# Policy values

Chapter 7: Policy Values

7.1 Summary 7.2 Assumptions

7.3 Policies with annual cashflows

7.4 Policy values with other discrete cashflows

7.5 Policy Values with continuous cash flows

7.6 Policy alterations

7.7 Retrospective policy values

7.8 Negative policy values

7.9 Deferred Acquisition Expenses

Note: additions/updates will be made to posted notes as we work through the material

AS3429/9429a Ch.7 Lecture notes (W2018)

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7.1 Summary

• chapter focuses on determination of policy values

which is a fundamental tool in insurance risk management

• policy values used to determine capital requirements and also to determine profit/loss of a given company over a period of time

• focus will be on policies with discrete cashflows, but will also look at the case of continuous cashflows

7.2 Assumptions

• The Standard Select Model(SSSM)’ continues as the text default model for example and text exercises

o Appendix D (some factors)

o also excel WS

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7.3 Policies with annual cashflows

Topics covered include;

(i) Future Loss Random Variables (ii) Policy Values (iii) Recursive Formulas for Policy Values (iv) Profit (v) Asset Shares

(i) Future Loss Random Variable

• extend Ch.6 future loss random variable definitions to consider times beyond inception

t nL = PV[future benefits]at time t – PV[future net premiums]at time t

t gL = PV[future benefits]at time t + PV[future expenses]at time t –

PV[future gross premiums]at time t

• Lt notation used for either if it’s clear what is meant.

• Note: Lt is defined only if contract is still in force(I/F) at that time

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Example 1 (text 7.1) Consider a 20-year endowment Insurance policy purchased by a life aged 50. Level premiums (P) are payable annually throughout the term of the policy and the sum insured, $500,000, is payable at the end of the year of death or at the end of the term, whichever is sooner. The basis used by the insurance company for all calculations is the Standard Select Survival Model, 5% per

year it and there no allowance for expenses.

(a) should be able to show(equivalence principle) P = $15,114.33

(b) Calculate E[Lnt ]for t = 10 and t = 11 in both cases just before the premium due at time t is paid.

a&& 60:10 =7.955550 a&& 61:9 = 7.328229 a&& [50]:20=12.845595 a&& [50]:10=8.05565

20|:[50]A =0.388305 10|:[50]A = 0.61635 10|:60A = 0.6211667

‘

10|:[50] A = 0.014387 10E[50] = 0.601963 20E[50] = 0.348322

will show 10V= E[L

n 10 ]= 190,339 and 11V= E[L

n 11 ] 214,757

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Example 1 (working page)

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20 year Endowment Insurance(Figure 7.1)

• invest excess in early years to build up assets required in later years

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20 year Term Insurance(Figure 7.2)

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Working with Example 1(Endowment Insurance Policy)

•••• Now assume N independent and identical policies (to that in Ex.1) are issued. Also assume experience of this group of insureds is exactly as what the insurer

assumed in determining the premium

•••• Look at insurers fund build up at end of 10 years(for these N policies), and then amount of this fund per surviving insured

AV(Fund)=AV(all premiums received)–AV(all claims paid)at end of 10 yrs

AVpremiums = NP(1.05) 10+NP(1.05)9p[50]+ NP(1.05)

8 2p[50]+…+ NP(1.05)9p[50]

= NP(1.05)10[1+ vp[50] +v 2 2p[50] +…+ v

9 9p[50] ]

= NP(1.05)10[ a&& [50]:10]

AVclaims paid = 500,000N[(1.05) 9 q[50 + (1.05)

8 1│q[50+(1.05)

7 2│q[50+ …+ 9│q[50]

= 500,000N(1.05)10[vq[50 + v 2 1│q[50 +v

3 2│q[50+ … + v

10 9|q[50]

= 500,000N(1.05) 10[

1

10|:[50] A ]

AV(Fund)= N(1.05)10[P a&& [50]:10 – 500,000 1

10|:[50] A ]=186,634N (using given values)

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Still working with Example 1(Endowment Insurance Policy)

AV(Fund)= (1.05)10[P a&& [50]:10 – 500,000 1

10|:[50] A ]= 186,634N

Amount per survivor =AV(Fund)/(N10p[50]) = 186,634N/(N(.9805343))

= 190,339 =E[Ln10]

• can also show the above by working with the equivalence EOV – i.e. equivalence principle EOV that was used to determine P (separate working

page provided for this)

Why is the amount per survivor here exactly equal to E[Ln10]?

(a) Premium(P) was calculated using the equivalence principle

(b) E[Ln10] was calculated using the premium basis(i.e. using the same assumptions that were used to determine the premium)

(c) we assumed experience was exactly what was assumed in the premium basis

In practice (a) or (b) may not apply, (c) rarely applies

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Example 1-working page

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(ii) Policy Values: (Policies with Annual Cash flows)

tV ≡ Policy Value t years after a policy is issued

tV = E[Lt]

tV+ EPV(Future Premiums) t = EPV(future benefits + expenses) t

•••• a positive policy value means future premium to be received is insufficient to cover future benefits and expenses (in terms of EPV) – The amount needed to cover this shortfall comes from the excess of the

premium over the EPV of the benefit in early years of the policy(e.g Ex.1)

•••• Ongoing Financial management includes determination of the total of all policy values (for all policies in force at that time), and comparing this amount to the company’s investments. For a company to be financially sound, total investments should be greater than the total policy value. This process is called the valuation of a company, and is done at least annually.

Note: Other texts denote tV = Reserve at time t (i.e. assume policy value = reserve)

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General Policy Value Definitions (not just for 7.3)

•••• 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 a specified basis(called policy value basis).The premiums used in the calculation are the actual premiums payable under the contract.

•••• Net 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 net future loss random variable on a specified basis or policy basis which makes no allowance for expenses. The premiums used in the calculation are the net premiums calculated on the policy value basis using the equivalence principle, not the actual premiums payable.

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Notes on Policy Value Definitions: (1) Assumptions used to calculate policy value are called the policy value basis.

Assumptions used to calculate the premium are called the premium basis. Policy basis assumptions may differ from the premium basis

(2) Net premium policy value is special case of gross premium policy value

(3) if policy value basis differs from the premium basis, the net premium policy value requires you to recalculate the premium(will see this in Example 2)

(4) Standard practice to consider any premiums (& premium related expenses) at time t in tV calns as future payments and any benefits at time t as past payments for life insurance. Need to state explicitly for annuity.

(5) nV = 0 for n-year term/endowment insurance products. For all policies, if premium is determined using equivalence principle and the policy value basis is same as premium basis, then 0V=E[L]=0.

(6) Endowment Insurance policy still in force at maturity date, n-V = S, where n- means the moment before n

(7) insurers build up reserves (policy values) for policies still in force by accumulating premiums paid and deducting claims (similar to example1, just not with the same precision)

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Example 2 (Text 7.2) An whole life insurance policy is issued to a life aged 50. The sum insured of $100,000 is payable at the end of the year of death. Level premiums of $1,300 are payable annually in advance throughout the term of the contract.

(a) Calculate gross premium policy value 5 years after issue of the contract, assuming that the policy is still in force, using the following basis: Survival model: Standard Select Survival Model Interest: 5% per year effective Expenses: 12.5% of each premium

(b) Calculate net premium policy value 5 years after issue of the contract, assuming that the policy is still in force, using the following basis: Survival model: Standard Select Survival Model Interest: 4% per year

Factors provided: a&& 55 = 16.05987 A55=0.23524 (select tables, i=5%)

a&& [50] = 19.35185 A[50] =0.255698 (select tables,i=4%)

a&& 55 = 18.054359 A55 =0.305602 (select tables,i=4%)

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Example 2 (working page)

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Example 3 (Text 7.3)

A woman aged 60 purchases a 20-year endowment insurance with sum insured of $100000 payable at the end of the year of death or on survival to age 80, whichever occurs first. An annual premium of $5,200 is payable for at most 10 years. The insurer uses the following basis for the calculation of policy values:

Survival model: Standard Select Survival Model Interest: 5% per year effective Expenses: 10% 1st premium, 5% of subsequent premiums $200 on payment of sum insured

Using the factors provided, calculate 0V, 5V, 6V and 10V, that is, the gross premium policy values at times t = 0,5, 6 & 10.

a&& [60]:10 = 7.9601 A[60]:20 = 0.41004 A70:10 = 0.63576

a&& 65:5 = 4.4889 A65:15 = 0.5114 A66:14 = 0.53422 a&& 66:4 = 3.6851

L0 =(100,000+200)(v) min(K +1,20) + .05P +P(.05-1) a&& |)10,1Kmin( + , K=K[60]

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Example 3 (working page)

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Example 4 (Text 7.4) 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

Calculate gross premium policy values for this policy at t=0, t=5, and at the end of the 15th year, just before and just after the annuity payment and expense due at that time.

Note: Ex. 4 required Insurance/annuity factors to be provided in class

AS3429/9429a Ch.7 Lecture notes (W2018)

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