A. Prerequisite Review
CALCULUS REVIEW
ddxex = ex∫ex dx = ex + c
D {f(g(x))} = f ′(g(x)) · g′(x)
(uv)′ = u′ · v + u · v′∫udv = uv −
∫vdu
DISCRETE RANDOM VARIABLES∑all x
p(x) = 1
E[X] =∑all x
xp(x)
E[g(X)] =∑all x
g(x) · p(x)
Var[X] = E[X2]− (E[X])2
F (x) = Pr[X ≤ x] =∑y≤x
p(y)
Bernoulli Shortcut for the RV:
Y =
{a p(a) = p
b p(b) = 1− p
Var[Y ] = (a− b)2 · p(1− p)
CONTINUOUS RANDOM VARIABLES∫x f(x) dx = 1
E[g(X)] =∫x g(x) f(x) dx
F (x) = Pr[X ≤ x] =∫ x−∞ f(y) dy
f(x) = ddxF (x)
UNIFORM DISTRIBUTION
X ∼ uniform a to b
f(x) = 1b−a
F (x) = x−ab−a
E[X] = a+b2
Var[X] =(b−a)2
12
median = mean
EXPONENTIAL DISTRIBUTION
Exponential distribution is memoryless
f(x) = 1µe− 1
µx
F (x) = 1− e−1µx
E[X] = µ
Var[X] = µ2
MIXED RANDOM VARIABLES
Example:
X = a p(a)
X = b p(b)
a < X < b f(x)
p(a) +∫ ba f(x) dx+ p(b) = 1
E[Xk] = ak · p(a) +∫ ba x
k f(x) dx+ bk · p(b)
COVARIANCE
Var(aX + bY ) = a2Var(X) + b2Var(Y )+2ab Cov(X,Y )
Cov(X,Y ) = E(XY )− E(X)E(Y )
If X and Y are independentthen covariance is 0.
CONDITIONAL PROBABILITY
Pr(A|B) =Pr(A ∩B)
Pr(B)
fX(x|y) =f(x, y)
fY (y)
fY (y) =∫∞−∞ f(x, y) dx
E[X] = EY [EX [X|Y ]]
Var[X] = EY[VarX [X|Y ]
]+ VarY
[EX [X|Y ]
]
NORMAL APPROXIMATION
Central Limit Theorem:
Y = X1 +X2 + · · ·+Xn
Xi’s are iid RVs(µx, σ2x)
Y ≈ Normal(E[Y ] = nµx,Var[Y ] = nσ2x)
95th percentile of Y :
y = µy + σy × Φ−1(0.95)
CONVERT BETWEEN i, d, and δ
0
d
1
i
(1)
(2)
d
1− d= i (1)
i
1 + i= d (2)
δ = ln(1 + i)
ACCUMULATING
1 + i = eδ
(1 + i)2 = e2δ
(1 + i)n = enδ
DISCOUNTING
v = (1 + i)−1 = e−δ
v2 = (1 + i)−2 = e−2δ
vn = (1 + i)−n = e−nδ
DOUBLE THE FORCE OF INTEREST
1 + i→ (1 + i)2
v → v2
i→ 2i+ i2
d→ 2d− d2
iδ→ 2i+i2
2δ
ANNUITY CERTAINS
an =1− vn
δ=i
δan
(Ia)n =an − nvn
i
(Da)n =n− an
i
a∞ =
∫ ∞0
vt dt =1
δ
(I a)∞ =
∫ ∞0
tvt dt =1
δ2
(I a)n =
∫ n
0tvt dt =
an − nvn
δ
(Da)n =
∫ n
0(n− t)vt dt =
n− anδ
c©2012 The Infinite Actuary
B. Survival Models and Life Tables
RANDOM VARIABLES
Tx ∼ future lifetime of (x)
Kx ∼ number of completed future years by(x) prior to death.
Kx = bTxc
Wx ∼ future lifetime or n years whichever isless.
Wx = Tx ∧ n
CUMULATIVE DISTRIBUTION
Probability of (x) dying before age x+ t.
Fx(t) = Pr(Tx ≤ t)
SURVIVAL DISTRIBUTION
Probability of a (x) attaining age x+ t.
Sx(t) = Pr(Tx > t) = 1− Fx(t)
ACTUARIAL NOTATION
tpx − probability that (x) will attain agex+ t.
tpx = Sx(t)
t+upx = tpx · upx+t
tqx − probability that (x) dies within t year
tqx = Fx(t)
tqx = 1− tpx
t|uqx − probability that (x) will survive t yearsand die within the following u years.
t|qx is the PDF for Kx
t|uqx = Sx(t)− Sx(t+ u)
= tpx − t+upx
= t+uqx − tqx
= tpx · uqx+t
LIFE TABLES
lx − expected number of survivors at (x)
ndx − expected number of deaths betweenages x and x+ n
FORCE OF MORTALITY
Intuitively, µx+t dt is the probability that a lifeage x+ t will die in the next instant.
µx+t =fx(t)Sx(t)
=ddt tqx
tpx=− d
dt`x+t
`x+t
npx = exp(−∫ n0 µx+t dt
)If µ∗x+t = k µx+t :
np∗x = (npx)k
ddt t
qx = tpx µx+t
ddt t
px = −tpx µx+t
PROBABILITY DENSITY FUNCTION
Intuitively, the probability that (x) dies at agex+ t.
fx(t) = F ′x(t) = −S′x(t) = tpx µx+t
CURTATE EXPECTATION OF LIFE
ex = E[Kx] =∞∑k=1
kpx
For integral x, ex is the expected number offuture birthdays.
Recursion
ex = px(1 + ex+1)
COMPLETE EXPECTATION OF LIFE
Average future lifetime of (x).
◦ex= E[Tx] =
∫ ∞0
tpx dt
E[T 2x ] = 2
∫ ∞0
t tpx dt
TEMPORARY LIFE EXPECTANCY
◦ex:n = E[Wx] =
∫ n
0tpx dt
ex:n = E[Kx ∧ n] =n∑k=1
kpx
E[W 2x ] = 2
∫ n
0t tpx dt
Recursion
◦ex=
◦ex:n +npx
◦ex+n
◦ex:m+n =
◦ex:m +mpx
◦ex+m:n
◦ex≈ px
(1+◦ex+1
)+ qx
(12
)
CONSTANT FORCE
Age doesn’t matter.
Future lifetime is distributed exponentiallywith mean 1/µ.
µx+t = µ for all x
npx = e−nµ = (px)n
◦ex=
1µ
Var[Tx] =1µ2
Sx(t) = e−µt
`x+t = `xe−µt
◦ex:n =
◦ex (1− npx)
ex:n = ex (1− npx)
Discrete Constant Force−Kx ∼ geometric
fKx (k) = pk · q
ex = E[Kx] =pq
Var[Kx] =pq2
DE MOIVRE’S LAW
Out of ω births, one dies every year until theyare all dead.
Future lifetime is uniformly distributed from 0to ω − x.`x = k (ω − x)
µx = 1ω−x
Sx(t) =ω−x−tω−x
qx = 1ω−x = µx
nqx = nω−x
n|mqx = mω−x
npx = ω−x−nω−x
◦ex=
ω−x2
half way to omega
◦ex:n = npx (n) + nqx
(n2
)a(x) = 1
2
Discrete DML
k|qx = 1ω−x
MODIFIED DML
µx = aω−x
`x = (ω − x)a
Sx(t) =(ω−x−tω−x
)anpx =
(ω−x−nω−x
)a◦ex=
ω−xa+1
FRACTIONAL AGES
Uniform Distribution of Deaths (UDD)
- Use linear interpolation.
Constant Force
- Use exponential interpolation.
c©2012 The Infinite Actuary
SELECT MORTALITY
[age] = age selected
Read across and then down the table.
GOMPERTZ LAW
µx = Bcx c > 1, B > 0
tpx = exp[−(Bln c
)cx(ct − 1)
]
MAKEHAM LAW
µx = A+Bcx c > 1, B > 0, A ≥ −B
tpx = exp(−At) · exp[−(Bln c
)cx(ct − 1)
]
c©2012 The Infinite Actuary
C. Insurance Benefits
PURE ENDOWMENT
MLC discount factor
Z 1x:n =
{0 0 ≤ Tx ≤ n
vn Tx > n
nEx = A 1x:n = vn npx
Var[Z 1x:n ] = 2
nEx − (nEx)2
= v2n npx nqx Bernoulli shortcut
WHOLE LIFE INSURANCE
Zx = vTx , Tx ≥ 0
Ax =
∫ ∞0
vt tpx µx+t dt
Zx = vKx+1, Kx = 0, 1, 2, . . .
Ax =
∞∑k=0
vk+1k|qx
N-YEAR DEFERRED WHOLE LIFE
n|Zx =
{0 0 ≤ Tx ≤ n
vTx Tx > n
n|Ax =
∫ ∞n
vt tpx µx+t dt
n|Ax = nEx · Ax+n
n|Zx =
{0 Kx = 0, 1, . . . , n− 1
vKx+1 Kx = n, n+ 1, . . .
n|Ax = nEx ·Ax+n
INCREASING WHOLE LIFE(IZ)x
= Tx vTx , Tx ≥ 0(IA)x
=
∫ ∞0
tvt tpx µx+t dt
(IZ)x = (Kx + 1)vKx+1, Kx = 0, 1, 2, . . .
(IA)x = Ax + 1|Ax + 2|Ax + · · ·
TERM INSURANCE
Z 1x:n =
{vTx 0 ≤ Tx ≤ n
0 Tx > n
A1x:n =
∫ n
0vt tpx µx+t dt
Ax = A1x:n + n|Ax
n|mAx =
∫ n+m
nvt tpx µx+t dt
Z 1x:n =
{vKx+1 Kx = 0, 1, . . . , n− 1
0 Kx = n, n+ 1, . . .
A1x:n =
n−1∑k=0
vk+1k|qx
Ax = A1x:n + n|Ax
n|mAx =
n+m−1∑k=n
vk+1k|qx
VARYING TERM INSURANCE(IA)1x:n =
∫ n
0tvt tpx µx+t dt(
DA)1x:n =
∫ n
0(n− t)vt tpx µx+t dt(
IA)1x:n +
(DA
)1x:n = nA1
x:n
(IA) 1x:n + (DA) 1
x:n = (n+ 1)A1x:n
ENDOWMENT INSURANCE
Zx:n =
{vTx 0 ≤ Tx ≤ n
vn Tx > n
Ax:n = A1x:n + nEx
Zx:n =
{vKx+1 Kx = 0, 1, . . . , n− 1
vn Kx = n, n+ 1, . . .
Ax:n = A1x:n + nEx
RECURSION
Ax = vqx + vpx Ax+1
Ax:n = vqx + vpx Ax+1:n−1
(IA)x = vqx + vpx (Ax+1 + (IA)x+1)
(IA)x = Ax + vpx(IA)x+1
(IA) 1x:n = vqx
+vpx[(IA) 1
x+1:n−1+A 1
x+1:n−1
](DA) 1
x:n = nvqx + vpx(DA) 1x+1:n−1
CONSTANT FORCE
nEx = e−n(µ+δ)
Ax = µµ+δ
Ax = vqvq+d
A1x:n = Ax (1 − nEx) A1
x:n = Ax (1 − nEx)
n|Ax = e−n(µ+δ)(
µµ+δ
)(IA)x = 1
µ
(Ax)2
(IA)x = 1vq
(Ax)2
DE MOIVRE’S LAW
APV is an annuity-certain for the number ofyears of insurance remaining divided by ω−x.All of the following formulas follow that.
Ax =aω−xω − x
Ax =aω−xω − x(
IA)x
=
(I a)ω−x
ω − x(IA)x =
(Ia)ω−xω − x
A1x:n =
an
ω − xA1x:n =
an
ω − x(IA)1x:n =
(I a)n
ω − x(IA) 1
x:n =(Ia)n
ω − x(DA
)1x:n =
(Da)n
ω − x(DA) 1
x:n =(Da)n
ω − x
PERCENTILES
Draw a pain curve.
For constant force:
Pr(SBP is sufficient) =(
µµ+δ
)µ/δ
CONTINUOUS VS. DISCRETE
Assuming UDD:
Ax = iδAx
A1x:n = i
δA1x:n
Ax:n = iδA1x:n + nEx
A(m)x = i
i(m)Ax(IA)x
= iδ
(IA)x(IA)1x:n = i
δ(IA) 1
x:n(DA
)1x:n = i
δ(DA) 1
x:n(IA)x
= iδ
(IA)x − iδ
(1d− 1δ
)Ax
Assuming death occurs in the middle of theyear:
Ax = (1 + i)1/2 ·Ax
A(m)x = (1 + i)(m−1)/2m ·Ax
c©2012 The Infinite Actuary
D. Life Annuities
WHOLE LIFE ANNUITY
Twin – Whole Life Insurance
Yx = aTx=
1− vTx
δ
ax =1− Axδ
ax =
∫ ∞0
tEx dt
Ax = 1− δax
Var[Yx]
=2Ax −
(Ax)2
δ2
Yx = aKx+1 =1− vKx+1
d
ax =1−Axd
ax =
∞∑k=0
kEx
Ax = 1− dax
Var[Yx]
=2Ax − (Ax)2
d2
Yx = aKx= aKx+1 − 1⇒ ax = ax − 1
N-YEAR TEMPORARY ANNUITY
Twin – N-Year Endowment Insurance
Yx:n =
{aTx
0 ≤ Tx ≤ nan n < Tx
=1− Zx:n
δ
ax:n =1− Ax:n
δax:n =
∫ n
0tEx dt
Ax:n = 1− δax:n
Var[Yx:n
]=
2Ax:n −(Ax:n
)2δ2
A1x:n = 1− δax:n − nEx
Yx:n =1− Zx:n
d
ax:n =1−Ax:n
dax:n =
n−1∑k=0
kEx
ax:n = ax − nEx ax+n
Var[Yx:n
]=
2Ax:n − (Ax:n )2
d2
ax:n = ax:n − 1 + nEx
N-YEAR DEFERRED WLA
No Twin
n|Yx =
{0 0 ≤ Tx ≤ naTx
− an n < Tx
n|ax = ax − ax:n
n|ax = nEx · ax+n
n|ax = nEx · ax+n
DEFERRED TEMPORARY ANNUITY
No twin.
n|max =
∫ n+m
nvt tpx dt
= ax:n+m − ax:n
= n|ax − n+m|ax
= nEx · ax+n:m
For annuity-due same 3 formulas as above butreplace a with a.
N-YR CERTAIN AND LIFE AFTER
Yx:n =
{an 0 ≤ Tx ≤ naTx
n < Tx
ax:n = an + n|ax
ax:n = an + n|ax
INCREASING WHOLE LIFE(IY)x
=(I a)Tx
Tx ≥ 0(I a)x
=ax −
(IA)x
δ(IA)x
= ax − δ(I a)x
(Iax)− (Ia)x = ax
VARYING TEMPORARY(I a)x:n
=
∫ n
0tvt tpx dt(
Da)x:n
=
∫ n
0(n− t)vt tpx dt(
I a)x:n
+(Da)x:n
= nax:n
(Ia)x:n + (Da)x:n = (n+ 1)ax:n
RECURSION
ax = 1 + vpx ax+1
ax = vpx ax+1
ax = ax:1 + vpx ax+1
ax:n = ax:1 + vpx ax+1:n−1
ax:n = 1 + vpx ax+1:n−1
(Ia)x = 1 + vpx (ax+1 + (Ia)x+1)
(Ia)x = ax + vpx (Ia)x+1
(Ia)x:n = 1 + vpx(ax+1:n−1 + (Ia)x+1:n−1
)(Ia)x:n = ax:n + vpx (Ia)x+1:n−1
(Da)x:n = n+ vpx (Da)x+1:n−1
ADJUSTED FORCE OF MORTALITY
If µ∗x+t = µx+t + c and δ∗ = δ − c, then
nE∗x = nEx
a∗x = ax
ACCUMULATED APV
sx:n =ax:n
nExsx:n =
ax:n
nExsx:n =
ax:n
nEx
CONSTANT FORCE
ax =1
µ+ δax =
1
vq + d
ax:n = ax (1− nEx) ax:n = ax (1− nEx)(I a)x
= (ax)2 (Ia)x = (ax)2
Pr(SBP is insufficient) =
(µ
µ+ δ
)µ/δ
DE MOIVRE’S LAW
Use the most important identity.
ANNUITIES PAYABLE M-THLY
Always true:
A(m)x = 1− d(m)a
(m)x
A(m)x:n = 1− d(m)a
(m)x:n
Assuming UDD:
symbol with (m) = α(m)· symbol w/o (m)
−β(m) (“start” - “end”)
a(m)x = α(m) ax − β(m) (0Ex −∞Ex)
a(m)x:n = α(m) ax:n − β(m) (0Ex − nEx)
n|a(m)x = α(m) n|ax − β(m) (nEx −∞Ex)
α(m) ≈ 1 β(m) ≈ m−12m
Woolhouse (3 terms)
a(m)x ≈ ax − m−1
2m− m2−1
12m2 (µx + δ)
−for other annuities, write them in terms ofa whole life annuity
c©2012 The Infinite Actuary
E. Premium Calculation
LOSS AT ISSUE
L ∼ loss at issue random variable
L = PV of benefits− PV of premiums
E[L] = APVFB−APVFP
EQUIVALENCE PRINCIPLE
E[L] = 0 ⇒ APVFB = APVFP
benefit premium =APV benefit
APV annuity
Whole Life and Endowment Insurance:
Var[L] =Var[Z]
(1− SBP)2
KEY PREMIUM IDENTITIES
Whole Life and Endowment Insurance Only:
P(Ax)
=Ax
axPx:n =
Ax:n
ax:n
P(Ax)
=1
ax− δ Px:n =
1
ax:n− d
P(Ax)
=δAx
1− AxPx:n =
dAx:n
1−Ax:n
ax =1
P(Ax)
+ δax:n =
1
Px:n + d
Ax =P(Ax)
P(Ax)
+ δAx:n =
Px:n
Px:n + d
WHOLE LIFE
Fully Continuous:
E[L] = SAx − P ax
Var[L] =(S + P
δ
)2 (2Ax −
(Ax)2)
Fully Discrete:
E[L] = SAx − P ax
Var[L] =(S + P
d
)2 (2Ax − (Ax)2
)
ENDOWMENT INSURANCE
Fully Continuous:
E[L] = SAx:n − P ax:n
Var[L] =(S + P
δ
)2 (2Ax:n −
(Ax:n
)2)Fully Discrete:
E[L] = SAx:n − P ax:n
Var[L] =(S + P
d
)2 (2Ax:n − (Ax:n )2
)
N-YEAR TERM INSURANCE
P(A1x:n
)=A1x:n
ax:n
P(A1x:n
)= P
(Ax:n
)− P
(A 1x:n
)P 1x:n =
A1x:n
ax:n
N-PAY WHOLE LIFE
nP(Ax)
=Ax
ax:n
nPx =Ax
ax:n
N-YEAR DEFERRED WLA
P(n|ax
)=
n|ax
ax:n
P(n|ax
)=
n|ax
ax:n
COMPARE VARIANCE
Two identical policies (whole life or endow-ment insurance)
P1 → L1
P2 → L2
Var[L2]
Var[L1]=
(P2 + d
P1 + d
)2
SEMICONTINUOUS INSURANCE
Benefit paid at the moment of death and pre-miums paid at beginning of the year.
P(Ax)
=Ax
ax
P(A1x:n
)=A1x:n
ax:n
P(Ax:n
)=Ax:n
ax:n
PERCENTILE PREMIUMS
Group of policies:
E[S] + Φ−1(1− x%)√
Var[S] = 0
Single policy:
Find the policy you must “win” on and thensolve for the premium it will take to win onthat death.
FRACTIONAL PREMIUMS
Premiums paid m-thly instead of annually.
P(m)x =
Ax
a(m)x
P(m)x:n =
Ax:n
a(m)x:n
nP(m)x =
Ax
a(m)x:n
(P-P) / P
nPx − P 1x:n
P 1x:n
= Ax+n
Px:n − nPx
P 1x:n
= 1−Ax+n
Px:n − P 1x:n
P 1x:n
= 1
GROSS PREMIUMS
Lg = PV of future benefits
+ PV of future expenses
− PV of future premiums
E[Lg ] = APVFB + APVFE−APVFP
New Equivalence Premium:
E[Lg ] = 0 ⇒ APVFB + APVFE = APVFP
G =APV of Benefits + APV of Expenses
APV Annuity
Var[Lg ] =(S + E + G−er
d
)2Var(Z)
CONSTANT FORCE
P(Ax)
= µ Px = vq
P(A1x:n
)= µ P 1
x:n = vq
Var[L] = 2Ax Var[L] = p · 2Ax
Fully Continuous Whole Life:
Pr(benefit prem is sufficient) =(
µµ+δ
)µ/δ
DE MOIVRE’S LAW
Use key premium identities and most impor-tant identity.
c©2012 The Infinite Actuary
F. Reserves
#1 PROSPECTIVE METHOD
All Policies and All Premiums. All other meth-ods only work when premiums are determinedusing the equivalence principle.
Reserve = APVFB−APVFP
Example – fully continuous whole life:
tV(Ax)
= Ax+t − P(Ax)ax+t
#2 RETROSPECTIVE METHOD
All Policies under EP.
Reserve = AAVPP−AAVPB
Example – fully discrete endowment ins.:
tVx:n = Px:n sx:t − tkx
ACCUMULATED COST OF INS
tkx =A1x:t
tEx
Thinking in terms of life tables:
tkx =dx(1+i)t−1+dx+1(1+i)t−2+···+dx+t−1
lx+t
For one year:
1kx = qxpx
#3 DIFFERENCE IN PREMIUM
Whole Life, Endowment Insurance, Term In-surance and Limited Pay Whole Life Only
Reserve = accumulated difference in premiumyou want to charge and premiumyou are actually charging
Example – fully continuous whole life:
tV(Ax)
=[P(Ax+t
)− P
(Ax)]ax+t
#4 PAID UP
Whole Life, Endowment Insurance, Term In-surance and Limited Pay Whole Life Only
Reserve = APV of the future benefit timesthe percentage of the future ben-efit that you are not funding withfuture premiums
Example – full continuous whole life:
tV(Ax)
= Ax+t
[1−
P(Ax)
P(Ax+t
)]
#5 ANNUITY FORM
Whole Life and Endowment Insurance Only
Reserve = 1 minus (the annuity at the dot di-vided by the annuity at issue)
Example – fully discrete endowment ins.:
tVx:n = 1−ax+t:n−tax:n
#6 LIFE INSURANCE FORM
Whole Life and Endowment Insurance Only
Reserve = insurance at the dot minus the in-surance at issue divided by one mi-nus the insurance at issue
Example – fully discrete whole life:
tVx =Ax+t −Ax
1−Ax
#7 BENEFIT PREMIUM FORM
Whole Life and Endowment Insurance Only
Reserve = premium at the dot minus the pre-mium at issue divided by the pre-mium at the dot plus delta (or d)
Example – fully discrete whole life:
tVx =Px+t − PxPx+t + d
#8 (P-P) / P
Accumulated Difference in Premiums equalsDifference in Reserves
Remember 1
P 1x:n
= sx:n
Example
Px:n − PxP 1x:n
= nVx:n − nVx = 1− nVx
#9 ANNUITY FORM SPINOFF
Whole Life and Endowment Insurance Only
Example – fully discrete whole life:
a+bVx = 1− (1− aVx) (1− bVx+a)
#10 RESERVE RECURSION
Start with the terminal reserve from the pre-vious year, collect the premium and put thosein the bank. If the policyholder dies pay thedeath benefit. If the policyholder lives setupthe next reserve.
Example - fully discrete 20-year endowment in-surance on (40):(10V40:20 + P40:20
)(1+i) = q50+p50 11V40:20
NAR version: = q50
(1− 11V40:20
)+ 11V40:20
#11 RESERVE CREATION FORMULA
Useful when death benefit is a function of thereserve.
Each premium accounts for two items:
1. cost of providing ensuing year’s DB
2. rest is for reserve creation
tV = P st
−t−1∑h=0
vqx+h (bh+1 − h+1V ) (1 + i)t−h
DB = Reserve: tV = P st
DB = 1 + Reserve:
tV = P st −t−1∑h=0
qx+h(1 + i)t−h−1
DB = 1 + Reserve and qx+h = q :
tV = (P − vq) st
VARIANCE LOSS
For Whole Life and Endowment Insurance:
Var(tL) =(S + P
δ
)2Var(vU )
EP ⇒ =Var(vU )
(1− SBP)2
Example – fully discrete endowment ins.:
Var(tL) =(S + P
d
)2(
2Ax+t:n−t −(Ax+t:n−t
)2)
EP ⇒ =
2Ax+t:n−t −(Ax+t:n−t
)2
(1−Ax:n )2
CONSTANT FORCE
Under EP, level premium whole life and terminsurance reserves are 0.
c©2012 The Infinite Actuary
GROSS PREMIUM RESERVES
To find the gross premium, G, solve new EP
APVFP = APVFB + APVFE
Expense Reserves:
kVe = APVFE−APVFL
kVe = AAVPL−AAVPE
Gross Premium Reserve:
kVg = kV
n + kVe
Can also use recursion for expense and grosspremium reserves.
SEMI-CONTINUOUS RESERVES
Assuming UDD
tV(Ax)
= iδ tVx
tV(A1x:n
)= i
δ tV 1x:n
tV(Ax:n
)= i
δ tV 1x:n + tV 1
x:n
INTERIM BENEFIT RESERVES
Exact Method:
(hV + πh) (1 + i)s = sqx+h · v1−s
+spx+h · h+sV
Approximation: use linear interpolation be-tween initial reserve and the terminal reserve.
THIELE’S DIFFERENTIAL EQN
ddt t
V = δt ·tV +Gt−et−(St + Et − tV )µ[x]+t
tV ≈t+hV − h
(Gt − et − (St + Et)µ[x]+t
)1 + h
(µ[x]+t + δt
)
MODIFIED RESERVES
α− lower modified prem in the first year
β−higher modified prem in the renewal years
FPT Reserves:
α = A1x:1
β = benefit premium for insurance issuedto (x+ 1)
1Vm = 0
tVm = (t− 1) reserve for insurance issuedto (x+ 1)
GAIN
Gain = Actual Profit− Expected Profit
Profit = (tV +Gt − et) (1 + i)
−qx+t (St + Et)− px+t · t+1V
Analysis of Surplus:
expenses: (et − e′t) (1 + i) + (Et − E′t) qx+t
interest: (i′ − i) (tV +Gt − e′t)
mortality:(qx+t − q′x+t
)(St + E′t − t+1V )
Order matters. Use actual experience if youhave already accounted for that source.
POLICY ALTERATIONS
Reduced Paid-Up:
RPU = tCVxAx+t
Extended Term (solve for n):
tCVx = A 1x+t:n
For endowment insurance if the cash value islarge enough to cover the remaining term, thensolve for reduced pure endowment benefit, PE
tCVx = A 1x+t:n−t + PE · n−tEx+t
c©2012 The Infinite Actuary
G. Markov Chains
NOTATION
tpijx − probability that a subject in state i at
time x will be in state j (where j mayequal i) at time x+ t
tpiix − probability that a subject in state i attime x stays in state i continuously un-til time x+ t
tpiix ≤ tpiix
DISCRETE MARKOV CHAINS
P(t) is the transition matrix at time t
tpijx = ij entry of P(x) ·P(x+1) · · ·P(x+t−1)
tpiix = piix · piix+1 · · · piix+t−1
CONTINUOUS MARKOV CHAINS
µijx = limh→0
hpijx
hfor i 6= j
tpiix = exp(−∫ t0 µ
i•x+s ds
)Probability of transition from one state to an-other depends on the model.
PERMANENT DISABILITY MODEL
Healthy0
Disabled1
Dead2
µ01x
µ02x µ12x
tp01x =∫ t0 sp
00x µ01x+s t−sp
11x+s ds
MULTIPLE DECREMENT MODEL
active0
exit1
exit2
exitm
tp0jx =
µ0jxµ0•x
(1− tp00x
)tp00x = exp
(−∫ t0 µ
0•x+s ds
)
KOLMOGOROV’S FORWARD EQNS
ddt t
pijx =∑
k=0,k 6=j
(tpikx µ
kjx+t − tp
ijx µ
jkx+t
)ddt t
pijx =prob “move
into j”at time x+ t
−prob “move
out of j”at time x+ t
Euler’s method turns a continuous MarkovChain into a discrete chain with time incre-ments of h and transition “probabilities”
hpijx+t =
{hµijx+t i 6= j
1− hµi•x+t i = j
PREMIUMS
benefit premium =APV of benefit
APV of annuity
ANNUITIES
Pay 1 per year continuously while in state j:
aijx =
∫ ∞0
e−δttpijx dt
If payable at the start of the year:
aijx =
∞∑k=0
vk kpijx
INSURANCE
Pay 1 on each future transfer into k:
Aikx =
∫ ∞0
e−δt∑j 6=k
tpijx µ
jkx+t dt
CONSTANT FORCE AND NO RE-ENTRY
If all transition forces are constant and no re-entry into a state
aiix =1
µi• + δ
For the multiple decrement model with con-stant transitions:
A0jx =
µ0j
µ0• + δ
RESERVES
tV (i)− reserve at duration t for a subject instate i at that time
B(i)t − rate of payment of benefit while the
policyholder is in state i
S(ij)t − lump sum benefit payable instanta-
neously at time t on transition fromstate i to state j
THIELE’S DIFFERENTIAL EQN
ddt t
V (i) = δt tV (i) −B(i)t
−n∑
j=0,j 6=iµijx+t
(S(ij)t + tV
(j) − tV(i))
Euler’s method:
t−hV(i) = tV (i) (1− δt h) + hB
(i)t
+hn∑
j=0,j 6=iµijx+t
(S(ij)t + tV
(j) − tV(i))
c©2012 The Infinite Actuary
H. Multiple Decrement Models
MULTIPLE DECREMENT MODEL
active0
exit1
exit2
exitm
q(j)x − prob. of decrement due to cause #j
q(τ)x − prob. of decrement due to any cause
tq(τ)x =
m∑j=1
tq(j)x
A(j)x =
∞∑k=1
vkk−1p(τ)x qjx+k−1
FORCES OF DECREMENT - 1
active0
exit1
exit2
exitm
µ01x+t
µ02x+t
µ0mx+t
0 is redundant so we use:
µ0jx+t = µ(j)x+t
FORCE OF DECREMENT - 2
µ(τ)x+t =
ddt t
q(τ)x
tp(τ)x
µ(j)x+t =
ddt t
q(j)x
tp(τ)x
µ(τ)x+t = µ
(1)x+t + µ
(2)x+t + · · ·µ(m)
x+t
tp(τ)x = exp
(−∫ t0 µ
(τ)x+s ds
)tq
(j)x =
∫ t0 sp
(τ)x · µ(j)x+s ds
Pr(J = j |T = t) =µ(j)x+t
µ(τ)x+t
If all forces are a constant multiple of the totalforce for all t
tq(j)x = µ
(j)x
µ(τ)x
(1− tp
(τ)x
)A
(j)x =
∫∞0 vt tp
(τ)x µ
(j)x+t dt
CONSTANT TRANSITION FORCES
tq(j)x = µ(j)
µ(τ)
(1− tp
(τ)x
)tq
(j)x = tq
(j)x
tq(τ)x
(1−
(p(τ)x
)t)A
(j)x =
µ(j)
µ(τ) + δ
ax =1
µ(τ) + δ
A(j)1x:n
= µ(j)
µ(τ)+δ
(1− nE
(τ)x
)
ABSOLUTE RATE OF DECREMENT
tq′(j)x = 1− tp′
(j)x = 1− exp
(−∫ t0 µ
(j)x+s ds
)tp
(τ)x = tp′
(1)x · tp′
(2)x · · · tp′
(m)x
ABSOLUTE RATE OF DECREMENT
Each cause is independent of the other causes.
Probabilities of Decrement
alive0
cause1
cause2
cause3
q(1)x
q(2)x
q(3)x
Rates of Decrement
alive0
cause1
alive0
cause2
alive0
cause3
q′(1)x
q′(2)x
q′(3)x
Notice how in the rate of decrement “worlds”there is only one cause of decrement. For therates of decrement we have 3 tables and for themultiple decrement “world” we have one tablewith 3 competing decrements.
tq′(j)x ≥ tq
(j)x
MDT vs. AST
Super secret approximation:
q′(j)x =d(j)x
`(τ)x − 1
2
∑i6=j
d(i)x
Constant Force and UDDMDT:
p′(j)x =(p(τ)x
)q(j)x /q(τ)x
UDDAST:
For 2 UDDASTs use the midpoint:
q(1)x = q′(1)x
(1− 1
2q′(2)x
)For 3 or more UDDASTs use integration:
tp′(j)x · µ
(j)x+t = q′(j)x - factor out of integral
For 3 UDDASTs (if you like memorizing):
q(1)x = q′(1)x
(1− 1
2
(q′(2)x + q′(3)x
)+ 1
3q′(2)x q′(3)x
)Hybrids:
Draw a picture. For 2 competing UDDASTsuse the midpoint of the interval.
ASSET SHARE
Recursion:
[kAS +G(1− ck)− ek] (1 + i)
= q(d)x+k + q
(w)x+k · k+1CV + p
(τ)x+k · k+1AS
n-th Asset Share using Asset Share Creation:
n−1∑k=0
G(1− ck)− ek − vq(d)x+k − vq
(w)x+k · k+1CV
n−kE(τ)x+k
For additional premium of ∆G:
∆AS =
n−1∑k=0
∆G(1− ck)
n−kE(τ)x+k
=
n−1∑k=0
∆G(1− ck)kp(τ)x vk
nE(τ)x
c©2012 The Infinite Actuary
I. Multiple Life Models
JOINT-LIFE STATUS
Txy = min[Tx, Ty ]
Failure on the first death.
tqxy − probability first death of (x) and (y)occurs within t years
tpxy − probability that (x) will attain x + tand (y) will attain y + t
tpxy = tpx · tpy (work with p’s)
tqxy = 1 − tpxy
fxy(t) = tpxy · µx+t:y+t
µx+t:y+t = µx+t + µy+t
nqxy =
∫ n
0tpxy µx+t:y+t dt
◦exy =
∫ ∞0
tpxy dt
exy =∞∑k=1
kpxy
exy = pxy (1 + ex+1:y+1)
axy =
∫ ∞0
vt tpxy dt
Axy =
∫ ∞0
vt tpxy µx+t:y+t dt
Axy = 1 − δaxy
Pxy = 1axy
− d
LAST-SURVIVOR STATUS
Txy = max[Tx, Ty ]
Txy = Tx + Ty − Txy
that for (xy) = that for (x) + that for (y)
−that for (xy)
tqxy − probability that last death happenswithin t years
tqxy = tqx · tqy (work the the q’s)
tpxy − probability at least one of (x) or (y)survives for t years
tpxy = 1 − tqxy
tpxy = tpx + tpy − tpxy
fxy(t) = tpx µx+t + tpy µy+t − tpxy µx+t:y+t
n|qxy = n+1qxy − nqxy
n|qxy = n|qx + n|qx − n|qxy
◦exy =
◦ex +
◦ey − ◦exy
axy = ax + ay − axy
Axy = Ax + Ay − Axy
A 1xy:n = A1
x:n +A1y:n −A 1
xy:n
EXACTLY ONE STATUS
tp[1]xy − probability exactly 1 of (x) and (y) live
t years
tp[1]xy = tpx + tpy − 2 tpxy
a[1]xy = ax + ay − 2axy
CONTINGENT PROBABILITIES
nq1xy − probability (x) dies before (y) and be-fore n years from now
nq1xy =
∫ n
0
(∫ ∞s
fxy(s, t) dt
)ds
nq 2xy − probability that (y) dies after (x) and
before n years from now
If (x) and (y) independent:
nq1xy =
∫ n
0tpx µx+t · tpy dt
nq 2xy =
∫ n
0tpy µy+t · tqx dt
nqy = nq 1xy + nq 2
xy
nq1xy = nq 2xy + nqx · npy
Probability (x) dies more than n years afterdeath of (y):
npx ·∞q 2x+n:y
CONTINGENT INSURANCE
A1xy =
∫ ∞0
vt · tpx µx+t · tpy dt
A 2xy =
∫ ∞0
vt · tpy µy+t · tqx dt
A 2xy =
∫ ∞0
vt · tpx µx+t · tpy Ay+t dt
Ay = A 1xy + A 2
xy
SBP for a payment of 1 at the death of (x) ifhe dies more than n years after the death of(y):
nEx · A 2x+n:y
REVERSIONARY ANNUITY
(y) gets money after (x) dies:
ax|y = ay − axy
Other versions:
ax|y:n = ay:n − axy:n
ax:n |y = ay − axy:n
In other words:
au|v = av − auv
COMMON SHOCK
Find the total force for each status:
total force on (x) = µ∗x + λ
total force on (y) = µ∗y + λ
total force on (xy) = µ∗x + µ∗y + λ
Pr[(x) dies first] =µ∗x
µ∗x+µ∗
y+λ
Pr[(y) dies first] =µ∗y
µ∗x+µ∗
y+λ
Pr[Tx = Ty ] = λµ∗x+µ∗
y+λ
Ax =µ∗x+λ
µ∗x+λ+δ
Ay =µ∗y+λ
µ∗y+λ+δ
CONSTANT FORCE
Axy =µx+µy
µx+µy+δ
axy = 1µx+µy+δ
nq1xy = µxµx+µy
nqxy
A1xy = µx
µx+µy+δ
c©2012 The Infinite Actuary
DE MOIVRE’S LAW
◦exx = ω−x
3
◦exy = y−xpx·
◦eyy +y−xqx·
◦ey
nq 2xy = 1
2 nqxy
A1xy =
ay:ω−x
ω − x
c©2012 The Infinite Actuary
J. Other Topics
PENSION MATHEMATICS
Replacement ratio:
R =pension income in year after retirement
salary in the year before retirement
Salary Scale:
sy
sx=
salary received in year of age y to y + 1
salary received in year of age x tp x+ 1
INTEREST RATE RISK
v(t) − current market price on a t year zero-coupon bond that pays $1 at time t.
− think of it as the discount function.
yt − t year spot rate of interest
v(t) =1
(1 + yt)t
f(t, t+k)− forward rate from time t to t+ k
(1 + f(t, t+ k))k =v(t)
v(t+ k)
A(x)y =
∫ ∞0
v(t) tpx µx+t dt
a(x)y =
∞∑k=0
kpx v(k)
A risk is diversifiable if:
limN→∞
√Var[
∑Ni=1Xi]
N= 0
ANNUAL PROFIT
The profit for year 0 is the negative of the ex-penses incurred at time 0.
Annual Profit =
Previous Rsv accumulated with interest
+ Premium Collected
− Expenses Incurred
+ Interest Earned on Premium less Expenses
− Expected Cost of Benefits
− Expected Cost of Ending Rsv
PROFIT SIGNATURE
Πt =
{Profit(0) t = 0
t−1px · Profit(t) t > 0
where t−1px is the probability that policy is inforce at beginning of year t.
INTERNAL RATE OF RETURN
Solve for j such that:n∑
t=0
Πt vtj = 0
PROFIT MARGIN
Profit Margin =NPV
P ax:n
DISCOUNTED PAYBACK PERIOD
Find the smallest m such that:m∑t=0
Πt vtr ≥ 0
UNIVERSAL LIFE - Notation
FA− face amount
AVt − account value at end of year t
CVt − cash value at end of year t
DBt − death benefit for year t
ADBt − additional death benefit for year t
CoIt − cost of insurance for year t
SCt − surrender charge for year t
ict − credited interest rate in year t
It − credited interest in year t
vq − discount factor used in CoI calc
q∗x+t−1−mort. rate used in CoI calc for year t
ECt − expense charge for year t
CVt = max[AVt − SCt, 0]
UNIVERSAL LIFE - COI
DBt = AVt +ADBt
Type A:
ADBt = FA−AVt
DBt = FA
Type B:
ADBt = FA
DBt = FA+AVt
CoIt = ADBt A 1x+t−1:1
= ADBt · vq q∗x+t−1
Type A:
CoIAt =
[FA−(AVt−1+Pt−ECt−CoIAt )(1+ict )]vq q∗x+t−1
CoIAt =[FA−(AVt−1+Pt−ECt)(1+ict )]vq q∗x+t−1
1−vq q∗x+t−1(1+ict )
Type B:
CoIBt = FA · vq q∗x+t−1
UNIVERSAL LIFE - Roll Forward
Account Value Roll Forward:
Starting AV (AVt−1)
+ Premium (Pt)
− Expense Charge (ECt)
− Mortality Charge (CoIt)
+ Credited Interest (It)
= Ending Account Value (AVt)
AV At =
(AVt−1+Pt−ECt−FAvq q∗x+t−1)(1+ict )
1−vq q∗x+t−1(1+ict )
AV Bt = same as numerator for AV A
t
UNIVERSAL LIFE - Annual Profit
The profit for year 0 is the negative of the ex-penses incurred at time 0.
Annual Profit =
Previous Rsv (usually AV)
+ Premium Collected
− Expenses Incurred
+ Interest Earned on
Prev Rsv plus Premium less Expenses
− Expected Cost of Benefits
− Expected Surrender Benefits
− Expected Cost of Ending Rsv
CORRIDOR FACTORS
To qualify as life insurance the death benefitmust be at least a certain multiple (γt for yeart) of the account value.
ADBct = (γt − 1)AVt
ADBft = FA−AVt
ADBt = max[ADBct , ADB
ft ]
CoIt = ADBt · vq q∗x+t−1
c©2012 The Infinite Actuary