R Programming for Actuarial Science

6 Multivariate Distributions, and Sums of Random Variables

Peter McQuire

library(MASS); library(psych); library(mvtnorm); library(mnormt)

6.1 Multivariate Distributions – Examples in Finance

Actuaries will frequently be interested in the joint behaviour of multiple variables, and in particular, the distributions of the sums of these variables. For example:

• we may wish to understand the distribution of investment returns from an asset portfolio, which consists of the individual returns from a number of individual assets, and to understand the benefits of diversification (see Chapters 7 and 8);
• in Chapter 10 Section 7, we look at the sum of asset values and liability values to determine the likelihood of an entity’s future insolvency, and hence develop an appropriate investment strategy to manage this risk;
• similarly, in Chapter 11 we look at the sum of price changes of an asset and a hedging instrument with the aim of finding a combination such that the net price change is as stable as possible;
• in Chapter 15 we model price changes of a bond portfolio, allowing for correlations between credit ratings of the individual constituent bonds;
• we may wish to understand how likely it is that an insurer’s total claims will exceed a particular level; the total claims will consist of claims incurred across individual policies (see Chapter 24 and 25), and also under various types of policies (e.g. motor, buildings, marine insurance). The study of multivariate distributions is therefore ...