The numerics of Premium Bonds

Premium Bonds sold by the UK National Savings and Investments (NS&I) agency are the possibly most popular example of lottery bonds. Premium Bonds holders renounce interest payments but instead participate in a lottery which distributes the equivalent of aggregate interest payments among them. While the random mechanism used in the lottery is well-documented the details of how to determine the probability distribution of a single bond holder's lottery prizes seem to be less well-known. We observe that the lottery prizes distribution is a multivariate hypergeometric distribution and discuss how to exactly calculate its probability masses as well as how to approximate the distribution by means of the Panjer recursion and Fourier transforms. We find that there are good reasons to prefer the approximations based on Panjer recursionor Fourier transforms to exact calculation of the lottery prize value distribution.

EXPANSIONS FOR MOMENTS OF COMPOUND POISSON DISTRIBUTIONS

Expansions for moments of $\overline{X}$, the mean of a random sample of size n, are given for both the univariate and multivariate cases. The coefficients of these expansions are simply Bell polynomials. An application is given for the compound Poisson variable SN, where $S_{n} = n \overline{X}$ and N is a Poisson random variable independent of X1, X2, …, yielding expansions that are computationally more efficient than the Panjer recursion formula and Grubbström and Tang's formula.

MODEL COMPOUNDS DALAM MENGHITUNG AGGREGATE LOSS

Penelitian ini bertujuan untuk menentukan taksiran distribusi aggregate loss. Dalam hal ini, aggregate loss merupakan total kerugian dalam periode satu tahun yang dialami oleh pemegang polis yang ditanggung suatu perusahaan asuransi. Dalam menentukan taksiran fungsi peluang aggregate loss, akan dibentuk model distribusi compound. Untuk menyelesaikan distribusi compound, ada beberapa metode yang digunakan antara lain metode Panjer Recursion, Fourier Inversion, dan Fast Fourier Transform. COMPOUNDS MODEL TO DETERMINE AGGREGATE LOSSABSTRACTThis study aims to determine the estimated aggregate loss distribution. In this case, the aggregate loss is a total loss within one year period experienced by the policyholder who paid an insurance company. In determining the estimated aggregate loss function , will be established distribution model compound. To determine the compound distribution, there are several methods used, namely, Panjer Recursion, Fourier Inversion, dan Fast Fourier Transform.

Computation of Compound Distributions II: Discretization Errors and Richardson Extrapolation

The standard methods for the calculation of total claim size distributions and ruin probabilities, Panjer recursion and algorithms based on transforms, both apply to lattice-type distributions only and therefore require an initial discretization step if continuous distribution functions are of interest. We discuss the associated discretization error and show that it can often be reduced substantially by an extrapolation technique.

Computation of Compound Distributions I: Aliasing Errors and Exponential Tilting

Numerical evaluation of compound distributions is one of the central numerical tasks in insurance mathematics. Two widely used techniques are Panjer recursion and transform methods. Many authors have pointed out that aliasing errors imply the need to consider the whole distribution if transform methods are used, a potential drawback especially for heavy-tailed distributions. We investigate the magnitude of aliasing errors and show that this problem can be solved by a suitable change of measure.