The log-normal model

Under the standard formula, agregation of risk depend on variation coefficients \(\sigma\)’s that are use to asses the variability of next year risks. In our case, this parameter can be understood as the ratio between the standard variation of next-year boni-mali’s divided by this year Best estimate. The calculation by an insurance compagny of an USP for reserve risk is ruled by one of 2 models, we will here focus on the first one.

This model supposes (on top of a log-normal hypohtesis) a linear relation between 2 factors :

• The Best estimate at end of the year, denoted here \(x_t\) for year \(t\)
• The same Best estimate plus paiements done during the year, denoted here \(y_t\)

and this relation should hold \(\forall t \in \{1,..,T\}\). We will also denote :

• \(\sigma\), the variation coefficient of the risk, which is the parameter we want to assess.
• \(\delta \in [0,1]\), mixing parameter
• \(\beta\), the expected ,
• \(\gamma = ln(\frac{\sigma}{\beta})\).

The log-normal model proposed by Solvency 2 states under thoose notations that :

• \((H1)\) : \(\mathbb{E}(y_t) = \beta x_t\), i.e losses are proportional to premiums through a coefficient \(\beta\).
• \((H2)\) : \(\mathbb{V}(y_t) = \sigma^2\left((1-\delta)\bar{x}x_t + \delta x_t^2\right)\), where \(\bar{x} = \sum_{t=1}^{T} x_t\)
• \((H3)\) : \(y_t \sim log\mathcal{N}(\mu_t,\omega_t^2)\), \(y_t\) follows a log-normal.

Under \((H123)\), we are looking for a maximum likelyhood estimate of \(\sigma\). The likelyhood of the model is here given by :

\[\mathcal{L}^1(y_1,...,y_T,\beta,\sigma) = \prod\limits_{t=1}^{T} \frac{1}{y_t \omega_t \sqrt{2\pi}} exp\left\{\frac{-(ln(y_t)-\mu_t)^2}{2\omega_t}\right\}\]

Maximising the log of this likelyhood, we can found an estimate for the \(\sigma\) paramter. Note that this parameter can be extracted from the log-likelyhood. Indeed, if we write :

\[\pi_t\left(\hat{\delta},\hat{\gamma}\right) = \frac{1}{ln\left(1+e^{2\hat{\gamma}}\left((1+\hat{\delta})*\frac{\bar{x}}{x_t}+\hat{\delta}\right)\right)}\]

and if we estimate \(\hat{\delta},\hat{\gamma}\) by maximising the log-likelyhood, Roos (2015) prooved that this was equivalent to minimise the function proposed by Solvency II in Commission-Européenne (2014) , given by :

\[\sum\limits_{t=1}^{T} \pi_t\left(\hat{\delta},\hat{\gamma}\right)\left(ln\left(\frac{y_t}{x_t}\right) + \frac{1}{2\pi_t\left(\hat{\delta},\hat{\gamma}\right) } - \frac{\frac{T}{2} + \sum\limits_{t=1}^{T} \pi_t\left(\hat{\delta},\hat{\gamma}\right) ln(\frac{y_t}{x_t})}{\sum\limits_{t=1}^{T} \pi_t\left(\hat{\delta},\hat{\gamma}\right)}\right)^2 - \sum\limits_{t=1}^{T} ln\left(\pi_t\left(\hat{\delta},\hat{\gamma}\right)\right)\text{,}\]

We then found our maximum likelyhood estimate for \(\sigma\) in \(\hat{\sigma}\left(\hat{\delta},\hat{\gamma}\right)\), defined as :

\[\hat{\sigma}\left(\hat{\delta},\hat{\gamma}\right) = exp\left\{\hat{\gamma} + \frac{\frac{T}{2} + \sum\limits_{t=1}^{T} \pi_t\left(\hat{\delta},\hat{\gamma}\right) ln(\frac{y_t}{x_t})}{\sum\limits_{t=1}^{T} \pi_t\left(\hat{\delta},\hat{\gamma}\right)}\right\}\text{.}\]

For th sake of simplicity, we’ll take here data that are old enough (15points) to have the needed credibility so that this estimator is our USP. We refere to Rouchati (2016),Gauville (2017) and to the standard formula for more details about this.

A funny fact : Finaly, the delegated regulation give us a “unbiaising” formula :

\[\hat{\sigma} = \hat{\sigma}\left(\hat{\delta},\hat{\gamma}\right) \sqrt{\frac{T+1}{T-1}}\]

But as remarked Roos (2015) (see appendix A), this unbiaising factor is not optimal under model hypothesis ! Furthermore, a right factor can be calulated.

If you want more details about the theory underlying this model, e.g proof f the estimators, see BEAUNE (2016) or Roos (2015).

Generating dummy dataset.

We will here work on a simultaed dataset of best estimates \(x_t\) and \(y_t\) on a a window of 15years. Let’s generate them not too far away from the model :

df <- 
  data.frame(year = 1:15) %>% 
  mutate(
    x = seq(10000,20000,length=15)+rnorm(15,0,500),
    y = 1.05*x + exp(rnorm(15,0,1))
  )

Checking model hypohtesis.

The model has fundamentaly 2 hypothesis :

• Log-normality of \(y_t\) serie,
• Linearity between \(y_t\) and \(x_t\)

Results from the model

Since the hypohtesis have been checked, let’s now calculate the \(\sigma\) parameter under theese hypothesis. The application of maximum likelyhood estimation is done through the optim function in R, via the Newton-Raphson algorythme. The algorythme converge realy fast because we dont have much data. Here is the result :

Note that \(\hat{\delta}\) is borned between \(0\) and \(1\).

Although we took stupid inputs and therefore results are awfull, this model is very satisfying beacause it’s high level, in the sense of programing level : it uses directly best estimates from the compagny and therefore takes into account everything that was taken into acount during calulation of thoose best-estimates, including the whole reserving process of the compagny.

It’s therefore a great method to juge the quality of reserving from a compagny. However, the method takes into account very few data points and is therefore unprecise..

Maybe some time we’ll talk about better method to assess reserving risk !