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)

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 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

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 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

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.

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.

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.

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.

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 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.