ACTS 4301
FORMULA SUMMARY
Lesson 1: Probability Review
1. Var(X)= E[X 2 ]- E[X]2
2. V ar(aX + bY ) = a2 V ar(X) + 2abCov(X, Y ) + b2 V ar(Y )
3. V ar(X̄) =
V ar(X)
n
4. EX [X] = EY [EX [X|Y ]] Double expectation
5. V arX [X] = EY [V arX [X|Y ]] + V arY (EX [X|Y ]) Conditional variance
Distribution, Density and Moments for Common Random Variables
Continuous Distributions
Name
f (x)
F (x)
Exponential
e− θ /θ
x
1 − e− θ
Uniform
1
b−a , x ∈ [a, b]
x
xα−1 e− θ /Γ(α)θα
Gamma
E[X] Var(X)
x
x
θ,
x ∈ [a, b]
Rx
0
f (t)dt
θ
θ2
a+b
2
(b−a)2
12
αθ2
αθ
Discrete Distributions
Name
f (x)
e−λ λx /x!
Poisson
Binomial
m
x
px (1
−
p)m−x
E[X]
Var(X)
λ
λ
mp
mp(1 − p)
2
Lesson 2: Survival Distributions: Probability Functions, Life Tables
1. Actuarial Probability Functions
t px - probability that (x) survives t years
t qx - probability that (x) dies within t years
t|u qx - probability that (x) survives t years and then dies in the next u years.
t+u px
t|u qx
=
=
=
=
t px ·u
px+t
t px ·u qx+t =
t px −t+u px =
t+u qx −t qx
2. Life Table Functions
dx = lx − lx+1
n dx = lx − lx+n
dx
qx = 1 − px =
lx
lx+t
t px =
lx
lx+t − lx+t+u
t|u qx =
lx
3. Mathematical Probability Functions
t px
= ST (x) (t)
t qx
= FT (x) (t)
t|u qx
= P r (t < T (x) ≤ t + u) = FT (x) (t + u) − FT (x) (t) =
= P r (x + t < X ≤ x + t + u|X > x) =
Sx (t + u)
Sx (u)
S0 (x + t)
Sx (t) =
S0 (x)
F0 (x + t) − F0 (x)
Fx (t) =
1 − F0 (x)
Sx+u (t) =
FX (x + t + u) − FX (x + t)
sX (x)
3
Lesson 3: Survival Distributions: Force of Mortality
f0 (x)
d
= − ln S0 (x)
S0 (x)
dx
fx (t)
d
µx+t =
= − ln Sx (t)
Sx (t)
dx
Z t
Z t
Z
Sx (t) =t px = exp −
µx+s ds = exp −
µx (s) ds = exp −
µx =
0
0
d lnt px
dt px /dt
=−
µx (t) = −
dt
t px
fT (x) (t) = fx (t) =t px · µx (t)
Z t+u
q
=
s px · µx (s) ds
t|u x
t
Z t
t qx =
s px · µx (s)ds
0
If µ∗x (t) = µx (t) + k for all t, then s p∗x = s px e−kt
If µx (t) = µ̂x (t) + µ̃x (t) for all t, then s px = s p̂xs p̃x
If µ∗x (t) = kµx (t) for all t, then s p∗x = (s px )k
x
x+t
µs ds
4
Lesson 4: Survival Distributions: Mortality Laws
Exponential distribution or Constant Force of Mortality
µx (t) = µ
t px
= e−µt
Uniform distribution or De Moivre’s law
1
, 0≤t≤ω−x
ω−x−t
ω−x−t
, 0≤t≤ω−x
t px =
ω−x
t
, 0≤t≤ω−x
t qx =
ω−x
u
, 0≤t+u≤ω−x
t|u qx =
ω−x
Beta distribution or Generalized De Moivre’s law
α
µx (t) =
ω−x−t
ω−x−t α
, 0≤t≤ω−x
t px =
ω−x
Gompertz’s law:
µx (t) =
µx = Bcx , c > 1
Bcx (ct − 1)
t px = exp −
ln c
Makeham’s law:
µx = A + Bcx , c > 1
Bcx (ct − 1)
p
=
exp
−At
−
t x
ln c
Weibull Distribution
µx = kxn
n+1 /(n+1)
S0 (x) = e−kx
5
Lesson 5. Survival Distributions: Moments
Complete Future Lifetime
Z ∞
tt px µx+t dt
e̊x =
Z0 ∞
e̊x =
t px dt
0
1
e̊x = for constant force of mortality
µ
ω−x
e̊x =
for uniform (de Moivre) distribution
2
ω−x
e̊x =
for generalized uniform (de Moivre) distribution
α+1
Z ∞
h
i
tt px dt
E (T (x))2 = 2
0
1
V ar (T (x)) = 2 for constant force of mortality
µ
(ω − x)2
for uniform (de Moivre) distribution
V ar (T (x)) =
12
(ω − x)2
V ar (T (x)) =
for generalized uniform (de Moivre) distribution
(α + 1)2 (α + 2)
n-year Temporary Complete Future Lifetime
Z
n
e̊x:n =
tt px µx+t dt + nn px
Z0 n
t px
e̊x:n =
dt
0
e̊x:n =n px (n) +n qx (n/2) for uniform (de Moivre) distribution
e̊x:1 = px + 0.5qx for uniform (de Moivre) distribution
1 − e−µn
for constant force of mortality
µ
Z n
h
i
2
E (T (x) ∧ n) = 2
tt px dt
e̊x:n =
0
Curtate Future Lifetime
∞
X
ex =
k k| qx
ex =
k=1
∞
X
k px
k=1
ex = e̊x − 0.5 for uniform (de Moivre) distribution
∞
h
i X
E (K(x))2 =
(2k − 1)k px
k=1
V ar (K(x)) = V ar (T (x)) −
1
for uniform (de Moivre) distribution
12
6
n-year Temporary Curtate Future Lifetime
ex:n =
ex:n =
n−1
X
k=1
n
X
k k| qx + nn px
k px
k=1
ex:n = e̊x:n − 0.5n qx for uniform (de Moivre) distribution
n
h
i X
E (K(x) ∧ n)2 =
(2k − 1)k px
k=1
7
Lesson 6: Survival Distributions: Percentiles, Recursions, and Life Table Concepts
Recursive formulas
e̊x = e̊x:n +n px e̊x+n
e̊x:n = e̊x:m +m px e̊x+m:n−m , m < n
ex = ex:n +n px ex+n = ex:n−1 +n px (1 + ex+n )
ex = px + px ex+1 = px (1 + ex+1 )
ex:n = ex:m +m px ex+m:n−m =
= ex:m−1 +m px (1 + ex+m:n−m ) m < n
ex:n = px + px ex+1:n−1 = px 1 + ex+1:n−1
Life Table Concepts
∞
Z
lx+t dt, total future lifetime of a cohort of lx individuals
Tx =
0
Z
n Lx =
Z
Yx =
n
lx+t dt, total future lifetime of a cohort of lx individuals over the next n years
0
∞
Tx+t dt
0
e̊x =
Tx
lx
e̊x:n =
n Lx
lx
Central death rate and related concepts
n mx
=
n dx
n Lx
qx
for uniform (de Moivre) distribution
1 − 0.5qx
n mx = µx for constant force of mortality
Lx − lx+1
a(x) =
the fraction of the year lived by those dying during the year
dx
1
a(x) =
for uniform (de Moivre) distribution
2
mx =
8
Lesson 7: Survival Distributions: Fractional Ages
Function Uniform Distribution of Deaths Constant Force of Mortality Hyperbolic Assumption
lx+s
lx − sdx
lx psx
lx+1 /(px + sqx )
s qx
sqx
1 − psx
sqx /(1 − (1 − s)qx )
1 − sqx
psx
px /(1 − (1 − s)qx )
s px
1−
psx
s qx+t
sqx /(1 − tqx ), 0 ≤ s + t ≤ 1
sqx /(1 − (1 − s − t)qx )
µx+s
qx /(1 − sqx )
− ln px
qx /(1 − (1 − s)qx )
s px µx+s
qx
−psx (ln px )
px qx /(1 − (1 − s)qx )2
mx
qx /(1 − 0.5qx )
− ln px
qx2 /(px ln px )
Lx
lx − 0.5dx
−dx / ln px
−lx+1 ln px /qx
e̊x
ex + 0.5
e̊x:n
ex:n + 0.5n qx
e̊x:1
px + 0.5qx
9
Lesson 8: Survival Distributions: Select Mortality
When mortality depends on the initial age as well as duration, it is known as select mortality,
since the person is selected at that age. Suppose qx is a non-select or aggregate mortality and
q[x]+t , t = 0, · · · , n − 1 is select mortality with selection period n. Then for all t ≥ n, q[x]+t = qx+t .
10
Lesson 9: Insurance: Payable at Moment of Death - Moments - Part 1
Z
∞
Āx =
e−δt t px µx (t) dt
0
Actuarial notation for standard types of insurance
Name
Present value random variable Symbol for actuarial present value
A¯x
vT
Whole life insurance
Term life insurance
vT
0
T ≤n
T >n
Ā1x:n
Deferred life insurance
0
vT
T ≤n
T >n
n |Āx
Deferred term insurance
Pure endowment
Endowment insurance
0
vT
0
T ≤n
n<T ≤n+m
T >n
0 T ≤n
vn T > n
vT
vn
T ≤n
T >n
1
n |Āx:m
=n |m Āx
Ax:n1 or n Ex
Āx:n
11
Lesson 10: Insurance: Payable at Moment of Death - Moments - Part 2
Actuarial present value under constant force and uniform (de Moivre) mortality for insurances payable
at the moment of death
Type of insurance
APV under constant force
APV under uniform (de Moivre)
Whole life
µ
µ+δ
āω−x
ω−x
µ
µ+δ
n-year term
1 − e−n(µ+δ)
ān
ω−x
n-year deferred life
µ −n(µ+δ)
µ+δ e
e−δn ā
ω−(x+n)
ω−x
n-year pure endowment
e−n(µ+δ)
e−δn (ω−(x+n))
ω−x
j-th moment
Multiply δ by j
in each of the above formulae
Gamma Integrands
Z
∞
Z0 u
0
tn e−δt dt =
te−δt dt =
n!
δ n+1
ā − uv u
1 u
−δu
=
1
−
(1
+
δu)e
δ2
δ
Variance
If Z3 = Z1 + Z2 and Z1 , Z2 are mutually exclusive, then
V ar(Z3 ) = V ar(Z1 ) + V ar(Z2 ) − 2E[Z1 ]E[Z2 ]
12
Lesson 11: Insurance: Annual and m-thly: Moments
Ax =
∞
X
k+1
=
k |q x v
∞
X
0
k px qx+k v
k+1
0
Actuarial present value under constant force and uniform (de Moivre) mortality for insurances payable
at the end of the year of death
Type of insurance
APV under constant force APV under uniform (de Moivre)
q
q+i
Whole life
n-year term
q
q+i
(1 − (vp)n )
aω−x
ω−x
an
ω−x
n-year deferred life
q
n
q+i (vp)
vn a
ω−(x+n)
ω−x
n-year pure endowment
(vp)n
v n (ω−(x+n))
ω−x
13
Lesson 12: Insurance: Probabilities and Percentiles
Summary of Probability and Percentile Concepts
• To calculate P r(Z ≤ z) for continuous Z, draw a graph of Z as a function of T . Identify
parts of the graph that are below the horizontal line Z = z, and the corresponding t’s. Then
calculate the probability of T being in the range of those t’s.
Note that
ln z ∗
ln z ∗
ln z ∗
P r(Z < z ∗ ) = P r(v t < z ∗ ) = P r t > −
=t∗ px , where t∗ = −
=−
δ
δ
ln(1 + i)
The following table shows the relationship between the type of insurance coverage and corresponding probability:
Type of insurance
P r(Z < z ∗ )
Whole life
t∗ px
n-year term
t∗ px
n-year deferred life
n qx
n-year endowment
n px
z∗ ≤ vn
z∗ > vn
+t∗ px z ∗ < v n
1
z∗ ≥ vn
0
z∗ < vn
∗
n
t∗ px z ≥ v

 n qx +n+m px z ∗ < v m+n
v m+n ≤ z ∗ < v n
n-year deferred m-year term
n qx +t∗ px

1
z∗ ≥ vn
• In the case of constant force of mortality µ and interest δ, P r(Z ≤ z) = z µ/δ
• For discrete Z, identify T and then identify K + 1 corresponding to those T . In other words,
P r(Z > z ∗ ) = P r (T < t∗ ) =k qx , where k = bt∗ c - the greatest integer smaller than t∗
P r(Z < z ∗ ) = P r (T > t∗ ) =k+1 px , where k = bt∗ c - the greatest integer smaller or equal to t∗
• To calculate percentiles of continuous Z, draw a graph of Z as a function of T . Identify where
the lower parts of the graph are, and how they vary as a function of T.
For example, for whole life, higher T lead to lower Z.
For n-year deferred whole life, both T < n and higher T lead to lower Z.
Write an equation for the probability Z less than z in terms of mortality probabilities
expressed in terms of t. Set it equal to the desired percentile, and solve for t or for ekt for any
k. Then solve for z (which is often v t )
14
Lesson 13: Insurance: Recursive Formulas, Varying Insurances
Recursive Formulas
Ax = vqx + vpx Ax+1
Ax = vqx + v 2 px qx+1 + v22 px Ax+2
Ax:n = vqx + vpx Ax+1:n−1
A1x:n = vqx + vpx A1x+1:n−1
n |Ax
Ax:n1
= vpxn−1 |Ax+1
1
= vpx Ax+1:n−1
Increasing and Decreasing Insurance
Z ∞
n
tn e−δt dt = n+1
δ
Z0 u
ā − uv u
1 nδ
te−δt dt = 2 1 − (1 + δu) e−δu =
δ
δ
0
µ
for constant force
I¯Ā x =
(µ + δ)2
2µ
for Z a continuously increasing continuous insurance, constant force.
E[Z 2 ] =
(µ + 2δ)3
1
1
I¯Ā x:n + D̄Ā x:n = nĀ1x:n
1
1
I Ā x:n + DĀ x:n = (n + 1)Ā1x:n
(IA)1x:n + (DA)1x:n = (n + 1)A1x:n
Recursive Formulas for Increasing and Decreasing Insurance
(IA)1x:n = A1x:n + vpx (IA)1x+1:n−1
(IA)1x:n = A1x:n + vpx (IAA)1x+1:n−1
(DA)1x:n = nA1x:n + vpx (DA)1x+1:n−1
(DA)1x:n = A1x:n + (DA)1x:n−1
15
Lesson 14: Relationships between Insurance Payable at Moment of Death and Payable
at End of Year
Summary of formulas relating insurances payable at moment of death to insurances payable at the
end of the year of death assuming uniform distribution of deaths
i
Āx = Ax
δ
i
Ā1x:n = A1x:n
δ
i
n |Āx = n |Ax
δ
1
i
I Ā x:n = (IA)1x:n
δ
1
i
I D̄ x:n = (ID)1x:n
δ
i 1
Āx:n = Ax:n + Ax:n1
δ
i
A(m)
= (m) Ax
x
i
2i
+ i2 2
2
Ax
Āx =
2δ
1
1
1 1
1
¯
I Ā x:n = I Ā x:n − Āx:n
−
d δ
Summary of formulas relating insurances payable at moment of death to insurances payable at the
end of the year of death using claims acceleration approach
Āx = (1 + i)0.5 Ax
Ā1x:n = (1 + i)0.5 A1x:n
n |Āx
= (1 + i)0.5 n |Ax
Āx:n = (1 + i)0.5 A1x:n + Ax:n1
A(m)
= (1 + i)(m−1)/2m Ax
x
2
Āx = (1 + i)2 Ax
16
Lesson 15: Annuities: Continuous, Expectation
Actuarial notation for standard types of annuity
Name
Payment per
annum at time t
Present value
random variable
Symbol for actuarial
present value
Whole life
1t≤T
āT
a¯x
annuity
Temporary life
1 t ≤ min(T, n)
0 t > min(T, n)
āT
ān
T ≤n
T >n
āx:n
annuity
Deferred life annuity
0 t ≤ n or t > T
1 n<t≤T
0
T ≤n
āT − ān T > n
n |āx
Deferred temporary
0 t ≤ n or t > T
1 n < t ≤ n + m and t ≤ T
0 T >n+m
0
T ≤n
āT − ān
n<T ≤n+m
ān+m − ān T > n + m
n |āx:m
life annuity
Certain-and-life
1 t ≤ max(T, n)
0 t > max(T, n)
ān
āT
Relationships between insurances and annuities
1 − Āx
δ
Āx = 1 − δāx
1 − Āx:n
āx:n =
δ
Āx:n = 1 − δāx:n
Āx:n − Āx
n |āx =
δ
āx =
General formulas for expected value
∞
Z
āx =
an t px µx+t dt
Z0 ∞
āx =
v t t px dt
0
Z
n
v t t px dt
āx:n =
Z0 ∞
n |āx
=
n
Formulas under constant force of mortality
v t t px dt
T ≤n
T >n
āx:n
17
1
µ+δ
1 − e−(µ+δ)n
āx:n =
µ+δ
−(µ+δ)n
e
n |āx =
µ+δ
āx =
Relationships between annuities
āx = āx:n +n Ex āx+n
āx:n = ān +n |āx
18
Lesson 16: Annuities: Discrete, Expectation
Relationships between insurances and annuities
1 − Ax
d
Ax = 1 − däx
1 − Ax:n
äx:n =
d
Ax:n = 1 − däx:n
äx =
A1x:n = vä1x:n − a1x:n
(1 + i)Ax + iax = 1
Relationships between annuities
äx:n = än +n |äx
äx:n = äx −n Ex äx+n
n |äx =n Ex äx+n
äx = äx:n +n |äx
ax = äx − 1
äx:n = ax:n + 1 −n Ex = ax:n−1 + 1
Other annuity equations
äx:n =
äx:n =
n−1
X
k=1
n−1
X
äk k−1 px qx+k−1 + än n−1 px
k
v k px , ax:n =
k=0
n
X
v k k px
k=1
1+i
äx =
, if qx is constant
q+i
Accumulated value
äx:n
n Ex
= s̈x+1:n−1 + 1
s̈x:n =
sx:n
1
n−1 Ex+1
1
+1−
n Ex
sx:n = sx+1:n−1 +
sx:n = s̈x:n
mthly annuities
ä(m)
x
∞
X
1 mk
=
v px
m mk
k=0
19
Lesson 17: Annuities: Variance
General formulas for second moments
Z ∞
2
E[Ȳx ] =
ā2t t px µx+t dt
0
E[Ÿx2 ] =
2
E[Ÿx:n
]
∞
X
äk2 k−1 |qx
k=1
n
X
=
k=1
ä2k k−1 |qx
+
2
n px än
=
n−1
X
ä2k k−1 |qx + n−1 px än2
k=1
Special formulas for variance of whole life annuities and temporary life annuities
2(āx −2 āx )
− (Āx )2
=
− (āx )2
δ2
δ
2 Ā
2
2(āx:n −2 āx:n )
x:n − (Āx:n )
V ar(Ȳx:n ) =
=
− (āx:n )2
δ2
δ
2 A − (A )2
2(äx −2 äx ) 2
x
x
V ar(Ÿx ) = V ar(Yx ) =
=
+ äx − (äx )2
d2
d2
2A
2
x:n − (Ax:n )
V ar(Ÿx:n ) = V ar(Yx:n−1 ) =
d2
V ar(Ȳx ) =
2 Ā
x
20
Lesson 18: Annuities: Probabilities and Percentiles
• To calculate a probability for an annuity, calculate the t for which āt has the desired property.
Then calculate the probability t is in that range.
• To calculate a percentile of an annuity, calculate the percentile of T , then calculate āT .
• Some adjustments may be needed for discrete annuities or non-whole-life annuities
• If forces of mortality and interest are constant, then the probability that the present value of
payments on a continuous whole life annuity will be greater than its actuarial present value is
µ/δ
µ
P r(āT (x) > āx ) =
µ+δ
21
Lesson 19: Annuities: Varying Annuities, Recursive Calculations
Increasing/Decreasing Annuities
1
¯
, if µ is constant
Iā
=
x
(µ + δ)2
¯
Iā
+ D̄ā x:n = nāx:n
x:n
(Ia)x:n + (Da)x:n = (n + 1)ax:n
Recursive Formulas
äx = vpx äx+1 + 1
ax = vpx ax+1 + vpx
āx = vpx āx+1 + āx:1
äx:n = vpx äx+1:n−1 + 1
ax:n = vpx ax+1:n−1 + vpx
āx:n = vpx āx+1:n−1 + āx:1
n |äx
= vpxn−1 äx+1
n |ax = vpxn−1 ax+1
n |āx = vpxn−1 āx+1
äx:n = 1 + vqx än−1 + vpx äx+1:n−1
ax:n = v + vqx an−1 + vpx ax+1:n−1
āx:n = ā1 + vqx ān−1 + vpx āx+1:n−1
22
Lesson 20: Annuities: m-thly Payments
In general:
a(m)
= ä(m)
−
x
x
1
m
!m
!−m
i(m)
d(m)
1+i= 1+
= 1−
m
m
1
i(m) = m (1 + i) m − 1
1
d(m) = m 1 − (1 + i)− m
For small interest rates:
m−1
2m
m−1
≈ ax +
2m
ä(m)
≈ äx −
x
a(m)
x
Under the uniform distribution of death (UDD) assumption:
m−1
2m
m−1
= ax +
2m
= α(m)äx − β(m)
ä(m)
= äx −
x
a(m)
x
ä(m)
x
(m)
äx:n = α(m)äx:n − β(m)(1 −n Ex )
(m)
n |äx
= α(m)n |äx − β(m)n Ex
āx = α(∞)äx − β(∞)
1
1
(m)
(m)
+ n Ex
ax:n = äx:n −
m m
(m)
1 − Ax
ä(m)
=
x
d(m)
Similar conversion formulae for converting the modal insurances to annuities hold for other types of
insurances and annuities (only whole life version is shown).
α(m) =
i − i(m)
i(m) d(m)
= d(∞) = ln(1 + i) = δ
β(m) =
i(∞)
id
i(m) d(m)
23
(m)
Woolhouse’s formula for approximating äx :
m − 1 m2 − 1
(µx + δ)
−
2m
12m2
1
1
āx ≈ äx − − (µx + δ)
2 12
m−1
m2 − 1
(m)
äx:n ≈ äx:n −
(µx + δ −n Ex (µx+n + δ))
(1 −n Ex ) −
2m
12m2
m−1
m2 − 1
(m)
≈n |äx −
n |äx
n Ex −
n Ex (µx+n + δ)
2m
12m2
1
1
e̊x = ex + − µx
2 12
When the exact value of µx is not available, use the following approximation:
1
µx ≈ − (ln px−1 + ln px )
2
ä(m)
≈ äx −
x
24
Lesson 21: Premiums: Fully Continuous Expectation
The equivalence principle: The actuarial present value of the benefit premiums is equal to the
actuarial present value of the benefits. For instance:
whole life insurance Āx = P̄ Āx āx
n − year endowment insurance Āx:n = P̄ Āx:n āx:n
1
1
n − year term insurance Āx:n
= P̄ Āx:n
āx:n
n − year deferred insurance n |Āx =n P̄ n |Āx āx:n
n − pay whole life insurance Āx =n P̄ Āx āx:n
n − year deferred annuity n |āx = P̄ (n |āx ) āx:n
1 − δāx
1
P̄ Āx =
=
−δ
āx
āx
Āx
δ Āx
P̄ Āx =
=
(1 − Āx )/δ
1 − Āx
1
−δ
P̄ Āx:n =
āx:n
δ Āx:n
P̄ Āx:n =
1 − Āx:n
For constant force of mortality, P̄ Āx and P̄ Āx:n are equal to µ.
Future loss formulas for whole life with face amount b and premium amount π:
π π
1 − vT
T
T
T
b
+
=
v
−
L
=
bv
−
πā
=
bv
−
π
0
T
δ
δ
δ
π π
E[0 L] = bĀx − πāx = Āx b +
−
δ
δ
Similar formulas are available for endowment insurance.
25
Lesson 22: Premiums: Net Premiums for Discrete Insurances Calculated from Life
Tables
Assume the following notation
(1) Px is the premium for a fully discrete whole life insurance, or Ax /äx
1 is the premium for a fully discrete n-year term insurance, or A1 /ä
(2) Px:n
x:n
x:n
1
1
(3) Px:n is the premium for a fully discrete n-year pure endowment, or Ax:n /äx:n
(4) Px:n is the premium for a fully discrete n-year endowment insurance, or Ax:n /äx:n
26
Lesson 23: Premiums: Net Premiums for Discrete Insurances Calculated from
Formulas
Whole life and endowment insurance benefit premiums
1
Px =
−d
äx
dAx
Px =
1 − Ax
1
Px:n =
−d
äx:n
dAx:n
Px:n =
1 − Ax:n
For fully discrete whole life and term insurances:
1 = vq .
If qx is constant, then Px = vqx and Px:n
x
Future loss at issue formulas for fully discrete whole life with face amount b:
Px
Px
Kx +1
−
L0 = v
b+
d
d
Px
Px
−
E[L0 ] = Āx b +
d
d
Similar formulas are available for endowment insurances.
Refund of premium with interest
To calculate the benefit premium when premiums are refunded with interest during the deferral
period, equate the premiums and the benefits at the end of the deferral period. Past premiums are
accumulated at interest only.
Three premium principle formulae
n Px
1
− Px:n
= Px:n1 Ax+n
Px:n −n Px = Px:n1 (1 − Ax+n )
27
Lesson 24: Premiums: Net Premiums Paid on an m-thly Basis
If premiums are payable mthly, then calculating the annual benefit premium requires dividing by an
mthly annuity. If you are working with a life table having annual information only, mthly annuities
can be estimated either by assuming UDD between integral ages or by using Woolhouse’s formula
(Lesson 20). The mthly premium is then a multiple of the annual premium. For example, for h-pay
whole life payable at the end of the year of death,
Ax
h Px ax:h
(m)
= (m) =
h Px
(m)
ax:h
ax:h
28
Lesson 25: Premiums: Gross Premiums
The gross future loss at issue Lg0 is the random variable equal to the present value at issue of benefits
plus expenses minus the present value at issue of gross premiums:
Lg0 = P V (Ben) + P V (Exp) − P V (P g )
To calculate the gross premium P g by the equivalence principle, equate P g times the annuity-due for
the premium payment period with the sum of
1. An insurance for the face amount plus settlement expenses
2. P g times an annuity-due for the premium payment period of renewal percent of premium expense,
plus the excess of the first year percentage over the renewal percentage
3. An annuity-due for the coverage period of the renewal per-policy and per-100 expenses, plus the
excess of first year over renewal expenses
29
Lesson 26: Premiums: Variance of Loss at Issue, Continuous
The following equations are for whole life and endowment insurance of 1. For whole life, drop n .
2 P 2
2
1+
V ar(L0 ) = Āx:n − Āx:n
δ
2
2 Ā
x:n − Āx:n
V ar(L0 ) =
2 , If equivalence principle premium is used
1 − Āx:n
µ
V ar(L0 ) =
, For whole life with equivalence principle and constant force of mortality only
µ + 2δ
For whole life and endowment insurance with face amount b:
2 P 2
2
b+
V ar(L0 ) = Āx − Āx
δ
2
P 2
2
b+
V ar(L0 ) = Āx:n − Āx:n
δ
If the benefit is b instead of 1, and the premium P is stated per unit, multiply the variances by b2 .
For two whole life or endowment insurances, one with b0 units and total premium P 0 and the other
with b units and total premium P , the relative variance of loss at issue of the first to second is
((b0 δ + P 0 ) / (bδ + P ))2 .
30
Lesson 27: Premiums: Variance of Loss at Issue, Discrete
The following equations are for whole life and endowment insurance of 1. For whole life, drop n .
P 2
2
2
1+
V ar(0 L) = Ax:n − (Ax:n )
d
If equivalence principle premium is used: V ar(0 L) =
− (Ax:n )2
(1 − Ax:n )2
2A
x:n
For whole life with equivalence principle and constant force of mortality only: V ar(0 L) =
q(1 − q)
q +2 i
If the benefit is b instead of 1, and the premium P is stated per unit, multiply the variances by b2 .
For two whole life or endowment insurances, one with b0 units and total premium P 0 and the other
with b units and total premium P , the relative variance of loss at issue of the first to second is
((b0 d + P 0 ) / (bd + P ))2 .
31
Lesson 28: Premiums: Probabilities and Percentiles of Loss at Issue
• For level benefit or decreasing benefit insurance, the loss at issue decreases with time for whole
life, endowment, and term insurances. To calculate the probability that the loss at issue is
less than something, calculate the probabiitiy that survival time is greater than something.
• For level benefit or decreasing benefit deferred insurance, the loss at issue decreases during
the deferral period, then jumps at the end of the deferral period and declines thereafter.
– To calculate the probability that the loss at issue is greater then a positive number,
calculate the probability that survival time is less than something minus the probability
that survival time is less than the deferral period.
– To calculate the probability that the loss at issue is greater than a negative number,
calculate the probability that survival time is less than something that is less than the
deferral period, and add that to the probability that survival time is less than something
that is greater then the deferral period minus the probability that survival time is less
than the deferral period.
• For a deferred annuity with premiums payable during the deferral period, the loss at issue
decreases until the end of the deferral period and increases thereafter.
• The 100pth percentile premium is the premium for which the loss at issue is positive with
probability p. For fully continuous whole life, this is the loss that occurs if death occurs at
the 100pth percentile of survival time.