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ACTS 4301
FORMULA SUMMARY
Lesson 1: Probability Review
1. Var(X)= E[X2]- E[X]2
2. V ar(aX + bY ) = a2V ar(X) + 2abCov(X,Y ) + b2V ar(Y )
3. V ar(X) = V ar(X)n
4. EX [X] = EY [EX [X|Y ]] Double expectation5. 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) E[X] Var(X)
Exponential e−xθ /θ 1− e−
xθ θ θ2
Uniform 1b−a , x ∈ [a, b] x
θ , x ∈ [a, b] a+b2
(b−a)2
12
Gamma xα−1e−xθ /Γ(α)θα
∫ x0 f(t)dt αθ αθ2
Discrete Distributions
Name f(x) E[X] Var(X)
Poisson e−λλx/x! λ λ
Binomial(mx
)px(1− p)m−x mp mp(1− p)
2
Lesson 2: Survival Distributions: Probability Functions, Life Tables
1. Actuarial Probability Functions
tpx - probability that (x) survives t years
tqx - probability that (x) dies within t years
t|uqx - probability that (x) survives t years and then dies in the next u years.
t+upx = tpx ·u px+t
t|uqx = tpx ·u qx+t =
= tpx −t+u px =
= t+uqx −t qx2. Life Table Functions
dx = lx − lx+1
ndx = lx − lx+n
qx = 1− px =dxlx
tpx =lx+t
lx
t|uqx =lx+t − lx+t+u
lx3. Mathematical Probability Functions
tpx = ST (x)(t)
tqx = FT (x)(t)
t|uqx = Pr (t < T (x) ≤ t+ u) = FT (x)(t+ u)− FT (x)(t) =
= Pr (x+ t < X ≤ x+ t+ u|X > x) =FX(x+ t+ u)− FX(x+ t)
sX(x)
Sx+u(t) =Sx(t+ u)
Sx(u)
Sx(t) =S0(x+ t)
S0(x)
Fx(t) =F0(x+ t)− F0(x)
1− F0(x)
3
Lesson 3: Survival Distributions: Force of Mortality
µx =f0(x)
S0(x)= − d
dxlnS0(x)
µx+t =fx(t)
Sx(t)= − d
dxlnSx(t)
Sx(t) =t px = exp
(−∫ t
0µx+s ds
)= exp
(−∫ t
0µx(s) ds
)= exp
(−∫ x+t
xµs ds
)µx(t) = −dtpx/dt
tpx= −d lnt px
dt
fT (x)(t) = fx(t) =t px · µx(t)
t|uqx =
∫ t+u
tspx · µx(s) ds
tqx =
∫ t
0spx · µx(s)ds
If µ∗x(t) = µx(t) + k for all t, then sp∗x = spxe
−kt
If µx(t) = µx(t) + µx(t) for all t, then spx = spxspx
If µ∗x(t) = kµx(t) for all t, then sp∗x = (spx)k
4
Lesson 4: Survival Distributions: Mortality Laws
Exponential distribution or Constant Force of Mortality
µx(t) = µ
tpx = e−µt
Uniform distribution or De Moivre’s law
µx(t) =1
ω − x− t, 0 ≤ t ≤ ω − x
tpx =ω − x− tω − x
, 0 ≤ t ≤ ω − x
tqx =t
ω − x, 0 ≤ t ≤ ω − x
t|uqx =u
ω − x, 0 ≤ t+ u ≤ ω − x
Beta distribution or Generalized De Moivre’s law
µx(t) =α
ω − x− t
tpx =
(ω − x− tω − x
)α, 0 ≤ t ≤ ω − x
Gompertz’s law:
µx = Bcx, c > 1
tpx = exp
(−Bc
x(ct − 1)
ln c
)Makeham’s law:
µx = A+Bcx, c > 1
tpx = exp
(−At− Bcx(ct − 1)
ln c
)Weibull Distribution
µx = kxn
S0(x) = e−kxn+1/(n+1)
5
Lesson 5. Survival Distributions: Moments
Complete Future Lifetime
ex =
∫ ∞0
ttpxµx+t dt
ex =
∫ ∞0
tpx dt
ex =1
µfor constant force of mortality
ex =ω − x
2for uniform (de Moivre) distribution
ex =ω − xα+ 1
for generalized uniform (de Moivre) distribution
E[(T (x))2
]= 2
∫ ∞0
ttpx dt
V ar (T (x)) =1
µ2for constant force of mortality
V ar (T (x)) =(ω − x)2
12for uniform (de Moivre) distribution
V ar (T (x)) =(ω − x)2
(α+ 1)2(α+ 2)for generalized uniform (de Moivre) distribution
n-year Temporary Complete Future Lifetime
ex:n =
∫ n
0ttpxµx+t dt+ nnpx
ex:n =
∫ n
0tpx dt
ex:n =n px(n) +n qx(n/2) for uniform (de Moivre) distribution
ex:1 = px + 0.5qx for uniform (de Moivre) distribution
ex:n =1− e−µn
µfor constant force of mortality
E[(T (x) ∧ n)2
]= 2
∫ n
0ttpx dt
Curtate Future Lifetime
ex =
∞∑k=1
kk|qx
ex =
∞∑k=1
kpx
ex = ex − 0.5 for uniform (de Moivre) distribution
E[(K(x))2
]=∞∑k=1
(2k − 1)kpx
V ar (K(x)) = V ar (T (x))− 1
12for uniform (de Moivre) distribution
6
n-year Temporary Curtate Future Lifetime
ex:n =n−1∑k=1
kk|qx + nnpx
ex:n =n∑k=1
kpx
ex:n = ex:n − 0.5nqx for uniform (de Moivre) distribution
E[(K(x) ∧ n)2
]=
n∑k=1
(2k − 1)kpx
7
Lesson 6: Survival Distributions: Percentiles, Recursions, and Life Table Concepts
Recursive formulas
ex = ex:n +n pxex+n
ex:n = ex:m +m pxex+m:n−m, m < n
ex = ex:n +n pxex+n = ex:n−1 +n px(1 + ex+n)
ex = px + pxex+1 = px(1 + ex+1)
ex:n = ex:m +m pxex+m:n−m =
= ex:m−1 +m px(1 + ex+m:n−m) m < n
ex:n = px + pxex+1:n−1 = px
(1 + ex+1:n−1
)Life Table Concepts
Tx =
∫ ∞0
lx+tdt, total future lifetime of a cohort of lx individuals
nLx =
∫ n
0lx+tdt, total future lifetime of a cohort of lx individuals over the next n years
Yx =
∫ ∞0
Tx+tdt
ex =Txlx
ex:n =nLxlx
Central death rate and related concepts
nmx =ndx
nLx
mx =qx
1− 0.5qxfor uniform (de Moivre) distribution
nmx = µx for constant force of mortality
a(x) =Lx − lx+1
dxthe fraction of the year lived by those dying during the year
a(x) =1
2for uniform (de Moivre) distribution
8
Lesson 7: Survival Distributions: Fractional Ages
Function Uniform Distribution of Deaths Constant Force of Mortality Hyperbolic Assumption
lx+s lx − sdx lxpsx lx+1/(px + sqx)
sqx sqx 1− psx sqx/(1− (1− s)qx)
spx 1− sqx psx px/(1− (1− s)qx)
sqx+t sqx/(1− tqx), 0 ≤ s+ t ≤ 1 1− psx sqx/(1− (1− s− t)qx)
µx+s qx/(1− sqx) − ln px qx/(1− (1− s)qx)
spxµx+s qx −psx(ln px) pxqx/(1− (1− s)qx)2
mx qx/(1− 0.5qx) − ln px q2x/(px ln px)
Lx lx − 0.5dx −dx/ ln px −lx+1 ln px/qx
ex ex + 0.5
ex:n ex:n + 0.5nqx
ex: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 andq[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
Ax =
∫ ∞0
e−δttpxµx(t) dt
Actuarial notation for standard types of insurance
Name Present value random variable Symbol for actuarial present value
Whole life insurance vT Ax
Term life insurancevT T ≤ n0 T > n
A1x:n
Deferred life insurance0 T ≤ nvT T > n n|Ax
Deferred term insurance0 T ≤ nvT n < T ≤ n+m0 T > n
n|A1x:m =n |mAx
Pure endowment0 T ≤ nvn T > n
A 1x:n or nEx
Endowment insurancevT T ≤ nvn T > n
Ax: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 payableat the moment of death
Type of insurance APV under constant force APV under uniform (de Moivre)
Whole life µµ+δ
aω−xω−x
n-year term µµ+δ
(1− e−n(µ+δ)
) anω−x
n-year deferred life µµ+δe
−n(µ+δ)e−δna
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 ∫ ∞0
tne−δt dt =n!
δn+1∫ u
0te−δt dt =
1
δ2
(1− (1 + δu)e−δu
)=au − uvu
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 =∞∑0
k|qxvk+1 =
∞∑0
kpxqx+kvk+1
Actuarial present value under constant force and uniform (de Moivre) mortality for insurances payableat the end of the year of death
Type of insurance APV under constant force APV under uniform (de Moivre)
Whole life qq+i
aω−xω−x
n-year term qq+i (1− (vp)n)
anω−x
n-year deferred life qq+i(vp)
nvna
ω−(x+n)
ω−x
n-year pure endowment (vp)n vn(ω−(x+n))ω−x
13
Lesson 12: Insurance: Probabilities and Percentiles
Summary of Probability and Percentile Concepts
• To calculate Pr(Z ≤ z) for continuous Z, draw a graph of Z as a function of T . Identifyparts of the graph that are below the horizontal line Z = z, and the corresponding t’s. Thencalculate the probability of T being in the range of those t’s.
Note that
Pr(Z < z∗) = Pr(vt < z∗) = Pr
(t > − ln z∗
δ
)=t∗ px, where t∗ = − ln z∗
δ= − ln z∗
ln(1 + i)
The following table shows the relationship between the type of insurance coverage and corre-sponding probability:
Type of insurance Pr(Z < z∗)
Whole life t∗px
n-year term
{npx z∗ ≤ vnt∗px z∗ > vn
n-year deferred life
{nqx +t∗ px z∗ < vn
1 z∗ ≥ vn
n-year endowment
{0 z∗ < vn
t∗px z∗ ≥ vn
n-year deferred m-year term
nqx +n+m px z∗ < vm+n
nqx +t∗ px vm+n ≤ z∗ < vn
1 z∗ ≥ vn
• In the case of constant force of mortality µ and interest δ, Pr(Z ≤ z) = zµ/δ
• For discrete Z, identify T and then identify K + 1 corresponding to those T . In other words,
Pr(Z > z∗) = Pr (T < t∗) =k qx, where k = bt∗c - the greatest integer smaller than t∗
Pr(Z < z∗) = Pr (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 wherethe 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 anyk. Then solve for z (which is often vt)
14
Lesson 13: Insurance: Recursive Formulas, Varying Insurances
Recursive Formulas
Ax = vqx + vpxAx+1
Ax = vqx + v2pxqx+1 + v22pxAx+2
Ax:n = vqx + vpxAx+1:n−1
A1x:n = vqx + vpxA
1x+1:n−1
n|Ax = vpxn−1|Ax+1
A 1x:n = vpxA
1x+1:n−1
Increasing and Decreasing Insurance∫ ∞0
tne−δtdt =n
δn+1∫ u
0te−δtdt =
1
δ2
(1− (1 + δu) e−δu
)=an δ − uvu
δ(IA)x
=µ
(µ+ δ)2for constant force
E[Z2] =2µ
(µ+ 2δ)3for Z a continuously increasing continuous insurance, constant force.(
IA)1x:n
+(DA
)1x:n
= nA1x:n(
IA)1x:n
+(DA
)1x:n
= (n+ 1)A1x:n
(IA)1x:n + (DA)1
x:n = (n+ 1)A1x:n
Recursive Formulas for Increasing and Decreasing Insurance
(IA)1x:n = A1
x:n + vpx (IA)1x+1:n−1
(IA)1x:n = A1
x:n + vpx (IAA)1x+1:n−1
(DA)1x:n = nA1
x:n + vpx (DA)1x+1:n−1
(DA)1x:n = A1
x:n + (DA)1x:n−1
15
Lesson 14: Relationships between Insurance Payable at Moment of Death and Payableat End of Year
Summary of formulas relating insurances payable at moment of death to insurances payable at theend of the year of death assuming uniform distribution of deaths
Ax =i
δAx
A1x:n =
i
δA1x:n
n|Ax =i
δn|Ax(
IA)1x:n
=i
δ(IA)1
x:n(ID)1x:n
=i
δ(ID)1
x:n
Ax:n =i
δA1x:n +A 1
x:n
A(m)x =
i
i(m)Ax
2Ax =2i+ i2
2δ2Ax(
IA)1x:n
=(IA)1x:n− A1
x:n
(1
d− 1
δ
)Summary of formulas relating insurances payable at moment of death to insurances payable at theend of the year of death using claims acceleration approach
Ax = (1 + i)0.5Ax
A1x:n = (1 + i)0.5A1
x:n
n|Ax = (1 + i)0.5n|Ax
Ax:n = (1 + i)0.5A1x:n +A 1
x:n
A(m)x = (1 + i)(m−1)/2mAx
2Ax = (1 + i)2Ax
16
Lesson 15: Annuities: Continuous, Expectation
Actuarial notation for standard types of annuity
Name Payment per Present value Symbol for actuarialannum at time t random variable present value
Whole life 1 t ≤ T aT ax
annuity
Temporary life1 t ≤ min(T, n)0 t > min(T, n)
aT T ≤ nan T > n
ax:n
annuity
Deferred life annuity0 t ≤ n or t > T1 n < t ≤ T
0 T ≤ naT − an T > n n|ax
Deferred temporary0 t ≤ n or t > T1 n < t ≤ n+m and t ≤ T0 T > n+m
0 T ≤ naT − an n < T ≤ n+m
an+m − an T > n+mn|ax:m
life annuity
Certain-and-life1 t ≤ max(T, n)0 t > max(T, n)
an T ≤ naT T > n
ax:n
Relationships between insurances and annuities
ax =1− Axδ
Ax = 1− δax
ax:n =1− Ax:n
δAx:n = 1− δax:n
n|ax =Ax:n − Ax
δ
General formulas for expected value
ax =
∫ ∞0
antpxµx+t dt
ax =
∫ ∞0
vttpx dt
ax:n =
∫ n
0vttpx dt
n|ax =
∫ ∞n
vttpx dt
Formulas under constant force of mortality
17
ax =1
µ+ δ
ax:n =1− e−(µ+δ)n
µ+ δ
n|ax =e−(µ+δ)n
µ+ δ
Relationships between annuities
ax = ax:n +n Exax+n
ax:n = an +n |ax
18
Lesson 16: Annuities: Discrete, Expectation
Relationships between insurances and annuities
ax =1−Axd
Ax = 1− dax
ax:n =1−Ax:n
dAx:n = 1− dax:n
A1x:n = va1
x:n − a1x:n
(1 + i)Ax + iax = 1
Relationships between annuities
ax:n = an +n |axax:n = ax −n Exax+n
n|ax =n Exax+n
ax = ax:n +n |axax = ax − 1
ax:n = ax:n + 1−n Ex = ax:n−1 + 1
Other annuity equations
ax:n =n−1∑k=1
akk−1pxqx+k−1 + ann−1px
ax:n =n−1∑k=0
vkkpx, ax:n =n∑k=1
vkkpx
ax =1 + i
q + i, if qx is constant
Accumulated value
sx:n =ax:n
nExsx:n = sx+1:n−1 + 1
sx:n = sx+1:n−1 +1
n−1Ex+1
sx:n = sx:n + 1− 1
nExmthly annuities
a(m)x =
∞∑k=0
1
mvkmkm
px
19
Lesson 17: Annuities: Variance
General formulas for second moments
E[Y 2x ] =
∫ ∞0
a2t tpxµx+t dt
E[Y 2x ] =
∞∑k=1
a2kk−1|qx
E[Y 2x:n] =
n∑k=1
a2kk−1|qx + npxa
2n =
n−1∑k=1
a2kk−1|qx + n−1pxa
2n
Special formulas for variance of whole life annuities and temporary life annuities
V ar(Yx) =2Ax − (Ax)2
δ2=
2(ax −2 ax)
δ− (ax)2
V ar(Yx:n) =2Ax:n − (Ax:n)2
δ2=
2(ax:n −2 ax:n)
δ− (ax:n)2
V ar(Yx) = V ar(Yx) =2Ax − (Ax)2
d2=
2(ax −2 ax)
d2+2 ax − (ax)2
V ar(Yx:n) = V ar(Yx:n−1) =2Ax:n − (Ax:n)2
d2
20
Lesson 18: Annuities: Probabilities and Percentiles
• To calculate a probability for an annuity, calculate the t for which at 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 aT .• 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
Pr(aT (x) > ax) =
(µ
µ+ δ
)µ/δ
21
Lesson 19: Annuities: Varying Annuities, Recursive Calculations
Increasing/Decreasing Annuities(I a)x
=1
(µ+ δ)2, if µ is constant(
I a)x:n
+(Da)x:n
= nax:n
(Ia)x:n + (Da)x:n = (n+ 1)ax:n
Recursive Formulas
ax = vpxax+1 + 1
ax = vpxax+1 + vpx
ax = vpxax+1 + ax:1
ax:n = vpxax+1:n−1 + 1
ax:n = vpxax+1:n−1 + vpx
ax:n = vpxax+1:n−1 + ax:1
n|ax = vpxn−1ax+1
n|ax = vpxn−1ax+1
n|ax = vpxn−1ax+1
ax:n = 1 + vqxan−1 + vpxax+1:n−1
ax:n = v + vqxan−1 + vpxax+1:n−1
ax:n = a1 + vqxan−1 + vpxax+1:n−1
22
Lesson 20: Annuities: m-thly Payments
In general:
a(m)x = a(m)
x − 1
m
1 + i =
(1 +
i(m)
m
)m=
(1− d(m)
m
)−mi(m) = m
((1 + i)
1m − 1
)d(m) = m
(1− (1 + i)−
1m
)
For small interest rates:
a(m)x ≈ ax −
m− 1
2m
a(m)x ≈ ax +
m− 1
2m
Under the uniform distribution of death (UDD) assumption:
a(m)x = ax −
m− 1
2m
a(m)x = ax +
m− 1
2m
a(m)x = α(m)ax − β(m)
a(m)x:n = α(m)ax:n − β(m)(1−n Ex)
n|a(m)x = α(m)n|ax − β(m)nEx
ax = α(∞)ax − β(∞)
a(m)x:n = a
(m)x:n −
1
m+
1
mnEx
a(m)x =
1−A(m)x
d(m)
Similar conversion formulae for converting the modal insurances to annuities hold for other types ofinsurances and annuities (only whole life version is shown).
α(m) =id
i(m)d(m)
β(m) =i− i(m)
i(m)d(m)
i(∞) = d(∞) = ln(1 + i) = δ
23
Woolhouse’s formula for approximating a(m)x :
a(m)x ≈ ax −
m− 1
2m− m2 − 1
12m2(µx + δ)
ax ≈ ax −1
2− 1
12(µx + δ)
a(m)x:n ≈ ax:n −
m− 1
2m(1−n Ex)− m2 − 1
12m2(µx + δ −n Ex(µx+n + δ))
n|a(m)x ≈n |ax −
m− 1
2mnEx −
m2 − 1
12m2 nEx(µx+n + δ)
ex = ex +1
2− 1
12µx
When the exact value of µx is not available, use the following approximation:
µx ≈ −1
2(ln px−1 + ln px)
24
Lesson 21: Premiums: Fully Continuous Expectation
The equivalence principle: The actuarial present value of the benefit premiums is equal to theactuarial present value of the benefits. For instance:
whole life insurance Ax = P(Ax)ax
n− year endowment insurance Ax:n = P(Ax:n
)ax:n
n− year term insurance A 1x:n = P
(A 1x:n
)ax:n
n− year deferred insurance n|Ax =n P(n|Ax
)ax:n
n− pay whole life insurance Ax =n P(Ax)ax:n
n− year deferred annuity n|ax = P (n|ax) ax:n
P(Ax)
=1− δaxax
=1
ax− δ
P(Ax)
=Ax
(1− Ax)/δ=
δAx1− Ax
P(Ax:n
)=
1
ax:n− δ
P(Ax:n
)=
δAx:n
1− Ax:n
For constant force of mortality, P(Ax)
and P(Ax:n
)are equal to µ.
Future loss formulas for whole life with face amount b and premium amount π:
0L = bvT − πaT = bvT − π(
1− vT
δ
)= vT
(b+
π
δ
)− π
δ
E[0L] = bAx − πax = Ax
(b+
π
δ
)− π
δSimilar formulas are available for endowment insurance.
25
Lesson 22: Premiums: Net Premiums for Discrete Insurances Calculated from LifeTables
Assume the following notation
(1) Px is the premium for a fully discrete whole life insurance, or Ax/ax(2) P 1
x:n is the premium for a fully discrete n-year term insurance, or A1x:n/ax:n
(3) P 1x:n is the premium for a fully discrete n-year pure endowment, or A 1
x:n/ax:n
(4) Px:n is the premium for a fully discrete n-year endowment insurance, or Ax:n/ax:n
26
Lesson 23: Premiums: Net Premiums for Discrete Insurances Calculated fromFormulas
Whole life and endowment insurance benefit premiums
Px =1
ax− d
Px =dAx
1−Ax
Px:n =1
ax:n− d
Px:n =dAx:n
1−Ax:n
For fully discrete whole life and term insurances:If qx is constant, then Px = vqx and P 1
x:n = vqx.
Future loss at issue formulas for fully discrete whole life with face amount b:
L0 = vKx+1
(b+
Pxd
)− Px
d
E[L0] = Ax
(b+
Pxd
)− Px
d
Similar formulas are available for endowment insurances.
Refund of premium with interestTo calculate the benefit premium when premiums are refunded with interest during the deferralperiod, equate the premiums and the benefits at the end of the deferral period. Past premiums areaccumulated at interest only.
Three premium principle formulae
nPx − P 1x:n = P 1
x:nAx+n
Px:n −n Px = P 1x:n(1−Ax+n)
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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 anmthly annuity. If you are working with a life table having annual information only, mthly annuitiescan 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-paywhole life payable at the end of the year of death,
hP(m)x =
Ax
a(m)
x:h
=hPxax:h
a(m)
x:h
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Lesson 25: Premiums: Gross Premiums
The gross future loss at issue Lg0 is the random variable equal to the present value at issue of benefitsplus expenses minus the present value at issue of gross premiums:
Lg0 = PV (Ben) + PV (Exp)− PV (P g)
To calculate the gross premium P g by the equivalence principle, equate P g times the annuity-due forthe premium payment period with the sum of
1. An insurance for the face amount plus settlement expenses2. 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 percentage3. 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
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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.
V ar(L0) =(
2Ax:n −(Ax:n
)2)(1 +
P
δ
)2
V ar(L0) =2Ax:n −
(Ax:n
)2(1− Ax:n
)2 , If equivalence principle premium is used
V ar(L0) =µ
µ+ 2δ, For whole life with equivalence principle and constant force of mortality only
For whole life and endowment insurance with face amount b:
V ar(L0) =(
2Ax −(Ax)2)(
b+P
δ
)2
V ar(L0) =(
2Ax:n −(Ax:n
)2)(b+
P
δ
)2
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 b′ units and total premium P ′ and the otherwith b units and total premium P , the relative variance of loss at issue of the first to second is((b′δ + P ′) / (bδ + P ))2.
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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.
V ar(0L) =(
2Ax:n − (Ax:n)2)(
1 +P
d
)2
If equivalence principle premium is used: V ar(0L) =2Ax:n − (Ax:n)2
(1−Ax:n)2
For whole life with equivalence principle and constant force of mortality only: V ar(0L) =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 b′ units and total premium P ′ and the otherwith b units and total premium P , the relative variance of loss at issue of the first to second is((b′d+ P ′) / (bd+ P ))2.
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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 wholelife, endowment, and term insurances. To calculate the probability that the loss at issue isless 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 probabilitythat 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 thedeferral period, and add that to the probability that survival time is less than somethingthat is greater then the deferral period minus the probability that survival time is lessthan the deferral period.
• For a deferred annuity with premiums payable during the deferral period, the loss at issuedecreases 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 atthe 100pth percentile of survival time.