PSTAT 172 A HOMEWORK: PREMIUMS March 8 2018
Ian Duncan, FSA FIA FCIA FCA CSPA MAAA
Duncan@pstat.ucsb.edu
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1. Forafully-discretewholelifeinsuranceof100,000on(35)youaregiven:
a. Percentofpremiumexpensesare10%peryear.
b. Perpolicyexpensesare$25peryear.
c. Per 1000 face amount expenses are 2.50 per year.
d. Allexpensesarepaidatthebeginningoftheyear.
e. 1000P35=8.36.
Calculate the level annual premium using the equivalence principle. Solution
2. Gistheannualexpense-loadedpremiumpayableforlifefor$1,000of insurance for a whole life policy to (x). Expenses are:
a. Commissions:20%ofGinFirstYear;5%ofGinrenewalyears.
b. Settlementcost:$5per$1000.
If the policy is modified so that first year commission is 35%, while all other expenses are unchanged, the new gross premium (G!) is 2% higher than G. If d = 0.06, find G.
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SOLUTION
G ä 1 0 0 5 A 0 . 1 5 G 0 . 0 5 G a G xxx
G!ä 1005A 0.35G! 0.05G!a G! xxx
0.95äx – 0.15 =1.02 0.95äx – 0.30
äx = 8.2105 => Ax = 0.50737 => G= 66.66
1005Ax 0.95ä – 0.15
x
1005Ax 0.95ä – 0.30
x
3. Forann-year,fully-discreteendowmentinsuranceof$100,000issuedto(x) the benefit (net) premium is $2,061. Expenses are: $25 per year plus 20% of the expense-loaded premium G, plus settlement expenses of $A.
If σ(0 L e)/ σ(0 L ) = 1.0005, find G.
Note: σ(0 L e) is s.d. of the loss-at-issue of the gross benefits (including expenses) and premiums; σ(0 L ) is the s.d. of the loss-at-issue of the benefits and net premiums.
SOLUTION
G äx:n| 100,000 A Ax:n| 0.2 G 25 äx:n| G = (100,000+A) P + 0.2 G +25
0.8 G – 25 = (100,000+A) P (where P is the net premium) 0 L e = (100,000 + A) Z + (0.2G + 25 – G) (1 –Z)/d
Where Z is the p.v.r.v. for an n year endowment insurance. =(100,000+A)Z -P(100,000+A)(1-Z)/d
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= (100,000 + A)(1 +P/d)Z – (100,000 +A) P/d
0Le=100,000Z -100,000P(1–Z)/d =100,000(1+P/d)Z–100,000P/d
σ(0Le) =(100,000+A)(1+P/d) =1 +A/100,000=1.0005 σ (0 L ) (100,000)(1 + P/d)
A = 50.
G = (100,050)P x:n + 31.25 = 2609
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4. A fully-discrete n-payment whole life insurance policy is issued to (x). The expenses on the policy are as follows:
Expense Type
First Year
Renewal
% of Gross Premium Commissions
6%
6%
Tax % of Gross Premium
4%
4%
Per policy
$30
$5
Settlement Expense
$15 per policy
!
Given that A 0.3; A 0.1; A 0.4 and i= 0.06, what is the expense-loaded
x x: n| x n
premium G, for a policy with face amount of 10,000?
SOLUTION
Note that per policy expenses are incurred for the whole life of the policy whereas the commissions and taxes, because they apply to premiums, only apply for the n years of premium payment.
Gä 10,00015A255a0.1Gä
x:n| x x x: n|
G 10,000 150.3 25 5äx 0.9 äx: n| 0.9 äx: n| (0.9)äx: n|
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We know that nEx =0.5
AA! E0.6 xn xn n x
ä 1Axn 7.067andä1Ax 0.7 12.367
x:n|
d
x
d 0.05661
A 0.3 A! E A
x x:n| n x xn
G = (10,000 +15) (0.3) + (0.9)(7.067)
= 486.04
__25____ + 5(12.367) (0.9)(7.067) (0.9)(7.067)
As a check: Ax = 0.3 and äx:n| 7.067 => Net Annual Premium = 424.5083.
Percentage of premium expenses are (0.06+0.04) or 10%; so this raises the net premium by 424.51/0.9 = 471.58. We add the $5 annual fee to this (the first year fee is amortized over a number of years) = 476.58. Finally the first year expense is amortized as well: 25/7.067 = 3.54. This all adds to: 480.22.
Close enough.
5. Fora10-payment,fullydiscrete20-yearterminsuranceof1000on(40),you are given:
i = 0.06
Mortality follows the ILT.
Calculate the expense-loaded premium using the equivalence principle. 5
Expense Type
First Year
Renewal
% of Gross Premium Commissions
25%
5%
Tax % of Gross Premium
4%
4%
Face amount
$10 per $1,000
$1 per $1,000
Per policy
$10
$5
SOLUTION
6. Foraspecial3-yearterminsuranceon(30)youaregiven:
a. MortalityaccordingtoILT
b. Deathsareuniformlydistributedwithintheyearofage.
c. Interest at 6%
d. Premiumsarepayablesemi-annuallyinthefirstyearonly.
e. Benefits,payableattheendoftheyearofdeathare:
i. $1000 (year 1)
ii. $500 (year 2)
iii. $250 (year 3)
Calculate the semi-annual benefit premium for this insurance.
Note that this is a 3-year insurance question, so we will probably solve by writing out each year’s term.
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7.
SOLUTION
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SOLUTION
8. Forafully-discretewholelifeinsuranceof1000on(60)youaregiventhe following expenses, payable at the beginning of the year:
a. LevelgrosspremiumisG
b. i =0.05
c. If the insured dies in the 3rd year, the value of the expense-augmented loss
variable, 0Le is 768.59. Calculate G.
Expense Type
First Year
Renewal
% of Gross Premium Commissions
20%
6%
Per policy
$8
$2
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SOLUTION
9. Forafully-discrete5-payment,10-yeardeferred,20-yearterminsuranceof 1000 on (30) you are given the following expenses:
a. Expensesarepaidatthebeginningofthepolicyyear.
b. Grosspremiumisdeterminedusingtheequivalenceprinciple.
Expense type
Year 1
Year 2-10
% Premium
Per policy
% Premium
Per policy
Taxes
5
0
5
0
Commission
25
0
10
0
Policy Maintenance
0
20
0
10
Find G, assuming ILT at 6%.
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SOLUTION
By the equivalence principle, PV [Gross Premiums] = PV [Benefits + Expenses]
PV[GrossPremiums]= Ga 30:5|
G(15.8561(0.74091)(15.3926)) 4.45157G = 1000[0.54733(0.16132 (0.36913)(0.27414))]
PV [Benefits] = 1000 E 10 30
= 32.90915
PV [Expenses] = 0.05Ga
= G[0.15(4.45157) 0.15] 10 10[15.8561 14.8166(0.54733)]
= 0.81774G + 10 +10(7.74653) = 0.81774G + 87.4653
4.45157G-0.81774G = 32.90915+ 87.4653
3.63383G = 120.37445
G= 33.13.
A! 40:20|
0.10Ga 0.15G 10a 10 30:5| 30:5| 30:10|
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10. Find the gross premium for the following example assuming that δ=0.05 and μ=0.03. Note that the question doesn’t state whether the premium should be calculated as a continuous premium, or as a semi-continuous premium. I will treat the problem as continuous; the semi-continuous solution will be similar.
SOLUTION
Ax:20| / ax:10|
Hence a
Also: a 1e20() 1e20(0.08) 9.976 => Hence A 10.05(9.976)0.5012
We need to calculate these quantities; we can use the following:
1 e10( ) 1 e10(0.08) 6.883 0.08
x:10|
x:20|
Hence: A x:20|
0.08
x:20|
/ a
x:10|
0.5012 0.073 6.883
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We will also need to calculate ax:20| for later use:
a 1 1 1 13.007 and a isalsoequalto13.007
x 1 e( ) 1 e(0.08) 1 0.9231 x20 (because of the constant force of mortality assumption).
a a -e20( ) a =13.007(1-e1.6 )=13.007(1-0.202)=10.381
x:20|
(as a check, this seems reasonable related to ax:20| )
Gross premium:
Gax:10| 1000Ax:20| 0.09Gax:10| 0.20G55ax:20|
Note that the policy maintenance expense is incurred for all the 20 years but is
due at the beginning of the year, hence ax:20| not ax:20| .
We can re-state the equivalence statement above as:
G(6.883) 501.20 0.09 G(6.883) 0.20 G 5 5(10.381) G(6.883)(0.91)0.20G506.2051.95 558.15 G(6.064) 558.15 G 92.04
Check: Net premium (above) is 0.073 = $73.00 for a $1,000 policy. If we load this 9% (for commissions and taxes) we get $80.22 (approximately). We add another $5 annually for policy maintenance, except that we need to first accumulate it and then amortize over 10 years (because premiums are only paid for 10 years). This means adding about $7.50 = $87.72. The additional load of 20% first year commission amounts to about $18.00 which we need to amortize over the 10 years = 18/6.883 annually or $2.60. Adding the pieces we get $91.30. This will underestimate the true premium a bit because the last 2 components need to be grossed-up (9% load) so we seem to be in the right ballpark.
x x20
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You can also treat this as a semi-continuous problem, for which you will need ax:10| which you calculate similarly to the above. You can check your own answer if you
solved this question semi-continuously. 11.
Solution
The above solution in incorrect; the return of premium in part (b) should be 10G/2. Then:
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+ 10G/2 x 0.33037 6.6184G-0.5G-0.56184G-1.652 G=300,010+2,809.2
3.905 G=302819 and G = 76,833.
As a check: without the return, premium is 54,498
With the return of premium we add 5G as a pure endowment. Present value of this is 5G(0.33037) or 1.652G. We need to amortize this over 10 years with an annuity factor of 6.6184. 1.652G/6.6184 => 0.25 G. Hence without loads we would have to increase the premium 25%. We have to gross this up (10%) for the renewal load, which gives about 28%. We also load the additional 40% commission on the increased premium, which increases the cost by 10%. In total we are looking at close to a 40% increase, or another 20,000, which brings us to about 75,000.
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