Chapter 10
Nonforfeiture Benefits and Dividends
10.1 Nonforfeiture Benefits and Dividends
Refers to those benefits a policyholder is entitled to if he/she:
a. Stops paying premiums before the end of the premium paying period. b. Wishes to terminate the policy.
Types of nonforfeiture benefits available are:
1. cash surrender value 2. paid-up insurance
3. extended term insurance 4. automatic premium loan
10.1.1 Cash Values
Let {x dies in the coming year}
{x withdraws in the coming year}
{x does not withdraw or die and persists to the coming year}
cash surrender value paid at the end of year t for withdrawal in year t.
death benefit paid at the end of year t for death in year t
Then,
appropriate benefit reserve to hold=(death benefit reserve) + (withdrawal benefit reserve)
i.e. satisfies
where , and
If death benefit reserve at end of year t,
then and
………(1)
Similarly, net premium for withdrawal benefit reserve and satisfies
withdrawal benefit reserve at end of year t.
Then and
………..(2)
total reserve held at end of year t
total net premium
Then (1) + (2) gives
Suppose we set
Then (3) becomes
which is the formula connecting successive terminal reserves under a single decrement of death only.
i.e., if cash value upon withdrawal equals terminal reserve under single decrement (death only) model, then single decrement death benefit reserve are equal to double decrement (death and withdrawal) death benefit reserves plus withdrawal reserves.
Clearly single decrement reserves will be sufficient if withdrawal benefit is less than terminal reserve, since for single decrement reserves,
if
10.1.2 Minimum Cash Values
The Standard Nonforfeiture Law (SNL) prescribes minimum cash values the following way:
minimum cash value at end of year t
= modified terminal reserve at end of year t for a “special” modified reserve syetem with modification period equal to premium paying period.
Recall for a general modified reserve system, if actuarial p.v. of death benefits
then and modified premiums satisfy
i.e.
where extra first year expense
= first year expense allowance
where -(“expense” reserve at end of year t)
= unamortized first year expense allowance
= surrender charge at end of year t
For minimum cash values, we denote renewal modified premiums by with first year expense allowance satisfying
where net level premium
This is base on the 1980 SNL,
Under the 1941 SNL,
where renewal modified premium (or adjusted premium) for given plan
renewal modified premium for a whole life policy on (x)
where
when ,
and
When
and
i.e.
Note
This calculation is done by trial and error in that if the assumption results in
, use the formula for .
Then one proceeds to calculate for assuming either or and recalculating if necessary.
Generally for most limited pay whole life plans, and .
Then
and
For an h pay whole life plan,
and for
for
Example 10.1.1 A fully discrete whole life insurance of one is issued to 50.
You are given:
(i)
(ii) (iii)
Calculate based on the 1980 nonforfeiture law
Solution
i.e.
Example 10.1.2 For a fully discrete whole life insurance of one issued to (x), you are given:
(i) the adjusted premiums under the 1941 and the 1980 nonforfeiture laws are equal. They are each less than 0.04
(ii)
Calculate
Solution
1941 SNL:
since
Then
i.e.
1980 SNL: and
Then and
Given: under 1980 SNL under 1941 SNL
i.e.
i.e.
since and
i.e.
Solving,
10.2 Minimum Cash Values for Non-Level Benefits
and Premiums
Consider an h-paying, n-year benefit policy with gross premiums , payable
at the beginning of the year and benefits payable at the end of the year.
The 1980 SNL defines minimum cash values as follows:
1. Define AAI = average amount of insurance
for
2. Define net level premium P where
Note: Any pure endowment benefits are included in the calculation of P.
3. Then first year expense allowance is given by if
if
4. The adjusted premiums are where satisfies
Note: Endowment benefits are included in calculating
5. Then minimum cash value at end of year t is
Example 10.2.1 Consider the following premiums and benefit payments for a policy on (x)
t q GP b
1 0.1 2,795 8,000
2 0.2 8,600 20,000
3 0.5 10,750 32,000
The policy endows at the end of year three for 8,000.
Assuming , calculate the minimum cash value under the 1980 SNL at the end of year one and year two.
Solution
8,000
x 0.1 x+1 0.2 x+2 0.5 x+3
[Note: Endowment benefit is excluded]
Actuarial present value of benefits
[Note: Endowment benefit is included]
Net level premium P satisfies
i.e.
i.e.
Adjustment factor satisfies
i.e.
i.e.
i.e.
Example 10.2.2 A whole life insurance issued to (x) provides a death benefit for death in
year j of payable at the end of the year. Level annual premiums are payable for life.
You are given:
(i)
(ii)
(iii) the interest rate is
Calculate the first year expense allowance for this policy under the 1980 Standard Nonforfeiture Law.
Solution
Net level premium P satisfies
i.e.
10.2.1 Paid-up Insurance
For an n-year endowment with cash value at end of year k, the amount of paid-up
insurance satisfies
i.e.
For a unit of insurance, if ,
See text Table 15.4 page 440 for other plan of insurance.
Example 10.2.3 An individual age 30 purchases a fully continuous 50,000 20 year endowment policy. At the end of 10 years, she surrenders the policy in return for reduced paid-up insurance.
You are given:
(i) (ii)
(iii) (iv)
(v) Cash values are equal to the net level premium reserves
Calculate the reserve on this reduced amount of insurance five years after the original policy was surrendered.
Reduced paid-up insurance b satisfies
i.e.
We want reduced paid up reserve in duration 5
i.e. reserve
10.2.2 Extended Term Insurance
For a unit of insurance with cash value of at the end of k years, extended term insurance solves for the length of the term period s for a unit of insurance such that
For an n-year endowment insurance, if , then we solve for the pure endowment amount such that
i.e.
If at time of surrender, the policy had an outstanding loan of per unit of insurance, the extended term is determined by
10.2.3 Automatic Premium Loan
When premium payment stops at end of year k, gross premium are paid by “borrowing” from the outstanding cash value where
outstanding cash value at (k+t) when premium default occurs at k
where gross premium
x x+1……..x+k x+k+1……….x+k+t-1 x+k+t x+k+t+1
The maximum length of the premium loan period t satisfies:
At (x+k+t), the outstanding cash value could be surrendered for cash or used to purchase extended term insurance.
Example 10.2.4 A fully discrete whole life insurance of 20,000 issued to (x) is surrendered after 20 years. There is an outstanding policy loan of 2,500 which is not repaid. The policy owner elects settlement under the extended term insurance option.
(i) The cash value at the end of the 20th year = 4,390
(ii)
(iii)
(iv)
(v)
Calculate the extended term benefit period
Solution
Extended benefit period solves
i.e.
0.11
0.108
0.1
11 s 12 s
Example 10.2.5 An individual age 30 purchases a fully continuous 20 year endowment insurance of 10,000. At the end of 10 years, the policy is surrendered for extended term insurance and a pure endowment payable on the original maturity date.
You are given:
(i) (ii)
(iii) (iv)
(v) Cash value are equal to the net premium reserves
(vi) The outstanding policy loan at the end of 10 years is 2,000
Calculate the amount of pure endowment
Solution
Outstanding cash value
n.s.p. for 10 year extended term
Amount of pure endowment E satisfies
i.e.
For a life insurance policy which pays out cash value upon withdrawal, the generalized pricing formula becomes
actuarial p.v. of gross premium = actuarial p.v. of death benefits
+ actuarial p.v. of withdrawal benefits
+ actuarial p.v. of expenses
For a life insurance policy which pays out death benefit at the end of year ,
, withdrawal benefit at the end of year t, incurs per policy expenses at
the beginning of year , then the sequence of gross premium expenses at the beginning
of year , then the sequence of gross premiums payable at beginning of the year must satisfy
………..(1)
where dies in the coming year
withdraws in the coming year
and
Note
For a life insurance policy which pays out dividends at the end of the year for policyholders who persist, we have an additional term
Recall for net premium reserves, we showed that given the basic net premium equation
where
We define net premium reserve at end of year ,
[prospective loss random variable]
We then showed the basic recursive formula holds:
Similarly, starting with price equation (1), we define
asset share at end of year
[prospective generalized pricing loss random variable]
We then have the recursive equation
……….(2)
Note
10.3.1 Relationship to Income Statement and Balance Sheet
If initial assets = 0, actual experience equals pricing experience assumption in equation (1)
then asset share at end of year
¿ assets at end of year
persisting policyholders at end of year
Check:
(Assets)0 = 0
(Assets)1 = (cash flow)1
from equation (2) for
In general,
(Assets)t+1 = (Assets)t + (cash flow)t+1
from equation (2)
Example 10.3.1 Consider the following 20 year endowment policy of 10,000 issued to (x).
Gross premium charged = 500
Per premium expenses: 40% of gross premium in year one
10% of gross premium in renewal years
Per policy expenses: 100 in year one
50 in renewal years
Asset share at end of year nine = 3,000
10th year cash value = 1,000
, ,
Calculate the asset share at the end of the 10th year.
Solution
i.e.
10.4 Retrospective Formulas for Asset Shares
Recall for the net premium reserves, retrospective reserve = prospective reserve
i.e.
Similarly, one can show that asset share at end of year
= prospective gross valuation reserve
= retrospective gross valuation reserve
10.4.1 Pricing to Achieve a Given Asset Share Goal
Using the retrospective formula, we can solve for the gross premium which results in a
given asset share at the end of n years.
Method:
Start off with an arbitrary gross premium .
Use the recursive equation
Suppose asset share at the end of n years starting with gross premium .
Then retrospectively,
- “terms not involving gross premium”
We want gross premium such that
- “terms not involving gross premium”
i.e.
Solving,
Example 10.4.1 For a fully discrete whole life insurance of 10,000 issued to (x), the asset share goal at the end of 20 years is . A trial gross premium results in an asset share at
the end of 20 years equal to .
You are given:
(i)
(ii) (iii)
(iv)
(v)
(vi)
is the gross premium that produces an asset share at the end of 20 years equal to .
Solution
at ;
i.e.
10.4.2 Experience Adjustment
The recursive asset share formula
can be written as
…..(1)
Starting with , if actual withdrawal, death, interest and expense experience varies from pricing and is denoted by primed symbols, we have
experience asset share at the end of the year
………..(2)
(2) – (1) gives
………..(a)
………(c)
……….(d)
total experience gain over pricing
(a) = gain from interest (b) = gain from expenses (c) = gain from mortality (d) = gain from withdrawal
If a policy pays out all experience gains in the form of dividends to persisting policyholders, the asset share formula becomes:
i.e.
………..(3)
Given: ……..(4)
(3) – (4):
Denote by , set and
Then
………..(5)
10.4.3 Variations
A. Dividends paid out to deaths and withdrawals as well. Then (5) becomes
Similar adjustment if dividends payable to persistent policyholders and deaths only, etc.
B. Dividends paid out at death and withdrawal and . Then (5) becomes
C. Same as (B) and set
3 factor dividend formula
Example 10.4.3 A fully discrete participating whole life insurance of 1,000 is issued to (35)
You are given:
(i) The fund share is equal to the cash value.
(ii) Dividends are paid at the end of each year up to and including the year of death and withdrawal.
(iv)
(v) ;
(vi) ; (vii)
(viii)
Calculate
Solution
i.e. since
i.e.
Given:
i.e.
Substituting,
Example 10.4.4 For a fully discrete whole life insurance of 1,000, you are given:
(i)
10 100.00 10 55
11 144.55 10 55
(iii)
(iv) Actual expenses equal the expected expenses
(v) All dividends are paid at the end of the year. Dividends are paid on survival, death and withdrawal.
(vi)
(vii)
(viii) is the highest rate of interest for which
Calculate
Solution
Since
i.e.
Solving
