Here, we discuss in detail how to formulate composite distributions. Composite distributions join together two weighted distributions at a given threshold value. In statistical terms, let $X$ be a random variable and let $f_1(.)$ be the pdf of the first distribution and the pdf of the second distribution be denoted by $f_2(.)$ of the same random variable. Let $F_1$ and $F_2$ be the corresponding cdf’s of the random variables. The pdf of the composite model can then be expressed as:

$f\left({\alpha }_1,{\alpha }_2,\theta ,\varphi \right)=\{\begin{array}{ll}\frac{1}{1+\varphi }{f}_1^{*}\left(x/{\alpha }_1,\theta \right),\hfill & \text{if}0<x\le \theta \hfill \\ \frac{\varphi }{1+\varphi }{f}_2^{*}\left(x/{\alpha }_2,\theta \right),\hfill & \text{if}0<x<\infty \hfill \end{array}$ (2.0)

The continuity condition and the differentiability conditions are imposed at the threshold $\theta$, such that in the limiting terms

$f^{+}\left({\alpha }_1,{\alpha }_2,\theta ,\varphi \right)=f^{-}\left({\alpha }_1,{\alpha }_2,\theta ,\varphi \right)$ (2.1)

$f^{\prime }\left({\alpha }_1,{\alpha }_2,\theta ,\varphi \right)=f^{\prime }\left({\alpha }_1,{\alpha }_2,\theta ,\varphi \right)$ (2.2)

where ${\alpha }_1$ and ${\alpha }_2$ are the parameters of the two pdfs on the two different intervals $\left(0,\theta \right]$ and $\left(\theta ,\infty \right)$, respectively. The differentiable and continuity conditions ensure $\theta$ and $\varphi >0$ are defined as functions of the parameters ${\alpha }_1$ and ${\alpha }_2$. In addition, the mixing weights for the two density functions are given by $\frac{1}{1+\varphi }$ and $\frac{\varphi }{1+\varphi }$. These mixing weights are defined as functions of ${\alpha }_1,{\alpha }_2,\theta$. This can be written in closed form in terms of the cumulative density function as:

$\varphi =-\frac{\frac{\text{d}\ln F_1\left(\theta /{\alpha }_1\right)}{\text{d}\theta }}{\frac{\text{d}\ln \left(1-F_2/{\alpha }_2\right)}{\text{d}\theta }}=\frac{\frac{f_1\left(\theta /{\alpha }_1\right)}{F_1\left(\theta /{\alpha }_1\right)}}{\frac{f_2\left(x/{\alpha }_2\right)}{1-F_2\left(\theta /{\alpha }_2\right)}}$ (2.3)

Substituting the expression for $\varphi$ into the differentiability conditions simplifies to

$\frac{\text{d}}{\text{d}\theta }\text{ln}\left(\frac{f_1\left(\theta /{\alpha }_1\right)}{f_2\left(\theta /{\alpha }_2\right)}\right)=0$

$\frac{{f^{\prime }}_1\left(\theta /{\alpha }_1\right)}{f_1\left(\theta /\alpha 1\right)}=\frac{{f^{\prime }}_2\left(\theta /\alpha 2\right)}{f_2\left(\theta /{\alpha }_2\right)}$ (2.4)

The functions $f_1^{*}\left(x/{\alpha }_1,\theta \right)$ and $f_2^{*}\left(x/{\alpha }_2,\theta \right)$ are truncated pdfs. In terms of their associated pdfs and cdfs is given by:

$f_1^{*}\left(x/{\alpha }_1,\theta \right)=\frac{f_1\left(x/{\alpha }_1\right)}{F1\left(\theta /\alpha 1\right)}$ (2.5)

$f_2^{*}\left(x/{\alpha }_2,\theta \right)=\frac{F_2\left(x/{\alpha }_2\right)}{1-F_2\left(\theta /{\alpha }_2\right)}$ (2.6)

and also

$F\left({\alpha }_1,{\alpha }_2,\theta ,\varphi \right)=\{\begin{array}{ll}\frac{1}{1+\varphi }\frac{F_1\left(x/{\alpha }_1\right)}{F_1\left(\theta /{\alpha }_1\right)},\hfill & \text{if}0<x\le \theta \hfill \\ \frac{1}{1+\varphi }\left[1+\varphi \frac{F_2\left(x/{\alpha }_2\right)-F_2\left(\theta /{\alpha }_2\right)}{1-F_2\left(x/{\alpha }_2\right)}\right],\hfill & \text{if}0<x<\infty \hfill \end{array}$ (2.7)

There are 16 loss distributions in the R-software package “actuar”, which is generally accepted for modelling losses. We fitted 240 composite distributions from these loss distributions in “actua” by taking two distributions at a time, making 16C2 × 2 = 240 in total. The results of the top 5 composite distributions are presented based on the three goodness of fit criteria: AIC, BIC, and log-likelihood.

According to [4] , Value at Risk (VaR) is a statistical measure that defines the worst expected loss over a specific time horizon under normal market conditions at a certain confidence level. It is commonly used in risk management to quantify the potential loss on an investment or portfolio. In simpler terms, VaR represents the maximum amount of money that could be lost on a portfolio within a set period of time with a specified level of confidence. The VaR at a 95% security level is calculated as follows:

${\text{VaR}}_{\alpha }\left(X\right)={\pi }_{\alpha }=F^{-1}\left(\alpha \right)$ (2.8)

[5] and [6] defined the theoretical estimate for the TVaR of the random variable, $X$ as:

$\begin{array}{l}{\text{TVaR}}_{\alpha }\left(X\right)\\ =\{\begin{array}{ll}\frac{1}{1-\alpha }\left[{\displaystyle {\int }_{{\pi }_{\alpha }}^{\theta }x{f}_1\left(x\right)\text{d}x}\frac{1}{F_1\left(\theta \right)}+{\displaystyle {\int }_{\theta }^{\infty }x{f}_2\left(x\right)\text{d}x}\frac{1}{1-F_2\left(\theta \right)}\right],\hfill & \text{if}0<\alpha \le \frac{1}{1+\varphi }\hfill \\ \frac{1}{1-\alpha }\frac{1}{1-F_2\left(\theta \right)}\left[{\displaystyle {\int }_{{\pi }_{\alpha }}^{\infty }x{f}_2}\right],\hfill & \text{if}\frac{1}{1+\varphi }<\alpha <1\hfill \end{array}\end{array}$ (2.9)

Secondary data obtained from an insurance company was used for this work. It consists of one year of claim data, consisting of 11,892 data points.

The descriptive statistics of the data are presented in Table 1 below:

Several single continuous distributions were fitted, but results from only the top five (5) are presented in Table 2 below. The table gives a summary of the parameter estimates and the AIC values. The AIC value for Lognormal was 5178.87, which was the lowest, indicating the best fit among the five distributions considered. This was closely followed by Gamma with an AIC of 5325.02 and Exponential distribution with an AIC of 5396.66. This explains the suitability of the Lognormal distribution as the appropriate model for the comprehensive data set.

Table 3 above shows the results of the top five composite models out of the 240 composite models fitted to the data.

From the three criteria used in choosing the best candidate model, lognormal-burr was demonstrated to be the best candidate model for the data with an AIC of 1091.2, which is the least. This means that for the claims data, the body is best fitted with a lognormal distribution and the tail fitted with a burr distribution.

Table 4 below shows the threshold and mixing weights values for the top five composite distributions. The mixing weights which are given by $\frac{1}{1+\varphi }$ and $\frac{\varphi }{1+\varphi }$ with $\varphi >0$ determines the appropriate segment of the data set which will be fitted to the body and tail distributions, respectively. For the best composite model, which is lognormal-Burr, 90.74% of the data points were fitted with the body distribution (Lognormal), whereas only 9.26% were fitted with the tail distribution. This clearly shows that the greater part of the losses was modelled with lognormal, whereas the remaining part was modelled by the burr distribution.

For the threshold and in a more general sense, given that $X_i$ are the losses and $\theta$ is the threshold value, then $X_i<\theta$ are fitted by the body (head) distribution whereas $X_i\ge \theta$ are fitted by the tail distribution. The best composite distribution, lognormal-burr, estimated that losses up to GH 36,553.9 can be modelled by the head distribution (lognormal) and losses greater than GH 36,553.9 can be modelled by the tail distribution (Burr).

Using the top five composite models, we now estimate the associated VaR and TVaR estimates at the security levels 95% and 99%. VaR is the worst possible loss an insurance company is likely to pay on any given trading day. The measure, when estimated, can be reliably used as an estimate for reserves. The right estimation of reserves is critical as it is necessary to ensure that only the right amount is set aside to pay claims, in order to invest any leftover money to generate investment income.

From Table 5 , the Lognormal-Burr distribution estimated that at a security level of 99%, a typical insurance company can make a loss of GH 30116.00 or GH 33210.00, respectively, for VaR and TVaR on any given day. This amount is substantially huge, and as such, knowing this amount can help in properly reserving it in order to meet all obligations of paying claims.

Now, given the differences in the VaR and TVaR values, it is essential to possibly find an average of the two risk measures to obtain an average risk estimate, which, to the best of our knowledge, may be more reliable. From the table, we realize that on average, from the two risk measures, a typical insurance company can pay claims of GH 31,663.00 on any given day. It is, however, important to note that this value is random.

There are two folds in a comparison of the best single and best composite distribution. We compare their AIC values and probability plots (Q-Q plot) of the two distributions. From Table 2 and Table 3 , the AIC values of lognormal and Lognormal-Burr distributions are respectively 5178.87 and 1091.2. A comparison of the values clearly shows that the composite Lognormal-Burr was a better fit than just using a lognormal distribution to fit the entire data.

Again, a comparison of the Q-Q plot of both the single distribution (Lognormal) in Figure 1 and the composite distribution (Lognormal-Burr) in Figure 2 shows that the composite distribution fitted the data better than the single distribution. This further supports the argument that composite distributions give a much better fit to claims data than a single distribution [5] [7] [8] .