### Abstract:

The study was conducted on the topic: Minimizing the Probability of Ultimate Ruin
by Proportional Reinsurance and Investments. The purpose of the study was to
determine the role of investments in minimizing the probability of ultimate ruin of an
insurance company, to assess the impact of proportional reinsurance on the survival
of insurance companies as well as to determine the optimal reinsurance percentage b ∈ (0,1].
The study considered a risk process comprising a diﬀusion-perturbated insurance
process and a diﬀusion-perturbated investment generating process in which invest
ments were modelled as a Black-Scholes model. The Hamilton-Jacobi-Bellman (HJB)
equation for this problem was derived, as well as its corresponding Volterra integro
diﬀerential equation which was then tranformed into a linear Volterra integral equa
tion of the second kind. This integral equation was then solved using the block-by
block numerical method for the retention percentage that minimizes the probability
of ultimate ruin. The major ﬁndings of this study were as follows:
1. That, as expected, the higher the investment rate, the lower the ruin probabil
ity. Furthermore, the study has revealed that volatility of stock prices results
in higher ruin probabilities.
2. That for a given initial surplus, the ruin probabilities keep reducing as the value
of the retention level b reduces. After a certain b, however, the ruin probabilities
begin rising again, giving an indication of the location of the optimal retention
percentage b∗.
3. That the optimal retention level, given certain assumptions regarding the ﬂow
of premium incomes, is b∗ = 0.315034 for the small claim case and b∗ = 0.461538
in the case of large claims.
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Some recommendations have also been made with regard to strategies that could be
used by insurance companies to minimize their ultimate ruin probabilities. The study
has recommended that in order to minimize their ruin probabilities insurers should
invest their surplus in both risky and risk-free assets. It has also been recommended
that insurers buy reinsurance as it helps in reducing the probability of ultimate
ruin for insurance companies. But, given certain assumptions regarding the ﬂow of
premium incomes, insurers can only reinsure optimally when b∗ = 0.315034 for small
claims and b∗ = 0.461538 for large ones.