Essay InstructionThis is going to be 4 Short Essays.Essays sho.docx

Essay Instruction
This is going to be 4 Short Essays.
Essays should be 500-750 words. For each essay, you must cite
at least two readings from the text book that the essay topic
addresses. I POSTED A LIST OF ALL POSSIBLE READINGS
TO CHOOSE FROM.
The textbook is “Introduction to women's gender and sexuality
studies by Ayu saraswati, Barbara Shaw, and Heather Rellihan”
The prompts are:
1. Understanding Privilege and Oppression in GWS Essay
How do systems of privilege and oppression function in our
society? How do we combat these systems? Why is it important
to recognize patriarchy as a system and not an individual
identity?
2. Historical Perspectives in GWS Essay
How has the feminist movement changed, morphed, and grown
between the nineteenth and twenty-first century? Why are these
changes important?
3. Science, Technology, and the Digital World Essay
How do racism, sexism, ableism, classism, and heterosexism
intersect with science, technology, and the digital world?
4. Activism Essay
What is the role of art in creating change? What art forms are
most effective for creating change? Why?

AS3429 Ch.6 Lecture notes (W2018)
1
Ch. 6 Premium Calculations
6.1 Summary
6.2 Preliminaries
6.3 Assumptions
6.4 Present Value of future loss random variable
6.5 Equivalence Principle
6.6 Gross Premium Calculation
6.7 Profit
6.8 Portfolio Percentile Premium Principle
6.9 Extra Risks
Note: additions//updates will be made to posted lecture notes
as we work through the material

2
6.1 Summary
• Chapter focus is on how to determine premium calculations
for
life insurance and life annuities, with a greater focus on life
insurance
• Two different premium principles considered
o Equivalence principle (most common for traditional policies)
o Portfolio percentile principle
6.2 Preliminaries
(i) Premium Types
(a) Net Premium
o considers benefit costs only, excludes expenses (and profit)
(b) Gross Premium

o explicitly allows for expenses
3
6.2 Preliminaries(continued)
(ii) Premium Frequency
Life Insurance
o Typically periodic premium payments(eg annually, monthly,
bi-weekly)
o Periodic premiums are usually a level $ amount but they can
be
varying (e.g. step rate)
Life Annuities
o Whole/temporary life annuities usually single premium
purchase
o Deferred life annuities may be purchased with single premium
or
periodic premiums during deferred period (pension funding
applications)
(iii) Premium Timing and duration
• Premiums are payable in advance with first premium payable
when

policy is purchased
• Periodic premiums can be paid for the life or term of an
Insurance
policy or shorter(e.g. Limited Pay Whole Life), and cease to be
payable
on death of the insured
4
6.3 Assumptions and Table sources
• The Standard Select Survival Model(SSSM)’ is default model
for
text examples and exercises. The model assumes i=5% and;
µx =A+Bc
x
, A=0.00022, B=2.7x10
-6
, C=1.124
µ[x]+s = 0.9
2-s
µx+s for 0 ≤ s ≤ 2

l20=100,000(radix)
• as noted previously, copy of Appendix D(SSSM) is posted
- it also includes several values(e.g. select Insurance factors,
pure
endowment factors, second moments for whole life insurance)
- SOA LTAM tables now used are ultimate part of SSSM (copy
posted)
• Standard Select Survival model(SSSM) Excel Worksheets
- required for some text questions(as well as to understand some
text examples)
- recommend you set up own SSSM worksheet for your own use
and/or use the
worksheet posted on OWL
- author spreadsheet(SSSM) referenced earlier (Life Cons
review notes) includes
several more factors than Table D
• ILT (copy posted)& other tables may be used for some class
examples
5
6.4 P.V. of the Future Loss Random Variable

•••• Cash flows associated with life insurance/annuity policies
are
generally life contingent
o This includes expenses and premiums as well as benefits
•••• Can model (future outgo-future income) with the random
variable
that represents the present value(P.V.) of future loss
o Net Future loss- excludes expenses
o Gross Future loss- includes expenses
o Future loss random variable depends on time of death
Future Loss Random Variable Notation:
0
nL
= PV benefit outgo – PV Net premium income
0
gL
= PV benefit outgo + PV expenses – PV Gross premium
income
6

Example 1(6.4)
A whole life insurance policy is issued to [45]. The sum
insured is
$25,000 payable immediately on death. Interest is at i =5%. A
premium of $500 is paid at the beginning of each year for the
policy.
The policyholder dies at the end 25.8 years. Calculate the
future loss
at issue.
7
Example 2
insured is
$200,000 payable immediately on death. Premiums of amount P
are
payable annually in advance, ‘ceasing at age 65 or on earlier of
death’
(20 Pay Whole Life Policy).What is the future loss random
variable for this policy?

•••• Distribution of future loss random variable used to
determine
Premiums for given benefit/benefit for given premiums, using a
Premium principle
o Equivalence Principle (“benchmark principle”)
o Portfolio Percentile Principle
o Cash flow Profit Criterion Method (Ch.12)
8
•••• Under the equivalence principle, premiums are determined
such
that Expected value of the future loss is zero at contract issue
•••• Equivalence principle most common method for traditional
insurances and is text default method for these policies (e.g
Ch.6,7)
(i) Net Premiums
•••• Ignore expenses
•••• Benefits include either and/or both of death benefit/survivor
benefits

•••• Equivalence principle means that E[L
n
0] =0 , and since
L
n
0 = PV benefit outgo – PV Net premium income, then under
equivalence principle EPVbenefit outgo= EPVnet premium
income
or PVFB = PVFP (at policy
issue)
9
6.5 Example 1 (SSSM is text default model)
Consider a 20 pay whole life insurance with annual premiums
issued
to [50]. The death benefit(sum insured) is $500,000 payable at
end of
year of death.
(a) What is an expression for net future loss random variable?

(b) Calculate the net annual premium using the SSSM
10
Example 2 (a few working pages are attached)
Consider an endowment insurance with term n years and sum
insured S payable at the earlier of the end of the year of death
or at
maturity, issued to a select life aged x. Premiums of amount P
are
payable annually throughout the term of the insurance (same as
saying
P’s are paid for the life of the policy).
(A) Derive expressions in terms of S, P and standard actuarial
functions for;
(i) The net future loss, Ln 0
(ii) The mean of Ln 0,
(iii) The variance of Ln 0
(B) Assume S=150,000; n=20, [x]=50 and SSSM
(iv) Calculate the annual premium (equivalence principle )
(v) Suppose now the premium is payable quarterly. What is
the quarterly premium? Assume UDD within each year of

age and you are provided factors for α(4) and β(4).
11
Example 2(working page)
12
13
Example 2–annuity factor calculation for B (v)
)4(
|20:]50[
a&&
=
)4(

]50[a&&
)4(
70]50[20 aE &&−−−−
)m(
|n:]x[
a&&
=
)m(
]x[a&&
)m(
n]x[]x[n aE +− &&
Under UDD : )m(Ba)m(a ]x[
)m(
]x[ −α≈
&&&&
or use: ]x[)m(
)m(
]x[ A
i

i
A ≈
in identity:
)m(
m
]x[)m(
]x[
d
)A1(
a
−
=&&
Also, can show under UDD;
≈
)m(
|n:]x[
a&& )E1)(m(Ba)m( ]x[n|n:]x[ −−α &&
)4(

|20:]50[
a&& 598403.12)E1)(4(Ba)4( ]50[20|20:]50[ =−−α≈ &&
14
Example 3 (Text 6.4)
An insurer issues a “regular premium deferred annuity contract”
to a select life aged x. Premiums are payable monthly
throughout the deferred period. The annuity benefit of X per
year is payable monthly in advance from age x+n for the
remainder of the life of (x).
(a) Write down the net future loss random variable in terms
of lifetime random variables for [x].
(b) Derive an expression for the monthly net premium.
(c) Assume now that, in addition, the contract offers a death
benefit of S
payable immediately on death during the deferred period. Write
the net future
loss random variable for the contract, and derive an expression
for the
monthly net premium

15
An insurer issues a regular premium deferred annuity contract
remainder of the life of x.
Ln0 = 0 – 12P |12/1K
)12(
a +&& for T[x] ≤ n or K[x]
(12)< n K ≡ K[x]
(12)
Ln0 = |n12/1K
)12(naXv −+&& – 12P |n
)12(
a&& for T[x] >n or K[x]
(12) ≥ n

benefit of S
the net future
for the
monthly net premium
16
17
Example 4
You are given an extract(see below) from a life table with a
four
year select period. A three year term insurance with death
benefit of $S is purchased by [41] with a net premium of $350
payable annually. Assume i=6% and that the death benefit is
payable at end of year of death.
Calculate
(i) S(sum insured) assuming the equivalence principle

(ii) standard deviation of L0
(iii) Pr [L0 >0]
[[[[x] ] ] ] l[[[[x]]]] l [[[[x]+]+]+]+1
l[[[[x]+2]+2]+2]+2 l[[[[x]+3]+3]+3]+3 lx++++4444
x++++4444
[40] 100,000 99,899 99,724 99,520 99,288 44
[41] 99,802 99,689 99,502 99,283 99,033 45
[42] 99,597 99,471 99,268 99,030 98,752 46
18
19
Example 4(working page
(ii) standard deviation of L0 or √Var[L0] calculations :
(using S=216,326.38 from (i))
k L0 Pr(K41=k)
0

49.731,203a350Sv
1
=− &&
q[41] = 0.0011322 =1— l[41]+1/ l[41]
1
52.849,191a350Sv
2
2
=− &&
1│q[41]= 0.0018737
2
11.640,180a350Sv
3
3
=− &&
2│q[41] = 0.0021943

≥3
69.991a3500
3
−=− && 3p[41] = 0.9947997
Var[L0]=E[L0
2
]–[E[L0]]
2
=E[L0
2
], when equivalence principle used (to determine P)
can show Var[L0] =188,537,738 � √Var[L0] = 13,731
Table above provides Lo values for given values of k. General
expression for Lo is;
Ln0 = Sv
K+1 – 350 |1Ka ++++&& for K[41]< 3 K ≡ K[41]
Ln0 = 0 – 350 |3a&& for K[41]
≥ 3

20
Example 5
A select life aged 45 buys a policy with a single premium(P).
The policy provides an annuity of $30,000 per year payable
annually in advance from age 65. In the event of death before
age 65, the premium is returned at the end of year of death.
Assume SSSM applies (used Appendix D for calculations)
(a) Give an expression for Ln0
(b) Calculate P assuming the equivalence principle
(c) Now assume annuity is guaranteed to be paid for at least 5
years if
insured survives to 65. How much does P increase by?
21
Example 5 continued

The policy
provides an annuity of $30,000 per year payable annually in
advance from
age 65. In the event of death before age 65, the premium is
returned at the
end of year of death. Assume SSSM applies ( Appendix D
used)
(a) Ln0 = Pv
K+1 – P for K[45] < 20 (K=K[45])
Ln0 = |201K
20 av000,30
−−−−++++
&& – P for K[45] ≥ 20
(c ) Now assume annuity is guaranteed to be paid for at least 5
years if insured survives to 65. How much does P increase by?
22

23
Example 6
A 20-pay $250,000 whole life insurance policy is purchased on
(45) where the premiums are payable monthly and the death
benefit is payable at moment of death.
Assume that i=6% and ILT mortality applies. Also assume
UDD
within each year of age. Calculate the monthly premium(P).
Answer: P = $384.34
24
Example 6 (working page)
25

Notation used for net annual premiums for fully discrete
Insurances (IAN)
x
x
x
a
A
P
&&
====
|n:x
x
xn
a
A
P
&&
====

|n:x
|n:x
|n:x a
A
P
1
1
&&
====
,
|m:x
|n:x
|n:x
m
a
A
P
1
1
&&
====

|n:x
a&&
1
|n:x1
|n:x
A
P ====
|n:x
a&&
|n:x
|n:x
A
P ====
|m:x
|n:x
|n:x
m

a
A
P
&&
====
Default assumption: Periodic Premiums are payable for the life
of the policy
Terminology
“Fully discrete” : both premiums at benefits payable at
discrete time points
“semi-continous” : DB paid at moment of death, but
Premiums paid at discrete points
“Fully continuous” : premiums paid continuosly, DB paid at
moment of death
AS3429/9429a Ch.7 Lecture notes (W2018)
22
(iii) Recursive Formulas- 1st revisit previous examples-working
page

(a) Example 1 revisited:
• 20 year endowment policy, no expenses
• premium basis = policy basis
• P=15,114.33
• showed 10V=190,339 and 11V=214,757
• work with tV to generate a recursive expression
−=
−+ |:[50]
500000A
t20tt
V
|:[50]
aP
t20t −+
&&

23
Recursive Formulas- 1st revisit previous examples-working
page
(c) Example 3 revisited
• also 20 year endowment policy, but Premiums payable for at
most 10
years, and P was given (P=5200), and there were expenses
• showed 5V=29,067.11 and 6V=35,324.45
24
(c ) Example 4 (Text 7.4) REVISITED
A man aged 50 purchases a deferred annuity policy. Annual
payments will be
payable for life, with the first payment on his 60th birthday.
Each annuity payment
will be $10,000. Level premiums of $11,900 are payable
annually for at most 10
years. On death before age 60, all premiums paid will be
returned, without interest,

at the end of the year of death.
The following basis is used for policy value calculations:
Survival model: Standard Select Survival Model
Interest : 5% per year
Expenses : 10% of the first premium, 5% of subsequent
premiums, $25 each time an annuity payment
is paid, and $100 when a death claim is paid
Expression for 5V in terms of 6V?
5V expression(Ex.4) is below, use this to get expression for 6V
|:55|:5560|:55|:55
AAA
5
1
5555
1
5
1
55
aP95.100aE025,10)I(PP5V &&&& −−−−++++++++++++====

25
Example 4 REVISITED (working page)
from example 4:
|:55|:5560|:55|:55
AAA
5
1
5555
1
5
1
55
aP95.100aE025,10)I(PP5V &&&& −−−−++++++++++++====
Similarly
|:56|:5660|:56|:56
AAA

4
1
4564
1
4
1
46
aP95.100aE025,10)I(PP6V &&&& −−−−++++++++++++====
26
(iii) Policy Value Recursive Formulas
•••• look at formulas that express tV in terms of t+1V
•••• revisited some previous examples to illustrate idea and
work
towards text general formula
o Ex.1 for recursive formula when no expenses

o Ex.3 for recursive formula (expenses)
o Ex.4 for recursive formula when Death Benefit varies(plus
there
are expenses)
•••• there are some cases where policy values are easier to
calculate
using a recursive formula(Example 7.7 text)
27
Recursive Formula-General Case
•••• Consider a policy issued to (x) where all cash flows can
occur only at the start or end of a year. Suppose this policy
has been in force for t years, where t is a non-negative integer.
Consider (t+1)st year, and let
P t ≡ the premium payable at time t
et ≡ the premium-related expense payable at time t
St+1 ≡ sum insured payable at t+1 if insured dies in the year

E t+1 ≡ the expense of paying the sum insured at time t+1
tV ≡ gross premium policy value for policy in force at time
t,
t+1V ≡ gross premium policy value for policy “ “ “ time t+1.
it ≡ the rate of interest assumed earned in the year
Note: et , Et+1 q [x]+t and it all as assumed in the policy value
basis.
( tV ++++Pt −−−− et)(1 ++++ it) = = = = q[[[[x]+]+]+]+t
(St++++1 ++++ Et++++1) + + + + p[[[[x]+]+]+]+t (t++++1V)
(1) or
( tV ++++Pt −−−− et)(1 ++++ it) = = = = t++++1V +
q[[[[x]+]+]+]+t (St++++1 ++++ Et++++1 ---- t++++1V)
(2)
•••• shaded term called Net Amount at Risk(NAR), Death Strain
at
Risk(DSAR) or Sum at risk
28

Formal derivation of general case recursive formula;
Lt= (1+it)
-1 (St+1+ Et+1) – Pt +et if K[x]+t=0 with probability q[x]+t
Lt= (1+it)
-1 Lt+1 – Pt +et if K[x]+t ≥1 with probability
p[x]+t
tV = E(Lt)
=q[[[[x]+]+]+]+t(1+it)
-1(St+1+Et+1)–(q[[[[x]+]+]+]+t+p[[[[x]+]+]+]+t )(Pt –et
)+p[[[[x]+]+]+]+t (1+it)
-1E[Lt+1]
(tV+ Pt –et ) = q[[[[x]+]+]+]+t(1+it)
-1(St+1+Et+1)+ p[[[[x]+]+]+]+t (1+it)
-1E[Lt+1]
(tV+ Pt –et )(1+ it) = q[[[[x]+]+]+]+t(St+1+Et+1)+
p[[[[x]+]+]+]+t (t+1 V) t+1V = E[Lt+1]
( tV ++++Pt −−−−et )(1 ++++ it ) ====
q[[[[x]+]+]+]+t (St++++1 ++++ Et++++1 ) ++++
p[[[[x]+]+]+]+t (t++++1V) (1)

29
Example 5 (text 7.7)
Consider a 20-year endowment policy purchased by a life aged
50. Level
premiums of $23,500 per year are payable annually throughout
the term of
the policy. A sum insured of $700,000 is payable at the end of
the term if the
life survives to age 70. On death before age 70 a sum insured is
payable at
the end of the year of death equal to the policy value at the start
of the year in
which the policyholder dies. The policy value basis used by the
insurance
company is as follows:
Survival model: Standard Select Survival Model(SSSM)
Interest : 3.5% per year
Expenses : nil
Calculate 15V the policy value for a policy in force at the start
of the 16th
year.
Why use recursive approach?
SSSM values used

q69 = .009294 q68 = .008297 q67 = .007409
q66 = .006619 q65 = .005915
30
Example 5 working page
31
(iv) Annual Profit
• recursive formulas for policy values show that if all cashflows
between t and t+1 are as assumed (i.e. policy value basis
assumptions are met), then insurer will be in a break-even
position
at time t+1, given they were in a breakeven position at time t
o unlikely that all assumptions(interest, mortality, expenses and
other
assumptions) met in any one year
o insurers assets could be more or less than what is needed to
match
actual t+1V values

• Analysis of surplus-break down profit/loss into its component
parts
o this analysis is done as a part of any valuation exercise
o analysis helps to determine if assumptions need to be changed
amongst other things
• Sources of profit depend on assumption category and product
o actual expenses < assumed expenses is a source of profit
o actual interest < assumed interest is a source of loss
o Actual Mortality < assumed mortality is a source of profit for
insurance,
but a source of loss for annuities
32
An insurer issued a large number of policies identical to the
policy in
Example 3 to women aged 60. Five years after they were issued,
a total of
100 of these policies were still in force. In the following year,
• expenses of 6% of each premium paid were incurred,
• interest was earned at 6.5% on all assets,
• one policyholder(p/h) died, and
• a $250 expenses was incurred on payment of sum insured for
the 1 p/h who died.

(a) Calculate profit or loss on this group of policies for this
year.
(b) Determine how much of this profit/loss is attributable to
profit/loss from interest, mortality, and from expenses.
( tV+Pt −−−− et)(1 + it )=q[x]+t (St+1 + Et+1)+p[x]+t (t+1V)
(breakeven)
Total Gain/Loss=(tV +Pt ― e’t)(1 +i’t)―[q’[x]+t (St+1 +
E’t+1) +p’[x]+t (t+1V)]
where ’ denotes actual experience
Note: Use 5V= 29,067.51, 6V= 35,324.17 (vs. Ex. 3 value
caln’s)
33
Example 6 (working pages)
Total “Profit (loss)” ≡ total Gain/(Loss)
Total “Gain (loss)” = 100(5V+.94P)(1.065) – (1)100,250 –
(99)(6V)

Total “Gain (loss)” = $18,918.99
Sources of Gain/Loss:
1. Interest: gain
2. Expenses: loss
3. Mortality: loss
How much gain/loss is attributable to each assumption?
• Look at each factor. At each step, assume factors not yet
considered are as specified in policy value basis, whereas
factors already considered are as actually occurred
34
Example 6 (working pages)
(b) sources of gain(loss) summary:
(i) Interest : 51,011
(ii) Expenses : (5,568)

(iii) Mortality : (26,524)
Total Gain(Profit) : 18,919
Note: if assumption order is different, #s change a bit by
source. For
example looking first at expenses, interest, then mortality
Sources of gain(loss) summary:
(i) expenses : (5,490)
(ii) Interest : 50,933
(iii) Mortality : (26,524)
Total Gain(Profit) : 18,919
35
Example 6 (working pages) _-another approach
100(tV+P-e)(1+i)-100(q[x]+t(St+Et)+p[x]+tt+1V)
(assumption notes)
Interest actual 51,011 (actual i, tV basis for
E&q)
expected 0 (all policy basis(tV) assms)

Gain(A-E): 51,011
Expenses: Actual 45,443 (actual i,E, tV basis for
q
Expected
51,011 (actual i, tV basis for E&q)
Gain(A-E): -5,567.57
Mortality Actual 18,919 (actual i, E, and q
Expected 45,443 (actual i,E, tV basis for q
Gain(A-E) -26,524.47
Net G/L 18,919.22
36
(v) Asset Shares
• Asset share is the share of the insurer’s assets attributable to
each policy in force at any given time

ASt ≡ Asset share per policy surviving at time t
(does NOT include/reflect Ps and related expenses
due at time t
so AS0=0))
• Asset Share(ASt) versus Policy value(tV)
o ASt is the amount the insurer actually has with respect to each
surviving insured
o tV is the amount the insurer needs to have with respect to
each
surviving insured
• calculate ASt using actual experience and by assuming the
given
policy is one of a large number of N identical policies
o accumulate Premiums received less claims and expenses paid
to time t and divide by number of surviving policies to get ASt
37
Example 7(text 7.9)
Consider a policy identical to the policy studied in Example 7.4

and
suppose that this policy has now been in force for five years.
Suppose that
over the past five years the insurer’s experience in respect of
similar
policies has been as follows.
• Annual interest earned on investments has been as follows;
Year 1 2 3 4 5
Interest % 4.8 5.6 5.2 4.9 4.7
• Expenses at the start of the year in which a policy was issued
were 15% of the
premium
• Expenses at the start of each year after the year in which a
policy was issued
were 6% of the premium.
• The expense of paying a death claim was, on average, $120.
• The mortality rate, q[50]+t, for t=0,1,…,4 has been 0.0015.
Calculate the asset share for the policy at the start of each of
the first six
years.

38
39
Table 7.1. Asset share calculation for Example 7.9
Fund Death
Fund Cashflow at eoy Claims Fund
at
Yr at b.oy. at b.oy. before claims & expenses at
eoy Survivors ASt
1 0 10115N 10601N 18N 10582N 0.9985 N
10,598
2 10582N 11169N 22970N 36N 22934N 0.99852N
23,003
3 22934N 11152N 35859N 54N 35805N
0.99853N 35,967

4 35805N 11136N 49241N 71N 49170N
0.99854N 49,466
5 49170N 11119N 63123N 89N 63034N
0.99855N 63,509
can extend class approach used for AS1 and AS2 to generate
rest of AS values
Note: AS5=63,509 versus 5V= 65,470
40
Recall Policy Value Recursive formula
( tV ++++Pt −−−− et)(1 + + + + it) = = = = q[[[[x]+]+]+]+t
(St++++1 ++++ Et++++1) + + + + p[[[[x]+]+]+]+t (t++++1V)
(1)
Analagous recursive asset share formula
(ASt ++++Pt −−−− e”t)(1 ++++ i”t) ====q”[x]+]+]+]+t
(St++++1 ++++ E”t++++1) ++++p” [[[[x]+]+]+]+t(ASt+1)
(i)
where ” indicates actual experience

Rearranging (i) gives
ASt+1 ====(1/(1/(1/(1/p” [[[[x]+]+]+]+t))))[[[[((((ASt
++++Pt −−−−e”t)(1 ++++ i”t) − q”x]+]+]+]+t (St++++1 ++++
E”t++++1)]
Text used modification of (i)
l”[x]+]+]+]+t (ASt ++++Pt −−−− e”t)(1 ++++ i”t) ====
d”[x]++++t (St++++1 ++++ E”t++++1) ++++llll”
[[[[x]+]+]+]+t+1(ASt+1)
where l”[x]+]+]+]+t ==== N N N N((((ttttpppp”xxxx) e) e) e)
etctctctc
41
7.4 Policy Values for other discrete cashflows
• Section looks at determining tV when discrete cashflows are
other than annual. General definitions below still hold,
calculations
can be more complex
� Gross premium policy value for a policy in-force at duration
t(≥0)years after it was purchased is the expected value at that

time of
the gross future loss random variable on specified basis. The
premiums
used in the calculation are actual premiums payable under the
contract.
� Net premium policy value for a policy in force at duration t
years
after it was purchased is the expected value at that time of the
net
future loss random variable on a specified basis (which makes
no
allowance for expenses). Premiums used in the calculation are
the net
premiums calculated on the policy value basis using the
equivalence
principle, not the actual premiums payable
• by example will show tV in theory can be calculated for any t
(and recursive formulas can be used), and linear interpolation is
useful
approximation when t between premium payment dates (7.4.2)
42
Example 8(text 7.10++)
A life aged 50 purchases a 10-year term insurance with sum
insured
$500,000 payable at the end of the month of death. Level
quarterly

premiums, each of amount P=$460 are payable for at most five
years.
(a) Calculate the (gross premium) policy values at durations
2.75, 3 and
6.5 years using the following basis.
Mortality : Standard Select Survival Model
Expenses : 10% of each gross premium
will show 2.75V = 3,090.30 and 3V=3,358.15
(b) How would you use recursive formulas to get 2.75V from
3V?
(c) Determine the policy value at each of (i) the end of 2 years
10 months
and (ii) 2 years and 9.5 months, assuming the policy is still in-
force at
each of these times(will look at this exactly and by
approximation(linear interpolation)
any required annuity/Insurance factors will be provided
43

44
Policy values between premium date approximation(working
page attached)
• interpolate between policy value just after previous premium
and policy value just before the next premium
o text formula: t+k+sV ≈ ( t+kV + Pt+k− Et+k )(1-s/k)+(
t+2kV)(s/k)
45
46

• interpolate between policy value just after previous premium
and policy value just before the next premium
Using t+k+sV ≈ (t+kV +Pt+k− Et+k )(1-s/k) + (t+2kV)(s/k)
and, from (a): 2.75V=3,090.30, 3V=3,358.15, (P2.75 -
E2.75) =(.9)(460)
(ii)policy value at 2 years, 9.5 months (or t = 2.79167)
k=3/12, s=0.5/12 so s/k=.5/3=.1667
2.79167V ≈ (2.75V+P2.75−E2.75 )(1-
.1667)+(3V)(.1667)
2.79167V ≈3,480.55 versus Exact: $3,481
(i)policy value at 2 years, 10 months (or t = 2.8333)
k=3/12, s=1/12 so (s/k)=1/3
2.833V ≈(2.75V+P2.75−E2.75 )(1-1/3) +(3V)(1/3)
2.833V ≈3,456.06 versus Exact: $3,456.73

47
7.5 Policies with continuos cashflows
• Recall, for policies with discrete cashflows:
( tV +Pt − et)(1 + it ) = q[x]+t (St+1 + Et+1) + p[x]+t (t+1V)
(1) or
( tV ++++Pt −−−− et)(1 ++++ it) = = = = t++++1V +
q[[[[x]+]+]+]+t (St++++1 ++++ Et++++1 ---- t++++1V)
(2)
• Can rewrite (2) as
(t+1V −−−− tV) ==== tV it + (Pt −−−− et)(1
++++ it ) −−−− q[[[[x]+]+]+]+t (St++++1 ++++
Et++++1 −−−− t++++1V) (1)
change in Increase due increase due to decrease
due to
policy value to interest (Prems-expenses)
mortality
• there’s a continuos version of the above formula called
Thiele’s differential
equation formula

48
• discrete version
(t+1V − tV) =tV it + (Pt − et)(1 + it ) − q[x]+t (St+1 +Et+1 −
t+1V) (1)
• continuous version or Thiele’s differential equation
(d/dt)(tV) ==== tV δt + (Pt −−−− et) −−−−
µ[[[[x]+]+]+]+ t (St ++++ Et −−−− tV), (2)
where
Pt ≡ annual rate of premium payable at time t
et ≡ annual rate premium-related expense payable at time t
St ≡ sum insured payable at exact time t if insured dies at exact
time t
Et ≡ expenses of paying the sum insured at exact time t
δt ≡ force of interest per year assumed earned a time t
tV ≡ policy value a policy in force at time t
all are continuous functions of t and as assumed in policy value

basis
49
(t+1V −−−− tV) ====tV it + (Pt −−−− et)(1 + + + + it )
−−−− q[[[[x]+]+]+]+t (St++++1 ++++Et++++1 −−−−
t++++1V) (1)
• Thiele’s differential equation
(d/dt)(tV) = tV δt + (Pt −−−− et) −−−− µ[x]+ t (St +
Et −−−− tV) (2)
• to apply (2) write (d/dt)(tV) ≈ ( t+hV −−−− tV)/h and
assume small h
(t+hV −−−− tV) ≈ tV δt h + ( Pt −−−− et)h −−−−
hµ[[[[x]+]+]+]+ t (St ++++ Et −−−− tV) , or
(tV)(1+ δt h) + ( Pt − et)(h ) ≈ t+hV+ hµ[x]+ t (St +Et −
tV)
• this method for numerical solution of a differential equation is
known as Euler’s method (which is a continuous time version of
discrete recursive method)
o the smaller the value of h, the better the approximation
o start with the policy value at end of policy term(or just before
it-
endowment insurance and work backwards, will show by
example)

50
Example 10(Text Example 7.12)
Consider a 20-year endowment insurance issued to a life aged
30.The
sum insured, $100,000, is payable immediately on death, or on
survival to the end of the term, whichever occurs sooner.
Premiums
are payable continuously at a constant rate of $2,500 per year
throughout the term of the policy. The policy value basis uses a
constant force of interest, δ, and makes no allowance for
expenses.
(a) Evaluate 10V.
(b) Use Euler’s method with h = 0.05 years to calculate 10V.
Perform the calculations on the following basis:
Interest: δ=0.04 per year
Use W3 formula where applicable (annual factors used provided
in class)
(a) |:40|:4010
AV
1010
a2500100000 −=

51
(a) 10 V calculations
Assumptions: δ =0.04, and ‘Standard Select survival Model’:
µx =A+Bc
x, A=0.00022, B=2.7x10-6, C=1.124
µ [x]+s = 0.9
2-s µx+s for 0 ≤ s ≤ 2 , l20=100,000(radix)
W3:
)m(
xa&& ≈ xa&& – (m–1)/2m – (m2–1)/12m2[δ+µx ]
|:40|:4010
AV
1010
a2500100000 −=

21671.8aEaa 401010 =−= 5040|:40
,61285.20a =40 )3W(6358.18a =50
Used SSSM excel WS: i = 0.040810774 = eδ — 1 to
calculate:
,11623.21a40 =&& ,13923.19a50 =&&
66518.0E4010 =
52
53
(b) Use Euler’s method with h ====0.05 years to calculate 10V.
• rearrange t+hV — tV and then can generate 10V recursively

• will do first iteration by hand, rest was done in excel
• result is 10V= 46,635 using Eulers method and h=.05
(versus answer of 46,591 in (a))
• if reduce size of h, answer will be even closer to that in (a))
54
Example 7.12 (h=.05)
x+t t ux+t tV
50 20.00 100,000.00
49.95 19.95 0.001147 99,675.67
49.9 19.90 0.001142 99,352
49.85 19.85 0.001136 99,029
49.8 19.80 0.001131 98,707
.
.
.
40.1 10.10 0.000513 47,069
40.05 10.05 0.000511 46,852

40.00 10.00 0.000510 46,635.12 (b)
55
Example 11
You are given the following information on a fully continuous
term
insurance policy that was issued to (25):
The death benefit is $100,000 payable at moment of death
Premiums are payable continously at a rate of $700 per year
The force of interest is δ = 6%
The force of mortality at age 40 is 0.008
There are no expenses
(d/dt)(tV) at t = 15 is equal to -4.80

Calculate the policy reserve at time 15.
56
7.6 Policy Alterations
• In practice, not uncommon, after the policy has been in force
for a
period of time, for the policyholder to request a change in terms
of
the policy(e.g. a policy alteration)
• Alteration examples include;
o policy cancellation: lapse or surrender
o changing benefit amount or reducing premiums
o converting from one type of coverage to a different type of
coverage
(e.g whole life to term conversion)
• Policy Surrenders or lapse
o when a policy is cancelled at request of policy holder before
end of
policy term, it is said to lapse or be surrendered
o a cash surrender value(CSV or Ct) is what policyholder would
be
entitled to if policy lapses(CSV generally 0 for term insurance

products)
o there is some regulation on minimum surrender values (non-
forfeiture
laws), which requires insurers to offer surrender values on
certain
contract types once they’ve been in-force a minimum number of
years
57
• Reasons the CSV(or Ct) is often less than either of tV or ASt
;
o anti-selection
o associated expenses
o liquidity risk
• Ct is often used towards the costs of alterations(other than
surrenders), and, when it is, you would have following E.O.V
for
altered benefits;
Ct+(EPV future Ps, altered contract)t = EPV(future
benefits+expenses, altered contact)t
• Paid-up Sum insured (use CSV as single P to purchase this)
o any policy where no premiums are payable is said to be paid
up

o a policyholders that doesn’t want to pay more premiums but
doesn’t
want to cancel policy can look at a policy alteration that results
in a
reduced sum insured
o equation highlighted above would be used to determined
reduced paid
up amount (but there would be no future P’s in this case)
58
Example 12
Consider the policy in Examples 7.4 and 7.9(Ex. 4& 7 in lecture
notes).
You’re given that the insurer’s experience in the 5 years
following the issue
of this policy is as in Example 7.9. At the start of the 6th year,
before paying
the premium then due, the policyholder requests the policy be
altered in one
of the following three ways.
(a) The policy is surrendered immediately.
(b) No more premiums are paid and a reduced annuity is
payable from age
60.In this case, all premiums paid are refunded at the end of the

year of death
if the policyholder dies before age 60.
(c) Premiums continue to be paid, but the benefit is altered from
an annuity
to a lump sum (pure endowment) payable on reaching age 60.
Expenses and
benefits on death before age 60 follow the original policy terms.
There is an
expense of $100 associated with paying the sum insured at the
new maturity
date.
Calculate the surrender value(a above), the reduced annuity (b
above) and
the sum insured (c above) using the assumptions below(see next
page)
59
Example 12(continued)
Calculate the surrender value(a), the reduced annuity (b) and
the sum
insured (c) using the Ct assumptions below and life insurance
and annuity
factors as provided(from Example 4)
(i) 90% of the asset share less a charge of $200, or
(ii) 90% of the policy value less a charge of $200

together with the assumptions in the policy value basis when
calculating
revised benefits and premiums
Notes:
– required life insurance/annuity factors provided(used
previously)
– previously calculated values: 5V=65,470 (notes Ex. 4),
AS5=63,509 (Ex.7)
(a) Surrender Value (policy is surrendered -calculate CSV)
(i) C5 = (.9)(63,509) – 200 = $56,958
(ii) C5= (.9)(65,470) – 200 = $58,723
60
Ex4(for reference): A man aged 50 purchases a deferred annuity
policy. Annual payments will be payable for life, with the first
payment
on his 60th birthday. Each annuity payment will be $10,000.

Level premiums of $11,900 are payable annually for at most 10
years. On death
before age 60, all premiums paid will be returned, without
interest, at the end of the year of death.
The following basis is used for policy value calculations:
Expenses : 10% of the first premium, 5% of subsequent
premiums, $25 each time an annuity payment
is paid, and $100 when a death claim is paid
61
Example 13(text exercise 7.11)
A10-year endowment insurance is issued to a life aged 40. The
sum
insured is payable at the end of the year of death or on survival
to the
maturity date. The sum insured is $20,000 on death, $10,000 on
survival to age 50. Premiums are paid annually in advance.
(a) The premium basis is:
Expenses : 5% of each gross premium including the
first
Interest : 5%

Show that the gross premium is $807.71.
(b) Show that 4V=3,429.68 (policy value just before 5
th P is paid)
(c)Just before the fifth premium is due the policyholder requests
that all
future premiums, including the fifth, be reduced to one half
their
original amount. The insurer calculates the revised sum insured
– the
maturity benefit still being half of the death benefit – using the
policy
value in part (b) with no extra charge for making the change.
Calculate
the revised death benefit
62
(b) Example 13 (working page)
10E[40] = 0.60929 6E44 = 0.7422401
a&& [40]:10 = 8.087046 a&& 44:6 = 5.319477

A[40]:10 = 1― d a&& [40]:10 A44:6 = 1― d a&&
44:6
63
Example 14 –Ch.7 Par Insurance example(text exercise 7.10)
A life aged 50 buys a 10,000 participating whole life insurance
policy.The sum
insured is payable at the end of the year of death. The premium
is payable annually
in advance. Profits are distributed through cash dividends paid
at each year end to
policies in force at that time. The premium basis is:
Initial expenses: 22% of the annual gross premium plus $100
Renewal expenses: 5% of the gross premium plus $10
Interest: 4.5% Survival model: Standard Select Survival
Model
a) Show that the annual premium, calculated with no allowance
for future bonuses, is
$144.63 per year.

b) Calculate the tV at each year end using the premium basis.
c) Assume earns insurer earns i= 5.5% each year. Calculate DIV
payable each year
assuming
(i) the policy is still in force at the end of the year,
(ii) experience other than interest exactly follows the
premium basis, and
(iii) that 90% of the profit is distributed as dividends to
policyholders.
d) Calculate the expected present value of the profit to the
insurer per policy issued,
using the same assumptions as in (c).
(e) What would be a reasonable surrender benefit for lives
surrendering their
contracts at the end of the first year
64
t p[50]+t tV Profitt DIVt
0 0.9989670 0
1 0.9987360 3.061 0.128 $ 0.1154
2 0.9985310 123.847 1.305 $ 1.1756
3 0.9983770 248.226 2.512 $ 2.2645
4 0.9982030 376.909 3.756 $ 3.3861

5 0.9980072 509.948 5.043 $ 4.5469
6 0.9977876 647.389 6.373 $ 5.7476
7 0.9975408 789.276 7.748 $ 6.9886
8 0.9972635 935.634 9.167 $ 8.2704
9 0.9969519 1086.477 10.630 $ 9.5935
10 0.9966018 1241.804 12.139 $10.9583
.
.
.
.
.
.
.
.
.
.
.

65
66
7.7 Retrospective Policy Values
• tV = E[Lt] is considered a prospective policy value
• Retrospective policy value (looks at past) equals:
accumulated
Premiums less accumulated benefits for a large group of
identical
policies and dividing this amount equally amongst survivors
o it is a “theoretical asset shared based on a different set of
assumptions
(i.e. is based on policy value assumptions)”
• retrospective policy value is not used much anymore
o seldom equals prospective value, due mostly to fact that
experience
often differs from assumptions and policy basis assumptions
often differ

from premium basis assumptions
– tV
Retro = tV
Prospective only if (i) contract P determined using equivalence
principle and (ii) same basis is used for P and tV
Retro , tV
Prospective
o retrospective policy values not used by countries that use
gross
premium policy values reserves,
67
When are retrospective Policy values useful?
• key retrospective measure is Asset share which measures the
accumulated
contributions of each surviving policy to the insurer’s funds
• however, when insurer uses net premium policy value,
retrospective
policy value may be useful

o recall (Ch. 7.2), net premium policy value uses premium
based on
valuation basis(policy basis) (regardless of true premium)
o IF P is calculated using the equivalence principle the
retrospective
and prospective net premium policy values will be the same
o retrospective policy value can be easier to calculate for
complicated
benefit or premium structures
o in U.S. some policies still valued using net premium policy
values and
in these cases retrospective formula is sometimes used
68
Defining Retrospective Net premium policy value
Define;
tV
P = Prospective net premium policy value (tV

P = E[Lt])
tV
R = Retrospective net premium policy value
(accumulated Prem less accumulated benefits for a
large group of
identical policies and divides this amount equally
amongst survivors)
Also define;
L0,t = (PV, at issue, of future benefits payable up to time t −
(PV,at issue of future net premiums payable up to time t)
tV
R ≡ -E[L0,t](1+i)
t/tpx = -E[L0,t]/tEx
Assuming P is calculated using equivalence principle, same
basis is used for
tV
R and tV
P , can show;
0 = E[L0]
0 = E [L0,t] + v

t
tpx E[Lt] , and rearranging gives
(0- E [L0,t])/( v
t
tpx ) = E[Lt]
tV
R = tV
P
69
Example 1:
A whole life insurance with sum insured of $10,000 is issued to
a life
age 30. The Net annual Premium(P) is determined using the
equivalency principle and the death benefit is paid at the end of
the
year of death.
Express the net premium policy value at time t in terms of
standard
actuarial functions using (i) the prospective and (ii) the
retrospective
method.
Verify that these expressions are equal.

70
Example 2:
A whole life insurance policy is issued to (40). The death
benefit in the
first five years of the contract is 5,000 and if death occurs after
five
years the death benefit is $100,000. Net Annual premiums are
payable
for a maximum of 20 years. Premiums are level for the first
five years
(P) then increase by 50% (1.5P).
(a) Provide an expression for net policy values: 4V
R and 4V
P
(b) Provide an expression for net policy values: 20V
R and 20V
P
Note: should be able to show that, say 4V
R =4V

P (same with t=20)
– as per Example 1 approach
– start with equation for 0V=E[L0]=0 (equivalence principle)
– alegbraically break up(1st 4 year versus the rest), and
rearrange
71
7.8 Negative Policy Values
• can have negative policy values in early durations but in
practice they
would be set to 0
• negative policy values can reflect poor policy design
72
7.9 Deferred Acquisition Expenses & Modified
Premium reserves
• the most common application of policy value calculations are
for reserve

determination (i.e. actual capital held in respect of a policy)
• in reserve determination, a net premium approach with
modification
may be used to approximate a gross premium policy value (will
look at
impact of acquisition costs on policy values first)
• Denote tV
n and tV
g as gross and net premium policy values using the
equivalence principle and using original premium interest
&mortality basis
tV
n = EPVfuture benefits – EPV future net premiums
tV
g = EPVfuture benefits + EPV future expenses – EPV future
gross premiums
0V
n = 0V
g =0
• Rewrite tV
g as

tV
g =EPVfuture benefits+EPVfuture expenses – (EPVfuture net
premiums+EPVfuture expense loadings)
= tV
n + ( EPVfuture expenses – EPVfuture expense loadings)
tV
g = tV
n + tV
e
where tV
e = (EPVfuture expenses – EPVfuture expense loadings)
73
tV
g = tV
n + tV
e
where tV
e = (EPVfuture expenses – EPVfuture expense loadings)
• Note that tV
e is generally negative � tV

n > tV
g
(same assumptions for
both), and this reflects the impact of the initial(acquisition)
expenses;
o Pg – Pn = Pe where Pe is called the expense loading(or
expense premium)
• note that if only had renewal expenses then Pe would equal the
expenses,
but when have acquisition/initial expenses Pe is greater as it has
to fund
both renewal and initial expenses
Example 1:
Calculate the expense loading and 10V
n, 10V
g
and 10V
e for a $100,000
whole life insurance policy issued to a life aged 50. Premiums
are payable
annually in advance throughout the term of the contract. The
following
assumptions are used for both premium and policy value
calculations:
Initial expenses: 50% of gross premium plus $250
Renewal expenses: 3% of the gross premium plus $25

Interest: 4%
Survival model: Standard Select Survival Model (required
factors below)
a&&
[50] =19.35185, A[50] =0.255698, a&& 60 =16.562066, A60
=0.362997 (i=4%)
74
Example 1 (working page):
aP
n
&&
[50]
A100000=
[50] 31.321,1P
n
======>==>==>==>
aP
g
&&
[50]
A100000=

[50]
)P03.25(225P47.0 gg ++++ a&&
[50]
89.435,1P
g
======>==>==>==>
aP
e
&&
[50]
)P03.25(225P47.0 gg +++= a&&
[50]
58.114Pe ===>
58.114PPP
Nge
====−−−−====
So, the expense loading portion(of gross premium) is $114.58
Note: renewal expenses = 68.08 = 0.03*PG+25
acquisition expenses = 967.95 =(0.5Pg )+250

so (114.58-68.08)=46.50 of the expense loading is used
to
amortize the acquisition expenses
75
Example 1 (working page):
10V calculations using P
n =1,321.31, Pg =1,435.89, Pe = 114.58
60
n
10 A100000V = 60
naP &&− 08.416,14=
60
G
10 A100000V = 60a25&&+ 60
gaP97.0 &&− 88.645,13=
=
e
10V 60a25 && 60

GaP03.0 &&+ 60
eaP &&− 19.770−=
10V
G = 10V
n + 10V
e , as expected
Notes:
negative expense reserve referred to as DAC
tV
n is conservative as doesn’t allow for DAC reimbursement
76
7.9Deferred Acquisition Expenses & Modified Premium
reserves(continued)
• negative expense reserve is referred to deferred acquisition
cost(DAC)
• net premium reserve can be viewed as being overly
conservative but can
look at a modification of net premium reserve to approximate
gross

premium reserves without having to directly account for
expenses
o Full Preliminary Term(FPT) is a modified premium reserve
method and the one that is most commonly used
• FPT method (intended to approximate tV
G esp. in early years)
o FPT uses net premium policy value approach, allows for a
lower initial
premium to implicitly consider DAC
o FPT considers policy as two policies-a 1 year term and a
separate
contract issued to same life 1 year later, if living
o Notation used
1P[x] ≡ single premium to fund year 1 benefits (cost of
insurance (COI))
Pn[x]+1 ≡ net premium in all subsequent years
o under FPT 1V = 0

77
Example 1 revisited:
(a) Calculate the modified premiums for this policy
(b) Compare net premium policy value, gross premium policy
value,
and FPT reserve at durations 0,1,2, and 10
Net Policy value (tV
n) Gross Policy Value (tV
g) FPT (tV
FPT)
time
0 0 0 0
1 1,272.15 383.67 0
2 2,573.97 1,697.21 1,318.50
10 14,416.08 * 13,645.88 * 13,313.21
* previously calculated
additional factors provided (to those in Example 1)
q[50]= 0.0010333
a&&

[50]+1 =19.105668, A[50]+1 =0.2651666, a&& 52
=18.853734, A52 =0.274856
78
Example 1 revisited(working page)
1
Ch. 6 Premium Calculations
6.1 Summary
6.2 Preliminaries
6.3 Assumptions
6.4 Present Value of future loss random variable
6.6 Gross Premium Calculation

6.7 Profit
6.8 Portfolio Percentile Premium Principle
6.9 Extra Risks
Note: additions//updates will be made to posted lecture notes
as we work through the material
2
6.1 Summary
• Chapter focus is on how to determine premium calculations
for
life insurance and life annuities, with a greater focus on life
insurance
• Two different premium principles considered
o Equivalence principle (most common for traditional policies)
o Portfolio percentile principle
6.2 Preliminaries

(i) Premium Types
(a) Net Premium
o considers benefit costs only, excludes expenses (and profit)
(b) Gross Premium
o explicitly allows for expenses
3
6.2 Preliminaries(continued)
(ii) Premium Frequency
Life Insurance
o Typically periodic premium payments(eg annually, monthly,
bi-weekly)
o Periodic premiums are usually a level $ amount but they can
be
varying (e.g. step rate)
Life Annuities

o Whole/temporary life annuities usually single premium
purchase
o Deferred life annuities may be purchased with single premium
or
periodic premiums during deferred period (pension funding
applications)
(iii) Premium Timing and duration
• Premiums are payable in advance with first premium payable
when
policy is purchased
• Periodic premiums can be paid for the life or term of an
Insurance
policy or shorter(e.g. Limited Pay Whole Life), and cease to be
payable
on death of the insured
4
6.3 Assumptions and Table sources
• The Standard Select Survival Model(SSSM)’ is default model
for
text examples and exercises. The model assumes i=5% and;

µx =A+Bc
x, A=0.00022, B=2.7x10-6, C=1.124
µ[x]+s = 0.9
2-s µx+s for 0 ≤ s ≤ 2
l20=100,000(radix)
• as noted previously, copy of Appendix D(SSSM) is posted
- it also includes several values(e.g. select Insurance factors,
pure
endowment factors, second moments for whole life insurance)
- SOA LTAM tables now used are ultimate part of SSSM (copy
posted)
• Standard Select Survival model(SSSM) Excel Worksheets
- required for some text questions(as well as to understand some
text examples)
- recommend you set up own SSSM worksheet for your own use
and/or use the
worksheet posted on OWL
- author spreadsheet(SSSM) referenced earlier (Life Cons
review notes) includes
several more factors than Table D
• ILT (copy posted)& other tables may be used for some class
examples

5
6.4 P.V. of the Future Loss Random Variable
•••• Cash flows associated with life insurance/annuity policies
are
generally life contingent
o This includes expenses and premiums as well as benefits
•••• Can model (future outgo-future income) with the random
variable
that represents the present value(P.V.) of future loss
o Net Future loss- excludes expenses
o Gross Future loss- includes expenses
o Future loss random variable depends on time of death
Future Loss Random Variable Notation:
0
nL
= PV benefit outgo – PV Net premium income
0
gL
= PV benefit outgo + PV expenses – PV Gross premium
income

6
Example 1(6.4)
insured is
$25,000 payable immediately on death. Interest is at i =5%. A
premium of $500 is paid at the beginning of each year for the
policy.
The policyholder dies at the end 25.8 years. Calculate the
future loss
at issue.
7
Example 2
insured is
$200,000 payable immediately on death. Premiums of amount P
are
payable annually in advance, ‘ceasing at age 65 or on earlier of
death’
(20 Pay Whole Life Policy).What is the future loss random
variable for this policy?

•••• Distribution of future loss random variable used to
determine
Premiums for given benefit/benefit for given premiums, using a
Premium principle
o Equivalence Principle (“benchmark principle”)
o Portfolio Percentile Principle
o Cash flow Profit Criterion Method (Ch.12)
8
•••• Under the equivalence principle, premiums are determined
such
that Expected value of the future loss is zero at contract issue
•••• Equivalence principle most common method for traditional
insurances and is text default method for these policies (e.g
Ch.6,7)
(i) Net Premiums
•••• Ignore expenses

•••• Benefits include either and/or both of death benefit/survivor
benefits
n
0] =0 , and since
L
n
0 = PV benefit outgo – PV Net premium income, then under
equivalence principle EPVbenefit outgo= EPVnet premium
income
or PVFB = PVFP (at policy
issue)
9
6.5 Example 1 (SSSM is text default model)
Consider a 20 pay whole life insurance with annual premiums
issued
to [50]. The death benefit(sum insured) is $500,000 payable at
end of
year of death.

(a) What is an expression for net future loss random variable?
(b) Calculate the net annual premium using the SSSM
10
Example 2 (a few working pages are attached)
Consider an endowment insurance with term n years and sum
insured S payable at the earlier of the end of the year of death
or at
maturity, issued to a select life aged x. Premiums of amount P
are
payable annually throughout the term of the insurance (same as
saying
P’s are paid for the life of the policy).
(A) Derive expressions in terms of S, P and standard actuarial
functions for;
(i) The net future loss, Ln 0
(ii) The mean of Ln 0,
(iii) The variance of Ln 0
(B) Assume S=150,000; n=20, [x]=50 and SSSM
(iv) Calculate the annual premium (equivalence principle )

(v) Suppose now the premium is payable quarterly. What is
the quarterly premium? Assume UDD within each year of
age and you are provided factors for α(4) and β(4).
11
12
13
Example 2–annuity factor calculation for B (v)
)4(
|20:]50[
a&&

=
)4(
]50[a&&
)4(
70]50[20 aE &&−−−−
)m(
|n:]x[
a&&
=
)m(
]x[a&&
)m(
n]x[]x[n aE +− &&
Under UDD : )m(Ba)m(a ]x[
)m(
]x[ −α≈
&&&&
or use: ]x[)m(
)m(

]x[ A
i
i
A ≈
in identity:
)m(
m
]x[)m(
]x[
d
)A1(
a
−
=&&
Also, can show under UDD;
≈
)m(
|n:]x[
a&& )E1)(m(Ba)m( ]x[n|n:]x[ −−α &&

)4(
|20:]50[
a&& 598403.12)E1)(4(Ba)4( ]50[20|20:]50[ =−−α≈ &&
14
An insurer issues a “regular premium deferred annuity contract”
remainder of the life of (x).
benefit of S
the net future
for the
monthly net premium

15
An insurer issues a regular premium deferred annuity contract
remainder of the life of x.
Ln0 = 0 – 12P |12/1K
)12(
a +&& for T[x] ≤ n or K[x]
(12)< n K ≡ K[x]
(12)
Ln0 = |n12/1K
)12(naXv −+&& – 12P |n
)12(
a&& for T[x] >n or K[x]
(12) ≥ n

benefit of S
the net future
for the
monthly net premium
16
17
Example 4
You are given an extract(see below) from a life table with a
four
year select period. A three year term insurance with death
benefit of $S is purchased by [41] with a net premium of $350
payable annually. Assume i=6% and that the death benefit is
payable at end of year of death.
Calculate

(i) S(sum insured) assuming the equivalence principle
(ii) standard deviation of L0
(iii) Pr [L0 >0]
[[[[x] ] ] ] l[[[[x]]]] l [[[[x]+]+]+]+1
l[[[[x]+2]+2]+2]+2 l[[[[x]+3]+3]+3]+3 lx++++4444
x++++4444
[40] 100,000 99,899 99,724 99,520 99,288 44
[41] 99,802 99,689 99,502 99,283 99,033 45
[42] 99,597 99,471 99,268 99,030 98,752 46
18
19
(ii) standard deviation of L0 or √Var[L0] calculations :
(using S=216,326.38 from (i))

k L0 Pr(K41=k)
0
49.731,203a350Sv
1
=− &&
q[41] = 0.0011322 =1— l[41]+1/ l[41]
1
52.849,191a350Sv
2
2
=− &&
1│q[41]= 0.0018737
2
11.640,180a350Sv
3
3
=− &&

2│q[41] = 0.0021943
≥3
69.991a3500
3
−=− && 3p[41] = 0.9947997
Var[L0]=E[L0
2]–[E[L0]]
2=E[L0
2], when equivalence principle used (to determine P)
can show Var[L0] =188,537,738 � √Var[L0] = 13,731
Table above provides Lo values for given values of k. General
expression for Lo is;
Ln0 = Sv
K+1 – 350 |1Ka ++++&& for K[41]< 3 K ≡ K[41]
Ln0 = 0 – 350 |3a&& for K[41]
≥ 3

20
Example 5
The policy provides an annuity of $30,000 per year payable
annually in advance from age 65. In the event of death before
age 65, the premium is returned at the end of year of death.
Assume SSSM applies (used Appendix D for calculations)
(a) Give an expression for Ln0
(c) Now assume annuity is guaranteed to be paid for at least 5
years if
insured survives to 65. How much does P increase by?
21
Example 5 continued
The policy
provides an annuity of $30,000 per year payable annually in

advance from
age 65. In the event of death before age 65, the premium is
returned at the
end of year of death. Assume SSSM applies ( Appendix D
used)
(a) Ln0 = Pv
K+1 – P for K[45] < 20 (K=K[45])
Ln0 = |201K
20 av000,30
−−−−++++
&& – P for K[45] ≥ 20
(c ) Now assume annuity is guaranteed to be paid for at least 5
years if insured survives to 65. How much does P increase by?
22

23
Example 6
A 20-pay $250,000 whole life insurance policy is purchased on
(45) where the premiums are payable monthly and the death
benefit is payable at moment of death.
Assume that i=6% and ILT mortality applies. Also assume
UDD
within each year of age. Calculate the monthly premium(P).
Answer: P = $384.34
24
25
Notation used for net annual premiums for fully discrete
Insurances (IAN)

x
x
x
a
A
P
&&
====
|n:x
x
xn
a
A
P
&&
====
|n:x
|n:x

|n:x a
A
P
1
1
&&
====
,
|m:x
|n:x
|n:x
m
a
A
P
1
1
&&
====

|n:x
a&&
1
|n:x1
|n:x
A
P ====
|n:x
a&&
|n:x
|n:x
A
P ====
|m:x
|n:x
|n:x
m
a

A
P
&&
====
Default assumption: Periodic Premiums are payable for the life
of the policy
Terminology
“Fully discrete” : both premiums at benefits payable at
discrete time points
“semi-continous” : DB paid at moment of death, but
Premiums paid at discrete points
“Fully continuous” : premiums paid continuosly, DB paid at
moment of death
26
6.6 Gross Premiums
•••• apply equivalence principle to determine gross premiums,
gross
premiums consider insurers expenses
•••• Types of insurer expenses that need to be considered

(a) Initial expenses, which are incurred when policy is issued.
Examples
include agents commission and underwriting expenses
(b) Renewal expenses are normally incurred each time premium
is
payable: can include cost of premium processing, commissions
etc
(c) Termination expenses are incurred when a policy expires,
typically on
death of a policyholder (or annuitant) or on the maturity date of
a term
insurance policy. Termination expenses are relatively small
•••• Timing and format of expenses
o Convenient to assume expenses incurred at exact time as a
premium or
benefit payment is made(in reality usually incurred slightly
before/after)
o Expenses can be proportional to premiums, proportional to
benefits, or a
‘per policy’ expense

27
•••• While GP calculations assumes expenses are known,
expense
allocation is a complicated process
Gross Premiums
g
0] = 0 , and since
L
g
0 = PV benefit outgo + PV expenses – PV Gross premium
income,
then under the equivalence principle
EPVbenefit outgo + EPVexpenses = EPVgross premium
income
or PVFB + PVFE = PVFGP (at policy issue)

28
An insurer issues a 25-year annual premium endowment
insurance with sum insured $100 000 to a select life aged 30.
The insurer incurs initial expenses of $2,000 plus 50% of the
first premium, and renewal expenses of 2.5% of each
subsequent premium. The death benefit is payable immediately
on death.
(a) Write down the gross future loss random variable.
(b) Calculate the gross premium assuming
– Standard Select Survival Model and i=5%(Appendix D)
– UDD within each year of age
29

30
30.295,2$
475.a975.0
2000A100000
P
25:]30[
25:]30[
====
−−−−
++++
====
&&

Appendix D values used (i =5%) : 384.19a ]30[ ====&& ,
060.16a55 ====&& 37256.0E ]30[20 ==== 77772.0E505
====
31
Calculate the monthly gross premium for a 10-year term
insurance with sum insured $50 000 payable immediately on
death, issued to a select life aged 55, using the following basis:
Survival model: Standard Select Survival Model(SSM)
I assumed UDD for fractional ages
Initial Expenses : $500+10% of each monthly premium in the
1st year
Renewal Expenses: 1% of each monthly premium in the second
and
subsequent policy years

Factors provided:
α(m) = i d/ i(m)d(m) � α(12) = 1.000197
B(m) = (i– i (m))/i(m)d(m) B(12) = 0.466502
024955.0=1
|10:[55]
A 97723.0a
)12(
|1:]55[
=&&
83389.7a )12(
|10:]55[
=&&
32

33
Example 2
•••• In this Example, Year one premium income is not enough
to
cover year 1 expenses (‘new business strain’)
o Annual Premium amount =12P= 12(18.99)= $ 227.88
o 1st year expenses= ($ 500+ .10(12P) >> $ 227.88
•••• New business strain is common
o Insurer needs to ensure enough funds to sell polices and
may have to borrow to write new business
o Early expenses gradually paid off by expense
loadings(loading=gross less net premium) in future
premiums and the part of the premium that funds initial
expenses is called deferred acquisition costs
34
Example 3
A Whole Life insurance policy is purchased on (35). The death

benefit is
paid at the end of year of death and is $ 300,000 if death occurs
within the
first 30 years, and 75,000 thereafter.
Gross Premiums are payable annually, where the initial annual
premium is
level for the first 30 years, and thereafter annual premiums are
1/3 of the
initial annual premium.
Expenses are 40% of the first year premium and 5% of all
subsequent
premiums. All expenses are payable at the beginning of the
year.
Calculate the initial gross annual premium, assuming mortality
follows
the Illustrative Life Table (ILT) and that i=0.06.
Solution
:
Set P = “Initial Gross Annual Premium”

EPVpremiums= EPVBenefits +EPVexpenses (or E[L
G
0]=0)
consider info provided in ILT to streamline calculations
35
36

Calculate the gross single premium(P) for a deferred annuity of
$80,000 per year payable monthly in advance, issued to [50]
with the first annuity payment on the life’s 65th birthday.
Allow for initial expenses of $1,000 and renewal expenses on
each anniversary of issue date, provided policyholder is alive.
Assume the renewal expense will be $20 on the first
anniversary of the issue date, and that expenses will increase
with inflation from that date at a compound rate of 1% per year.
Assume Standard Select Survival Model(SSSM) and i=5%.
The following SSSM values are provided (i=5%):
15E[50] =0.4616267
)12(

65a&& =13.08696 (used W3 approxm) )
37
38
Example 5
A $250,000 Whole Life insurance policy is purchased on (35)
with annual
gross premiums(P) payable for a maximum of 10 years. The
death benefit is
paid at the end of year of death. You are given:

(i) Expenses of $200 are payable at end of each year, including
year of
death
(ii) i = 6% and mortality follows the Illustrative Life
table(ILT)
Calculate P.

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