Parameter Risk

Study guides: Exam 7, Exam CP-351

Parameter risk is the uncertainty in a reserve that comes from not knowing the model's parameters: development factors, an expected loss ratio or growth-curve parameters are estimated from limited, noisy data. Together with process risk, the randomness of outcomes when the parameters are known, it makes up the prediction error. Model (specification) risk sits on top of both.

$\mathrm{Var}(R) = \underbrace{E\big[\mathrm{Var}(R \mid \theta)\big]}_{\text{process}} + \underbrace{\mathrm{Var}\big(E[R \mid \theta]\big)}_{\text{parameter}}$

$\theta$ is the parameter vector and $R$ the unpaid amount. In a Bayesian model (Verrall, Meyers' MCMC) the split is literal, with $\theta$ drawn from its posterior. In a frequentist model, the second term is the sampling variance of $\hat\theta$ carried through to $\hat R$ .
Where each Exam 7 model puts it.
- Mack: the $1/\sum_j C_{j,k}$ term of the standard error.
- Clark: $(\partial R/\partial\theta)^{\top}\,\Sigma\,(\partial R/\partial\theta)$ , where the covariance matrix $\Sigma$ comes from the inverse information matrix of the maximum likelihood fit.
- ODP bootstrap: the spread of reserves across refitted pseudo-triangles, before gamma process noise is added.
- MCMC: the spread of the posterior draws.
Parameter percentiles are percentiles of the simulated or posterior parameter set, such as the 5th to 95th percentile of simulated ELRs or development factors. A reserve computed at the 95th-percentile parameter is not the 95th percentile of unpaid claims. Process risk sits on top, so the outcome percentile is wider (see Unpaid Claim Distribution).
It grows with the number of parameters relative to the data. A chain ladder fits a factor for every age. Clark finds parameter variance far larger than process variance when his LDF method fits 12 parameters (an ultimate for each of ten years plus the two growth-curve parameters) to 55 data points. Replacing the ultimates with one Cape Cod ELR leaves 3 parameters, and the parameter standard deviation of the total reserve falls from $4{,}688{,}826$ to $3{,}143{,}967$ ; the total variance is cut in half. It concentrates in the most recent accident years, where a large CDF multiplies a thin diagonal. That is why a bootstrap CoV can turn up again for the latest year.
It does not diversify with volume. Process CoV shrinks roughly like $1/\sqrt{\text{volume}}$ . A misestimated factor, however, is applied to every year, so a large book still carries it in full. In Marshall et al.'s framework, the random part of parameter risk is independent risk, while parameter selection error is internal systemic risk, a model risk (see Risk Margin).

Process and Parameter Variance in a Cape Cod Fit A Clark Cape Cod fit gives $\widehat{\text{ELR}} = 0.70$ with standard error $0.05$ . On-level premium is $10{,}000$ ($000s), the fitted growth curve says $40\%$ of ultimate is still unreported, and the ODP scale is $\sigma^2 = 50$ . Ignore uncertainty in the curve parameters.

Split the reserve's variance.
Answer
 $\begin{align*} \hat R &= 0.70 \times 10{,}000 \times 0.40 \\ &= 2{,}800 \\ \text{Process} &= \sigma^2 \hat R = 50 \times 2{,}800 \\ &= 140{,}000 \\ \text{Parameter} &= (10{,}000 \times 0.40)^2 \times 0.05^2 \\ &= 40{,}000 \\ \mathrm{PE} &= \sqrt{180{,}000} \\ &= 424.3 \end{align*}$ 
The CoV is $15.2\%$ and parameter risk is $22\%$ of the variance. Suppose the book doubled with the same ELR uncertainty. Process variance would double, but parameter variance, which scales with $\hat R^2$ , would quadruple. That is why parameter risk dominates for large books: their process risk has largely diversified away.

Parameter Percentile Versus Unpaid Percentile A Bayesian model gives a posterior for the ELR that is approximately normal with mean $0.70$ and standard deviation $0.05$ . Given the ELR, unpaid claims are approximately normal with mean $5{,}000 \times \text{ELR}$ and process standard deviation $300$ .

Compare the reserve at the 95th-percentile ELR with the 95th percentile of unpaid claims.
Answer At the parameter percentile:
 $\begin{align*} \text{ELR}_{0.95} &= 0.70 + 1.645(0.05) \\ &= 0.7822 \\ 5{,}000 \times 0.7822 &= 3{,}911 \end{align*}$ 
Unpaid claims. The mean is $3{,}500$ , and the parameter standard deviation is $5{,}000 \times 0.05 = 250$ :
 $\begin{align*} \mathrm{SD}(R) &= \sqrt{250^2 + 300^2} \\ &= 390.5 \\ R_{0.95} &= 3{,}500 + 1.645(390.5) \\ &= 4{,}142 \end{align*}$ 
Setting the parameter to its 95th percentile understates the 95th percentile of the outcome by $231$ , because it leaves out process risk. A simulation draws the parameter and then the outcome for each iteration, and ranks the outcomes. Reading percentiles off the parameter draws alone measures only one of the two risks.