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PD-9:
TR-9 :
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An Introduction to the Munich Chain Ladder
Method
Introduction à la méthode du triangle de
Munich (Chain ladder)
MODERATOR/
MODÉRATRICE :
Claudette Cantin
SPEAKER/
CONFÉRENCIER :
Gerhard Quarg
?? = Inaudible/Indecipherable
ph = phonetic
U-M = Unidentified Male
U-F = Unidentified Female
Moderator Claudette Cantin: Good morning, everybody. It’s the first day of fall and it’s raining, so it’s a blah day, but
we’ve got the pleasure of having Gerhard Quarg with us. Gerhard came from Munich on Tuesday. I guess he went back
from Peturn [ph 00:21] to leave on Monday, went to Oktoberfest and then packed to come to Toronto. So that’s been a
very tough week for him.
I was very pleased when the Canadian Institute of Actuaries approached us to ask if Gerhard would come and do his
presentation on the Munich chain ladder. He didn’t hesitate and said yes. He’s in the central reserving division of
Munich Re in Munich, and he takes care of, he focuses on loss reserve adequacy within the group as well as looking at
improving method, developing loss reserve method and tools. He was born in, I won’t tell you which town in Germany
because I cannot say it, but he studied mathematics at the University of Regensburg. And in November 2001, he
obtained his PhD, with a thesis in the field of algebraic geometry.
He’s got three children, with the little baby being six weeks old on Monday. He is a member of the German Actuarial
Society and he lectures for the German Actuarial Academy. He won the Gauss Prize in 2003 for the Munich chain
ladder, a reserving method that reduces the gap between the IBNR projection using paid methods and using incurred
method. And as all of us know, this is a real challenge at times, to try to explain the differences in estimates when you use
the paid method or the incurred method. So hopefully this presentation will help you find a solution for that and he will
give us some concrete examples of how effective the Munich chain ladder can be. On that I’ll leave it to Gerhard.
Speaker Gerhard Quarg: Yes, good morning and thank you all for coming. I’m not a morning person so the jet lag
works in favour for me. In my time it’s like the afternoon, so I’m pretty well awake. And thank you all for the
opportunity to speak to you about this topic, which I like very much. I do not have a lot of formulas on the slides. I like
more to present you the basic ideas and concepts. If you’re interested I also brought a little booklet. It’s not $20, you can
have it for free. If anybody wants to give me $20 I take them, but . . . I brought some copies, they’re lying over there so if
you’re interested in more details and more formulas and all the backgrounds just go there and get them. Otherwise you
can drop me a note or Claudette and I can also send you an electronic copy because I think that won’t be enough for
everybody.
That’s the agenda of the day. I brought more slides than I can talk to you about. That’s not a problem because I think
we have enough to talk about with the first two talking about what’s the idea of Munich chain ladder, how do we get
from the basic chain ladder idea to a Munich Chain Ladder idea and what happens if you really perform a Munich
Chain Ladder calculation. If there is more time left we can talk about errors and about extending this idea to other
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methods. If you have questions I am willing to take them at any time. Just raise your hand or something but you can also
ask them and leave them for the end, that’s fine.
When I worked in reserving, and that’s a problem which all of you have had, to project a triangle to make a quadrangle
out of it, you have usually a paid triangle, you have an incurred triangle and when you do your projections then the
ultimate you get from the paid triangle, the ultimate you get from the integral triangle, they do not agree. Sometimes
they are close, but sometimes even if they agree in total, it’s like for individual accident years you see a huge difference.
Let’s jump to an example.
That’s an Asian proportional motor portfolio I came along so all the examples I have are real life examples. It’s figures
which I came across in my reserving work. And it doesn’t look like a triangle, I agree. What does it show? It shows the
paid to incurred ratios. We divide the paid triangle by the incurred triangle and we plant it in a way that on the right
hand side we show the development years, or the delay as it was called yesterday in Mark’s presentation. And on this side
the paid to incurred ratio. So what does it mean?
It means that for this triangle in the first accident year that’s a far spread of how much is paid, it goes from almost a little
bit above 0%, almost nothing paid until almost 60%. Going along the triangle, the blue line indicates the average. Going
along this triangle you’ll see that the average paid to incurred ratio is increasing. Well, as you would think it does. And
after 10–11–12–13 years we are at about 100%, which means no cases have left. The paid to incurred ratio is 100%.
Everything has been paid out and the triangle has calmed down. So it’s a fully developed triangle.
Now we can do on this triangle a paid chain ladder calculation and we can do on this triangle an incurred chain ladder
calculation. We can again build the paid to incurred ratios of the predicted values. They are coming here in red. I think
you’ll see this doesn’t look good. It’s kind of, well, it’s a nice picture maybe like this of going up but it’s not as we would
expect it to be because if you look here, for example, down here, there’s a point at around 60%. That means that the
ultimate projection of paid of this accident year is only 60% of the ultimate projection of incurred. So it’s a huge gap.
But there are other years where years like these are like 150%. This means that the paid projection for this accident year
is 50% higher than the paid projection for the incurred triangle. So even if the total sum maybe is not too far away from
the ultimate you have for individual accident year, these are huge differences.
And why is that the case? Because looking here, in the past this came up to 100%. And this now looks more like if I was
above average it stays above average. If the paid to incurred ratio was below average it stays below average. So that’s the
feeling we have. It looks kind of parallel how these lines work out. Now, and actually that’s what it really is. You can get
this into formulas and . . . So that’s the interpretation I gave already.
If you have it, the formulas, you can see what happens if you make a paid chain ladder calculation and an incurred chain
ladder calculations, the following, the individual paid to incurred ratio of accident year I of the K development years. If I
take the ratio to the average paid to incurred ratio of the K development years, it’s the same as the individual to the
average on the diagonal. Once again . . . I have an average paid to incurred ratio, which I calculate by sum of paid
divided by sum of incurred. And the ratio of an individual accident year if it’s twice the average at the diagonal, at the
most recent value we have, it will stay twice the average until the end. And if it’s half the average, it will in my projection
stay half the average until the end. But I mean that’s not what happened in reality or what happens in reality because we
know somehow if case reserves are paid out it has to get to 100% and it cannot get to 150 or to 50%.
How can the paid to incurred ratio if it’s below average get back to average? Now there’s only one way. Either the
development factors which are coming for paid have to be stronger, the paid has to speed up or incurred has to pull the
brakes. You know, if the average is too low then the paid needs to be quicker or incurred needs to brake. What would it
mean? It would mean that if I looked at development factors they should show a dependency on the paid to incurred
ratios. So for paid development factors, as I just said, maybe I should see for high paid to incurred ratios, I should see
small factors and for small paid to incurred ratios I should see high factors.
So, that’s again about the ratios, what I said. But in the past below average paid to incurred ratios were followed by high
paid or low incurred development factors and above average paid to incurred ratios were followed by low paid or high
incurred development factors. Well, that’s a lot of high/low and so on, so let’s look at a chart. I think it’s a lot easier.
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That’s a marine triangle. Transport triangle. Development year one to two, it shows the development factors on the Y–
Xs and it does show the paid to incurred ratios after one year. And we see a very clear dependence. We see that this
accident year had a paid to incurred ratio of about 43% and the factor which came was pretty high, relative to the others.
The black lines indicate the averages. And if I had a high paid to incurred ratio, then the factors which came after that
have been pretty low relative to the average. So it’s a very strong dependence. If you do a paid chain ladder calculation,
we are just taking the factor of the black line no matter what the paid to incurred ratio is. It’s always the same factor we
take. Now that was the paid triangle.
Let’s look at the incurred triangle. It looks like this. It looks like a little bit less correlation, but it’s all the question of the
scaling you do, but you still see that obviously high paid to incurred ratios had high factors, low paid to incurred ratios
had low factors. And going back that’s already the basic idea of Munich chain ladder: let’s not use the same development
factors, the average black line, if we have more information, and the more information is where is the paid to incurred
ratio. But let’s look at this chart. Well, and it’s really inviting you for drawing a regression line down here. Let’s make a
regression line. And if we know for the youngest accident year where we have to make a projection and we know the paid
to incurred ratio is like 40% then we use a development factor of 2.6 and not like the average of 2.2 and a little bit more.
And the same for incurred. And if the world would be nice we could stop here because that was the idea and you do this
and we are done. Unfortunately, there are always problems coming. I mean, I was looking at some data triangles and
looking for this dependence if it is really there or not and obviously if I show you a chart I don’t show you an arbitrary
bad chart, I take one of the best ones I found to convince you that this is really there. And so it doesn’t always look like
that. It can be a more like skewed picture and more volatile. So let’s look at the problems we come across.
One problem is given that chart, these are the paid development factors from the second to the third development year
of a non-proportional fire triangle. It’s not so important with triangle, but you see there is again the strong dependence
of the factors we have seen from the paid to incurred ratios, so at this point it’s fine. But would you try to draw a
regression line here? Or would you draw it? It’s rather difficult. If you do it on the first set of points you are coming
below zero then you have negative factors and maybe Excel is doing that but it doesn’t make any sense. Or you take into
account the points up there on the right, but then you miss the very high factors, which you had after very low paid
development factors. It doesn’t look like a line. It looks more like a hyperbola, something like that.
And how can we solve this problem? And this was an idea which Thomas Mack came around which said, “Well, don’t
look at the paid to incurred ratios, look at the incurred to paid ratios.” We are not used, or at least I am not used to
doing that. Our software shows the paid to incurred ratios and nice charts and so on but not the incurred to paid ratios.
But it means basically the same figures just presented in a different way. And if you make the same chart it looks like
this. Now the Y–X is the same, the X–X is instead of P divided by I, it’s I divided by P. The points which were on the far
left went out to the far right.
Does it look like it’s on a line now? Yes? No? Nodding, hmm, hmm. Well, I say it doesn’t have to be on a line, it has to
be randomly scattered around the line. Then I’m happy. And I think that’s the case here. If I draw a line and it doesn’t
even have to be homo stochastic, it can be further apart here. Closer here, yes, but it has to be scattered around a line and
I think if I put a line in here it looks pretty good again for the prediction. And the other big advantage of turning it
around that way is the whole situation gets more symmetric. You know how confusing it was before when I said high
paid to incurred ratio for pay is a low factor, no a high factor average, low or high, I don’t remember. Now it’s
symmetric because for paid we look at incurred to paid. For incurred we look at paid to incurred. And we always expect
increasing behaviour in these charts. So it’s a lot easier to the scope [ph 15:34] and the following calculations when you
formulate a model and try to do your calculations and all this, this also gets all simpler because it’s a symmetric situation,
you only have to do all your calculations once and then you switch the letters P and I and you have the calculation for
the other side.
OK, this was one problem, the first one. But actually it’s not the biggest problem. The biggest problem is if I go back to
this chart, it’s a rather large triangle, I think it’s 24 accident years which are in there. So it’s enough points to draw a
regression line. But in any triangle, even a large triangle, if you go to higher development years you will have less points.
So it’s going to be less and less. And everybody knows that if only three accident years are left then already calculating the
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average chain ladder factor is something which might produce unreasonable results. But if you have three values and you
want to draw a regression line, then it’s pretty sure that you are lost and it doesn’t make any sense to do that. Any result
can come out of it, it might be an increasing line, a decreasing line or whatsoever. So it’s not advisable to do the simple
method as I said, like take for each development year a regression line and we are done. We have to find some other way
where we get all development years into one trial.
Now, as development factors are not . . . So we have the high volatility. And as development factors are not comparable
from early years and later years, we need to pass over to residuals. Similarly calculated as the residuals yesterday, only I do
not have any formulas and no age at (ph) [17:30] adjusting matrix or something like that. The formulas are in the $20
booklet, which you can buy.
But the residuals, they simply measure the deviation of the average value in multiples of the standard deviation. So a
residual of one means it’s above average and it’s above it by one standard deviation. That’s the basic idea of that and if I
do that I can make the same chart as I had before, but I can put in all residuals, all developments of the whole triangle at
once. And that’s in the picture, that’s now a general liability triangle, a European proportional general liability triangle. It
basically shows the same thing as before. The Y–Xs, it’s not development factors anymore, but it’s residuals of paid
development factors. On the X–Xs it’s not incurred to paid ratios anymore but it’s residuals of incurred to paid ratios.
But as it’s residuals, they are normalized, they have sort of standard deviation one in this direction and in this direction
and the whole picture over this triangle is a nice cloud, and do you see any correlation in there? Any dependency? Well I
do. If you calculate the correlation it’s like almost, it’s 45%, I think, was the figure or something like that.
Now we can do the same. We can draw in a regression line here and read off the factor. So let’s look at the regression
line. The basic idea now is if I have to project and I need a development factor, I see what is my incurred to paid ratio. I
see what’s the residual of it. Let’s assume the residual is like one, which is around here. Then I do not take an average
development factor, which would have a residual of zero, but I look at my regression line here and I take a development
factor which has a residual of like one half. And from the residual of one half I have to calculate back to the factor. So
what we did before directly was we looked where is the incurred to paid ratio and we looked at our regression line and
took the development factor. Now we have to go one step further, we have to go from the incurred to paid ratio to the
respective residual, then to the regression line step and then go back to the factor.
By the way, who would have drawn the regression line the same way it’s drawn here? Or who would have drawn it
steeper? If you go back, if you think here and you have to draw a regression line there. Steeper? Same size? Steeper, yeah.
Everybody would do that and me too. But it’s still correct because what you are doing when you draw a regression line
with your eye is you’re minimizing the distance of the line to the points. Like with quadratic distances and so on you
have good eyes which can do that. But what we need here is something different. We need to minimize the differences
only in vertical direction.
Imagine I have the residual of, let’s say here, your regression line would be here and you would take a development factor
up here. But I mean, all these points were down here. It’s different, you get a flatter regression line when you only
minimize . . . when you always average like this. Imagine you only put two pieces of paper there, and you only look at
the stripe here and here. And in this stripe you have to do your regression line because we are in the situation that we give
the X–Xs and when we want to predict the Y–Xs. So that’s a different thing what we do.
OK, let’s look at the reported, which is basically the same. That’s the reported triangle and here we have a little less
correlation. And this is the whole Munich chain ladder idea. We predict the paid triangle and the incurred triangle not
by using the average development factors, but we use this residual chart to correct for where the incurred to paid or the
paid to incurred ratio is. We do it on the paid side and on the incurred side. So again, regression line here it’s rather
uncorrelated. The correlation is not so big so the regression line and the correction is a lot more flat. Now we come to
the point where some formulas are coming. Any questions by now? No.
Let’s go back to chain ladder. The basic chain ladder assumptions are written down here. They say what I expect for the
paid factor to come: it’s paid of accident year I after K plus one years divided by after K, so the development factor. If I
know the paid development up till point K, well that’s a figure that’s the paid development factor. And what I expect for
incurred, the factor to come, if I know my incurred (ph) [23:15] triangle until delay K, it’s the incurred development
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factors. These are the usual. I mean you can write it down different ways or so but these are the usual assumptions for the
chain ladder model. But you see on the paid side there is no I and on the incurred side there is no P. These assumptions
have nothing to do with each other. They are always designed for the projection of one triangle and if you read a paper
on this it always uses letter C, not P or I because it doesn’t specify which triangle you’re working on. It could also be a
number of loss triangle or a premium triangle or whatsoever. But it’s only designed for the projection of one triangle.
But we have two. We have the information of two. So what we really need is something different. We need to take these
systematic correlations which we now have seen in real data that they are there, we need to take them into account. So if
I write it as a formula, the only difference is we need to know what do we expect for paid as a development factor if we
know paid and incurred. So it’s a different conditional expectation. And we need to know what do we expect for
incurred if we know paid and incurred. Or which is just a way to recalculate or reformulate it.
The same one, instead of knowing the factor, we could ask ourselves what do we expect for the residual of the paid
development factor if we know paid and incurred? What do we expect for the residual of the incurred development
factor if we know paid or incurred? And the answer of that, we have given the answer in the charts before. Because we
said we’d go on the regression line and take the amount of the regression line. So if lambda is the slope of the regression
line, then what we expect for the paid residual, for the residual of the paid development factor, is just lambda times the
residual of the incurred to paid ratio. And what we expect for the residual of the incurred development factor is just
lambda times the residual of the paid to incurred value. It’s nothing but the chart we had before with the red line, yes.
What do I expect here if I have given the whole X–Xs? Well, it’s the amount on the regression line so it’s lambda, if
lambda is the slope, it’s lambda times the X value. That’s nothing but a direct translation of what we have done before
graphically and thought, “Oh, that’s a good idea.” A direct translation into mathematical formula.
With this formula we can play around a little bit and see, for example, what is lambda. So if you do some calculations
which you can find in the $20 booklet you’ll see that the correlations between the paid factor and the incurred to paid
ratio given paid is the lambda, lambda P. What does it mean? Imagine you are on the paid triangle and you work to the
diagonal and you want to look to the next paid value, that’s the paid to incurred ratio, but on the other hand you look to
the incurred triangle, what’s happening over there, that’s the incurred to paid ratio. And what’s happening here in the
next step and what happened there that’s correlated. It has to do something which each other and the correlation factor
is this lambda Pi. So these two figures they sort of describe together the interdependency of paid and incurred.
What does it mean if the lambda for paid is high and the lambda for incurred is zero? It means that in the development
of the triangle the paid development factors they show a high correlation to where incurred is. Yes. Paid will look at
incurred and adjust its behaviour. On the incurred side the correlation, as example, I said it’s zero so incurred doesn’t
care about paid and the incurred is just moving on. Could be a reasonable situation when you set up individual case
reserves and they are paid out sooner or later and if it’s sooner or later that means paid has to follow the case reserves and
in that regard you have this picture that you have a high paid correlation and no incurred correlation.
But there are examples where it’s all different. It could be that the paid correlation is zero and the incurred correlation is
high. What does it mean? Well if the paid correlation is zero it means like paid is just moving on, doesn’t care about the
case reserves. Things are paid. And case reserves are adjusted later to the payments. Is this a typical behaviour? Well
maybe not very typical, but if you have, for example, old accident years where there are case reserves sitting for years now
with some claims handled 20 years ago, made up, and you just leave them there and do not touch them, then maybe the
case reserves are not very predictive about what will be paid out and you have some regular payments going on. They do
not care. At some point in time you adjust your case reserves to the payments which you have seen and in this case it
would be a reasonable scenario where you would say, “Well, the incurred reserves would completely follow the paid
reserves.”
For most triangles I have seen it’s neither one nor the other. For most triangles you have correlations on both sides, so
you really have an interdependence. Paid is following incurred, incurred is following paid. And usually on the paid side
the correlation is a little bit higher than on the incurred side. But as I said, this differs from portfolio to portfolio from
triangle to triangle and that’s why I think it is important to measure this correlation in the data if possible and not to
assume some sort of correlation because this can be different. Good.
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Now we have the model. From the model you also need estimators because you need your projections, you need to
project something so you need estimators. And I don’t give you the details for the estimators here in the paper, but what
does it look like if you have then a formula for your projection? You get recursion formulas, very similar to that of, for
example, simple chain ladder. The next paid amount you estimate of the K plus one years is the factor times the paid
amount ahead. Yes, this would be the same as chain ladder is except that the factor is a little bit more complicated. It’s
not only like one estimate; it’s a longer term.
Now let’s look at this in different ways. Yes, I haven’t explained, mentioned all parameters. The lambda, as I said, is
estimating the slope of the regression line and as we have to calculate residuals, we need variant parameters. So we have
sigma for the residuals of the factors and the rho (ph) [30:34] for the residuals of the paid to incurred ratios and we have
our average incurred to paid ratio or our average paid to incurred ratio, which we also have to estimate. Or is something
showing up? One of the problems obviously is we are doing something much more sophisticated but we have suddenly
much more parameters which we have to estimate, which is again, well, not as good if these parameters cannot estimate
it in a stable way.
But let’s stick to the formula now and write it in a different way. It’s the same formula now here only for incurred as the
slide was too small to take it both. How does it look like? It looks like incurred equals and now we have the usual chain
ladder term, the chain ladder factor age to age ratio times the incurred amount we had estimated one development year
earlier and we have a correction term. The correction term is again a factor times the incurred amount and how does the
correction term look? Well, it looks like first of all we have the correlation. If there is no correlation on the incurred side
then there is no correction. Yes, that’s good and that’s plausible. Then we have a scaling factor which comes from
calculating the residuals back and forth. And then we have this term which is paid to incurred minus average paid to
incurred and we see this term is changing the sign depending if paid to incurred is above average or below average.
It’s exactly how we would expect it to be. If we have below-average paid to incurred ratios then this is negative and we
are correcting the factor downwards. And if it’s above-average then this correction term is positive and we are correcting
it upwards. So looking at it this way you can say it’s a chain ladder calculation where we correct the factors.
And another way to look at it that’s the way is like Thomas Mack I mentioned before. He worked with me on that issue,
likes to see it is the same formulas. It’s a little more complicated if you introduce new ladders. But you see, you can also
say it has the form of a double regression. It’s not important how these As and these Bs are exactly defined for now, it’s
just to see that the formula is like paid factor times I plus the factor times P. I is a factor times P plus a factor times I. So
you can consider this sort of a double regression if you want to handle it this way.
Another way to look at the formula, that’s one which, like Dave Clark pointed out to me from Munich or America, it’s
like you can write it down this way if you introduce another letter W, which is this long term. Then you have the new
incurred amount is the usual chain ladder term times one minus W plus this term times W. So it’s sort of a credibility
approach because this W is between zero and one. And depending on, let’s see, the lambda, when the lambda is zero
then the W is zero, then we have the usual chain ladder term. If the lambda is very large then the credibility weight is
more on this side, though less on the usual chain ladder side, but more on this corrected amount here on the right side.
Why corrected amount? Well, that’s the incurred amount and that’s the paid amount divided by the paid to incurred
ratio so that would be a good incurred amount. If the incurred amount would fit the paid amount perfectly then it
would be like this. So that’s credibility approach if you either take the incurred side or if we take the paid value and
produce an incurred amount out of the paid value.
OK, so much about the formulas. I think these were the last formulas to show. The idea was just to show that take from
the formulas the following. Again you have your paid triangle, you have the incurred triangle and as with chain ladder
you fill your triangle with a recursion formula, you fill it up, but you fill it both at the same time. That was this one
because in calculating the new paid amount you also need the incurred amount, yeah. So you are projecting two triangles
but you have to project them at the same time with a recursion formula. And this recursion formula might look pretty
awkward because it’s pretty big, but you can simplify it. You can either imagine it as adjusted to ladder factors. You can
look at it as sort of a double regression on both data sets you have or you can look at it as a credibility approach between
chain ladder and another approach. Yeah, that was what I would like you to take with you from these formulas.
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Now, let’s have a look at results if you really perform chain ladder calculations. Any questions on the formulas? Good.
What’s the minimum requirement if you want to do a Munich chain ladder calculation? Well, in my opinion the most
important factor is you need a paid triangle where a chain ladder is adequate. You need an incurred triangle where chain
ladder is adequate and the triangle has to be large enough. Now, for the first thing paid and incurred triangle it should be
adequate for chain ladder; we had it yesterday also in Mark’s presentation. It’s very important that you do not just apply
a method. It’s very important that you look at the assumptions of the models you are using if they hold in the triangle.
And if the triangle is not a chain ladder triangle because it behaves different, then it’s definitely not a Munich chain
ladder triangle. Paid and incurred have to look as the typical chain ladder behaviour, which means from one
development year to the next you expect an average factor. It’s sometimes higher, it’s sometimes lower, but it’s spread
around this average.
The other one is the triangle must be large enough. Why? Because we have to estimate some parameters. The most
important one is this correlation parameter lambda P and lambda I. And if you have only very few points in your triangle
then this residual chart, which I showed you which had like a hundred points or something like that, will have only very
few points and then you have to be aware that trying to estimate a correlation out of a few data points simply doesn’t
work. So the triangle has to be large enough.
The next point is that Munich chain ladder will project a paid ultimate and an incurred ultimate. So you do not get one
ultimate, you get two of them. They don’t have to be the same, but they will concur as much as can be expected from
the data you have. What does it mean? Let’s look at examples. Let’s go back to the initial example I had. You remember
that initial example, I showed the paid to incurred chart? And then we had the red dots and now you see the split by
accident year. We had one red dot which was only at 60%, that’s the one here. We show this one right here. These are
accident years and these are the paid to incurred ratios we have at the ultimate projection in the last column. So you see,
this accident year number nine we only had the 60% and there we had two accident years 14 and 15 where we had
almost 150% paid to incurred ratio, meaning that the incurred ultimate was here far below the paid projection for these
years.
Now if we do a Munich chain ladder calculation on that it looks different and it’s correcting for that. So you see we are
pretty close to 100% in all accident years. Here we are at, let’s say, 97% only in this accident year. This means we do not
have the same paid ultimate and incurred ultimate from our two projections. But as it’s 97% it means the difference now
is now a very small one. That’s, I think, something we can live with. Before we had been at 60% and now we are at 97%.
So it’s pretty close together.
If we look at the chart, as it looks like that’s the same as we had in the introductory example. That was what the red dots
looked like if you did the paid to incurred projection separate to ladder calculations on paid and on incurred side, and
that’s what it looks like with a Munich chain ladder calculation. So I think that’s a much more convincing picture. The
red dots now sort of behave as one would expect it. It’s not perfect because, as you see, for example, here are some points
where we come above 100% and we know that the real triangle will not show this behaviour; well, except you have
negative case reserves in your triangle which sometimes makes sense so it could be, but usually you don’t have that. It’s
not perfect. As you see in the end, we are not exactly at 100% with all, but in principle it’s a pretty good behaviour. At
least from the picture you cannot say, “Well, what we do here, that’s complete nonsense.”
Let’s look at another example. That’s the example where we had the residual plot from the general liability example.
Again we see the ultimate paid to incurred ratios for the different accident years in doing separate chain ladder
calculations, and you see for this triangle already it means they are between 92% and 102% so actually that’s not one of
the big trouble triangles. If both projections are rather close together like this with no more than less than 10% deviation
or so, usually we would be pretty happy about that. But nevertheless let’s use it as an example; it’s a pretty smooth
triangle. And let’s see what happens if we do a Munich chain ladder calculation down here. Again we see it stabilizes the
situation. The ratio is very similar for all of the years. But it’s not at 100%. It’s at 96%. So as the title of the presentation
talks about closing the gap between paid and incurred projection, the gap obviously is not closed. It’s now a smooth gap,
it’s a nice gap for all accident years but the gap is still there.
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Now why is it the case? Let’s look at the paid to incurred chart we have for this triangle. We see again it’s a 14
development year triangle and obviously those are 14 accident years. From the beginning the paid to incurred ratios are
very close together. It’s like after one year of development they are between 30 and 35. After 10 years of development
they are between 91 and 93 or so. But you clearly see it doesn’t look like in the history we have that this is going to be
100%. It’s a German triangle and in German, by German local gap or so, it’s often the case that primary insurance
companies leave some careful case reserves from very old years, which is unclear if they will be ever needed but they are
sitting there. So there is some sort of remaining gap after these 14 years of case reserves. For the future development it’s,
for now, totally unclear will they be needed, these case reserves. Will they be paid out or will they come down at one
point in time? That’s something which Munich chain ladder cannot do. It cannot look across further than the triangle
that we have here. That’s different from if you do a chain ladder calculation or whichever else calculation on your
triangle. The given history points towards 96%. That’s where we would come. And the Munich chain ladder calculation
does exactly that. For the given history we come at 96% and not further.
What do we have to do if we have this problem? How about an unfinished run-off as is the case here? Now first of all,
always in the latest development years, as I said, estimating these parameters, that’s a very volatile thing, but they are
important for the final result. That’s not Munich chain ladder-specific, that’s already with the chain ladder. If you take
your H to H ratio from the peak of the triangle and you just take it without looking at it, it might produce very
nonsensical results if you have a huge factor there and you apply this huge factor to all your later accident years, and
that’s something you don’t want to do. So the parameters of the latest development years are relevant for the final result,
but they are very volatile.
The paid and incurred prognosis we have of the Munich chain ladder, they will not get together if the paid to incurred
pattern does not reach 100%. So what’s the usual solution for that? The usual solution is, well, within the known data
triangle we have to smooth our parameters by applying a regression line and taking out the volatility of the parameters
we have estimated. And beyond the known triangle we have to extrapolate. That’s already a difficult task to do it for the
chain ladder where you have to estimate H to H factors of the later development years and beyond the triangle, and it’s
guesswork to apply a reasonable tail here. If you want to do this as a Munich chain ladder then you don’t only have the
factors, you have the sigma parameters, you have these rho parameters, you have these paid to incurred ratios, and these
are a lot of parameters which we have to extrapolate in the future.
In principle that’s perfectly possible. You can all of them extrapolate and do your recursion formulas as the Munich
chain ladder would say so. In reality you might come across the problems that there are then some unreasonable
parameters in there and they will spoil all your projections. So that’s a time-consuming act, it’s very volatile. But the
good thing is it’s often unnecessary. What I would recommend is if you do Munich chain ladder calculations then use
the Munich chain ladder correction terms only in as many development years as they are necessary that the paid and the
incurred differences are singled out, and that paid and the incurred projections are close together. And in later years use
only like tail factors or H to H factors as you are used to and don’t care about calculating a lot of other parameters where
the estimation of those is very difficult and very, they are subject to some [47:08] judgment. OK, that was about
unfinished run-offs.
Then let’s get to another problem: the correlation parameters. As I said before, when you apply a model you should
always try to look at the assumptions of the model fulfilled. We made one big assumption. We went to residuals and we
have put all residuals into one big chart, the residual chart, and took the regression line, and therefore we estimated our
correlation to lambda. And there’s a very big hidden assumption in that. The hidden assumption is that this correlation
behaviour between paid and incurred is the same for all development years because we already estimated one parameter
for all development years. But how do we know that?
We don’t know it so we need to check it. We assume the single correlation parameter for all development years. But we
check it by calculating development year lambdas. That is this residual chart, which we do for all development years
together, we can do it for individual development years. Then we have less points and it’s more volatile but we can do
that. And we can look at the lambdas we get from the individual development years. Now usually these development
year lambdas are very widely scattered. That’s the basic reason why we put all together in one chart, because the
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estimation of the individual ones is very unstable. So if you see a picture for your development lambdas, which are very
high and very small and it’s fluctuating strongly, that’s OK. That’s the reason why we are doing that.
But there should be no clear trend over development years. If the first development years show very small lambdas on the
paid side and then later development years show very high ones like a break like that, then this is not a good sign because
obviously, maybe, your dependent behaviour is changing over time. But you can check this with these charts. When you
look for trends you should not look at the development year lambda of the later development years because as I said, in
the peak of the triangle where only a few amounts are there and you estimate these residuals and you estimate these
lambdas, it’s so volatile this estimation process that it doesn’t have any meaning if this lambda is 60% or -40% or
something like that. If you estimate this out of three points, out of three amounts you won’t get a reasonable value. So
look for trends for the development year lambdas over time, but not for the last, for the high development years. Those
just to meet them. Good.
So once we have checked for the lambdas and done the calculations, let’s have a look at the sign of the correction terms.
As I said before, Munich chain ladder is we have the usual chain ladder factor and the correction term up or down. So it
is a plus sign or a minus sign. How should the signs behave? Now, depends on whether Munich chain ladder considers
the preceding paid to incurred ratio below or above average. Because if it’s below or above average we will correct in one
direction or we will correct in the other direction. So within one line, within one accident year it would not make much
sense if the paid to incurred ratio on the diagonal is very far below average. Then I go one step further and then it’s
above average and I go one step further and it’s below average. Because that’s not sort of the expected behaviour we
would think of. If something is below average it should come to the average from down there. So within one line the sign
shouldn’t change.
Now, how about a column? We take one development year and look at this one and then we go across various accident
years. It would be very strange if in development year seven all accident years would be above average. It would be
something which should vary at random. One accident year should be below average, the next above, below, below,
below, above. So the behaviour we would expect is—see it now more visually—if you project and we look at the sign of
the correction terms only, then we would within one accident year expect the sign here, always a plus, to stay the same.
But if we look at the development year it should be short of randomly scattered around. And that’s also a chart, a table,
which if you program like a Munich chain ladder or something like that you can, we will look at your correction terms
and you can highlight the sales in colour simply to see in a very simple way are there any special patterns which I’ve seen
and if they are then maybe the assumptions of the Munich chain ladder model do not hold. And if it looks like that,
then by line the colour stays the same. But by column these colours change or so then I know OK, here I’m fine, that’s
the picture how it should look like.
Now what is the right sign to note behaviour, OK? Well, within a line, small changes of sign do not matter. And know
that a mathematically correct sign can only change and that cannot be a small sign change. But what I mean if it’s a little
bit above average, a little bit below average, yes, this fluctuation basically it is in the average, the amount, the accident
years have been predicted to be in the average. Then it will fluctuate a little bit. It will be, so the colours will change a
little bit, but this doesn’t matter. That’s of no importance.
But it can be, for example, that the paid to incurred pattern is implausible or it doesn’t match the paid and incurred
development factors you have chosen. If, for example, you took your paid to incurred pattern from the whole triangle
and for your factors you thought, “I only take three-year averages”, and there are some trends in your triangle, then the
paid to incurred pattern and the factors you have chosen don’t fit together anymore. And if this is the case then it may be
that an idea might suddenly from below average or far below average jump too far above average, and then we have this
sign change, which is implausible. Or it could also be that the estimation of the variant parameters, but these are not
reasonable because variant parameters, if you don’t have enough data points to estimate them, then the estimation is
rather volatile and then it may be that the results which you get here are not reasonable.
The solution that often works is what I said before, use your Munich chain ladder correction terms only to a certain
point in time where you have enough data for stable estimations, and for later on, if it’s not stable enough to estimate all
this, leave it and do usual H to H factors. Now, within a column it’s basically the same. It can also be that the paid to
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incurred pattern is implausible and doesn’t match the factors. Or it can be that new accident years have a different paid
to incurred pattern than old accident years. So, if now payments are made quicker or reserves are set more scarce or less
reserves are scarce, then newer accident years will have higher paid to incurred ratios and the pattern might be different.
The bad news is that if this is really the case, then the assumptions of the Munich chain ladder model are violated and
it’s not reasonable to just use it. The good news is that if you look at it this way it’s one way to discover these sort of
trends and things. So it’s also a tool for your analysis that you know what’s going on in your triangle. And then you have
sort of to go back and think what’s going on in my triangle, what changed over the years and how do I have to adjust my
reserve calculation. But the bad news, as I said, is you cannot just use your model and just run it through because the
result will not be reasonable.
OK, these were some points on when you perform calculations some difficulties you come across and which from my
experience when using this model as something where you can try and how you can try to solve for use in these
problems.
So, I’m pretty well in time. I have a lot more slides, so no problem, but that was . . . Up till here this was the point which
I really wanted to get through. Do we have any questions?
U-M: This question is in regards to when you get a situation with a change in reserving pattern, you get a company that
actually increases it’s reserving pattern where Berquist/Sherman methodology would be the normal approach, adjust your
triangle, would you . . .
Speaker Quarg: It’s very hard to understand.
U-M: Excuse me. When you have a situation where you’ve got an increase in reserving pattern, the claim demonstration
process has increased your pattern, so a Berquist/Sherman approach would be a reasonable approach because you want to
adjust the historical adverse claims reserve. Would you do that change before going through your system, or let’s assume
your approach would actually correct that, correct for that dramatically? Well, no change in reserving pattern.
Speaker Quarg: So you know what’s going on in your triangle, you know there is a change.
U-M: You know there’s a change in reserving for loss of fee. The case reserves are higher than in the past. Would this
approach correct it dramatically or do you have to apply a Berquist/Sherman approach first before doing this?
Speaker Quarg: I think there is no best way to do it. If you can backwards adjust your case reserves to nowadays level,
which usually would mean a lot of work that you can sort of correct your triangle. The situation is you have a triangle
and the old part of the triangle is different from the young part because the cases of behaviour change. Now if you can
adjust the old part of your triangle that it’s similar to nowadays level, then this would be a good thing to do and you
could do it but this usually is not possible. That’s usually too much work. And then I would sort of try to restrict to that
part of the triangle which are reasonable, so leave the old parts away if this is possible.
U-F: So if you correct your triangle then you can apply the Munich chain ladder?
Speaker Quarg: If you correct your triangle then you could apply the Munich chain ladder or any other method and try
to apply it on the whole triangle. But you can only also apply it by, you don’t always need the whole triangle, you can
also use for example stripe, the last five or last 10 development years, calendar years, something like that.
U-M: So your approach would not correct that? That was my question. The approach would not . . . You have to correct
your triangle beforehand, that’s what you’re saying?
Speaker Quarg: Exactly. Correct the triangle beforehand or do not use the data which is not predictable for what you
are doing. Any more questions?
OK, so maybe I’ll tell you a little bit more about when you have a prediction and an ultimate projection, the next step;
first step is best estimate. The next step is always the error around the best estimate. So you need something like a
prediction error. Yesterday was a very interesting presentation on the bootstrap method and basically this is about having
a whole distribution around your point estimate. The Mack method is for a chain ladder. For example, it calculates a
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standard error for your triangle and for Munich chain ladder the same thing works very similar along the lines as the
Mack method. So you can also calculate a predict share for Munich chain ladder. In order to do that, you need again
some more assumptions. You need an assumption on the correlation of paid, one more assumption on the correlation of
paid and incurred.
Now you might say, “Oh, again one more assumption”, something like that. It’s not unusual if you do only your chain
ladder; you do not need all the assumptions which are, for example in the Mack model, the variance assumptions. But if
you want to calculate a standard error you also need some more assumptions on further behaviour. But if you do that the
work is very similar. The prediction error which you want to calculate, it splits into a random error and then estimation
error. So one is the variance of the real outcome or random error or process error, I think, or a process variance it’s also
called sometimes. And the one is the error that you have in your estimation error. And from these two parts you can split
it up and both of them you can calculate by a regression formula, which as you have more data and two triangles instead
of one is more complicated than it is with the usual chain ladder calculation.
Now I here only showed the recursion formulas for the process variance, and you’ll see it’s rather complicated formulas.
You see the As and Bs, they are coming from these double regression type which we had before, because that’s the easiest
way if you want to calculate the prediction error that you use this formulation. And you see you have one recursion
formula for the variance paid, you have one recursion formula for the variance of incurred and you have one recursion
formula for the co-variance of paid and incurred. That’s where we need this additional and new assumption. And these
three recursion formulas are connected and the variance formula for paid. You have the variance formula for incurred in
there. You have the variance formula for paid in there and the one for the co-variance in there and again for the other
three. So it’s really interconnected recursion formulas which will give you then the variance of that.
Does the structure have to be so complicated? Hmm, basically, yes. Because what we are doing is we make a prediction
of paid or incurred ultimate knowing the paid and the incurred triangle. So if it’s a good triangle then the paid and the
incurred ultimate should be equal, which we have seen in some examples, they are. But also the variance should be equal.
Because what we are calculating is, what’s the variance of my paid ultimate given the two triangles? And what’s the
variance of my incurred ultimate given the two triangles? Now, if the paid and incurred ultimate are equal, which we
know that really they are, then I’ll know the variance should be the same. And that’s the outcome of these interconnected
recursion formulas that if it’s a good triangle, like fully developed until the end, that paid and incurred ultimates really
agree then the output will also be that the variances we calculated for paid and for the incurred side, they are also again
very similar. And that’s, yeah, that’s a good sign that it is like this because otherwise the interpretation would be difficult.
I have one ultimate but sort of two variances of the same figure.
That was the first step: do paid and incurred predictions agree? And it said so, they come close due to the interacting
recursion formulas. Now, a question which is often asked is, is the Munich chain ladder prediction error smaller or larger
than the chain ladder prediction error? Well, I think that’s not a valid question. The chain ladder projections and the
Munich chain ladder projections they do not agree, so you have different ultimates and then asking how about the
variance. Uh, is the variance the same? First of all you would need that the ultimates are at least similar or comparable.
But in one case, when the correlation, let’s say, on the incurred side the correlation is zero. When I said the lambda
parameter for incurred is zero, then I said we have no correction term because in the correction term always this lambda
appears, then the results are the same. So how about the case when the lambda for incurred is zero? Chain ladder and
Munich chain ladder ultimates do agree for the incurred calculation. How about the prediction error, is it the same or is
it not?
Now, as the underlying model is the same it’s the random error, so the process variance which is estimated is actually
really the same. But the estimation error or parameter error is larger for Munich chain ladder than for the chain ladder
estimation. Now is this reasonable? We get the same result, we have the larger one. Yes, it is reasonable because for chain
ladder we assume basically this correlation factor to be zero and for Munich chain ladder we estimate it to be zero but
Munich chain ladder takes into account that we could have misestimated this amount of zero. Basically the assumption
of lambda being zero is you only do chain ladder, it’s sort of hidden in the model assumption, yes, and then the error is
not measured. For Munich chain ladder it’s explicitly allowed that it would be different and if it is zero we assume,
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“Well, could be that we are wrong and therefore this also has a parameter error.” And therefore the outcome is a larger
error. So chain ladder hides this error in the model assumptions.
Can we apply the basic idea of Munich chain ladder also to something else? What was the basic idea again? The basic
idea was in the past we have seen inter-correlations between paid and incurred triangles. Speaking in the chain ladder
world, we have seen that the development factors for paid depend on the paid to incurred ratios. And that the chain
ladder factors for incurred depend on paid to incurred ratios. Can we take this idea over to other models which we use?
And the answer is yes, we can, because otherwise I wouldn’t have started this topic. Otherwise I could have just talked
about somebody else.
I wanted just to give you a sort of brief overview about another method where you do not use factors to get from one
stage to the next, but you take ratios, sort of loss ratios, an incremental loss ratio method. We also call it editive method or
whatsoever. A chain ladder assumes that from one year to the next you have an average factor. This method assumes that
from one year to the next you have an increment of on average 15% loss ratio. It could be above or it could be below. So
that’s a method which is quite, a little bit similar from a formalistic point of view. You know, the formulas that all look
similar. If you multiply in chain ladder you have to add in the incremental loss ratio method. If you divide in chain
ladder you have to subtract. But in the incremental loss ratio method you always have to look at an exposure measure,
which usually you can use the premium or an index premium or something else.
For chain ladder we looked at the beginning at the paid to incurred chart. Paid divided by incurred; this is if division is
negative paid minus incurred. And up to the sign these are the case reserves, so we have to look at the case reserves and
we always have to refer ourselves to the exposure, that’s why we look at the case reserve divided by this exposure letter B.
Yes, usually you could imagine being as the premium. We can do the same thing and look at the case reserve ratio. And
we see maybe a rather typical picture again, the case reserve ratio. Well, if it’s typical it’s hard to say. At the beginning it’s
an average like 20%, it increases to 40% to 50% almost on average. So we have 50% loss ratio sitting in our case reserves
and if the end of the triangle is fully developed then 0% of the loss ratios are sitting in our case reserves because all, now
all the loss ratios are paid out and no case reserves are left.
And without telling you in detail how this loss ratio method works, it’s very similar to chain ladder; we look at an
incremental loss ratio, we take an average incremental loss ratio and we add this average incremental loss ratio then year
by year to derive at our ultimates. But if you do this for this triangle then we have a very similar picture as we have seen
before. We can do this and the cases of ratios of the ultimates are not zero, they are above or below zero, meaning that
there are in our predictions either positive case reserves left or negative case reserves left, which both doesn’t make sense
because in the end they should agree. You can do the same thing, the formulation with what we had the paid to incurred
ratio being above average or below average and the ratio if it’s twice the average it will stay twice the average. If it’s half
the average it will stay half the average.
We can do the same thing here. Formula looks a little bit more complicated. Instead of the ratio to the average it’s now
the difference to the average. So it’s like the reserve ratio of an individual year, the difference to the average reserve ratio
will always stay the same as time goes on. So going back here, if I’m here 20% loss ratio below the average, I will be here
20% loss ratio below the average in my prediction. And I can do the same sort of similar picture as Munich chain ladder.
I have the same sort of dependence, I can calculate the residuals of the incremental loss ratios for paid, the residuals of
the incremental loss ratios for incurred. I can with the residual charts draw a regression line and have a lambda parameter
for doing that. And correct with it from the lambda parameter and I just want to go, “Here we are”, so I have the same
sort of residual charts which I can use. That’s the residual chart for the paid incremental loss ratios versus the reserve
ratios. So it’s very similar to what we had before.
That’s the residual chart for the incurred. Now that’s one where I think here you really do not see any sort of correlation.
Here the correlation, which is the measure that is basically zero, it looks very much like a cloud of points which is
arbitrarily spent there. And you can also do this formulation which we have for the standard. I won’t go into the details
of the formulas now, just make it clear to you that it’s the same sort of structure what we expect for the residual of the
incremental loss ratio, knowing both is the slope of the regression line times the residual of the reserved ratio which we
have. You are coming across the same sort of recursion formulas and for the final result you see that if you apply this to
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our initial example again, then we had here where paid to incurred ultimates would agree or not agree and if you apply
this, which I call them the Munich incremental loss ratio method, if you apply this then you are again much closer and
the corrections they take place and the corrections will look that you are coming to a zero percent loss ratio.
Basic message of that is yes, the basic idea being that you have this dependence. That if you pass the residuals you can
measure this dependence in your triangle. You can do this with any method where you can calculate the residuals and
then you can make this residual chart and take over this dependence in your projections.
OK, any more questions on the formulas or the charts?
Moderator Cantin: No questions, no comments?
U-F: Hi there. I just wondered if there were any Excel documents or anything that we could see with some sample
calculations of the formulas that you have? Not today but if you had anything that was available.
Speaker Quarg: I’m not completely sure about that. I know that also some consulting firms or so had already taken this
in their usual range of methods which they apply so they have programmed these methods also. But doing that and
programming this in Excel is really not very complicated, it’s rather straightforward. And in the $20 booklet which is
lying there, there are all the formulas in there, which I would say that, well, if you want a really comfortable Excel tool
you can spend as much time as you want, but if it’s just about to recalculate this and to see what’s coming out of that or
so I would say that’s not more than a day or so which you need. But there are already sort of tools around or it is
implemented in quite some tools.
André Racine: André Racine: I have a question about interpreting the results. Say you do the exercise and compare to
the chain ladder indication and what you get is closer to the incurred projection or closer to the paid projection, does it
mean something that you can derive about your data like change in the payment, speed of payment or change in the
reserving level? It’s a follow-up on the question that Pierrette had before.
Speaker Quarg: OK, I understood the first part of the question. It’s sort of is the new ultimate which you get of Munich
chain ladder is it closer to paid, is it closer to incurred or . . .
M. Racine: Well, depending on whether it’s closer to paid or to incurred, does it carry a different interpretation?
Speaker Quarg: It mainly depends on these correlation parameters measured. Because that’s the behaviour which is
coming out of the data and if the data, for example, tells you there’s a high correlation on the paid side and a very small
one on the incurred side, then your total result will be very similar to the incurred projection. So if the data tells you
there’s a correlation both sides so that both data types meet, then accident year by accident year you will meet, well, not
in the middle but somewhere in the middle.
The total outcome could then be a bit higher or smaller than these previous ultimates which you had. So it’s not that you
can always say it’s going one direction or the other direction.
M. Racine: OK, so it’s not at all related to the current state of reserving compared to settlement, it’s really just a matter
of how strong the correlations are?
Speaker Quarg: Can you say that again?
André Racine: I understand that from your . . .
Speaker Quarg: Sorry, it’s my German English which makes it difficult, you know.
André Racine: I know, sorry, and my English is not perfect—far from it. But . . .
Moderator Cantin: Do you want French, Gerhard?
Speaker Quarg: You can try in French, yes. I had four years’ French at school but I think . . .
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André Racine: Good. So I thought that where we would get would depend on the current spent of reserving or the
current timeliness of payment, but from your answer I gather that it only depends of how correlated they have been in
the past.
Speaker Quarg: Yes, it depends on the correlation parameters where it will shift, but on the diagonal if the current paid
to incurred ratio is very high or very low, that’s obviously giving the direction. If the current paid to incurred ratio is very
low then paid will be corrected upwards usually.
André Racine: OK.
Speaker Quarg: Which is completely correct, if the assumptions of the model are fulfilled then that’s what we would
expect. If the case is that like case reserving philosophy has changed and that’s the reason why the most recent accident
years, for example, all have very low paid to incurred ratios then you would see it on the one chart, if you look at the
signs of the correction terms, that all these signs are all positive or all negative and then it would not make sense to apply
just the method because the method assumes that these things have not changed.
But then you would know, “Oh, something has changed here, I have a new case reserving philosophy.” The old years,
what incurred has done in my old years is not predictive of what incurred is doing now and maybe this would lead to
either leave away the old years for incurred projection or maybe this would lead to saying like I do not believe my
incurred projection at all. I believe more my paid data and then a paid chain ladder might be the better answer than the
Munich chain ladder is because here you’ve tried to combine two data sets but if you do not believe one of the data sets
because you say, “Yes, things have changed here”, then you better go only with the one which is more predictive.
André Racine: OK, thank you.
Shawn Doherty: Shawn Doherty. Have you done any testing where you knocked out a couple of the calendar diagonals
and looked to see how well you were predicting data that then comes in?
Speaker Quarg: So, back testing that you leave something away and coming?
Shawn Doherty: Yes.
Speaker Quarg: Yes and no. Not in a formalized project that we took a hundred triangles and see how much closer we
get or how much we do not get. But for individual triangles, yes. And the answer, I would say, it’s similar to other
methods. If your triangles, if the most recent diagonals behave rather good then the results were good, and they were
even better than with chain ladder or with pure chain ladder calculations because they take these additional correlations
into account. If your triangles are spoiled by something that on the most recent calendar years you have certain calendar
year effects due to high inflation, low inflation, legal changes or whichever might drive these issues, then again this
method also fails as all the other methods also do.
William Shi: William Shi. My understanding Munich chain ladder based on both previous payment and incurred
information which I think is improvement, since it takes into consideration both previous payment incurred
interactively. But I assume the weight assigned to a payment and the incurred is the same. My question is, is it possible
in your Munich chain ladder to consider the different weight for payment and the incurred? Thank you.
Speaker Quarg: I wouldn’t say the weight is, sort of this . . .
William Shi: Because sometimes I think (off mic) a payment may be more reliable than incurred. So it will be much more
flexible ?? [1:23:40]
Speaker Quarg: Yes, I see what you mean. Hmm, I don’t see sort of a direct way where you can on a simple place input
a parameter like here one half and here two, and correct for that. But anyway, what you usually would do, you would
calculate several methods anyway. You would have your paid method and your incurred method and then you have here
something which is more in-between. So if you then at the end you do some weighting, apply some credibility to your
methods and blend those . . .
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William Shi: (off mic) I would not expect automatically the mathematical (ph) [1:24:24]… the consideration of their
weight. But it may be possible for the user to, you know, to give the weight manually.
Speaker Quarg: Hmm, hmm.
William Shi: So your systems will give the user a choice to use the manual weight. You know, I would like to give 25%
to the incurred information rather than the same weight for both payment and incurred.
Speaker Quarg: Yes, I see your point. I mean, the easiest answer would be like when blending the results, if you do that
at the end, that you take paid chain ladder results and Munich chain ladder results and blend them with giving you more
weight to paid to incurred side, that’s the easiest answer. A method which is doing that directly in the calculation, I have
not thought about that yet.
Joe Cheng: Joe Cheng. After last year’s presentation at the Oka (ph) [1:25:43] meeting, I put your method through 20
set of data, different type of business and I noticed one thing. First, if I take out all the unusual events like catastrophes
or very large shocks, the method improves quite a bit. But that is the same as a chain ladder method if I pull out the
unusual event. Another perspective is if I apply your method to the cumulative pay and incurred data versus if I divide
each row by their earned premiums so that now my input data is cumulative pay loss ratio and cumulative incurred loss
ratio and I apply the methodology, it produce slightly different answers. Would the latter be better because it reflect
growth by year and might be inflation as well?
Speaker Quarg: Hmm, hmm. So if instead of applying the whole method to the dollar amounts you are applying the
whole method to the loss ratios. So, per se, I would say . . . I’m winding out of this a little bit now with the typical
answer that you cannot say per se which would be better. I mean, it’s then two different models you apply. One you
apply your model assumptions to the dollar amounts and one to the loss ratio amounts and you would have to check
which looks better on the set. Is it more reasonable on the one side or on the other side taking in loss ratio? So that’s a
question which you don’t have to go to Munich chain ladder, you can ask it simply for chain ladder also. I apply chain
ladder to the dollar amounts or I apply chain ladder to the loss ratio amounts.
The idea of chain ladder is then to give more weight to high volume years and high volume years means then that you
have high losses, which can be due to bad loss ratios or high volume because you have more volume in the business.
If you do a chain ladder on the loss ratio side you give more weight to years with bad loss ratios. So that’s what I would
say. These are the two different assumptions you have. One is that you say the years are the more stable, the more
volume they have, chain ladder on amounts. The one is the years are the more stable the more predictive, the higher the
loss ratios are. That’s the amount if I apply it to the loss ratio. And now that’s the assumption you basically have to check
if your current situation of a triangle.
It might be that you say, “I’m back in a soft market period now and I again have relatively high loss ratios and I think
that the loss development of the previous soft market is more predictive than the loss development of the hard market, so
this method might produce more reasonable results.” Or it might be that you say, “The high loss ratio years, they have
been bad because some strange things are going on there and I had some very special losses which have a very strange
development pattern, and these years are rather less predictive to my current years and then I should not do it, then I
should go more with the dollar amounts.
So that would be the answer: go back, what are the assumptions and where do they fit better to my current task in
predicting? Does it answer your question? Thank you.
By the way, as you just said, you can apply it to loss ratios. I mean, in principle the chain ladder method is you have two
triangles where at the end you know they come together. So you can also apply it if you make a loss number calculation
and the incurred losses are all losses, a loss count triangle and the paid losses then for closed losses. So one is a subset of
the other and in the end all losses will be closed so in the end the number is equal. So all this methodology where the
letters P and I stand for you can use for closed losses and all losses, for example, and even if you do not use it for
predictions you can use it for analyzing and look at dependencies and if you see something special going on with the
number of closed losses compared to the number of all losses you have. So this would be another application.
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Moderator Cantin: On that we’ll close the session. Thank you very much Gerhard for coming and explaining the
method and again, for those of you who are interested and want all the formulas to program them, I have run across a
few people who have told me they have done this in Excel and it’s quite easy. So please feel free to come in and get a
book. Thank you so much.
(Applause)
[End of recording]
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