I was asked an interesting question a couple of weeks ago when talking through some modelling with a client.
​
We were modelling an airline account, and for various reasons we had decided to base our large loss modelling on a very basic top-down allocation method. We would take a view of the market losses at a few different return periods, and then using a scenario approach,  would allocate losses to our client proportionately. Using this method, the frequency of losses is then scaled down by the % of major policies written, and the severity of losses is scaled down by the average line size.

To give some concrete numbers (which I’ve made up as I probably shouldn’t go into exactly what the client’s numbers were), let's say the company was planning on taking a line on around 10% of the Major Airline Risks, and their average line was around 1%. We came up with a table of return periods for market level losses. The table looked something like following (the actual one was also different to the table below, but not miles off):

​Then applying the 10% hit factor if there is a loss, and the 1% line written, we get the following table of return periods for our client:

Hopefully all quite straightforward so far. As an aside, it is quite interesting to sometimes pare back all the assumptions to come up with something transparent and simple like the above. For airline risks, the largest single policy limit is around USD 2.5bn, so we are saying our worst case scenario is a single full limit loss, and that each year this has around a 1 in 50 chance of occurring. We can then directly translate that into an expected loss, in this case it equates to 50m (i.e. 2.5bn *0.02) of pure loss cost. If we don't think the market is paying this level of premium for this type of risk, then we better have a good reason for why we are writing the policy!

So all of this is interesting (I hope), but what was the original question the client asked me?

We can see from the chart that for the market level the highest return period we have listed is 1 in 50. Clearly this does translate to a much longer return period at the client level, but in the meeting where I was asked the original question, we were just talking about the market level. The client was interested in what the 1 in 200 at the market level was and what was driving this in the modelling.

The way I had structured the model was to use four separate risk sources, each with a Poisson frequency (lambda set to be equal to the relevant return period), and a fixed severity. So what this question translates to is, for small Lambdas $(<<1)$, what is the probability that $n=2$, $n=3$, etc.? And at what return period is the $n=2$ driving the $1$ in $200$?

Let’s start with the definition of the Poisson distribution:

Let $N \sim Poi(\lambda)$, then:

$$P(N=n) = e^{-\lambda} \frac{ \lambda ^ n}{ n !} $$

We are interested in small $\lambda$ – note that for large $\lambda$ we can use a different approach and apply sterling’s approximation instead. Which if you are interested, I’ve written about here:
www.lewiswalsh.net/blog/poisson-distribution-what-is-the-probability-the-distribution-is-equal-to-the-mean

$$ e^{-\lambda} = \sum_{i=0}^{\infty} \frac{\lambda^i}{ i!} = 1  - \lambda + \frac{\lambda^2}{2} + o(\lambda^2) $$
​
We can then examine the pdf of the Poisson distribution using this approximation:

$$P(N=1) =\lambda e^{-\lambda} = \lambda ( 1 – \lambda + \frac{\lambda^2}{2} + o(\lambda^2) ) = \lambda - \lambda^2 +o(\lambda^2)$$

I came across a useful way of looking at Rate on Lines last week, I was talking to a broker about what return periods to use in a model for various levels of airline market loss (USD250m, USD500m, etc.). The model was intended to be just a very high level, transparent market level model which we could use as a framework to discuss with an underwriter. We were talking through the reasonableness of the assumptions when the broker came out with the following:
 
'Well, you’d pay about 12.5 on line in the retro market at that attachment level, so that’s a 1 in 7 break-even right?'
 
My response was:

'ummmm, come again?'
 
His reasoning was as follows:
 
Suppose the ILW pays $1$ @ $100$% reinstatements, and that it costs $12.5$​% on line.
Then if the layer suffers a loss, the insured will have a net position on the contract of $75$%. This is the $100$% limit which they receive due to the loss, minus the original $12.5$% Premium, minus an additional $12.5$% reinstatement Premium. The reinsurer will now need another $6$ years at $12.5$% RoL $(0.0125 * 6 = 0.75)$ to recover the limit and be at break-even.
 
Here is a breakdown of the cashflow over the seven years for a $10m$ stretch at $12.5$% RoL:
 

So the loss year plus the six clean years, tells us that if a loss occurs once every 7 years, then the contract is at break-even for this level of RoL.

So this is kind of cool - any time we have a RoL for a retro layer, we can immediately convert it to a Return Period for a loss which would trigger the layer.
 
Generalisation 1 – various rates on line
 
We can then generalise this reasoning to apply to a layer with an arbitrary RoL. Using the same reasoning as above, the break-even return period ends up being:
 
$RP= 1 + \frac{(1-2*RoL)}{RoL}$
 
Inverting this gives:
 
$RoL = \frac{1}{(1 + RP)}$
 
So let's say we have an ILW costing $7.5$% on line, the break-even return period is:
 
$1 + \frac{(1-0.15)}{0.075} = 11.3$
 
Or let’s suppose we have a $1$ in $19$ return period, the RoL will be:
 
$0.05 = \frac{1}{(1 + 19)}$
 
Generalisation 2 – other non-proportional layers
 
The formula we derived above was originally intended to apply to ILWs, but it also holds any time we think the loss to the layer, if it occurs, will be a total loss. This might be the case for a cat layer, or a clash layers (layers which have an attachment above the underwriting limit for a single risk), or any layer with a relatively high attachment point compared to the underwriting limit.
 
Adjustments to the formulas
 
There are a few of adjustments we might need to make to these formulas before using them in practice.
 
Firstly, the RoL above has no allowance for profit or expense loading, we can account for this by converting the market RoL to a technical RoL, this is done by simply dividing the RoL by $120-130$% (or any other appropriate profit/expense loading). This has the effect of increasing the number of years before the loss is expected to occur.
 
Alternately, if layer does not have a paid reinstatement, or has a different factor than $100$%, then we would need to amend the multiple we are multiplying the RoL by in the formula above. For example, with nil paid reinstatements, the formula would be:
 
$RP = 1 + \frac{(1-RoL)}{RoL}$
 
Another refinement we might wish to make would be to weaken the total loss assumption.  We would then need to reduce the RoL by an appropriate amount to account for the possibility of partial losses. It’s going to be quite hard to say how much this should be adjusted for – the lower the layer the more it would need to be.

It's the second week of your new job Capital Modelling job. After days spent sorting IT issues, getting lost coming back from the toilets, and perfecting your new commute to work (probability of getting a seat + probability of delay * average journey temperature.) your boss has finally given you your first real project to work on.

You've been asked to carry out an annual update of the Underwriting Risk Capital Charge for a minor part of the company's Motor book. Not the grandest of analysis you'll admit, this particular class only makes up about 0.2% of the company's Gross Written Premium, and the Actuaries who reserve the company's bigger classes would probably consider the number of decimal places used in the annual report more material than your entire analysis. But you know in your heart of hearts that this is just another stepping stone on your inevitable meteoric rise to Chief Actuary in the Merger and Acquisition department, where one day you will pass judgement on billion dollar deals in-between expensive lunches with CFOs, and drinks with journalists on glamorous rooftop bars.

The company uses in-house reserving software, but since you're not that familiar with it, and because you want to make a good impression, you decide to carry out extensive checking of the results in Excel. You fire up the Capital Modelling Software (which may or may not have a name that means a house made out of ice), put in your headphones and grind it out.

Hours later you emerge triumphant, and you've really nailed it, your choice of correlation (0.4), and correlation method (Gaussian Copula) is perfect. As planned you run extracts of all the outputs, and go about checking them in Excel. But what's this? You set the correlation to be 0.4 in the software, but when you check the correlation yourself in Excel, it's only coming out at 0.384?! What's going on?

Simulating using Copulas

The above is basically what happened to me (minus most of the actual details. but I did set up some modelling with correlated random variables and then checked it myself in Excel and was surprised to find that the actual correlation in the generated output was always lower than the input.) I looked online but couldn't find anything explaining this phenomenon, so I did some investigating myself.

So just to restate the problem, when using Monte Carlo simulation, and generating correlated random variables using the Copula method. When we actually check the correlation of the generated sample, it always has a lower correlation than the correlation we specified when setting up the modelling.

My first thought for why this was happening was that were we not running enough simulations and that the correlations would eventually converge if we just jacked up the number of simulations. This is the kind of behaviour you see when using Monte Carlo simulation and not getting the mean or standard deviation expected from the sample. If you just churn through more simulations, your output will eventually converge. When creating Copulas using the Gaussian Method, this is not the case though, and we can test this.

I generated the graph below in R to show the actual correlation we get when generating correlated random variables using the Copula method for a range of different numbers of simulations. There does seem to be some sort of loose limiting behaviour, as the number of simulations increases, but the limit appears to be around 0.384 rather than 0.4.

The actual explanation

First, we need to briefly review the algorithm for generating random variables with a given correlation using the normal copula.

Step 1 - Simulate from a multivariate normal distribution with the given covariance matrix.

Step 2 - Apply an inverse gaussian transformation to generate random variables with marginal uniform distribution, but which still maintain a dependency structure

Step 3 - Apply the marginal distributions we want to the random variables generated in step 2

We can work through these three steps ourselves, and check at each step what the correlation is.

The first step is to generate a sample from the multivariate normal. I'll use a correlation of 0.4 though out this example. Here is the R code to generate the sample:

a <- library(MASS)
library(psych)

set.seed(100)

m <- 2
n <- 1000
sigma <- matrix(c(1, 0.4,
                  0.4, 1), 
                nrow=2)
z <- mvrnorm(n,mu=rep(0, m),Sigma=sigma,empirical=T)

​

And here is a Scatterplot of the generated sample from the multivariate normal distribution:
​

We now want to check the product moment correlation of our sample, which we can do using the following code:
​

cor(z,method='pearson')

Which gives us the following result:

> cor(z,method='pearson')
     [,1] [,2]
[1,]  1.0  0.4
[2,]  0.4  1.0

So we see that the correlation is 0.4 as expected. The Psych package has a useful function which produces a summary showing a Scatterplot, the two marginal distribution, and the correlation:

Let us also check Kendall's Tau and Spearman's rank at this point. This will be instructive later on. We can do this using the following code:

cor(z,method='spearman')

cor(z,method='Kendall')

Which gives us the following results:

> cor(z,method='spearman')
          [,1]      [,2]
[1,] 1.0000000 0.3787886
[2,] 0.3787886 1.0000000

> cor(z,method='kendall')
          [,1]      [,2]
[1,] 1.0000000 0.2588952
[2,] 0.2588952 1.0000000

Note that this is less than 0.4 as well, but we will discuss this further later on.

We now need to apply step 2 of the algorithm, which is applying the inverse Gaussian transformation to our multivariate normal distribution. We can do this using the following code:

u <- pnorm(z)

We now want to check the correlation again, which we can do using the following code:

cor(z,method='spearman')

Which gives the following result:

> cor(z,method='spearman')
          [,1]      [,2]
[1,] 1.0000000 0.3787886
[2,] 0.3787886 1.0000000

​Here is the Psych summary again:

u is now marginally uniform (hence the name). We can see this by looking at the Scatterplot and marginal pdfs above.

We also see that the correlation has dropped to 0.379, down from 0.4 at step 1. The Pearson correlation measures the linear correlation between two random variables. We generated normal random variables, which had the required correlation, but then we applied a non-linear (inverse Gaussian) transformation. This non-linear step is the source of the dropped correlation in our algorithm.

We can also retest Kendall's Tau, and Spearman's at this point using the following code:

cor(z,method='spearman')

cor(z,method='Kendall')

This gives us the following result:

> cor(u,method='spearman')
          [,1]      [,2]
[1,] 1.0000000 0.3781471
[2,] 0.3781471 1.0000000

> cor(u,method='kendall')
          [,1]      [,2]
[1,] 1.0000000 0.2587187
[2,] 0.2587187 1.0000000

Interestingly, these values have not changed from above! i.e. we have preserved these measures of correlation between step 1 and step 2. It's only the Pearson correlation measure (which is a measure of linear correlation) which has not been preserved.

Let's now apply the step 3, and once again retest our three correlations. The code to carry out step 3 is below:

x1 <- qgamma(u[,1],shape=2,scale=1)
x2 <- qbeta(u[,2],2,2)

df <- cbind(x1,x2)
pairs.panels(df)

The summary for step 3 looks like the following.

This is the end goal of our method. We see that our two marginal distributions have the required distribution, and we have a correlation between them of 0.37. Let's recheck our three measures of correlation.
​

cor(df,method='pearson')

cor(df,meth='spearman')

cor(df,method='kendall')

> cor(df,method='pearson')
          x1        x2
x1 1.0000000 0.3666192
x2 0.3666192 1.0000000

> cor(df,meth='spearman')
          x1        x2
x1 1.0000000 0.3781471
x2 0.3781471 1.0000000

> cor(df,method='kendall')
          x1        x2
x1 1.0000000 0.2587187
x2 0.2587187 1.0000000

So the Pearson has reduced again at this step, but the Spearman and Kendall's Tau are once again the same.

Does this matter?

This does matter, let's suppose you are carrying out capital modelling and using this method to correlate your risk sources. Then you would be underestimating the correlation between random variables, and therefore potentially underestimating the risk you are modelling.

Is this just because we are using a Gaussian Copula? No, this is the case for all Copulas. Is there anything you can do about it? Yes, one solution is to just increase the input correlation by a small amount, until we get the output we want.

A more elegant solution would be to build this scaling into the method. The amount of correlation lost at the second step is dependent just on the input value selected, so we could pre-compute a table of input and output correlations, and then based on the desired output, we would be able to look up the exact input value to use.   

The Reinstatement Premium payable following a loss to an Excess of Loss contract, is related to the ceded loss to the contract by a simple formula. Therefore, it seems reasonable that we should be able to come up with a simple formula relating the price charged for the Excess of Loss contract to the price charged for a Reinstatement Premium Protection (RPP) cover. I was in a meeting last week with two brokers who were trying to do just this. They had come up with an indicative price for an Excess of Loss programme and were trying to use this to price the equivalent RPP cover.

At the time I didn't have an answer for them, and when I did a quick Google, nothing came up. When thinking about it subsequently though, there are a couple of easy approximate methods we can use. Below I discuss three different methods for how you can price an RPP cover, two of which do not require any stochastic modelling assuming you already know the price of the Excess of Loss layers.

Let's quickly review what we mean by all these terms so that we are starting from the same point.

What is a Reinstatement Premium?

If you already understand how a Reinstatement Premium works, then feel free to skip this section.

Most Excess of Loss contracts will have some form of reinstatement premium. This is a payment from the Insurer to the Reinsurer to reinstate the protection in the event of a loss. In the London market, most contracts willl have either $1$, $2$, or $3$ reinstatements and generally these will be payable at $100 \%$.

What is a Reinstatement Premium Protection Cover?

The reinstatement premiums can be quite a large proportion of the overall premium paid for the Excess of Loss reinsurance. This is especially the case for lower level, working layers. Furthermore, the Reinstatement Premium will be payable when the insurer has just suffered a loss. From the point of view of the insurer, this additional payment comes at the worst possible time. The Insurer is being asked to fork over another large premium to the Reinsurer, just after having suffered a loss.

To address some of these concerns, Reinsurers developed a product called a Reinstatement Premium Protection cover (RPP cover). This cover pays the Insurer's Reinstatement Premium for them, which gives the insurer further indemnification in the event of a loss. Here's an example of how it works in practice:

Let's suppose we are considering a $5m$ xs $5m$ Excess of Loss contract, there is one reinstatement at $100 \%$ (written $1$ @ $100 \%$), and the Rate on Line is $25 \%$. The Rate on Line is just the Premium divided by the Limit. So here, the Premium can be found by multiplying the Limit and the RoL:

$$5m* 25 \% = 1.25m$$

So we see that the Insurer will have to pay the Reinsurer $1.25m$ at the start of the contract. Now let's suppose there is a loss of $7m$. The Insurer will recover $2m$ from the Resinsurer, but they will also have to make a payment  to cover the reinstatement premium of: $\frac {2m}  {5m} * (5m * 25 \% ) = 2m * 25 \% = 0.5m$ to reinstate the cover. So the Insurer will actually have to pay out $5.5m$. The RPP cover, if purchased by the insurer, would pay the additional $0.5m$ on behalf of the insurer

Now that we know how it works, how would we price the RPP cover?

Three methods for pricing an RPP cover

Method 1 - Full stochastic model

If we have priced the original Excess of Loss layer ourselves using a Monte Carlo model, then it should be relatively straight forward to price the RPP cover. We can just look at the expected Reinstatements, and apply a suitable loading for profit and expenses. This loading will probably be broadly in line with the loading that is applied to the expected losses to the Excess of Loss layer, but accounting for the fact that the writer of the RPP cover will not receive any form of Reinstatement for their Reinsurance.

What if we do not have a stochastic model set up to price the Excess of Loss layer? What if all we know is the price being charged for the Excess of Loss layer?

Method 2 - Simple formula

This was the situation I was in when I was asked to price the RPP cover last week. The broker had come up with some very approximate pricing for a programme of Excess of Loss layers. This pricing was driven by the burning cost, and other commercial factors rather than any actuarial modelling. The brokers wanted to come up with an approximate price for the RPP cover, just based on the price of the Excess of Loss layer. The two should be related, as they pay out dependant on the same underlying losses. So what can we say?

If we denote the Expected Losses to the layer by $EL$, then the Expected Reinstatement Premium should be:
$$EL * ROL $$
To see this is the case, I used the following reasoning; if we had losses in one year equal to the $EL$ (I'm talking about actual losses, not expected losses here), then the Reinstatement Premium for that year would be the proportion of the layer which had been exhausted $\frac {EL}  {Limit} $ multiplied by the Deposit Premium $Limit * ROL$ i.e.:

$$ RPP = \frac{EL}  {Limit} * Limit * ROL  = EL * ROL$$

Great! So we have our formula right? The issue now is that we don't know what the $EL$ is. We do however know the $ROL$, does this help?

If we let $DP$ denote the deposit premium, which is the amount we initially pay for the Excess of Loss layer and we assume that we are dealing with a working layer, then we can assume that:

Which is our first very simple formula for the price that should be charged for an RPP. Was there anything we missed out though in our analysis? 
​
Method 3 - A more complicated formula:

There is one subtlety we glossed over in order to get our simple formula. The writer of the Excess of Loss layer will also receive the Reinstatement Premiums during the course of the contract. The writer of the RPP cover on the other hand, will not receive any reinstatement premiums (or anything equivalent to a reinstatement premium). Therefore, when comparing the Premium charged for an Excess of Loss layer against the Premium charged for the equivalent RPP layer, we should actually consider the total expected Premium for the Excess of Loss Layer rather than just the Deposit Premium.

What will the additional premium be? We already have a formula for the expected Reinstatement premium:

$$EL * ROL $$

Therefore the total expected premium for the Excess of Loss Layer is the Deposit Premium plus the additional Premium:

$$ DP + EL * ROL $$

This total expected premium is charged in exchange for an expected loss of $EL$.

So at this point we know the Total Expected Premium for the Excess of Loss contract, and we can relate the expected loss to the Excess of Loss layer to the Expected Loss to the RPP contract. 

i.e. For an expected loss to the RPP of $EL * ROL$, we would actually expect an equivalent premium for the RPP to be:

$$ RPP =  (DP + EL * ROL) * ROL $$

This formula is already loaded for Profit and Expenses, as it is based on the total premium charged for the Excess of Loss contract. It does however still contain the $EL$ as one of its terms which we do not know.

We have two choices at this point. We can either come up with an assumption for the profit and expense loading (which in this hard market might be as little as only be $5 \% - 10 \%$ ). And then replace $EL$ with a scaled down $DP$:

$$RPP =  \frac{DP} {1.075} * ( 1 + ROL) * ROL $$

Or we could simply replace the $EL$ with the $DP$, which is partially justified by the fact that the $EL$ is only used to multiply the $ROL$, and will therefore have a relatively small impact on the result. Giving us the following formula:

$$RPP =  DP ( 1 + ROL) * ROL $$

Which of the three methods is the best?

The full stochastic model is always going to be the best in my opinion. If we do not have access to one though, then out of the two formulas, the more complicated formula we derived should be more accurate (by which I mean more actuarially correct). If I was doing this in practice, I would probably calculate both, to generate some sort of range, but tend towards the second formula.

That being said, when I compared the prices that the Brokers had come up with, which is based on what they thought they could actually place in the market, against my formulas, I found that the simple version of the formula was actually closer to the Broker's estimate of how much these contacts could be placed for in the market. Since the simple formula always comes out with a lower price than the more complicated formula, this suggests that there is a tendency for RPPs to be under-priced in the market.

This systematic under-pricing may be driven by commercial considerations rather than faulty reasoning on the part of market participants. According to the Broker I was discussing these contracts with, a common reason for placing an RPP is to give a Reinsurer who does not currently have a line on the underlying Excess of Loss layer, but who would like to start writing it, a chance to have an involvement in the same risk, without diminishing the signed lines for the existing markets. So let's say that Reinsurer A writes $100 \%$ of the Excess of Loss contract, and Reinsurer B would like to take a line on the contract. The only way to give them a line on the Excess of Loss contract is to reduce the line that Reinsurer A has. The insurer may not wish to do this though if Reinsurer A is keen to maintain their line. So the Insurer may allow Reinsurer B to write the RPP cover instead, and leave Reinsurer A with $100 \%$ of the Excess of Loss contract. This commercial factor may be one of the reasons that traditionally writers of an RPP would be inclined to give favourable terms relative to the Excess of Loss layer so as to encourage the insurer to allow them on to the main programme and to encourage them to allow them to wrte the RPP cover at all.

Moral Hazard

​One point that is quite interesting to note about how these deals are structured is that RPP covers can have quite a significant moral hazard effect on the Insurer. The existence of Reinstatement Premiums is at least partially a mechanism to prevent moral hazard on the part of the Insurer. To see why this is the case, let's go back to our example of the $5m$ xs $5m$ layer. An insurer who purchases this layer is now exposed to the first $5m$ of any loss. But they are indemnified for the portion of the loss above $5m$, up to a limit of $5m$. If the insurer is presented with two risks which are seeking insurance - one with a total sum insured of $10m$, and another with a total sum insured of $6m$, the net retained exposure is the same for both risks from the point of view of the insurer. By including a reinstatement premium as part of the Excess of Loss layer, an therefore ensuring that the insurer has to make a payment any time a loss ceded to the layer, the reinsurer is ensuring that the insurer keeps their financial incentive to not have losses in this range. 

By purchasing an RPP cover, the insurer is removing their financial interest in losses which are ceded to the layer. There is an interesting conflict of interest in that the RPP cover will almost always be written by a different reinsurer to the Excess of Loss layer. The Reinsurer that is writing the RPP cover is therefore increasing the moral hazard risk whichever Reinsurer has written the Excess of Loss layer. Which will almost always be business written by one of the Reinsurer's competitors! 

Working Layers and unlimited Reinstatements

Another point to note is that this pricing analysis makes a couple of implicit assumptions. The first is that there is a sensible relationship between the expected loss to the layer and the premium charged for the layer. This will normally only be the case for 'working layers'. These are layers to which a reasonable amount of loss activity is expected. If we are dealing with clash or other higher layers, then the pricing of these layers will be more heavily driven by considerations beyond the expected loss to the layer. These might be capital considerations on the part of the Reinsurer, commercial considerations such as 

Another implicit assumption in this analysis is that the reinstatements offered are unlimited,. If this is not the case, then the statement that the expected reinstatement is $EL * ROL$ no longer holds. If we have limited reinstatements (which is the case in practice most of the time) then we would expect the expected reinstatement to be less than or equal to this.
​

I was attempting to set up a Loss Model in VBA at work yesterday. The model was a Compound-Poisson Frequency-Severity model, where the number of events is simulated from a Poisson distribution, and the Severity of events is simulated from a Severity curve.

There are a couple of issues you naturally come across when writing this kind of model in VBA. Firstly, the inbuilt array methods are pretty useless, in particular dynamically resizing an array is not easy, and therefore when initialising each array it's easier to come up with an upper bound on the size of the array at the beginning of the program and then not have to amend the array size later on. Secondly, Excel has quite a low memory limit compared to the total available memory. This is made worse by the fact that we are still using 32-bit Office on most of our computers (for compatibility reasons) which has even lower limits. This memory limit is the reason we've all seen the annoying 'Out of Memory' error, forcing you to close Excel completely and reopen it in order to run a macro.

The output of the VBA model was going to be a YLT (Yearly Loss Table), which could then easily be pasted into another model. Here is an example of a YLT with some made up numbers to give you an idea:

It is much quicker in VBA to create the entire YLT in VBA and then paste it to Excel at the end, rather than pasting one row at a time to Excel. Especially since we would normally run between 10,000 and 50,000 simulations when carrying out a Monte Carlo Simulation. We therefore need to create and store an array when running the program with enough rows for the total number of losses across all simulations, but we won't know how many losses  we will have until we actually simulate them.
​
And this is where we come across our main problem. We need to come up with an upper bound for the size of this array due to the issues with dynamically resizing arrays, but since this is going to be a massive array, we want the upper bound to be as small as possible so as to reduce the chance of a memory overflow error.

Upper Bound

What we need then is an upper bound on the total number of losses across all the simulations years. Let us denote our Frequency Distribution by $N_i$, and the number of Simulations by $n$. We know that $N_i$ ~ $ Poi( \lambda ) \: \forall i$. 

Lets denote the total size of the YLT array by $T$. We know that $T$ is going to be:

$$T = \sum_{1}^{n} N_i$$
​We now use the result that the sum of two independent Poisson distributions is also a Poisson distribution with parameter equal to the sum of the two parameters. That is, if $X$ ~ $Poi( \lambda)$ , and $Y$ ~ $Poi( \mu)$, then $X + Y$ ~ $Poi( \lambda + \mu)$. By induction this result can then be extended to any finite sum of independent Poisson Distributions. Allowing us to rewrite $T$ as:

$$ T \sim Poi( n \lambda ) $$

$$ T \sim N ( n \lambda , n \lambda ) $$
Remember that $T$ is the distribution of the total number of losses in the YLT, and that we are interested in coming up with an upper bound for $T$.

Let's say we are willing to accept a probabilistic upper bound. If our upper bound works 1 in 1,000,000 times, then we are happy to base our program on it. If this were the case, even if we had a team of 20 people, running the program 10 times a day each, the probability of the program failing even once in an entire year is only 4%.

I then calculated the $Z$ values for a range of probabilities, where $Z$ is the unit Normal Distribution, in particular, I included the 1 in 1,000,000 Z value.

We then need to convert our requirement on $T$ to an equivalent requirement on $Z$.

$$ P ( T \leq x ) = p $$ 

If we now adjust $T$ so that it can be replaced with  a standard Normal Distribution, we get:

Now replacing the left hand side with $Z$ gives:

Hence, our upper bound is given by:

Dividing through by $n \lambda $ converts this to an upper bound on the factor above the mean of the distribution. Giving us the following:

We can see that given $n \lambda$ is expected to be very large and the $Z$ values relatively modest, this bound is actually very tight.

For example, if we assume that $n = 50,000$, and $\lambda = 3$, then we have the following bounds:

So we see that even at the 1 in 1,000,000 level, we only need to set the YLT array size to be 1.2% above the mean in order to not have any overflow errors on our array.


References
(1) Proof that the sum of two independent Poisson Distributions is another Poisson Distribution
math.stackexchange.com/questions/221078/poisson-distribution-of-sum-of-two-random-independent-variables-x-y

(2) Normal Approximation to the Poisson Distribution.
stats.stackexchange.com/questions/83283/normal-approximation-to-the-poisson-distribution

I came across this problem at work last week, and I don't think there's anything in the notes on how to deal with it, so I have written up an explanation of how I solved it.

Suppose we are carrying out a rating exercise on a number of lines of business. For two of the lines of business, Marine Hull and Marine Cargo, we have very few claims so we have decided to group these two lines together for the purpose of the Experience Rating exercise. 

We have been supplied with separate rate change indices for all the lines of business. How do we combine the Hull and Cargo Rate Change indices?

Firstly, let's review a few basics:

What is a rate change?

It is a measure of the change in price of insurance from one year to the next when all else is constant.

Why do we need a rate change?

We will often use historic premium as an exposure measure when experience rating a book of business. If we do not adjust for the change in price of insurance then we may under or over estimate the rate of historic losses. 

What do we mean when we say the rate change 'for 2018'?
​
This answer is debatable, and if there is any uncertainty it is always better to check with whoever compiled the data, but generally the '2018' rate change means the 2017 - 2018 rate change.

How do we combine the rate changes from two lines of business?

Let's work though an example to show how to do this. I am going to be using some data I made up. These figures were generated using the random number generator in excel, please don't use them for any actual work!

Suppose we are given the following Premium estimates:

And that we are also given the following rate changes:

Then first we need to adjust the rate changes so that they are in an index. We do this by setting 2018 to be equal to 100%, and then recursively calculating the previous years using:

$${Index}_n = {Index}_{n-1} (1 - {(Rate \: Change )}_{n-1} ) $$

We can then calculate our On-Levelled Premiums, by simply multiplying our Premium figures by the Rate Change Index.

$${ ( On \: Levelled \: Premium ) }_n = ({Premium}_n) \: ({Index}_n) $$

Using the combined on-levelled premiums, we can then calculate our combined Rate Index using the following formula:

$${Index}_n = \frac{ { ( On \: Levelled \: Premium ) }_n } { {Premium}_n }  $$
And our combined Rate Change using the following formula:
$${Rate Change}_{n-1} = \frac{ {Index}_n } { {Index}_{n-1}} -1  $$

Most of the time we will only be interested in the combined On-Levelled Premium. The combined Rate Change is only a means to an end to obtain the combined On-Levelled Premium. If we have a model though where we need to input a combined Rate Change in order to group the two classes of business, then the method above can be used to obtain the required Rate Changes.