LDF Curve-Fitting and Stochastic Reserving - Clark

Study Strategy¶

Read a little bit of theory (till parameterization), understand what Clark is aiming at. Then start with the excel questions.[^1]
- You need to learn Cape-cod method first
Go with the flow (follow checklist while solving questions)
For "Different Exposure Periods", draw on a piece of paper and try to make sense of each formula (LATER)

Checklist¶

My Notes¶

Introduction¶

We want a range of reserves
Two key elements from Clark's method
- Expected amount of loss to emerge
- Distribution of actual emergence around the expected value
Emergence pattern from 0 to 100% as \(t:0 \to \infty\)
Triangle \(\to\) Table
- AY \(\downarrow\) 2021 2022 2023→ 12mo 24mo 36mo
- to AY | From | To | Increment | Diagonal Age | AY Total

Parameterization¶

Parameterizations
- Loglogistic
  - \(G(x|\omega,\theta) = \dfrac{x^\omega}{x^\omega + \theta^{\omega}}\)
- Weibull
  - \(G(x|\omega,\theta) = 1- \exp(-(x/\theta)^{\omega})\)

Advantages & Disadvantages¶

Advantages of parameterized curves
1. estimate only 2 parameters
2. allow use of data not strictly from a triangle with evenly spaced evaluation dates
3. final indicated pattern is a smooth curve
Disadvantages
- Will not work if there is real expected negative development (salvage recoveries)

Recall

Paid + Reserves
Reported + IBNR

Calculating Ultimates¶

Options
- Loglogistic or Weibull
- Truncation point or without truncation
Use on-leveled premium.
Average age of 5 years = \(60\text{ mo} - 6\text{ mo} = 54\text{ months}\)

LDF Method¶

Steps
1. Find average age → Calc. \(G(x)\)
2. Calc. used-up premium using \(G(x)\) → Calc. Used-up loss ratio!
3. \(LDF = \dfrac{1}{\text{\% rept.}} = \dfrac{1}{G(x)}\)
4. Ultimate loss = \(\text{Cum Rept}\times LDF\)
Parameters
- One for each AY
- \(\theta\) and \(\omega\)
Highly leveraged LDFs
- Because growth curves extrapolates losses to full ultimate (paid at infinity)
- Two options
  1. Pick a truncation point (Clark says 20 years for his 10 year triangle) or,
  2. Choose a growth distribution with a smaller tail → Weibull

Cape Cod Method¶

Assumption

Constant loss ratio
- So trending LR defies it
Assumes known relationship between \(E(\text{ultimate})\) in each year

Table for estimation of CC ELR:

AY	x	G(x)	Used-up Prem	LR
2010	42	0.86	53,503	60%

Steps
1. ELR
  - Find average age → Calc. \(G(x)\)
  - Calc. used-up premium using \(G(x)\) → Calc. Used-up loss ratio!
    > Ensure that the LR is not trending up or down over the AYs as it would introduce bias into our model.
  - Select the ELR
2. IBNR
  - \(\text{\% unrept. }= 1-G(x)\) or \(G(x_{trunc})-G(x)\)
  - IBNR \(=\text{\% unrept.} \times\text{ELR} \times\text{Premium}\)
Parameters
- \(ELR\)
- \(\theta\) and \(\omega\)

Variance¶

\[ \text{Variance} = \text{Process Variance} + \text{Paramter Variance} \]

Process → Fluctuations caused by unpredictability of insurance (easy to calculate)
- If actual distribution of loss is given (say ODP, just use the formula \(\sigma^{2}\mu\))
Parameter (estimation error) → Uncertainty in our estimators because of our inability to reliably estimate expected reserves (difficult to calculate)
If we are given the \(\dfrac{\text{variance}}{\text{mean}}\) ratio, just multiply with expected losses and we are done.
If not,
1. Calc. incremental loss (ACTUAL)
2. Calc. expected loss = \((G(x) - G(x-1)) \times\text{Ult}\)
3. Calc. Error term \(=\dfrac{(\text{actual - expected})^{2}}{\text{expected}}\)
4. Specify
  - \(n\) = number of entries in triangle
  - \(p\) = number of params
    - \(3\) for Cape Cod
    - # AY + 2 for LDF
5. Calc. \(\sigma^{2} = \dfrac{1}{n-p}\times \sum\text{Error Term}\)
6. Process variance of reserves \(= \sigma^{2} \times\text{Sum(Reserves)}\)

Testing of Assumptions¶

\[ \text{Normalized Residuals} = \dfrac{\text{actual}-\text{expected}}{\sqrt{ \sigma^{2} \times \text{expected} }} = \dfrac{\text{residual}}{\text{Process SD}} \]

Expect residuals to be random around zero, amount of variability = roughly constant across the graph
- Counter example: All negative for one age, all positive for another

Why do residuals need to be (around) zero?
- As a prove that the model is indeed capturing all the systematic variation and that what is left is just random.

Muffs¶

WARNING While calculating variance: if the values of actual and expected are given in 1000 then be careful about what \(\sigma^{2}\) needs to be multiplied by.
- It should be multiplied by 1000 and not \(1000^{2}\) because, it is \(\dfrac{(\text{actual} - \text{expected})^{2}}{\text{expected}}\) so it would be in \(\dfrac{1000^2}{1000} = 1000s\)
When in the exam It is mentioned:
> Calculate a reserve estimate for the accident year by credibility-weighting the two estimates of ultimate loss in parts b. and c. above using the Benktander method.
- You are supposed to use the answers of (b) and (c). If you find the answer by recalculation, then credit will not be awarded.
- When explaining why the Total standard error of CC is lower than LDF method, you will have to
- Tell "LDF method uses more parameters"
- Describe the calculation for approximating \(\sigma^{2}\), particularly penalizing parameters, dividing by \((n-p)\) and how this is smaller for LDF \(\implies\) \(\sigma^{2}\) will be larger for LDF method than CC method

Insights¶

Spring 2014 Exam 7 - Q3: There are two things
- The losses incurred during an accident year (which can happen in any distribution. I think this is what loss modelling is about)
- How the losses incurred are reported in the future maturities. This is what Clark tries to model with \(G(x)\)
Spring 2014 Exam 7 - Q5: Cape cod is the same as BF
Spring 2016 Exam 7 - Q4: If you are already given an ELR, then you are using the BF method. Hence, no need to calculate the ELR