PART 6
Sample Essay Solutions
Fall 2002
21.
a) Link ratio estimate Æ y=cx where c=E(y)/E(x) = 1.346
Æ y = 1.346(60) = 80.77
Budgeted loss estimate Æ y=E(y)= 87.5
b) Least squares y = a+bx
b=(E(xy) – E(x)E(y))/(E(x^2)-E(x)^2)= 0.5
E(xy) = (65(90)+50(80)+70(85)+75(95))/4 = 5731.25
E(x^2) = (65^2+50^2+70^2+75^2)/4 = 4312.5
a=E(y)-bE(x)
= 87.5 - .5*65
= 55
Least Squares Estimates Æ y=55+.5(60) = 85
L(x) = Z(cx) + (1-Z)E(y) where Z = b/c = .5/1.346 = .3715
= .3715(80.77) + (1-.3715)(87.5) = 85 ties to least squares.
PART 6
Sample Essay Solutions
Fall 2002
22.
(a) IBNR @ 12/31/01 using case incurred loss development.
Case Incurred @
LDF to
12/31/01 (000’s)
Ultimate
IBNR Factor
IBNR @ 12/01
AY
98
100
1.080
0.080
8.00
99
1,000
1.134
0.134
134.00
00
900
1.247
0.247
222.30
01
600
1.559
0.559
335.40
Total
2,600
$699.70
$699,700 = IBNR @ 12/01
(b) IBNR reserve using B/F method.
AY
98
99
00
01
Total
EP (000’s)
200
1000
1500
1500
ELR
0.80
0.80
0.80
0.80
IBNR Factor
0.0741
0.1182
0.1981
0.3586
IBNR
@12/31/01
11.856
94.56
237.72
430.32
774.456
$774,456
(c) B/F method is preferred if there is little to no credible historical data. For example,
on a new line of business.
(d) When the loss development pattern in the data is consistent and if the loss ratio is
deteriorating.
(e)
AY
98
99
00
01
Total
EP (000’s)
200
1000
1500
1500
ELR
0.80
0.80
0.80
0.80
Factor for IBNR to emerge
during CY 2002
0.0741 = 0.0741
0.1182 – 0.0741 = 0.0441
0.1981 – 0.1182 = 0.0799
0.3586 – 0.1981 = 0.1605
IBNR to
emerge
during CY
2002
11.856
35.280
95.880
192.60
335.616
$335,616
PART 6
Sample Essay Solutions
Fall 2002
23.
Reliance on Non-Actuarial Expertise:
The loss event modules are non-actuarial in nature, and require understanding of
complicated events. A reluctance for actuaries to rely on non-actuaries for such an
important input.
Black Box:
These models are proprietary, and as such much is not disclosed because of the
proprietary nature. Actuaries reluctant to accept without understanding the process of the
model (complicated).
Cost/Benefit Analysis:
The models are expensive, so the benefit must outweigh the cost.
24.
Cumulative Closed Claims
Closed = (Cumulative Reported) - Open
0-12 12-24 24-36 36-48
1998 20
50
80
100
1999 21
53
86
2000 24
59
91
115
2001 25
Proportion Closed Ratios
0-12
12-24
24-36
36-48
25/120 = 0.208
(59-24)/(115-24) = 0.385
(86-53)/(110-53) = 0.579
1.000
0.579*(115-59)=32.4 32.4 + 59 = 91
Calculate Average Cost per Closed Claim for 2000
24-36: 4,725*1.05 = 4961.25
36-48: 7,500*(1.05^2) = 8,268.75
Accident Year 2000 Ultimate Loss =
1,653*24 + 3,308*(59-24) + 4,961.25*(91-59) + 8,268.75*(115-91) = 512,662
PART 6
Sample Essay Solutions
Fall 2002
25.
a) (i) I would expect the incurred loss method to produce a slightly overstated reserve.
The method is not affected by the deteriorating loss ratio, but the slip in reserve
adequacy and subsequent correction by claims would produce larger than
necessary development factors.
(ii) I would expect the expected loss method to understate reserves. Since the
actuaries have made no adjustment, the expected loss ratio used is lower than
what one would now expect.
(iii) I would also expect the Percentage of Premium method to understate reserves,
although I expect it to be a better estimate than the expected loss method. This
method would be relying on previous years of lower losses for the factors so
the reserves would be understated.
b)
Using paid losses to predict development. Once the correction is made by claims,
I would expect the paid development to be represented by past development.
PART 6
Sample Essay Solutions
Fall 2002
26.
Disposal Rate= Cumulative Closed Claims
Projected Ultimate Claims
AY
12
24
36
48
98
.325 .700 1.000 1.000
99
.350 .750 1.000
00
.375 .800
01
.375
Selected
.400 .800 1.000 1.000
(Increasing Trend in disposal rates)
Restated Closed Claims
Disposal Rate
.400 .800 1.000 1.000
AY
12
24
36
48
98
20000 40000 50000 50000
99
21000 42000 52500 52500
Restated Closed Claims for AY00 at 24 months = (55000)(.8)=44000
Restated Paid Loss (using linear interpolation w/ closed claims)
AY
@12
98
99
121875*(35000-20000)/(35000-16250) + 262500*(20000-16250)/(35000-16250) = 150000
137813*(39375-21000)/(39375-18375) + 295313*(21000-18375)/(39375-18375) = 157500.5
AY
@24
98
99
262500*(50000-40000)/(50000-35000) + 375000*(20000-35000)/(50000-35000) = 150000
295313*(52500-42000)/(52500-39375) + 393750*(42000-39375)/(52500-39375) = 157500.5
Development Factors:
AY
24-36
98
375000/300000=1.250
99
393750/315000.4=1.250
Selected
1.250
46-48
375000/375000=1.000
1.000
1.000
UltimateAY00 Loss = (330,000)(1.25)(1.00)
=412,500
PART 6
Sample Essay Solutions
Fall 2002
27.
Adjustment to Total Known Losses
a).
Ultimate loss estimates for AY99,00,01,02
Expected Losses - average then adjust
= 1/n(sum of Ka*) = 1/3(1400+1700+1200) = 2092,000
1/n(sum of 1/fa)
1/3(1/1.8+1/1.5+1/1.2)
AY
99
00
01
02
Expected Ult
Losses (000’s)
2,092
2,092
2,092
2,092
%
unreported
@ 12/31/01
16.7%
33.3%
44.4%
100%
IBNR
(000’s)
349
697
930
2,092
Known(000’s)
@12/31/01
1,400
1,700
1,200
0
Ultimate
Loss
1,748,667
2,397,333
2,129,778
2,092,000
% unreported = 1-1/age to ult factor
b). The average of the estimates of ultimate losses should be equivalent to the
expected ultimate losses used in the calculation
¼*(1,748,667+…+2,092,000) = 2,092,000
PART 6
Sample Essay Solutions
Fall 2002
28.
a)
Ultimate LDF for unlimited loss @ 12
= 2.004 x 1.498 x 1.335 x 1.245 x 1.098
= 5.4785
Ultimate loss for AY 2001
= $220,000 x 5.4785
= $1,205,272
Ultimate LDF for deductible loss @ 12
= 1.894 x 1.419 x 1.251 x 1.201 x 1.049
= 4.2358
Ultimate deductible loss for AY 2001
= $220,000 x 4.2358
= $931,882
The reported excess loss as of Dec 31, 2001 for AY 2001
= $220,000 - $220,000
=0
IBNR for excess loss for AY 2001
= $1,205,272 - $931,882
= $273,390
b)
Adv:
1) It estimates the IBNR for excess loss based on more stable loss development pattern
from the limited loss triangle.
2) It provides the estimate of excess loss at maturities where they have not emerged.
c)
Service revenue asset
= (Ultimate deductible loss – Ult losses excess of aggregate)
x loss multiplier – known recoveries
= ($931,882 - $256,944) (0.1) - $50,071
= $17,423
PART 6
Sample Essay Solutions
Fall 2002
29.
a) W2 is the weight given to the “experienced” adjustments for the policy year. Berry’s
formula weights together expected (a priori) deviation, and deviation implied by
experience to date. Since no substantial adjustments will have been paid by 21
months, Berry gives the “experience” portion of his formula no weight. Note that for
a 12-month policy, adjusted 6 months after termination, with adjustment taking 2-3
months to appear in the data, we should not have any substantial adjustments before
21 months.
b) W2 = 0 for 1 to 20 months, then increases linearly by 1/28 per month until reaching
1.00 at 48 months. Thus, at 33 months, W2 = 1/28 *(33 – 20) = 0.4643
c) Morrell believes that the informational value of actual paid deviations increases faster
than linearly. He suggests something that looks like:
W2
Months
d) It is assumed that at 48 months, no reserve is required, as adjustments beyond 48
months are likely to be small & are equally likely to be positive or negative.
e) % of 1st adjustments paid at 22 months = 1 / DPF at 22 mos. = 1 / 3.57 = 28%
f) Add’l premium from 2nd & later adj. As % of incurred losses = LPF
LPF at 46 months = 0.0005 = 0.05%
PART 6
Sample Essay Solutions
Fall 2002
30.
A. Test 1: UEPR must provide for premium return if cancellation. Assume $800,000 of
premium written at 1 July 2001, all policies are for 5 years, but only 6 months have
been earned at 31 Dec 2001. Therefore unearned premium = (60 – 6) / 60 * 800,000
= $720,000 for Test 1.
Test 2: UEPR = WP * a / b
a
=
=
=
b =
=
=
future losses and expenses
88,400 + 102,000 + 136,000 + 204,000 + 102,000
632,400
total loss and expense
632,400 + 47,600 + (.10)(800,000)
760,000
Therefore, UEPR = 800,000 * [632,400 / 760,000] = 665,684 for Test 2.
Test 3: UEPR = present value of future loss and expense – present value of future
guaranteed premiums (= 0)
Discount rate = min [ .04, .05 - .015 ] = 3.5%
Therefore, UEPR = (88,400)(1.035)-.5 + (102,000)(1.035)-1.5 + (136,000)(1.035)-2.5
+ (204,000)(1.035)-3.5 + (102,000)(1.035)-4.5
= 308,555 + 268,230
= 576,785
B. Company UEPR (580,000) fails Test 1 and Test 2. It passes Test 3.
31.
1. Uses created year statistics
- most companies don’t carry such data by created year
- use accident year data instead – more readily available
2. F-ratio is an abstract quantity that is difficult to explain because the ratio of paid
ALAE to losses disposed of has no real meaning.
- use the ratio of paid ALAE to paid loss instead
3. The loss disposed of ratio is artificial and may be unstable.
- the stability will depend on whether estimates of ultimate losses are revised
since losses disposed of = losses paid + changes in O/S, thus it may be unsuitable
for long tailed lines where ultimate loss estimates must be continually monitored.
- use paid losses instead of losses disposed of.
PART 6
Sample Essay Solutions
Fall 2002
32.
PDLD2 = [P2 – P1] / [L2 – L1] = [CL2 – CL1] / [L2 – L1] * TM * LCF
CL / L = 1 - χ - LER
CL1 / L1 = 1 – (.123 - .005) - .035 = .847
CL2 / L2 = 1 – (.134 - .004) - .035 = .835
PDLD2 = [ (CL2/L2 * ELR * % Loss2) – (CL1/L1 * ELR * % Loss1) ] / [ELR * % Loss2
– ELR * % Loss1] * TM * LCF
= [.835*.9 - .847*.8] / [.9 - .8] * 1.03 * 1.20 = .09134 / .10 = .9134.
33.
(a)
(1) The price derived by the pricing algorithm. This incorporates company specific
expense ratios, loss experience, investment performance and target market
percentile to develop an indicated rate.
(2) The market price. The price at which the company will be able to write its goal
amount of business given its underwriting selectivity. This is limited by the price
competitors are charging for coverage with a similar underwriting selectivity.
(b)
One must consider that a company’s price may decrease its allowable level of
underwriting selectivity, which will bring in business with higher loss levels. This
then increases the companies pricing formulas.
PART 6
Sample Essay Solutions
Fall 2002
34.
Reinsurer A:
95,000 / (450,000 + 76,000) = 18% < 20%
Therefore only reinsurance recoverable on paid losses & LAE > 90 days overdue
are subject to penalty.
Penalty = 95,000 x 0.20 = 19,000
Reinsurer B:
128,000 / (500,000 + 63,000) = 22.7% > 20%
Therefore all reinsurance recoverables are subject to penalty.
Penalty = (500,000 + 450,000 + 200,000 + 215,000 + 6,000 – 110,000) * 0.20
= 252,000
Total Penalty = 19,000 + 252,200 = 271,200
35.
a)
In SAP, policy acquisition costs are immediately expensed. In GAAP, they are matched
with revenue (ie – earned evenly in the period).
b)
Since GAAP matches revenue with expenses, a deferred PAC is created as an asset on the
expense not yet recognized. DPAC’s do not exist for SAP.
PART 6
Sample Essay Solutions
Fall 2002
36.
(a)
1. It is probable that an asset has been impaired or a liability has been incurred at the
date of the financial statement.
2. The amount of loss can be reasonably estimated.
(b)
Disclosure shall be noted in the financial statement. The estimated amount of loss shall
be disclosed. If the amount can not be estimated, it should state that the amount can not
be estimated.
37.
a)
250,000 written premium.
(250,000 x 0.75) = 187,500 = written premium ceded.
(187,500 x 0.30) = 56,250 = ceding commission.
(187,500 – 56,250) = 131,250 premium due reinsurer.
50,000 losses.
9,000 ALAE.
(59,000 x 0.75) = 44,250 ceded and due from reinsurer.
Premium – loss = 131,250 – 44,250 = 87,000 is due to the reinsurer.
b)
The purpose of the ceding commission is for the reinsurer to pay for the primary
insurer’s policy acquisition costs that were incurred in obtaining the business ceded to
the reinsurer. The ceding commission is adjusted as a way to price the reinsurance
contract.
PART 6
Sample Essay Solutions
Fall 2002
38.
Exposure Rate 50,000 XS 50,000
35,000
50,000
100,000
200,000
500,000
ILF
1.00
1.15
1.32
1.45
1.55
Premium
2,000,000
3,000,000
3,000,000
2,000,000
5,000,000
15,000,000
Exp Factor
0
0
17/132
17/145
17/155
Exp Prem
0
0
386,364
234,483
548,387
1,169,234
Prem = (1,169,234)(primary co. ELR)(ALAE loading)(Reinsurer exp & profit load)
=(1,169,234)(1-0.35)(1.15)(1.20)
=1,048,803
Exposure Rate = 1,048,803 / 15,000,000 = 0.0699 = 6.99%
39.
State
CA
FL
IL
GA
Total
(1)
Expected Loss = (Standard
Premium)x(ELR)
2,360,000 (0.75) = 1,770,000
1,200,000 (0.80) = 960,000
800,000 (0.70) = 560,000
640,000 (0.70) = 448,000
(2)
% of Loss in $700,000 XS
of $300,000
0.08 – 0.03 = 0.05
0.125 – 0.055 = 0.07
0.025 – 0.005 = 0.02
0.040 – 0.010 = 0.03
(3) =
(1) x (2)
Excess Loss
88,500
67,200
11,200
13,440
180,340
Therefore, loss cost for $700,000 XS $300,000 = 180,340/Total Premium = $180,340 /
5,000,000 = 3.61%
PART 6
Sample Essay Solutions
Fall 2002
40.
Before Loss Corridor
Range of Loss
Ratios
0-75%
75-85%
85%+
Avg. Loss Ratio in
Range before Loss
Corridor
65.0%
82.0%
93.5%
Probability that
Loss Ratio is in the
Range
60%
25%
15%
Reinsurer Loss Ratio = (.65)(.6) + (.82)(.25) + (.935)(.15) = 0.39 + 0.205 + 0.140 = 0.735
Reinsurer Loss Ratio without Loss Corridor = 73.5%
After Loss Corridor
Adjust Average Loss Ratio for Range 75-85% =
0.75+(0.82 - 0.75) * (1- 0.7) = 0.771 = 77.1%
Adjust Average Loss Ratio for Range 85% + =
0.75 + (0.85 – 0.75) * (1-0.7) + (0.935 – 0.85) = 0.865 = 86.5%
Reinsurer Loss Ratio After Loss Corridor = (0.65)(0.6) + (0.771)(0.25) + (0.865)(0.15) =
0.7125
Reinsurer Loss Ratio After Loss Corridor = 71.25%
PART 6
Sample Essay Solutions
Fall 2002
41.
Ceding Co.
Coverage
Limit
100,000
200,000
300,000
400,000
500,000
Direct
Premium
5,000,000
5,000,000
5,000,000
5,000,000
5,000,000
% of loss
falling under
ceding co.
Retention
100%
84.6%
73.6%
66.3%
61.5%
% of loss
falling under
retention &
treaty limit
100%
100%
100%
100%
96.1%
% of loss
covered by
reinsurer
0%
15.4%
26.4%
33.7%
34.6%
Company expected loss and ALAE ratio = 0.70 * 1.15 = 0.805
Ceding Co.
Coverage Lim.
100,000
200,000
300,000
400,000
500,000
Direct
Premium
5,000,000
5,000,000
5,000,000
5,000,000
5,000,000
Expected
Loss &
ALAE ratio
.805
.805
.805
.805
.805
Expected
Ult. Loss
& ALAE
4,025,000
4,025,00 0
4,025,000
4,025,000
4,025,000
% of loss
covered by
reinsurer
0%
15.4%
26.4%
33.7%
34.6%
Loss & ALAE
covered by
reinsurer
0
619,850
1,062,600
1,356,425
1,392,650
4,431,525
Exposure Rate = {(4,431,525/25,000,000) * 1.1 * [1/(1-0.2)]} – 1 = 0.2437
42.
A)
1) Both use an expected loss ratio to apply to premium.
2) Both use the unreported loss percentage to apply to the expected losses.
B)
1) Bornhuetter-Ferguson use unadjusted earned premium, while StanardBuhlmann use earned premium that is adjusted (adjusted for on-level premium
rates and trend).
2) Stanard-Buhlmann use data from all years to estimate an expected loss ratio,
while Bornhuetter-Ferguson use separate estimates for each year without
grouping all years together to estimate the loss ratios.
PART 6
Sample Essay Solutions
Fall 2002
43.
Advantages
1)
2)
can retain larger portion of premium on small profitable risks.
Helps stabilize underwriting results.
Disadvantages
1)
no surplus benefit
2)
increase in frequency of small losses can deteriorate the loss ratio.
44.
a)
1. The assuming company must assume significant insurance risk (both timing and
underwriting)
2. The assuming company must have a reasonable possibility of having a significant
loss
b)
If no significant risk, it is fine if substantially all of the risk is transferred. This
puts the reinsurer in essentially the same position as the ceding company.
PART 6
Sample Essay Solutions
Fall 2002
45.
First, estimate future paid losses using loss development method.
age-to-age
(3-yr wtd avg)
12 – 24
3200/1600
= 2.000
24 – 36
3000/2000
= 1.500
36 – 48
2000/1500
= 1.333
age-to-ult LDF
4.000
2.000
1.333
Projected Paid Claims Pattern
98
99
12
500
500
24
1000
1000
36
1500
1500
48
2000
2000
00
600
1200
1800
2400
01
750
1500
2250
3000
Claims Projected Paid by Calendar Year
2002:
2003:
2004:
500 + 600 + 750
600 + 750
750
= 1,850
= 1,350
= 750
Discounted Future Payments:
Commutation Price
= [1850/(1.05)^.5] + [1350/(1.05)^1.5] + [750/(1.05)^2.5]
= 3,724
b. Upon commutation, all reserves are released
Current reserves = 0 + 400 + 600 + 750 =
Since booked reserves < commutation price,
Reinsurer will realize an underwriting loss
1,750