The intention of this work is to find and investigate approximation methods for the distribution of the final wealth of a fund saving plan, as it is usually offered by banks or insurance companies. Of special interest are approximations of the cumulative distribution function, quantiles, truncated first moments and tail-values-at-risk. The final wealth of a stock saving plan is a sum of lognormally distributed random variables due to the modelling of stock price processes as geometric Brownian motions. It is well known that the distribution of such a sum can not be calculated explicitly. Although simulation is possible, mostly it is not practicable due to the large amount of saving plans which have to be managed by a bank or insurer.
In many articles it is claimed that the sum of lognormally distributed random variables is well approximated by a single lognormally distributed random variable having the same expectation and variance like the sum. This is mostly only based on simulations. We now present a result, which justifies this approximation: the suitably normalized sum converges in distribution to a lognormal distribution, when the variances/covariances of the summands tend to zero. We find that this is a special case of a result in Dufresne(2004) about the asymptotic distribution of continuous and discrete averages of geometric Brownian motions.
Relatively new is the idea to construct approximations to a sum distribution based on the concepts of comonotonicity and convex order. Approximating a sum of lognormally distributed random variables by its convex order bounds leads to easy computable upper and lower bounds for truncated first moments and tail-values-at-risk. Thus, this approach can be applied to many problems in insurance and finance.
In order to quantify the error when approximating the sum distribution by one of its convex order bounds, we construct an upper bound for the approximation error in terms of truncated first moments. One important feature of this upper bound, which is found on the basis of a geometric approach, is the following: the variance is the only characteristic of the sum distribution which is involved in its computation. The bound for the error of the approximation by the upper convex bound yields a new lower bound for truncated first moments of the sum distribution. Similarly, the bound for the error of the approximation by the lower convex bound yields a new upper bound for them. We compare the latter with those obtained by Nielsen and Sandmann (2003). Our ''geometrical'' bound can (in contrast to the latter) be easily computed, as long as any lower convex bound (with known distribution) for the sum distribution is available. Additionally, it is for certain truncated first moments sharper than the Nielsen-Sandmann bound. We also discuss, how in the special situation of sums of lognormally distributed random variables the convex order bounds have to be chosen in order to mimimize the approximation error in a specific sense.
At the end we consider as practical application a fund saving plan, where in each period the investment can be made in more than one stock. Unfortunately, the resulting final wealth is then no longer a sum of lognormally distributed random variables. However, we show that its summands can be assumed to be approximately lognormally distributed, justified by the above mentioned result about the asymptotic distribution of a normalized sum of lognormals. The quality of this approximation is very good, as we show by simulation in a specific setting. This quality carries over to the approximation of the final wealth. Furthermore, we compare all approximation methods for the final wealth (now assumed to be lognormally distributed) explained in this thesis. It turns out, that a convex combination of an upper and a lower convex bound having the same expectation and variance like the sum yields in many cases a very good approximation. The benefit of this approximation is that it allows at any time during the saving plan's duration an easy computation of relevant characteristics of the distribution of the final wealth.