We are fortunate to work in a profession with a thriving body of research. Every year, innovative new articles appear, introducing new methods and generalizing older techniques. With all this progress, it can be easy to lose sight of the work done by our “actuarial forefathers.” It’s worthwhile and illuminating to occasionally survey older actuarial publications; they contain a wealth of practical, understandable methods.
In this article, we’ll discuss a triangle reserving method introduced in 1978 by F.E. de Vylder [1]. The de Vylder method is intuitive and relatively simple to implement in Excel.
We begin with a simple chain-ladder example, which we assume requires no introduction (Figure 1). In this case, the selected LDFs are based on a straight average of all available years.
There’s a fair amount of variation here, making the LDFs difficult to select. In the 12-24 column, could the 1.125 factor be an outlier? How about the 1.750 factor in the 24-36 column? This is a simple example, but experienced actuaries know that these factors can sometimes be very difficult to select when the observed factors are volatile. We take various indications (weighted averages, averages excluding the highest and lowest points, etc.), and try to make a selection that balances the conflicting information.
The de Vylder method can be used to “balance the conflicting information” in a formal sense. We will present the method by working through an example using the triangle from Figure 1. First, we display the triangle data incrementally, as in Figure 2.
Note that the “100” amount in accident year 2010, age 24, has been deleted from the incremental view. This is because the incurred loss and ALAE at age 12 for this year is missing — therefore, we don’t know the incremental amount incurred between ages 12 and 24.
Now suppose the ultimate loss and ALAE for each accident year is given by U2010, U2011,…,U2014. Suppose there are factors p12, p24,…,p60 such that the incremental amount incurred at each development period, divided by the associated factor, equals the ultimate loss and ALAE. That is, reading from the top row of the triangle,
75 = U2010 * p36
25 = U2010 * p48
10 = U2010 * p60
200 = U2011 * p12
0 = U2011 * p24
50 = U2011 * p36
50 = U2011 * p48
200 = U2012 * p12
25 = U2012 * p24
55 = U2012 * p36
350 = U2013 * p12
25= U2013 * p24
400 = U2014 * p12
In practice this set of equations will be contradictory and have no solution. As a compromise, we can look for a set of ps and Us that minimize the mean squared error (MSE).
We have MSE= (p36 * U2010 – 75)2+(p48 * U2010 – 25)2+…+(p12 * U2014 – 400)2. We can simply set this equation up in Excel and use the Solver function to find optimal ps and Us. It’s also a good idea to add the constraint ∑p=1. This essentially implies that the tail factor is 1.000. In practice we’ve found that this method can yield strange results without this constraint; that is, it doesn’t appear to be an appropriate method for selecting the tail. The tail can be selected by any number of other methods and incorporated into the development pattern. The resulting pattern is displayed in Figure 3, converted into development factor notation (the sequence of ps corresponding to the pattern below is p12= 0.64, p24= 0.04, p36= 0.18, p48= 0.11, p60= 0.03). Note that the de Vylder solution for Age 48 is 1.028, whereas the only observed historical value is 1.050.
“The Excel Solver” approach works,1 but in de Vylder’s original paper he presents an interesting iterative approach to minimizing the MSE. We start by guessing a sequence of ps, say p12= 0.50, p24= 0.30, p36= 0.10, p48= 0.05, p60= 0.05. Then, in order to minimize the overall MSE, we must in particular minimize the following sum:
(p36*U2010 – 75)2+(p48*U2010 – 25)2+(p60* U2010 – 10)2.
We minimize this by setting the derivative in U2010 equal to 0, i.e., p236*U2010 – 75 p36+ p248*U2010 – 25 p48+ p260*U2010 – 10 p60= 0, which implies U2010= (75 p36+25 p48+10 p60)/( p236+ p248+ p260). Substituting in our initial guess, we get U2010= 617. By similar reasoning, we get U2011= 305, U2012= 323, U2013= 537, and U2014= 800.
But our minimal solution must also minimize the following sum:
( U2011*p12 – 200)2+ ( U2012*p12 – 200)2+( U2013*p12 – 350)2+ ( U2014*p12 – 400)2.
We again use calculus, this time differentiating with respect to p12 and obtain the equation U22011*p12 – 200 * U2011 + U22012*p12 – 200 * U2012+ U22013*p12 – 350 * U2013+ U22014*p12 – 400 * U2014= 0, which implies p12= (200 * U2011 +200 * U2012 +350* U2013 +400 * U2014)/( U22011+ U22012+ U22013+ U22014). Substituting in the Us that we derived above, we get p12= 0.56. By a similar calculation, we get p24= 0.04, p36= 0.14, p48= 0.06, and p60= 0.02. These don’t sum to 1.00, so we “normalize” them by dividing by their sum, and get p12= 0.68, p24= 0.05, p36= 0.17, p48= 0.08, and p60= 0.02. With this new set of ps, we start the process over again and repeat until we achieve “convergence,” i.e., until the output ps match the input ps (up to some rounding convention).
The iterative approach can also be easily implemented in Excel by setting up a straightforward VBA macro. In every case we have tried, the Solver and iterative approaches have yielded identical solutions.
We close by providing the iterative equations in a more general notation, as displayed in the original de Vylder paper. In the equations that follow, Ui is the ultimate loss for accident year i, cij is the incremental amount paid in accident year i and development period j, and pj is the incremental payment percentage as described above. Then Ui=∑cij* pj /∑p2j and pj = ∑cij*Uj / ∑U2j.
Reference:
De Vylder, F., “Estimation of IBNR Claims by Least Squares,” Mitteilungen der Vereinigung Schweizerischer Versicherungsmathematiker, 249-254.
Eric Blancke, ACAS, is an actuarial consultant, and Jeremy Smith, FCAS, CERA, is an actuarial director. Both work for CNA Insurance Companies in Chicago.
1 In practice it is often necessary to run the Solver multiple times, until the squared error stops changing with each successive run.
