Siewert: LDFs for high deductibles

Study Strategy¶

Checklist¶

My Notes¶

Why would corporate opt for a high-deductible policy? (Workers' comp)
- To have decreased premium "I can cover for smaller losses"
- "Just cover me for the larger ones"
- "Payment and Reimbursement" instead of seeking indemnity

Issues in Reserving for High-D policies

Losses may not develop above the deductible for many years. So in the earlier maturities no losses have developed (development techniques gone for a toss)
Not long enough development history FOR a relatively NEW PRODUCT
Deductibles size and mix vary (between policies)
LDFs between limited and excess need to be consistent
Need to account for inflation → Index deductible limits over time
Aggregate loss limit liabilities need to be calculated

The last (7) point is #doubt

Notation

\(LDF_{t}^L\) → LDF for limited losses under deductible \(L\), from \(t\) → Ult.
\(XSLDF_{t}^L\) → LDF for excess…
\(LDF_{t}\) → for Unlimited losses
\(R_{t}^L\) → Severity relativity for limited losses under \(L\), at \(t\)
\(R^L\) → Severity relativity… at Ult.

Loss Ratio Method¶

\(\chi\) → Per-occurrence excess charge → determine % of losses that exceeded the deductible
Better to determine account-based excess ratios (reflect unique state and hazard group¹ characteristics)

\[ L_{\text{xs}} = P \cdot E \cdot \chi \]

Workers' comp also have an aggregate limit → to prevent large accumulation below deductible. "I don't want to be responsible for that many claims"
Aggregate policy limit ensures we aren't paying more than that limit for smaller losses.
- Say, totally losses under deductible aggregated to 50M and aggregate limit = 5M, then we just need to pay for 5M and the rest of the 45M will be covered by the aggregate policy.
- Per aggregate charge → \(\phi\).

\[ L_{\text{agg}} = P \times E \times (1- \chi) \times \phi \]

\[ \text{Policy Losses} = L_{\text{xs}} + L_{\text{agg}} \]

My #doubt → Does the aggregate policy cover for everything above the limit? or is it a fixed percentage \(\phi\)?

Implied Development¶

\[ \text{Ult XS Loss} = \text{Ult Unlimited Loss} - \text{Ult Limited Loss} \]

Warning

"Full coverage" factor should be consistent with "limited loss" factor i.e.
Limited losses < unlimited loss
Lower limit losses < higher limit losses

Indexing deductibles to account for inflation¶

Logic

Inflation = 5% (say)
Prop of losses over 100k deductible (this year)
\(\equiv\) prop of losses over 105k deductible (next year)

How to determine the index?
- Fit a line to average severities over a long-term history
- choose index → reflect annual severity change movements
- adjust for large claims → prevents distortions

Direct Development¶

Find XS development factors → Calculate Ult XS losses (incorporating XS reported to date and the LDFs found)
Severity relativity \(R^L\) is given by \(\dfrac{\text{Severity for Limited Losses}}{\text{Severity for Unlimited Losses}}\)

Given severity relativities \(R^L_{t}\) (where \(R^L\) → is for ultimate losses)

\[ LDF^L = LDF_{t} \times \dfrac{R^L}{R_{t}^L} \]

\[ XSLDF^L = LDF_{t} \times \dfrac{1-R^L}{1-R_{t}^L} \]

\[ LDF_{t} = R_{t}^L \times LDF^L + (1-R_{t}^L)\times XSLDF^L \]

Notice how the first two formulae: \(LDF_{t}\) (maturity \(t\)) gets multiplied to \(R_{t}^L\) (maturity \(t\)).

Credibility Weighting¶

Weight #Loss Ratio Method and Development method (direct / implied)
Development → \(Z\) and Expected → \((1-Z)\)

\[ L = O_{t} \times LDF_{t} + Z + E\times (1-Z) \]

\(Z = \dfrac{1}{LDF}\) \(\implies\) Bornhuetter Ferguson method

Distributional Model¶

LDFs may not be consistent (when limited LDFs > unlimited LDFs)

\[ \text{Loss} \sim Weibull(\dots) \]

Time dependent parameters will be selected for this distribution
Can be solved for using
- MoM
- MLE
- Min. \(\chi\)-squared error

Summary of Methods¶

Method	Pros	Cons
Loss Ratio method	Works well for immature years	Ignore actual emerging experience
	More credible (industry data)	Not reflecting account characteristics properly (exposures written don't align with exposures to determine loss ratios and excess ratios)
	can be Consistently tied to pricing programs
Implied Development	Actual loss evergence incorporated	Misplaced focus → we actually prefer explicitly recognizing excess loss dev
	Estimate of XS even if they haven't emerged
	LDFs for limited losses are more stable than XS
	We also get ultimate limited losses → to calculate service revenue
Direct Development	Appropriate focus on the actual goal of XS losses	XS LDFs are highly leveraged + volatile
		If no XS losses emerged \(\implies\) impossible
Credibility Weighting (B-F)	Liabilities can be determined directly/indirectly	Ignores actual experience to the extent of complement of credibility.
	Estimates tied to pricing in early maturity years
	More stable estimates over time
Distributional Models	Ties relativities to severities which provides consistent LDFs	Need to estimate parameters
	Easy to interpolate between limits and years

Aggregate Limits¶

This limit applies after the per-claim deductible has been applied.
We are finding the amount of losses XS of aggregate
1. Find XS losses per claim \((A)\)
2. Then limited losses per claim (\(B\) = Claim size - \(A\))
3. Contribution to aggregate = \(C = B \times\text{\#claims}\)
4. Losses XS of aggregate = \(\max(C,\text{Aggregate Limit})\)
Trying to find LDFs for these XS of aggregate is mathematical hell (too complex) & not very credible LDFs (thin data)

Reserve Estimate for Agg-Lim¶

Collective risk model
- Weibull → severity
- Poisson → counts
BF method
NCCI method (Exam 8)
- More practical than (1) (PROS)
- Accuracy depends on proper insurance charge² (CONS)
NCCI method
1. Limited losses at \(j\) and Ult
2. Entry ratio³ \(=\dfrac{\text{aggregate limit}}{\text{limited loss}}\) (for each \(j\) and Ult)
3. Adjusted limited losses
  - \(\text{loss XS of d}\) = \(1 -\dfrac{\text{limited loss}}{\text{expected unlimited loss}}\)
  - Adjustment factor = \(\dfrac{1+0.8 \times \text{loss XS of d}}{1 - \text{loss XS of d}}\)
  - Adjusted limited loss = Adj factor \(\times\) Expected unlimited loss
4. Insurance charge ratio
  - adjusted limited loss → ELG, entry ratio, ELG → Insurance charge ratio (If entry ratio is in-between → just interpolate)
5. Calculate Insurance charge = \(\text{Ratio} \times\text{Limited Loss}\)
6. IBNR = CIC(Ult) - CIC(\(j\))
Adjustment factor →
- IC = E(Agg excess loss) without considering deductible
- And E(Agg with deductible) \(\lt\) E(Agg without deductible)⁴
- More the loss → lower the ELG → lower the IC
- The adjustment increase the expected aggregate losses
Aggregate Dev Factors
- Read #later

Service Revenue¶

Insurer pays for all claims (even those below deductible) and seeks reimbursement from insureds for losses below deductible, not subject to aggregate.
Service revenue = payment to insurer to cover cost of handling these claims (usually a flat \%), collected as losses are paid
If insurer isn't paid, they reflect this anticipated payment as an asset on their balance sheet.

\[ (E(X\wedge d) - L_{\text{agg}}) \times \text{Service Revenue \%} \]

Less any known recoveries → gives us Service Revenue asset

Workers' comp policies are classified into Hazard groups → having similar expected losses ↩
GIGO otherwise ↩
The influence of aggregate limit on losses below deductible ↩
"Why" is beyond the scope of the syllabus. ↩