Section A: Unpaid Claims for Layers

Question 1

Siewert: What are the benefits of high deductible programs?

Answer 1

• Price flexibility with the additional risk passed to larger insureds
• Improved the residual market charges and premium taxes in some states
• Cash flow advantages similar to those of paid loss retro policies
• Provides insureds a way to control losses while protecting against random large losses
• Allows for easier “self-insurance” for insureds

Question 2

Siewert: How do you estimate the overall reserve while reflecting the different the mix of deductibles and limits?

Answer 2

After selecting appropriate development factors, apply them at the account level using each account’s deductible and limits.

Then you can aggregate the estimated ultimate over all accounts.

Question 3

Siewert: In absence of credibility development histories, a common approach for determining liabilities is using loss ratios. How do you select a loss ratio for the Loss Ratio Method?

Answer 3

• Use the company experience by state and calculate the full-coverage loss ratio using an individual account’s premium distribution by state
• We might blend that experience with industry experience using credibility

Note: Loss experience should be developed to ultimate, brought on level and trended to the appropriate exposure period for calculating loss ratios.

Question 4

Siewert: Loss Ratio Method

Estimate of per-occurrence Excess Losses

Answer 4

Deductible Loss Charge = Prem • ELR • 𝛘

𝛘 = per occurrence charge (excess ratio)

Question 5

Siewert: Loss Ratio Method

Estimating the Aggregate Loss Charge

Answer 5

Aggregate Loss Charge = Prem • ELR • (1-𝛘) • ø

𝛘 = per occurrence charge (excess ratio)

ø = per-aggregate charge (aggregate ratio)

Question 6

Siewert: Loss Ratio Method

What are the advantages? Disadvantages?

Answer 6

Advantages

• Useful when no data is available or data is very immature
• Loss ratio estimates can be consistently tied to pricing, at least in the beginning they can be
• Relies on a more credible pool of company and/or industry experience

Disadvantages

• Ignores emerging experience
  
  • not very useful after several years of development
• May not properly reflect account characteristics and losses may develop differently due to the type of exposures written

Question 7

Siewert: What is the Implied Development Method?

Answer 7

Determine excess development implicitly by:

1. Develop full coverage losses to ultimate
2. Then, develop deductible losses to ultimate by applying development factors with inflation indexed limits
3. Take the difference between the full coverage ultimate and the limited ultimate losses to derive the excess ultimate losses

Ult_xs = Ult_unlimited - Ult_limited

Resv_xs = Ult_xs - (Loss_unlimited - Loss_limited)

Note:

• Unlimited loss tail factor should be consistent (higher)with limited tail
• Limited LDFs must be calculated to reflect inflation-indexed limits at different accident years

Question 8

Siewert: Implied Development Method

To calculate limited LDFs for deductible loss, the limits need to be indexed for inflation.

Why?

What are 2 ways to determine the index?

Answer 8

To calculate limited LDFs for deductible loss, index limits for inflation:

• This keeps the proportion of deductible/excess losses constant around the limit from year to year
• Otherwise, a constant deductible implies increasing excess losses

Possible ways to determine the index:

1. Fit a line to average severities over the long-term history
2. Use an index that reflects the change in annual severity

Question 9

Siewert: What are the advantages of Implied Development?

What are the disadvantages?

Answer 9

Advantages

• Provides an estimate for excess losses at early maturities, even when excess losses haven’t emerged
• LDFs for limited losses are more stable than LDFs for excess losses

Disadvantages

• Misplaced focus, because we would like to explicitly recognize excess loss development

Question 10

Siewert: What is the Direct Development Method?

Answer 10

This approach explicitly focuses on excess development.

• Given the limited and full coverage LDFs, there are XSLDFs that balance limited and excess development with full covreage development

Ult^excess = Loss^excess • XSLDF

Question 11

Siewert: What are the disadvantages of directly determining excess development factors and applying them to excess losses?

Answer 11

Disadvantages

• XSLDFs can be quite leveraged and volatile so they can be difficult to select
• If there is no excess loss emergence then we can’t estimate the ultimate

Question 12

Siewert: Formulas for the Credibility-Weighted method for determining excess liabilities.

Answer 12

Deductible Loss Charge = Prem • ELR • 𝛘

Z_BF = 1 / XSLDF

Ult^excess = Z x (Loss • XSLDF) + (1 - Z) x E[Loss]

Ult = Cred•Ult_{Direct Development} + (1-Cred)•Ult_{Loss Ratio Method}

Question 13

Siewert: What are the advantages & disadvantages of the credibility-weighted method for determining excess liabilities?

Answer 13

Advantages

• Ties with pricing estimates for immature years where excess losses have not emerged
• Estimates are more stable over time compared to direct development

Disadvantages

• Ignores actual experience for the complement of credibility
  
  • Might use alternative credibility-weights that are more responsive to actual experience if desired

Question 14

Siewert: What is Limited Severity Relativity?

Answer 14

Limited Relativity Severity

• Ratio between limited and unlimited severity

R_t^L = Severity Limited to limit L at age t / Unlimited Severity at age t

R^L = Severity Limited to limit L at Ultimate / Unlimited Severity at Ultimate

Question 15

Siewert: What is the relationship between the limited LDF, unlimited LDF and severity relativities?

Answer 15

Question 16

Siewert: What is the relationship between XSLDF, unlimited LDF and severity relativities?

Answer 16

Question 17

Siewert: What is the relationship between the limited excess and unlimited LDFs?

Answer 17

LDF_t = R_t^L • LDF_t^L + (1-R_t^L) • XSLDF_t^L

Question 18

Siewert: Describe the relationship between limited severity relativities over time in words and using a graph.

Answer 18

• Severity relativity should decrease as age increases
  
  • more losses are capped at the per-occurrence limit as age increases
• Severity relativity should be higher for higher limits and should decrease more slowly than you would see for lower limits
  
  • A higher limit means less of the loss is capped so the relativity is higher

Question 19

Siewert: One option for estimating reserves for an excess layer is to use a distributional model.

How does this model work?

Answer 19

• A distributional model works by modeling the development process by determining the distribution of parameters that vary over time.
• Once the parameters are determined, we can calculate severity relativities.
• Comparing these relativities over time results in development factors.

Question 20

Siewert: One option for estimating reserves for an excess layer is to use a distributional model.

Identify 3 methods for estimating the parameters of a distributional model.

Answer 20

• Method of Moments
• Maximum Liklihood
• Siewert’s Approach - minimize the chi-square between the actual and expected severity relativities around a particular deductible size

Question 21

Siewert: One option for estimating reserves for an excess layer is to use a distributional model.

What are two advantages of using a distributional model?

Answer 21

1. Provides consistent LDFs
2. Allows for interpolation among limits and years

Question 22

Siewert: What formula does Siewert derive that shows the expected development partitioned around the deductible limit?

Answer 22

Question 23

Siewert: What does the following graph show?

(Based upon Weibull Distribution)

Answer 23

The graph shows that as development age increases, an increasing proportion of development is the excess the deductible limit.

Question 24

Siewert: What is the aggregate loss limit?

Answer 24

Aggregate Loss Limit

Limits the total losses below the deductible that are paid by the insured

• similar to maximum premium

Question 25

Siewert: How does Siewert suggest coming up with LDF’s for aggregate limits?

Answer 25

Collective Risk Model

1. Model frequency & severity separately
   
   • Siewert uses Weibull for severity and Poisson for claim counts
2. Combine frequency & severity into a collective risk model
3. Sample from collective risk model to calculate development factors
   
   • Might improve by including parameter risk in the model
4. Use BF method with LDFs from model to estimate losses excess the aggregate limit

Question 26

Siewert: What observations did Siewert make regarding the development of losses excess of aggregate limits?

Answer 26

• Development for losses excess aggregate limits decreases more rapidly over time with smaller deductibles than larger ones
  
  • This is because most later development is in layers of loss above the deductible limit and not subject to the aggregate limit
• Development is more leveraged for larger aggregate limits
  
  • Takes longer for losses to breach the aggregate limit

Question 27

Siewert: What method does Siewert suggest to use to smooth out indications of ultimate liability for aggregate limits?

Answer 27

• Use the Bornhuetter-Ferguson method to smooth out indications of ultimate liability

Question 28

Siewert: Siewert mentions revenue item that should be reflected on the asset side of the balance sheet from a higher deductible program.

What is this item?

Answer 28

Service Revenue

Revenue for an insurer to service claims under a high deductible program

Question 29

Siewert: What are the Service Revenue Formulas?

Answer 29

Step 1: Determine ultimate deductible losses at the account level

Step 2: Find the Ultimate Net of aggregate limits

Ult_{Net of Aggregate} - Ult_Deductible - Ult_{XS Agg}

Step 3: Determine Ultimate Recoverable’s

Ult_Recoverables = Ult_{Net of Aggregate} • Loss Multiplier

Step 4: Determine the total asset

Service Revenue Asset = Ult_Recoverables - Known Recoveries

Question 30

Siewert: What are the two principal ways of handling allocated claim expense under a higher deductible program?

Answer 30

1. Account manages the expense
   
   • Development patterns used should be loss-only
2. Treated as loss and subject limits
   
   • Should use loss + ALAE for development patterns

Question 31

Siewert: What tail factor distribution does Siewert use? What are the advantages and disadvantages of this method?

Answer 31

Siewert uses an inverse power curve with 3 parameters:

y = 1 + a•(t+c)^-b

• Fit the above curve on unlimited age-to-age factors to project unlimited ultimate losses
• Select projection age that development should stop as power curve will go on forever
• Once ultimate age is selected, fit the curve to each deductible limit and extend that to a common maturity
  
  • must compound fitted factors to get the ultimate

Advantage - consistent for each limited and produces uniformly decreasing tail factors

Disadvantage - its bias due to extending each limit to the same maturity (lower limits should fully develop much sooner than higher limits)

Question 32

Sahasrabuddhe: When is it appropriate to use development patterns from unadjusted data and apply them to claims for all exposure periods?

Answer 32

• When claims data is ground-up unlimited, and
• When trend only acts in the accident year direction

Question 33

Sahasrabuddhe: What model does the paper use to derive the claim size?

Answer 33

Φ ~ EXP(𝜃)

LEV(X) = 𝜃 (1 - exp^-x/𝜃)

In other terms:

LEV(Limit) = Trended Mean x (1 - e^{-(Limit / Trended Mean)})

Question 34

Sahasrabuddhe: How do you adjust the observed data to a common cost level and to basic limits?

Answer 34

Use the following formula:

Adjusted Data_i,j = Raw Data_i,j • (LEV_n,j at BL / LEV_i,j at Raw Data Limit)

Note 1: the observed data is capped at the upper limit and you need to get this to the basic limit so you can calculate the LDFs at the basic limit. The reason we do this is because the data is more credible at the basic limit than the limit in the unadjusted data.

Note 2: Since data is trended, we are going from the raw limit at AY i to the basic limit at AY n

Question 35

Sahasrabuddhe: What formula is needed to calculate the development patterns for any layer and cost level?

Answer 35

Let X_i,j = Limit we are trying to get to for accident year i and development period j

Let Y_i,j = Basic limit at accident year i and development period j

F = Cumulative LDF or CDF

Question 36

Sahasrabuddhe: Simplified model for calculating development pattern for other layers and cost levels:

R_{AY, Ult} = ?

F_AY,k ^X = ?

Answer 36

R_AY,Ult = R_j(X, Y) = LEV(X_i,j) / LEV(Y_i,j)

with Decay

R_j(X,Y) = U + (1-U)*Decay Ratio

CDF

Question 37

Sahasrabuddhe: When do you use the simplified model?

Answer 37

When claim size models by development are unknown

Question 38

Sahasrabuddhe: Define R_j(X, Y)?

Answer 38

R_j(X, Y) is the ratio between the limited expected values for layer X and Y at the end of development interval j

Question 39

Sahasrabuddhe: What are the properties of R_j?

Answer 39

If the new layer, X, is lower than the basic limit, the R_j < 1 and we should see the following:

• R_j decreases as age increases - this is because there will be less development in the excess layer for earlier maturities
• R_j > U where U = lim (j→∞) R_j, calculated as the ratio of LEVs at ultimate between layer X and Y
  
  • U is simply R_j at ultimate
  • R_j should keep decreasing since more development at older maturities will be above layer X
• If the base CDFs are from the ground-up unlimited layer, then max(R_j) = U • CDF

Question 40

Sahasrabuddhe: Discuss the 5 assumptions that must be met in order to implement the standard reserving procedures described in the paper.

Answer 40

1. Must select a basic limit that is sufficiently credible for development factors
2. Procedure requires the use of a claim size model
3. Procedure requires that the data triangle be adjusted to a basic limit and common cost level
4. Requires claim size models at maturities prior to ultimate
5. Requires a triangle of trend indices

Question 41

Sahasrabuddhe: Briefly describe 3 problems with the current application of trend rates.

Answer 41

Current issues with how trends are applied or not applied:

1. Trend rates tend not to vary by AY
2. Trend that occurs in the development period or calendar period direction is often not considered
3. Trends don’t really vary by claims layer

This paper shows how to come up with trends in a triangle (AY and CY).

Question 42

Sahasrabuddhe: What are 2 assumptions that could become burdensome in the model?

Answer 42

• Claim size models for maturities prior to ultimate are generally not available but necessary for the method
• A table of trend indices is required for cumulative claim activity, but:
  
  • trend typically applies to incremental claims
  • impact of trend on reported incurred claims and the timing of trends effect on case reserves is difficult to ascertain

Question 43

Sahasrabuddhe: Given the development pattern of the limited factors, is it appropriate to use the pattern on all exposure periods?

Answer 43

• Limited development patterns are a function of maturity AND cost level so they shouldn’t be used for all exposure periods
• Adjusting the development patterns to different exposure periods may be immaterial when:
  
  • The development pattern is short
  • Trend rates are low
  • Limits are above the working layer

Question 44

Sahasrabuddhe: What is Sahasrabuddhe’s key finding?

Answer 44

Development factors at different cost levels and different layers are related to each other based on claim size models and trend.

Question 45

Sahasrabuddhe: Describe the steps underlying the basic reserving procedure.

Answer 45

1. Calculate the trend indices using AY and CY trends
2. Develop claim size model parameters at the latest exposure period cost level
3. Calculate the claim size model parameters for each exposures/development period combination by applying the trend indicates to the claim size parameters for the latest exposure period.
4. Calculate the limited expected values for the BL and limit in the triangle using the trended claim size model parameters
5. Determine the adjusted cumulative claims triangle
6. Derive the CDF’s from the adjusted cumulative claims triangle
7. Calculate the CDF’s at the various layers
8. Use the new development factors to derive the reserves by layer and AY

Question 46

Teng & Perkins:

What is the formula for premium on a retro policy?

Answer 46

On a retro policy, premium is calculated as a function of loss:

P_n = [BP + (CL_n x LCF)] x TM

Question 47

Teng & Perkins: Retro Formulas

PDLD in First Adjustment Period

Answer 47

PDLD₁ = [BP + CL₁ x LCF] x TM ÷ L₁

Can also be written as:

PDLD₁ = [BP/L₁ x TM] + [CL₁/L₁ x LCF x TM]

Question 48

Teng & Perkins: Retro Formulas

PDLD Formula for Second or Subsequent Retro Adjustment Periods

Answer 48

PDLD_n = (Change CL_n / Change L_n) • LCF • TM

(Change CL_n / Change L_n) ⇒ Incremental Loss Capping Ratio

Question 49

Teng & Perkins: What is a premium asset?

Answer 49

This is the premium that the insurer expects to collect based on the expected ultimate loss experience less the premium that the insurer already has booked.

Premium Asset = Ultimate Premium - Booked Premium

Ultimate Premium = ΣCPDLD_i*Expected Loss to Emerge_i + Premium booked at last Retro Adjustment_i

Question 50

Teng & Perkins: Why are Retro policies popular?

Answer 50

1. Promotes loss control and loss management - insured gets premium returned for good loss experience
   
   • good for insurer as well since this attracts preferred customers
2. Cashflow Advantage - allows insureds to pay premium as losses are reported or paid
3. Shifts Risk to Insured - premium varies with insured’s loss experience so the worse experience the higher the premium
   
   • increasing difficultly in estimating cost of insurance so this benefits the insurer as the risk is shifted to the insured

Question 51

Teng & Perkins: What is the meaning of the cumulative loss capping ratio?

Answer 51

• Represents capped losses that contribute to additional premium
  
  • this would be any loss that falls in a range: higher than the retro minimum but below the maximum
• The difference between the capped loss and the uncapped loss is the portion of the loss outside the boundaries of the retro minimum and maximum (loss elimination ratio which means 1-LER would be the capped loss)
• Loss capping ratio decrease as the data becomes more mature

Question 52

Teng & Perkins: What does it mean when the loss capping ratio decreases as the data matures?

Answer 52

• an increasing portion of the loss development occurs outside of the loss limitations, so outside the max and min retro limits
• this results in a lower loss capping ratio

Note: Careful here, this doesn’t say that less losses are capped, this says that losses outside the range are still developing and as they are more severe they take longer to develop. Losses that are capped are more likely to develop quicker so the ratio will decline as it focuses on capped losses only.

Question 53

Teng & Perkins: List the advantages and disadvantages of the Retro PDLD formula.

Answer 53

Advantages

• Responsive to changes in the retro rating parameters that are sold
  
  • if parameters change significantly, should give more weight to retro formula PDLD ratios that those from historical data

Disadvantages

• Must select retro rating parameters which can be difficult as parameters may vary between policies sold
  
  • Potential for bias may exist since parameters are changing and we are using average parameters
  • Can check for this by retrospectively testing PDLD ratios against actual emergence

Question 54

PDLD Ratios: Empirical Formulas for Period 1 and Subsequent

Answer 54

PDLD₁ = Prem^0-27 / Loss^0-18

PDLD_n = Change Prem / Change Loss

Question 55

Teng & Perkins: What does an upward trend mean when examining empirical PDLD ratios?

Answer 55

• More liberal retro rating parameters - such as a higher maximum, minimum or per accident limitation
• Improvement in loss experience which results in a larger portion of loss being within the boundaries of the retro maximum and the per accident limitation which drives additional premium
  
  • additional premium is charged when the insured has more losses in the boundaries as we can’t charge premium for losses that exceed the retro maximum

Question 56

Teng & Perkins: What causes the historical PDLD ratios to fluctuate significantly after the first retro adjustment?

Answer 56

• Development on a few policies can drive total incremental development (both premium and loss development)
• Negative PDLD ratios are possible - could see upward development in high loss layers AND downward development in layers within loss limitations
  
  • upward development in high loss layers doesn’t add premium
  • downward development in layers could result in return premium

Note: if fluctuations exist, average as many historical points as possible or use the formula approach

Question 57

Teng & Perkins: What could lead to divergence between the historical and formula PDLD ratios?

Answer 57

• Worse than expected loss experience could lead to a larger portion of the loss to be outside the boundaries of the retro maximum and the per accident limitation than the formula approach predicted (vice versa)
• Average retro parameters may be changing over time (remember parameters are based on an average)

Question 58

Teng & Perkins: CPDLD Formula

Answer 58

CPDLD_n = ΣPDLD_n • % Incremental Loss Emerged / Σ%Total Loss Emerged

• Weighted average of PDLD ratios with incremental loss emergence

Question 59

Teng & Perkins: What does the CPDLD ratio say according to Teng & Perkins (not Feldblum)?

Answer 59

Tells an insurer how much premium it can expect to collect for a dollar of loss that has yet to emerge

e.g. 1.63 can be seen as for every dollar of loss emerging, we can expected $1.63 in premium

Question 60

Teng & Perkins: Why is the first CPDLD ratio usually greater than 1?

Answer 60

• First retro premium computation includes the basic premium
• Only small portion of loss is limited at this point
• Application of LCF and TM results in higher dollar of premium per dollar of loss

→Subsequent adjustments should be less than 1 due to the retro max and per accident limitation

Question 61

Teng & Perkins: Premium Asset Formulas

Expected Future Loss Emergence_PY = ?

Expected Future Premium_PY = ?

Ultimate Premium = ?

Prem Asset = ?

Answer 61

Expected Future Loss Emergence_PY = Ult Loss_PY - Loss at Prior Adjustment_PY

Expected Future Premium_PY = Expected Future Loss_PY • CPDLD_n

Ultimate Premium = Expected Future Premium + Booked Premium at Prior Adjustment

Prem Asset = Ultimate Premium - Current Booked Premium

Question 62

Teng & Perkins: Graphical Representation of the Fitzgibbon method vs. the “Enhanced” PDLD method

Answer 62

Question 63

Teng & Perkins: Fitzgibbon Method Formulas

Answer 63

Question 64

Teng & Perkins: What are the advantages of the PDLD method?

Answer 64

1. Easy to explain since its based on the retro rating formula
2. Emphasis on premium sensitivity is similar to how loss-sensitive contracts are handled under Risk-Based Capital
3. Useful when indications from other methods are distorted due to changes in the retro rating plan parameters

Question 65

Teng & Perkins: What are the disadvantages of the Fitzgibbon Method?

Answer 65

Disadvantages

1. There is no way to correct the method if sensitivity of the retro-premium to date is less responsive than expected since the method estimates the ultimate premiums from cumulative losses to date
   
   • ignores emerging experience
2. Fitzgibbon uses a constant slope, but we’d expect the slope to decrease for more mature losses and higher overall loss ratios because of loss limits and maximum premiums

Question 66

Teng & Perkins: Why is the PDLD method better than the chain-ladder when it comes to estimating the Premium Asset?

Answer 66

1. Timeliness - can estimate ultimate loss sooner than retrospective premiums and update the premium asset estimate quarterly as new data becomes available
2. Retrospective premiums depend on the incurred losses

Question 67

Teng & Perkins: What are the PDLD assumptions?

Answer 67

• Premium responsiveness for future adjustment is independent of premium responsiveness of past adjustments
• The slope of the line segment (premium responsiveness) is independent of the beginning loss ratio and the beginning retro premium ratio

Question 68

Teng & Perkins: What are the general rules for premium responsiveness?

Answer 68

• Premium responsiveness decreases as the book matures
  
  • Larger percent of losses are excluded by loss limits and max premium
• Premium responsiveness decreases for higher overall loss ratios
  
  • Policies are more likely to hit max premium

Question 69

Teng & Perkins: What are 2 ways to interpret the Fitzgibbon graph of Y = A + B•X?

Answer 69

1. Relates the ultimate loss ratio and the ultimate retro premium ratio among different books of business or different years (Berry & Fitzgibbon)
2. Relates the reported loss ratio and the net EP at different points of time for a single book of business (Teng & Perkins)

Question 70

Teng & Perkins: The Teng & Perkins plot assumes the first line segment passes through the origin. What are the issues with this assumption? What does Felblum suggest to correct this?

Answer 70

• by assuming that the first line segment passes through the origin, Teng & Perkins are combining two separate items:
  
  • Basic premium ratio
  • True slope of first line segment (PDLD1)
• Problem 1 - cannot tell how much each item contributes to the total slope of the first line segment
• Problem 2 - results in confusing interpretations; e.g. say that we expect to get 1.42 of premium per emerging loss but that would include the basic premium which does not represent the emerging loss
• Problem 3 - harder to analyze changes over time

Feldblum suggest that we subtract the average basic premium charge from the first CPDLD ratio. This would give a flatter slope and positive y intercept.