actuarial liability - (aV)(t)
(aV)(t) =
∫ r
a
e−δ(r−x)PT (t+ r − x)M(x)dx (2.3.4)
Integrating (2.3.3) by parts obtain:
P(t) =
∫ r
a
e−δ(r−x)PT (t+ r − x)dM(x)
= e−δ(r−x)PT (t+ r − x)M(x)|rx=a −
δ
∫ r
a
e−δ(r−x)PT (t+ r − x)M(x)dx+
∫ r
a
e−δ(r−x) ∂
∂tPT (t+ r − x)M(x)dx
= PT (t)− δ(aV )(t) +d
dt(aV )(t),
or equivalently:
P(t) + δ(aV)(t) = PT (t) +d
dt(aV)(t).
2.3.2.5 Actuarial present value of future normal costs - (Pa)(t)
(Pa)(t) =
∫ r
a
e−δ(r−x)PT(t+ r − x)
∫ r
x
m(y)dydx (2.3.5)
or
(Pa)(t) =
∫ r
a
e−δ(r−x)PT(t+ r − x)(1−M(x))dx
From (2.3.2), (2.3.4) ir (2.3.5) we obtain:
(aV)(t) = (aA)(t)− (Pa)(t),
or
(aA)(t) = (aV)(t) + (Pa)(t).
20CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
2.3.3 Funding methods
2.3.3.1 Individual funding methods
Different ways to choose m(x) and M(x)
2.3.3.1.1 Accrued benefit cost method
m(x) =1
r − a; M(x) =
x− ar − a
.
or
m(x) =w(x)∫ r
aw(y)dy
; M(x) =
∫ xaw(y)dy∫ r
aw(y)dy
.
2.3.3.1.2 Entry age accrual cost method
π
∫ r
a
e−δ(y−a)s(y)dy = e−δ(r−a)s(r)ahr .
(aV )(x) = e−δ(r−x) s(r)
s(x)ahr −
π︷ ︸︸ ︷(e−δ(r−a)s(r)ahr∫ rae−δ(y−a)s(y)dy
)∫ r
x
e−δ(y−x) s(y)
s(x)dy
= (aA)(x)−
(aA)(x)︷ ︸︸ ︷e−δ(r−x) s(r)
s(x)ahr ·
eδ(r−x)e−δ(r−a)∫ rxe−δ(y−x)s(y)dy∫ r
ae−δ(y−a)s(y)dy
= (aA)(x)
(1−
∫ rxe−δys(y)dy∫ r
ae−δys(y)dy
)= (aA)(x)M(x),
so
M(x) = 1−∫ rxe−δys(y)dy∫ r
ae−δys(y)dy
=
∫ xae−δys(y)dy∫ r
ae−δys(y)dy
and:
m(x) =e−δxs(x)∫ raeδys(y)dy
.
If
2.3. FUNDING METHODS IN PENSION FUNDS 21
π
∫ r
a
e−δ(y−a)s(y)w(y)dy = (aA)(a) = e−δ(r−a)s(r)ahr ,
then
M(x) = 1−∫ rxe−δys(y)w(y)dy∫ r
ae−δys(y)w(y)dy
and
m(x) =e−δxs(x)w(x)∫ r
ae−δys(y)w(y)dy
.
2.3.3.2 Aggregate funding methods
(aU)(t) = (aV)(t)− (aF)(t). (2.3.6)
d
dt(aF)(t) = (aC)(t) + δ(aF)(t)−PT(t), aF)(0) = const (2.3.7)
Assume:
(aC)(t) = P(t) + λ(t)(aU)(t), (2.3.8)
Leta(t) =
(Pa)(t)
P(t)
and:
λ(t) = 1/a(t)
Then:
(aC)(t) = P(t) +(aV)(t)− (aF)(t)
a(t)
=(Pa)(t) + (aV)(t)− (aF)(t)
a(t)=
(aA)(t)− (aF)(t)
a(t),
because (aA)(t) = (Pa)(t) + (aV)(t), so
(aC)(t)a(t) = (aA)(t)− aF)(t).
22CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
From (2.3.7) and (2.3.8):
d
dt(aF)(t) = (aC)(t) + δ(aF)(t)−PT(t) (2.3.9)
= P(t) +(aU)(t)
a(t)+ δ(aF)(t)−PT(t)
Remember:
P(t) + δ(aV)(t) = PT (t) +d
dt(aV)(t)
so
d
dt(aV)(t) = P(t) + δ(aV)(t)−PT (t) (2.3.10)
Subtract (2.3.9) from (2.3.10):
− d
dt(aU)(t) = −δ(aU)(t) +
(aU)(t)/a(t)︷ ︸︸ ︷(aC)(t)− P (t)
= (1
a(t)− δ)(aU)(t),
sod
dt(aU)(t) = −(
1
a(t)− δ)(aU)(t).
Then: ∫ t
0
ddu
(aU)(u)
(aU)(u)= −
∫ t
0
(1
a(u)− δ)du
and finally:
(aU)(t) = (aU)(0) exp [−∫ t
0
(1/a(u)− δ)du],
or
(aF)(t) = (aV)(t)−
−((aV)(0)− (aF)(0)) exp [−∫ t
0
(1/a(u)− δ)du].
We need to show that exp [−∫ t
0(1/a(u)− δ)du]→ 0, t→∞.
2.3. FUNDING METHODS IN PENSION FUNDS 23
Suppose a(u) < a∞| = 1/δ, so
1/a(u)− δ ≥ ε > 0,
then:
exp [−∫ t
0
(1/a(u)− δ)du]→ 0, t→∞
and:
(aF)(t)→ (aV)(t).
2.3.3.3 Actuarial methods for retired members
(rA)(t) =
∫ ∞r
l(x, t− x+ a)fw(r)g(t− x+ r)ahxdx (2.3.11)
(rA)(t) =
∫ ∞r
n(t− x+ a)fw(r)g(t− x+ r)[
∫ ∞x
e−δ(y−x)h(y)s(y)dy]dx
(2.3.12)
(rV)(t) = (rA)(t)
B(t) =
∫ ∞r
l(x, t− x+ a)fw(r)g(t− x+ r)h(x)dx,
or:
B(t) =
∫ ∞r
n(x, t− x+ a)g(t− x+ r)fw(r)s(x)h(x)dx, (2.3.13)
24CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
d
dt(B(t)) =
∫ ∞r
fw(r)s(x)h(x)∂
∂t[n(x, t− x+ a)g(t− x+ r)]dx
= −∫ ∞r
fw(r)s(x)h(x)∂
∂x[n(x, t− x+ a)g(t− x+ r)]dx
= −fw(r)s(x)h(x)n(x, t− x+ a)g(t− x+ r)|x=∞x=r +∫ ∞
r
fw(r)n(x, t− x+ a)g(t− x+ r)[
s′(x) = −s(x)µx︷︸︸︷s′(x) h(x) + s(x)h′(x)]dx
=
"replacement effect"︷ ︸︸ ︷fw(r)l(r, t− r + a)g(t)−
∫ ∞r
fw(r)l(x, t− x+ a)g(t− x+ r)h(x)µxdx
"adjustment effect"︷ ︸︸ ︷+
∫ ∞r
fw(r)l(x, t− x+ a)g(t− x+ r)h′(x)dx =
= I − II + III
(rV)(t) = (rA)(t) =
∫ ∞r
n(t−x+a)fw(r)g(t−x+r)[
∫ ∞x
e−δ(y−x)h(y)s(y)dy]dx
Then:
d
dt(rV)(t) = fw(r)
∫ ∞r
[
∫ ∞x
e−δ(y−x)h(y)s(y)dy]∂
∂t[n(t− x+ a)g(t− x+ r)]dx
= −fw(r)
∫ ∞r
[
∫ ∞x
e−δ(y−x)h(y)s(y)dy]∂
∂x[n(t− x+ a)g(t− x+ r)]dx
= −fw(r)[
∫ ∞x
e−δ(y−x)h(y)s(y)dy][n(t− x+ a)g(t− x+ r)]|∞x=r +
fw(r)δ
∫ ∞r
[
∫ ∞x
e−δ(y−x)h(y)s(y)dy][n(t− x+ a)g(t− x+ r)]dx−
fw(r)
∫ ∞r
s(x)h(x)n(t− x+ a)g(t− x+ r)dx
= PT(t) + δ(rV)(t)−B(t)
so,
PT(t) + δ(rV)(t) =d
dt(rV)(t) + B(t) (2.3.14)
2.4. SAMPLE EXERCISES 25
2.3.3.4 Mutual functions for active and retired members
A(t) = (aA)(t) + (rA)(t)
V(t) = (aV)(t) + (rV)(t)
Remember that
P(t) + δ(aV)(t) = PT (t) +d
dt(aV)(t) (2.3.15)
from (2.3.14) and (2.3.15) we get:
P(t) + δV(t) = B(t) +d
dtV(t)
2.4 Sample exercises
2 Exercise. Assume pension fund was established 2 years ago at t = 0, sonow it is time moment t = 2 (end of 2nd year). At time t = 0 cohort ofemployees that joined pension plan was:
Age (x) No of employees30 100035 100040 1000
Assume that population is closed. For 30 ≤ x ≤ 65, then s(x) =e−µx(x−a); µx = 0, 0007+0, 00005·100.04·x. For 65 ≤ x ≤ 110, qx = GHx
1+GHx ;G =0.0000023;H = 1.134
Suppose at time t = 0: w(30) = 1000;w(35) = 1025;w(40) = 1050,annual salary increase is 0,5%. Let i = 0, 04, r = 65, initial pension benefit- 60% (of last salary), pension is indexed by 0,5% annually.
26CHAPTER 2. DEMOGRAPHICAL AND FINANCIAL MODELS OF PENSION SYSTEMS
Assume:
m1(x) =1
r − a
m2(x) =w(x)∫ r
aw(y)dy
m3(x) =e−δxs(x)∫ r
ae−δys(y)dy
m4(x) =e−δxs(x)w(x)∫ r
ae−δys(y)w(y)dy
Assume that pension is accrued by level percentage π of current salary.Find π and (aV (2)).
3 Exercise. There are three decrement causes from pension fund: 1 - death;2 - change of employer; 3 - disability. Assume:
µ1(x) = 0, 0007 + 0, 00005 · 100,04x
µ2(x) = 0, 005
µ3(x) = 0.00007x
1 Part. Construct multiple decrement table (calculate q(1)x ; q
(2)x ; q
(3)x ; d
(1)x ; d
(2)x ; d
(3)x
and l(τ)55 , if x = 55, 56, ...., 65, l(τ)
x = 1000):Age (x) q
(1)x q
(2)x q
(3)x l
(τ)x d
(1)x d
(2)x d
(3)x
55 100056...65
2 Part. Active members of pension fund are entitled for pension benefit.
Assume that a65 = 11, 45 (first payment at age 65). Pension benefit - 30%of last salary, initial salary w(55) = 12000, annual salary increase - 0.5%.
Method of accrual - Entry age accrual cost method, interest rate - i = 4%.Premiums into fund are paid at the end of year (so, first payment - at age56; last - at age 65).
Calculate:
2.4. SAMPLE EXERCISES 27
a) Percentage π1, if funding is based on level salary percentage throughoutcareer.
b) Percentage π1, if funding is based on level premiums (in monetaryterms) throughout career.
c) In both cases calculate P (x) and compare them.
Chapter 3
Pension annuities
3.1 Early actuarial modelsJan de Witt (1671) model:
ax = a1| · qx + a2| ·1| qx + a3| ·2| qx + ....
E. Halley (1693) model and mortality table of city of Breslau:
ax =ω−x−1∑t=1
(1 + i)−tLx+t
Lx
3.2 Life annuities
3.2.1 Life annuities: deterministic vs stochastic approach
See ([3]), p.p. 2-39.
3.2.1.1 Deterministic approach
Let S - annuity premium, b - amount of pension (paid at the end of eachyear), lx - number of individuals aged (x) at t = 0. Then, assuming thatlxV0 = lxS:
lx+tVt = lx+t−1Vt−1(1 + i)− lx+tbt, t = 1, 2, ...ω − x. (3.2.1)
Or:Vt =
lx+t−1
lx+t
Vt−1(1 + i)− b,
29
30 CHAPTER 3. PENSION ANNUITIES
and:
Vt − Vt−1 = Vt−1i+lx+t−1 − lx+t
lx+t
Vt−1(1 + i)− b.
So we obtain:
S =ω−x∑t=1
blx+t
lx(1 + i)−t.
Or, using survival probabilities:
S = bω−x∑t=1
tpx(1 + i)−t.
Finally:
S = bω−x∑t=1
at|tpxqx+t, (3.2.2)
where:at| =
1− (1 + i)−t
i
We may also write:
S = b · E(aKx) = bax,
where Kx = Int(Tx).
On the other hand we may show that:
Vt = bax+t.
Vt, depending on situation, may be regarded as assets or liabilities.
3.2.1.1.1 Mutuality effect From (3.2.1):
lx+t−1Vt−1(1 + i) = lx+tVt + lx+tbt, t = 1, 2, ...ω − x.
Moreover (3.2.1) may be presented:
Vt = Vt−1(1 + i)(1 +lx+t−1 − lx+t
lx+t
)− b
= Vt−1(1 + i)(1 + θx+t)− b,
3.3. INTRODUCING LONGEVITY RISK: COHORT VS PERIOD MORTALITY TABLES31
where:θx+t =
lx+t−1 − lx+t
lx+t
,
or:θx+t =
1
px+t−1
− 1.
θx+t is called (interest from mutuality, or mortality drag).
3.2.1.1.2 Solidarity
3.2.1.1.3 Tontine annuities
3.2.1.2 Stochastic approach
3.2.1.3 Random present value of life annuity
Y = aKx|
Possible values of such random variable:
y0 = 0
y1 = (1 + i)−1
....
yω−x = (1 + i)−1 + (1 + i)−2 + ...+ (1 + i)−(ω−x)
Then:P(aKx| = yh
)= P (Kx = h) .
3.2.1.4 Portfolio results: stochastic approach
3.3 Introducing longevity risk: Cohort vs Pe-riod mortality tables
See ([3]), p.p. 45-62Cohort tablesPeriod tablesSelect mortalitySurvival function, life expectancy, markers or life tables in age
continuous context.
32 CHAPTER 3. PENSION ANNUITIES
3.4 Forecasting mortality under longevity riskSee ([3]), p.p. 137-172.
3.4.1 Extrapolation of annual death probabilities
3.4.1.1 Reduction factors
Rx(t− t′) > 0; t′ - Reduction factors:
qx(t) = qx(t′) ·Rx(t− t′), t > t′.
3.4.1.2 Exponential formula
qx(t) = qx(t′)rt−t
′
x , t.y.
3.4.1.3 More general exponential formula
qx(t) = ax + bx · ctx,
3.4.2 Projected mortality tables
kp↗x (t) =
k−1∏j=0
(1− qx+j(t+ j)) ;
e↗x (t) =ω−x∑k=1
kp↗x (t) + 1/2.
e↑x(t) =ω−x∑k=1
kp↑x(t) + 1/2;
3.4.2.1 Single entry projected table
qxmin+h(τ)(τ+xmin+h(τ)), qxmin+h(τ)+1(τ+xmin+1+h(τ)), ...qx+h(τ)(τ+x+h(τ)).
h(τ) is called Rueff adjustment:
h(τ) =
≥ 0, τ < τ ;0, τ = τ ;≤ 0, τ > τ .
kp[τ ;h(τ)]↗x
k−1∏j=0
[1− qx+h(τ)+j(τ + x+ h(τ) + j)]
3.4. FORECASTING MORTALITY UNDER LONGEVITY RISK 33
3.4.3 Parametric methods
3.4.4 Exposure to risk and Crude mortality rate
ETRxt - "exposed to risk at age x in year t".
ETRxt ≈−Lxt · qx(t)ln(1− qx(t))
.
µx(t) =Dxt
ETRxt
= mx(t).
mx(t) - Crude death rate.
3.4.5 Lee Carter method
(1992)Let:
µx+ξ1(t+ ξ2) = µx(t); 0 ≤ ξ1, ξ2 ≤ 1.
then
mx = µx.
Lee Carter model:
ln(mx(t)) = ax + bxkt + εx,t; (3.4.1)
E(εx,t) = 0;D(εx,t) = σ2ε .
Constraints:tmax∑tmin
kt = 0;xmax∑xmin
βx = 1
ortmax∑tmin
kt = 0;xmax∑xmin
β2x = 1
Objective function
O =xm∑x=x1
tn∑t=t1
(ln mx(t)− αx − βxkt)2 (3.4.2)
34 CHAPTER 3. PENSION ANNUITIES
3.4.5.1 Singular value decomposition
tn∑t=t1
ln mx(t) = (tn − t1 + 1)αx + βx
tn∑t=t1
κt = (tn − t1 + 1)αx,
So
αx =1
tn − t1 + 1
tn∑t=t1
ln mx(t),
M =
mx1(t1) · · · mx1(tn)... . . . ...
mxm(t1) · · · mxm(tn)
Z = ln M − α =
ln mx1(t1)− αx1 · · · ln mx1(tn)− αx1
... . . . ...ln mxm(t1)− αxm · · · ln mxm(tn)− αxm
Minimize:
OLS(β, k) =xn∑x=x1
tn∑t=t1
(zxt − βxkt)2.
Using singular value decomposition obtain estimation of parameters
3.4.5.2 Newton - Rapson method
Differentiate O (3.4.2) and equate to zero. Then:
tn∑t=t1
(ln mx(t)− αx − βxkt) = 0;x = x1, x2, ..., xm
xm∑x=x1
βx(ln mx(t)− αx − βxkt) = 0; t = t1, t2, ..., tn
tn∑t=t1
kt(ln mx(t)− αx − βxkt) = 0;x = x1, x2, ..., xm
3.5. SAMPLE EXERCISES 35
Using Newton - Rapson procedure obtain estimates:
α(k+1)x = α(k)
x +
∑tnt=t1
(ln mx(t)− α(k)x − β(k)
x k(k)t )
tn − t1 + 1
k(k+1)t = k
(k)t +
∑xm
x=x1β
(k)x (ln mx(t)− α(k+1)
x − β(k)x k
(k)t )∑xm
x=x1(β
(k)x )2
β(k+1)x = β(k)
x +
∑tnt=t1
k(k+1)t (ln mx(t)− α(k+1)
x − β(k)x k
(k+1)t )∑tn
t=t1(k
(k+1)t )2
Finally
αx → αx + βxk
kt = (kt − k)∑
βx
βx =βx∑βx,
3.4.6 Mortality forecasts
Random walk with drift:
kt+1 = a+ kt + εt.
3.5 Sample exercises4 Exercise. Let qx = GHx
1+GHx ;G = 0.0000023;H = 1.134; 65 ≤ x ≤ 110
Assume L65 = 100. Find distributions of L70;L80;L100.
Find distribution of a65|, E[a65|] and V ar[a65|].
5 Exercise. Using real Lithuanian mortality data (for example, from www.mortality.org)find Lee Carter estimates of αx, βx and kt.
Chapter 4
Management of Longevity risk
See ([3]) 267 - 342.
4.1 Main types of mortality risk
Process (insurance) risk
Model and parametric risk
Catastrophic risk
4.1.0.1 Main types of actuarial modeling
Static vs dynamic modelling
4.2 Measuring Longevity risk in a static frame-work
Suppose we have several set of assumptions about future mortality, e.g. weconsider the set of possible mortality scenarios A(τ) (specific scenario fromthis set will be denoted by Ah(τ)).
Let Y (Π)t random present value of benefits paid from portfolio. Let Nt -
random number of pensioners from cohort N0 (concrete realization will bedenoted by nt (N0 = n0)). We will denote policy for j individual by indexj = 1, 2, ..., n0. Then total portfolio at time t is:
Πt = {j|T (j)x > t},
37
38 CHAPTER 4. MANAGEMENT OF LONGEVITY RISK
benefits:B
(Π)t =
∑j:j∈Πi
b(j); j = 1, 2, ...,
present value of benefits d.v. t = 1, 2, ...:
Y(j)t = b(j)a
K(j)x0
Y(Π)t =
∑j:j∈Πi
Y(j)t
Let b(j) = b; ∀j, then:B
(Π)t = bNt.
So:
Y Πt =
ω−x0∑h=t+1
BΠh (1 + i)−(h−t) =
ω−x0∑h=t+1
bNh(1 + i)−(h−t)
Suppose real mortality depend on mortality scenario Ah(τ). Then forhomogenous portfolio:
E(Y
(Π)t |Ah(τ), nt
)= ntE
(Y
(1)t |Ah(τ)
).
If future lifetimes are i.i.d. :
D(Y
(Π)t |Ah(τ), nt
)= ntD
(Y
(1)t |Ah(τ)
).
Coefficient of variation (CV), or Risk index:
CV(Y
(Π)t |Ah(τ), nt
)=
1√nt
√D(Y
(1)t |Ah(τ)
)E(Y
(1)t |Ah(τ)
) .
It is easy to see that:
limnt→∞
CV(Y
(Π)t |Ah(τ), nt
)= 0.
Now let assign "weights" ρh;∑ρh = 1 to mortality scenarios Ah(τ).
Then
E(Y
(Π)t |nt
)= Eρ
[E(Y
(Π)t |A(τ), nt
)]= ntEρ
[E(Y
(1)t |A(τ)
)]= nt
m∑h=1
E(Y
(1)t |Ah(τ)
)ρh = ntE(Y
(1)t ),
4.2. MEASURING LONGEVITY RISK IN A STATIC FRAMEWORK 39
E(Y(1)t ) =
∑mh=1E
(Y
(1)t |Ah(τ)
)ρh.
Moreover:
D(Y
(Π)t |nt
)= Eρ
[D(Y
(Π)t |nt, A(τ)
)]+Dρ
[E(Y
(Π)t |nt, A(τ)
)]= ntEρ
[D(Y
(1)t |A(τ)
)]+ n2
tDρ
[E(Y
(1)t |A(τ)
)]= nt
m∑h=1
D(Y
(1)t |Ah(τ)
)ρh + n2
tDρ
[E(Y
(1)t |A(τ)
)]= nt
m∑h=1
D(Y
(1)t |Ah(τ)
)ρh + n2
t
m∑h=1
[E(Y
(1)t |Ah(τ)
)− E(Y
(1)t )]2
ρh.
Then
CV(Y
(Π)t |nt
)=
√D(Y
(Π)t
)E(Y
(Π)t
)=
√√√√ 1
nt
Eρ[D(Y(1)t |A(τ))]
E2(Y(1)t )
+Dρ[E(Y
(1)t |A(τ))]
E2(Y(1)t )
=
=√I + II.
I - "pooling risk"; II- "non pooling risk".
It is easy to see that
limnt→∞
CV(Y
(Π)t |nt
)=
√√√√Dρ[E(Y(1)t |A(τ))]
E2(Y(1)t )
,
so non pooling risk does not decrease when size of portfolio increases.
40 CHAPTER 4. MANAGEMENT OF LONGEVITY RISK
4.3 Other methods for management of mortal-ity risk
4.3.1 Hedging
4.3.2 Reinsurance
4.3.3 Longevity bonds
4.4 Sample exercises
6 Exercise. Suppose annuity a65 is paid, interest rate is i, probabilities ofmortality - q65, q66, ...., q100 = 1 and w = 101. Assume that at t = 0 personis aged 65, so annuity is paid. Let Yt - present value of this annuity at timet. Provide formulas for Y0 and Y3 using probabilities qx and / or px. Provideformulas for E[Y0] and D[Y0].
7 Exercise. Suppose that possible mortality scenarios are:Scenario, Ah: I II III IV V
E(Y |Ah) 13, 514 13, 821 14, 271 14, 450 15, 112D(Y |Ah) 20, 114 25, 006 22, 714 21, 978 24, 215
probability: 0.1 0.1 0.6 0.1 0.1
Calculate E(Y0) and D(Y0), if n = 100
8 Exercise. Suppose mortality is modeled using Heligman - Pollard law,e.g.:
qx =G · (Kx)
1 +G · (Kx); ω = 110; x− age.
A1 A2 A3 A4 A5
G 6, 378E − 07 3, 803E − 06 2, 005E − 06 1, 060E − 06 3, 149E − 06H 1, 14992 1, 12347 1, 13025 1, 13705 1, 11962
"weights" 0.1 0.1 0.6 0.1 0.1
Using i = 4% calculate:e65 = E[K65|Ah] =
∑117−65k=0 k ·k|1 qx
and√D[K65|Ah].
For t = {0; 5}, h = 1, 2, 3, 4, 5, N0 = 1 and l b = 1 calculate:
4.4. SAMPLE EXERCISES 41
E(Y
(Π)t |Ah
); D
(Y
(Π)t |Ah
)kurY Π
t =117−65∑h=t+1
Nh(1 + i)−(h−t).
For t = 0 calculate:
E(Y
(Π)t
); D
(Y
(Π)t
).
For N0 = 1, 100, 1000 calculate Coefficient of variation:
CV(Y
(Π)t |nt
)=
√D(Y
(Π)t
)E(Y
(Π)t
)=
√√√√ 1nt
Eρ[D(Y (1)t |A(τ))]
E2(Y (1)t )
+Dρ[E(Y (1)
t |A(τ))]
E2(Y (1)t )
;
define pooling and non pooling risk.
9 Exercise. Suppose that mortality is described using III scenariu (see ex-ercise (8)).
Insurance company uses hedging, e.g. pension annuity is paid for (65)year old individual, moreover upon death of insured person (annuity) reserveis paid for beneficiaries. Assume that all deaths occur and benefits are paidat the end of each year. Other conditions as in exercise (8).
Calculate insurance premium and coefficient of variation. Compare bothinsurance premium and coefficient of variation with the case described inexercise (8), explain differences of product riskiness and marketability inboth cases.
Literature
[1] Barr, N. "Reforming Pensions: Myths, Truths and Policy Choices".International Social Security Review, Vol. 55 No. 2. (2002).
[2] Bowers, N.L.; Gerber, H.U.; Hickman, J.C. "Actuarial Mathematics".The Society of Actuaries, (1986).
[3] Pitacco, E., et. al. "Modelling Longevity Dynamics for Pension andAnnuity Business". Oxford University Press (2009).
[4] Werding, M. "After Another Decade of Reform: Do Pension Systemsin Europe Converge?". CESifo DICE Report No. 1 (2003).
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