Section A - Credibility Weighted Estimates

Question 1

Mack (2000): What is the Benktander method? Describe the method and the associated formula for the second iteration of the BF method.

Answer 1

• Benktander method derives the ultimate loss by credbility-weighting the chain ladder and expected loss ultimates
• Iteration 1:

Ult_BF = Loss + (1 - %Paid) x Prem x ELR

U_BF = C_k + q_kU₀

• Iteration 2:

U_GB = Loss + (1 - %Paid) x Ult_BF

U_GB = C_k + q_kU_BF

Question 2

Mack (2000): Benktander as a Credibility-Weighting of the Chain Ladder & Expected Loss Ultimates

U_GB = ?

Answer 2

Chain Ladder Ultimate = Loss x CDF

U_CL = C_k ÷ p_k

Benktander:

q_k = 1 - (CDF)^-1

U_GB = (1 - q_k²)U_CL + q_k²U₀

Ultimate_GB = (1 - %Unpaid²)xUlt_CL + %Unpaid² x Prem x ELR

Question 3

Mack (2000): Benktander as a Credibility-Weighting of the Chain Ladder & BF Reserves:

R_GB = ?

Answer 3

Reserve_GB = (1 - %Unpaid) x Reserve_CL + %Unpaid x Reserve_BF

R_GB = (1 - q_k) x R_CL + q_k x R_BF

• Esa Hovinen Reserve = R_EH = cR_CL + (1-c)R_BF
• when c = p_k then R_EH = R_GB

Question 4

Mack (2000): What happens when we iterate between reserves and ultimates indefinitely?

Answer 4

• As the number of iterations increases, the weight on the chain ladder method increases until it converges to the chain ladder method entirely

Note: We are iterating the BF method to infinity where q(∞) goes to zero so 100% weight on CL method

Question 5

Mack (2000): What are the advantages of the Benktander method?

Answer 5

• Lower MSE
• Better approximation of the exact Bayesian procedure
• Superior to chain ladder since more weight is given to the priori expectation of ultimate losses
• Superior to BF method since it gives more weight to actual loss experience

Question 6

Mack (2000): When referring to the chain ladder method, what average are you using to calculate the age-to-age factor?

Answer 6

• Always use the all year volume weighted average

Question 7

Mack (2000): Given the following information for AY 2012 at 12 months, which reserve has the smaller MSE?

c* = 0.32

C_k = $3,000

U_CL = $5,000

Answer 7

• Neuhaus showed that MSE of R_GB is almost as small as the optimal credibility reserve, c*, unless p_k is small at the same time c* is large
• R_GB has a smaller MSE than R_BF when c*>p_k/2 holds
  
  • makes sense as c* is closer to c=p_k than it is to zero
• 0.32>0.3 so R_GB has smaller MSE

Question 8

Hurlimann: What is the main difference between Mack (2000) and Hurlimann estimate of claim reserves?

Answer 8

• Hurlimann uses expected incremental loss ratios (m_k) to specify the payment pattern rather than LDFs from actual losses
• Hurlimann uses two reserving methods:
  
  • Individual Loss Ratio Reserve (R^ind) - CL Method
  • Collective Loss Ratio Reserve (R^coll) - BF method
• Key Idea: R^ind and R^coll represent extremes of credibility on the actual loss experience so that we can calculate a credibility-weighted estimate that will minimize the MSE of the reserve estimate
  
  • provides an optimal credibility weight for combining the CL or individual loss ratio reserve with the BF or collective loss ratio reserve

Question 9

Hurlimann: Describe the notation that Hurlimann uses.

p_i = ?

q_i = ?

U_i^BC = ?

U_i^coll = ?

U_i^ind = ?

U_i^(m) = ?

….

Answer 9

p_i = loss ratio payout factor (loss ratio lag factor); proportion of total ultimate claims from origin period i expected to be paid in development period n-i+1

q_i = 1 - p_i = loss ratio reserve factor

U_i^BC = U_i⁽⁰⁾ = burning cost of total ultimate claims from origin period i

U_i^coll = U_i⁽¹⁾ = collective total ultimate claims from origin period i

U_i^ind = U_i^(∞) = individual total ultimate of claims for origin period i

U_i^(m) = ultimate claim estimate at the mth iteration for origin period i

R_i^coll = collective loss ratio claims reserve for origin period i

R_i^ind = individual loss ratio claims reserve for origin period i

R_i^c = credible loss ratio claims reserve

R_i^GB = Benktander loss ratio claims reserve

R_i^WN = Neuhaus loss ratio claims reserve

R_i = i^th period claims reserve for origin period i

R = total claims reserve

m_k = expected loss ratio in development period k (incremental)

n = number of origin periods

V_i = premium in origin period i

S_ik = paid claims from origin period i as of k years development where 1≤i, k≤n

C_ik = cumulative paid claims from origin period i as of k years of development

Question 10

Hurlimann: State the formulas for the following:

Total Ultimate Claims =

Cumulative Paid Claims =

i-th Period Claims Reserve =

Total Claims Reserve =

Answer 10

Total Ultimate Claims = Σ_k=iⁿ S_ik

Cumulative Paid Claims = C_ik = ΣS_ij (sum over j = 1, 2, …,k)

i-th Period Claims Reserve = R_i =Σ S_ik (sum over k = n-i+2, ….., n)

Total Claims Reserve = R = ΣR_i (sum over i=2, …., n)

Question 11

Hulimann: Expected Loss Ratio

m_k =

ELR =

Answer 11

Question 12

Hurlimann: What are the forumlas for the loss ratio payout factor and the loss reserve factor?

Answer 12

• different from Mack (2000) as this paper uses the loss ratios for p_i not the LDF’s

Question 13

Hurlimann: What are the formulas for the individual loss ratio claims estimate?

Answer 13

Question 14

Hurlimann: What are the formulas for the collective loss ratio claims estimate?

Answer 14

Question 15

Hurlimann: What are the crediblity-weighted loss ratio claim reserve formulas for Banktender, Neuhaus and Optimal?

Answer 15

Question 16

Hurlimann: Formulas for Optimal Credibility Weights (Simplified)

Answer 16

• When f=1, we get the simplified version of t^opt = √p
• the optimal credibility weight, z^opt, is the weight that minimizes the MSE between the actual reserve and the credibility reserve

Question 17

Hurlimann: What is the basic form of t_i^opt?

Answer 17

t_i^opt = E[⍺_i²(U_i)] / (var(U_i^BC) + var(U_i) - E[⍺_i²(U_i)])

• this is the generalized version that is not used often
• Hurlimann starts off with this form and eventually gets to the ‘f’ form by using distributions to estimate the parameters in the above equation.

Question 18

Hurlimann: What is an advantage of the collective loss ratio claims reserve over the traditional BF reserve?

Answer 18

• Different actuaries come to the same result if the same premiums are used, because judgement isn’t used to select the ELR.

Question 19

Hurlimann: How do the collective and individual loss ratio claims reserve estimates represent two extremes?

Answer 19

R^ind - 100% credibility is placed on the cumulative paid claims (C_i) and ignores the burning cost estimate (U^BC)

R^coll - places 100% credibility on the burning cost and nothing on the cumulative paid losses

Question 20

Hurlimann: The mean squared error for the credible loss ratio reserve is given by:

mse(R_i^c) =

Answer 20

• For collective loss ratio MSE, set Z=0
• For individual loss ratio MSE set Z = 1

Question 21

Hurlimann: Under the following assumption, what are the optimal credibility weights (Z*) that would minimize the MSE of the optimal reserve (R_i^c)?

Assumption:

Answer 21

• Basic Form

Question 22

Hurlimann: Under the assumption 4.4 (conditional for loss ratio payout) in the previous slide, what are the MSE for the following:

mse(R_i^coll) = ?

mse(R_i^ind) = ?

mse(R_i^c) = ?

Answer 22

mse(R_i^coll) = E[⍺_i²(U_i)] * q_i*(1 + q_i/t_i)

mse(R_i^ind) = E[⍺_i²(U_i)] * q_i/p_i

mse(R_i^c) = E[⍺_i²(U_i)] * [Z_i²/p_i + 1/q_i + (1-Z_i)²/t_i] * q_i² (basic method - can be used for all methods)

Question 23

Hurlimann: How would you estimate reserves for the Optimal Cape Cod Method?

Answer 23

• Use the loss ratio to get ultimate for R^coll and to get p_k use LDFs
  
  • LR = ΣC_{i, n-i+1} / Σp_i^CLV_i
  • the loss ratio takes the total paid losses to date and divides the losses by the earned premium to date which means premium is adjusted to be aligned with losses seen to date
• Formulas are the same for R^ind, R^coll, and Z using t^1/2
• The credibility weighted reserve is not the Cape Cod method. It’s a weighted method between the chain ladder and cape cod.

Question 24

Hurlimann: How would you estimate reserves for the Optimal BF Method?

Answer 24

• Replace the loss ratio that Hurlimann uses to get p_k with the CDF
  
  • LR_i =should be given to you as this is some selected initial loss ratio for an origin period
• p_k is derived using LDFs
• Formulas are the same for R^ind, R^coll, and Z using t = √p
• if not optimal, then Z = p
• the BF method is a Benktander-type credibility mixture where the loss ratio is pre-selected
• in the Cape Cod method, the loss ratio is derived using all years
• the credibility mixture does not equal the BF method; the collective reserves are equal to the standard BF reserves which is weighted with the chain ladder reserves.

Question 25

Hurlimann: Briefly describe 3 differences between Hurlimann’s method and the Benktander method.

Answer 25

1. Hurlimann’s method is based on a full development triangle; whereas, the Benktander method is based on a single accident year
2. Hurlimann’s method requires a measure of exposure (e.g. premiums) for each accident year
3. Hurlimann’s method relies on loss ratios rather than link ratios to determine reserves

Question 26

Hurlimann: Briefly describe one similarity between Hurlimann’s method and the Benktander method.

Answer 26

• Similar to the GB method, Hurlimann’s method represents a credibility weighting between two extreme positions:
  
  • relies on cumulative paid claims (i.e. individual loss reserves)
  • versus completely ignoring cumulative paid claims (collective loss reserves)

Question 27

Hurlimann: Explain why t^*_i = √p_i is an appealing choice when calculating optimal credibility weights.

Answer 27

t^*_i = √p_i ​

• this assumption yields the smallest credibility weights for the individual loss reserves which means more emphasis is placed on the collective loss reserves
• this is useful when little has been paid to date and we want to rely on the a priori estimate more

Question 28

Brosius: What is the formula for the least-squares method?

Answer 28

• if using calculator, be sure to bring y to ultimate!!!! Then use calculator functions to get a and b.

Question 29

Brosius: What is the formula for Least Squares as a credibility-weighting of the link ratio and budgeted loss methods?

Answer 29

ŷ = Z*(x/d) + (1-Z)*E[y]

ŷ = Z * LDF * x + (1 - Z) * y-bar

where:

c = LDF = y-bar/x-bar

Z = b/c

Question 30

Brosius: Draw a graph illustrating the esimates of developed losses for each method in Brosius.

Answer 30

Question 31

Brosius: In what special cases is the LSM equivalent to the link ratio or budgeted loss method?

Answer 31

• Budgeted Loss Method

If developed losses (y) are uncorrelated with undeveloped losses (x) then b=0 and L(x) = a

• Loss Ratio Method

If the regression line fits through the origin, so a = 0, and L(x) = bx

Question 32

Brosius: Explain how the LSM is more flexible than the BF method.

Answer 32

BF Method

• Ultimate losses are estimated as expected unobserved loss + actual observed loss

L(x) = a + x,

• NOTE: b = 1 then LSM is BF method
• similar to LS method as it provides a compromise between the link ratio and budgeted loss methods

LS Method

• Allows b to vary according to the data

L(x) = a + bx

• b is not constrained to 1
• LS method allows for negative development
• more flexible than BF as the least squares method will apply more or less weight to the observed value of x as appropiate (credibility weight)

Question 33

Brosius: What situations result in problems for estimating parameters for the Least Squares method?

Answer 33

1. Significant changes to the nature of the loss experience in the book of business
2. Normal sampling error will lead to variance in a and b estimates

*LSM can handle random fluctuations but not systematic shifts

Question 34

Brosius: For the Least Squares method, what are the problems if a<0 or b<0? How can this be corrected?

Answer 34

a < 0

• Estimate of developed losses (y) will be negative for small values of x
  
  • substitute the link ratio method instead
  • as intercept is negative, set to zero and then left with bx which is just the link ratio method

b < 0

• Estimate of y decreases as x increases
  
  • substitute the budgeted loss method instead to avoid negative development
  • makes sense as b is zero and only a is left and it doesn’t change - we put our faith in the budget estimate

Question 35

Brosius: Based on Hugh White’s Question, if reported losses are higher than expected, what are 3 different responses to estimated loss reserves and the reasoning behind each response?

Answer 35

If x is greater than E[X], possible responses to the loss reserve estimate are:

1. Reduce the reserve by the corresponding amount
   
   • We believe the loss reporting is accelerating so we keep the ultimates the same which draws down the IBNR (Budgeted Loss method)
2. Leave the reserve at the same percent of expected losses
   
   • Believe there was random fluctuation (e.g. large loss) but the ratio of the bulk reserve (IBNR) remains the same to losses (Bornheutter-Fergueson method)
   • The reported loss to date will increase, the un-reported will remain unchanged (or increase slightly due to %unreported factor) and therefore the ultimate goes up.
3. Increase the reserve proportionally to the increase in actual reported losses
   
   • We are not confident in the expected losses, E[X], so we allow the ultimate to increase as both incurred losses and IBNR increase (Link-Ratio method)
   • Applying the CDF to losses which will increase the ultimate as the loss to which the CDF is applied is higher

Question 36

Brosius: When is the LSM appropiate to use? When is it inappropiate?

Answer 36

• Appropiate if we have a series of years of data where we can assume stable distributions for X and Y
  
  • assume fluctuations driven by random chance
• Inappropiate if year-to-year changes are due to systematic changes in the book of business (e.g. shift in the mix of business)
  
  • Other methods such as the Berquist-Sherman may be better

Question 37

Brosius: What adjustments should be made to data prior to using Least-Squares development?

Answer 37

• If using incurred losses, correct data for inflation to put losses on a constant-dollar basis
• If there is significant growth in the book, divide losses by an exposure basis to correct the distortion

Question 38

Brosius: List the formulas needed to determine the ultimate loss where the system is changing and the ultimates is based on Bayesian Credibility.

(Development Formula 2)

Answer 38

• estimating Y by observing X where X is a random variable that is differently distributed though related to Y
• in other words, it’s a _credibility weighte_d formula between the link-ratio (x/d) and the budgeted estimate E(Y)

Question 39

Brosius: Briefly describe the caseload effect.

Answer 39

• When the caseload is low, a claim is more likely to be reported in a timely manner than when caseloads are high
• Thus, we would expect the development ratio (d) to decrease as y increases:

Expected Development Ratio = E[X|Y=y] / y = dy + x₀

Therefore, EDR = d +x₀/y

→decreases as y increases

Question 40

Brosius: What is the credibility development formula that allows for the caseload effect?

(Development Formula 3)

Answer 40

Question 41

Brosius: Why would you use a linear approximation to the Bayesian credibility as described by Buhlmann?

Answer 41

Pros

1. simpler to compute
2. easier to understand and explain
3. less dependent upon underlying distributions

Cons

1. less accurate than Bayes

Question 42

Brosius: What is the best linear approximation, L(x), to the Bayesian estimate Q(x)?

(Development Formula 1)

Answer 42

Formula for the best linear approximation to Q(x), the Bayesian estimate:

Question 43

Brosius: Using the best linear approximation, Q(x), how does the relationship between Cov(X,Y) and Var(X) impact the response to loss reserves for a large reported loss?

Answer 43

A large reported loss (increasing x) changes the loss reserves according to the 3 different answers to Hugh White’s question.

For x > E[X]:

• Cov(X,Y) < Var(X) → loss reserve decreases
• Cov(X,Y) = Var(X) → loss reserve unaffected since ultimate loss increases by the increase to x
• Cov(X,Y) > Var(X) → loss reserve increases

Question 44

Brosius: There are a few models we can use to test the validity of the least-squares method. One of those tests is the simple model. Describe the steps used to derive the reserve.

Answer 44

Steps:

1. Calculate (numerator): P(Y=y, X=x)
2. Calculate the conditional probablity: P(Y|X=x) which is step 1 divided by P(X=x)

• sum up each row in the table where each row is x
• then divide step 1 by the row total

3. Calculate Q(x) - this is the E[Y|X=x]
4. Calculate R(x) - this is Q(x) - x or E[Y-X|X=x]

Question 45

Brosius: There are a few models we can use to test the validity of the least-squares method. One of those tests is the general Poisson-Binomial case. Describe the test.

Answer 45

Assume Y is Poisson distributed with mean µ where any given claim has probability of being reported, d, by year-end. (X is BIN(Y,d))

Q(x) = x + µ(1-d)

R(x) = µ(1-d)

• no x in R(x) formula
• notice that the link ratio doesn’t work for this method
• Therefore, since the expected number of outstanding claims, R(x), does not depend on the number of claims reported to date, the Bornhuetter Ferguson estimate is optimal in the Poisson-Binomial case

Question 46

Brosius: There are a few models we can use to test the validity of the least-squares method. One of those tests is the Negative Binomial-Binomial case. Describe the key findings of the test.

Answer 46

Assume Y is Negative Binomial distributed with (r, p) where any given claim has probability of being reported, d, by year-end. (X is BIN(Y, d))

• This is an increasing linear function of x (except when d=1)
  
  • since increasing this won’t be budgeted loss method or BF method
• An increase in reported claims results in an increase to our estimate of outstanding claims; however since this is not proportional, the link ratio method is not optimal
• Therefore, the least squares is a good model

Question 47

Brosius: There are a few models we can use to test the validity of the least-squares method. Discuss the fixed prior case and fixed reporting case tests.

Answer 47

Fixed Prior Case Test

• Suppose Y is not random and is equal to some value called k
• thus, Q(x) = k and R(x) = k-x
• this is the budgeted loss method

Fixed Reporting Case Test

• Suppose there is a number d≠0 such that the percentage of claims reported by year end is always d
• thus, Q(x) = x/d and R(x) = x/d -x
• this is the link ratio method as 1/d is the link ratio
• this corresponds to the link ratio method

Question 48

Brosius: Regarding the Negative Binomial-Binomial case, why does R(x) being an increasing function make sense?

Answer 48

• the NB has more variance than Poisson distribution with the same mean which means we have less confidence in the a priori estimate of expected loss, and therefore are more willing to increase our estimated ultimate claim count

Question 49

Brosius: What is EVPV and VHM?

Answer 49

• EVPV - expected value of the process variance is variability arising from the loss reporting process (distrust of the claims department?)
  
  • EVPV = E[σ_X/Y² * Y²] = Var(X/Y) *(Var(Y) + E(Y)²)
• VHM - variance of the hypothetical mean is the variability arising from the loss occurrence process (distrust of the UW? Depends on where we write business, type of business, etc.)
  
  • VHM = Var(E(X/Y)*Y) = E(X/Y)²*Var(Y)

Question 50

Brosius: Explain why R(x) decreases as x increases in the context of a non-linear Q(x).

Answer 50

• R(x) will decrease if we have more confidence in the prior estimate of expected losses as we are less likely to change our estimate based on what has emerged (i.e. reported) so far.
• Support: compare to poisson distribution and if Var(Y) is less than the variance from the Poisson distribution with the same mean then we will have more confidence in losses seen to date. This results in a decreasing R(x) as we continue to reach ultimate which is relatively unchanging.
• See question MP 7 b

Question 51

Brosius:

What is the problem with assuming that:

E[X|Y = y] = dy + x_o giving us

E[X|Y = 0] = x_o > 0 ?

Answer 51

• If no claims come in for the year we still get a number for x₀
• This doesn’t make sense

*(Development Formula 3)

Question 52

Patrik: Reinsurance loss reserving has many of the same issues as primary insurance loss reserving, and many of the same methods can be used. However, there are 7 major technical problems that make reinsurance loss reserving more complicated. List the 7 technical issues as described by Patrik.

Answer 52

1. Claim report lags to reinsurers are generally longer
   
   • ​​Longer reporting pipeline: Reported claim is perceived as reportable to reinsurer→Claim goes to reinsurance accounting→Claim goes to reinsurer→Claim is booked in reinsurer system
2. There is a persistent upward development of most claim reserves.
   
   • Due to economic/social inflation, underestimating ALAE for larger claims
3. Claims reporting patterns differs by reinsurance line, by type of contract and contract terms, by cedant, and possibly by intermediary. (Heterogeneity)
4. Industry statistics are not very useful.
   
   • Due to the heterogeneity
5. The reports the reinsurer receives may lack important information
   
   • Reinsurers receive summary claims data for proportional covers. Data is usually reported quarterly in arrears.
6. Reinsurers often have data coding and IT system problems
   
   • Due to heterogeneity in coverage and reporting requirements
7. The size of an adequate loss reserve compared to surplus is greater for a reinsurer.

Question 53

Patrik: What are the components of a reinsurer’s loss reserve?

Answer 53

1. Case reserves reported by the ceding companies
2. Reinsurer’s additional case reserves (ACR)
3. IBNER
4. IBNYR - pure IBNR
5. Discount for future investment income
6. Risk Load

Question 54

Patrik: List the important variables for partitioning the reinsurance portfolio.

Answer 54

1. LOB - property, casualty, bonding, etc.
2. Type of contract - facultative, treaty, finite
3. Type of reinsurance cover - quota share, surplus share, excess per-risk, excess per-occurrence, aggregate excess, catastrophe, etc
4. Primary line of business - for casualty
5. Attachment point - for casualty
6. Contract terms - flat rated, retro-rated, sunset clause, claims-made, occurrence coverage, etc.
7. Type of cedant - small, large, or E&S company
8. Intermediary

Question 55

Patrik: What are the disadvantages of using the chain ladder or BF methods for reinsurandce reserving?

Answer 55

Disadvantages of the CL Method

• Highly-leveraged LDFs for recent years for long tail lines result in extremely variable IBNR estimates due to few reported or paid losses-to-date

Disadvantages of BF Method

• Highly dependent on selected loss ratio
• The IBNR estimate for each AY doesn’t reflect actual loss experience unless the selected loss ratio is chosen to reflect it

Question 56

Patrik: What are the advantages and disadvantages of the Cape Cod method (Standard-Buhlmann)?

Answer 56

Advantages

• Ultimate ELR for all years combined is calculated from the overall experience, not judgementally selected
• The IBNR estimate is more stable for the most recent years compared to the CL method

Disadvantages

• IBNR is highly dependent upon rate-level adjusted premium by year

Question 57

Patrik: What adjustments are needed for the premium used in the Cape Cod method?

Answer 57

Earned Risk Pure Premium (ERPP)

• Premium net of reinsurance commission, brokerage fees, and internal expenses

Adjusted Premium

• ERPP adjusted to remove rate-level differences by year
• The SB method assumes a constant ELR by accident year
  
  • we need to use on-leveled adjusted premium

Question 58

Patrik: What are the reserve estimation methods for the different exposure categories?

Answer 58

Short-Tailed Exposures

1. Set IBNR to a percent of last year’s earned premium
   
   • useful for non-CAT exposures
2. Reserve up to a selected loss ratio

• e.g. treaty property proportional, treaty property catastrophe, treaty property excess

Medium-Tailed Exposures

1. Chain ladder can be a good method
   * e.g. treaty property excess higher layers, construction risks

Long-Tailed Exposures

1. BF method or Cape Cod method may be appropiate
   * e.g. treaty casualty excess, treaty casualty proportional, falcultative casualty, Asbestos, etc.

Question 59

Patrik:

ELR = ?

IBNR_AY = ?

ULR = ?

Answer 59

*When calculating the ultimate, you must add IBNR to reported loss to date - do NOT assume that ELR * Premium is the ultimate. CC is similar to BF in that = reported + unreported

Question 60

Patrik: Stanard-Buhlmann (Cape Cod) Method

IBNR_CL = ?

Z_AY =

IBNR_Cred =

Answer 60

Question 61

Patrik: What are the IBNR monitoring formulas?

Answer 61

Question 62

Patrik: What is the typical procedure for reinsurance loss reserving?

Answer 62

1. Partition the reinsurance portfolio into reasonably homogenous exposure groups that are relatively consistent over time with respect to mix of business
2. Analyze the historical development patterns
3. Estimate the future development. If possible, estimate the bulk reserves for IBNER and pure IBNR separately

Question 63

Patrik: When looking at the AvE and you notice that more claims emerged than what was expected, what would you conclude?

Answer 63

• Need to understand if its due to random fluctuations
• Is it consistent for all years and similar emergence across quarters? Could mean that IBNR was too low at year end
• Could also indicate a change in the development pattern
  
  • i.e. the lags were too short/high (so CLDF too low)