Clark Flashcards
Elements of a statistical loss reserving model
- The expected amount of loss to emerge in some time period
- The distribution of actual emergence around the expected value
Notation
G(x) = 1/ LDF_x = cumulative % paid/reported as of time x
where x represents the time (in months) from the “average” accident date to the evaluation date
Loglogistic
G(x|w,theta) = x^w / (x^w + theta^w)
theta - scale
w - shape
Weibull
G(x|w,theta) = 1 - exp(-(x/theta)^w)
Advantages to using parameterized curves to describe the emergence pattern
- Estimation is simple since we only have to estimate 2 parameters
- We can use data that is not from a triangle with evenly spaced evaluation data
- The final pattern is smooth and does not follow random movements in the historical age-to-age factors
LDF Method
Let mu_i;x,y = expected incremental loss dollars in AY between ages x and y
mu_i;x,y = ULT_AY * [G(y|w,theta) - G(x|w,theta)]
MLE estimate ULT_i = SUM(c_i,t) / SUM(G(x_t) - G(x_t-1)) = LDF * Paid Loss
Num of parameters p = # of AYs + 2 (each ULT_i, theta, w)
assumes the loss amount in each AY is independent from all other years
- tends to be over-parameterized when few data points exist
LDF Method Algorithm
- Calculate G(x) for each AY
- Calculate the LDF for each AY = 1/G(x) (if truncated, G(trunc)/G(x))
- Calculate the Ultimate Loss = LDF * Paid Loss
Cape Cod Method
Let mu_i;x,y = expected incremental loss dollars in AY between ages x and y
mu_i;x,y = P_i * ELR * [G(y|w,theta) - G(x|w,theta)]
MLE estimate ELR = SUM(c_i,t) / SUM(P_i * [G(x_t) - G(x_t-1)]) = Total Paid Losses / Total Used-Up Premium
where Used-Up Premium = P * G(x) to put premium on the same basis as paid losses
Num of parameters p = 3 (ELR, theta, w)
- premium should be on-leveled first
- ELR calculated BEFORE truncation
- preferred since data is summarized into a loss triangle with relatively few data points
- smaller parameter variance than LDF Method due to the additional information given by the exposure base and fewer parameters
Cape Cod Method Algorithm
- Calculate the estimated ELR before truncation
- Calculate the G(x) for each AY
- Calculate the % unpaid/unreported for each AY = 1 - G(x) (if truncated, G(trunc) - G(x))
- Calculate the Expected Loss for each AY = P * ELR
- Calculate the estimated Reserves for each AY = Expected Loss * % unpaid/unreported
Variance/Mean Ratio
The loss in any period has a constant ratio of variance/mean
Var/Mean = sigma^2 = 1/(n - p) * SUM[(c_AY,t - mu_AY,t)^2 / mu_AY,t] = 1/(n - p) * SUM[(actual - expected)^2 / expected]
n = # of data points in the traingle
p = # of parameters
Over-Dispersed Poisson (ODP) Distribution
Assume actual loss emergence c follows an ODP distribution with scaling factor/dispersion parameter sigma^2
E(c) = lambda * sigma^2 = mu
Var(c) = lambda * sigma^4 = mu * sigma^2
Advantages:
- Inclusion of scaling factors allow us to match the first and second moments of any distribution, allowing high flexibility
- Maximum likelihood estimation produces the LDF and CC estimates of ultimate losses, so the results can be presented in a familiar format
Process Variance
The random amount
Var(c) = Variance of actual loss emergence = sigma^2 * mu = sigma^2 * Estimated Reserves
Parameter Variance
The uncertainty of the estimator (estimation error)
Can be found using the information matrix (known as the Delta Method)
For each method, if we take the inverse of the information matrix, we obtain the covariance matrix. The diagonals of the covariance matrices provide the variances of each parameter.
Variance of the Reserves
Var(R) = Process Variance + Parameter Variance = sigma^2 * Estimated Reserves + Parameter Variance
Key Assumptions of the Model
- Incremental losses are iid (one period does not affect the surrounding and emergence pattern is the same for all AYs)
- The variance/mean scale parameter sigma^2 is fixed and known
- Variance estimates are based on an approximation to the Rao-Cramer lower bound
Growth Curve Extrapolation
Extrapolates losses to ultimate
- For curves with “heavy” tails (such as loglogistic), it may be necessary to truncate the growth curve at a finite point in time to reduce reliance on the extrapolation
- An alternative to truncation is using a growth curve with a “lighter” tail (such as Weibull)
- Clark only truncates growth curves when estimating reserves/ultimate losses - the untruncated growth curve should be used when calculating fitted values for the estimate of the variance/mean ratio
Truncated Growth Curve
- Add a “Trunc.” row to the top of the AYs with an average age = years * 12 - 6
- Calculate truncated LDFs/% unpaid for each AY
- G(trunc) / G(x) for LDF Method
- G(trunc) - G(x) for CC Method - Calculate ultimate losses and estimated reserves based on the truncated LDF/% unpaid
How to find fitted incremental paid losses
- Calculate the incremental paid loss triangle
- Calculate the fitted incremental paid loss triangle using the untruncated growth curve:
- LDF Method: mu_i = Ult * (G(x_t) - G(x_t-1))
- CC Method: mu_u = Expected Loss * (G(x_t) - G(x_t-1))
Normalized Residuals
r_i;x,y = (c_i;x,y - mu_i;x,y) / SQRT(sigma^2 * mu_i;x,y) = (actual - estimate) / SD of estimate
Should be randomly scattered around y = 0
Can plot against:
- AY age
- Expected loss in each increment – useful for testing if the variance/mean ratio is constant
- AY
- CY - useful for testing for diagonal effects
Testing the Constant Cape Cod ELR Assumption
Graph the CC expected ultimate LR by AY
CC Expected Ult LR = Loss / Used-Up Premium
Variance of the Prospective Losses
Prospective loss = OLP * ELR
Process Variance = sigma^2 * Prospective loss
Parameter Variance = Var(ELR) * OLP^2
Calendar Year Development
LDF Method: Multiply the difference in growth functions by the estimated Ult loss
CC Method: Multiply the difference in growth functions by the expected loss
Variability in Discounted Reserves
Discount the incremental losses at interest rate i
v = 1 / (1 + i)
R_d = SUM(ULT_i * v^k-0.5 * [G(x+k) - G(x+k-1)]
Process Var(R_d) = sigma^2 * SUM[ULT_i * v^2k-1 * (G(x+k) - G(x+k-1)]
ULT_i = estimated ultimate loss for LDF method or the expected loss from the CC method
