Banks issue mortgages with an embedded option for borrowers to prepay a part of the loan. However, this behaviour poses a risk to banks as it disrupts the level and timing of mortgage cash flows. From an earning perspective, when interest rates decrease, customers are financially incentivised to prepay their mortgages, resulting in a decrease in the bank’s income when the cash proceeds are reinvested at a lower rate. Conversely, from a value perspective, with an increase in interest rates, reducing the financial incentive to prepay, cash flows are moved further ahead in time, thereby increasing the duration of the mortgage. These two scenarios highlight the instability in the bank’s margin and value caused by prepayments. To address this risk, banks employ hedging strategies to mitigate the prepayment risk and achieve margin and value stability. This research aims to identify an effective hedging strategy that can accomplish both.

The research utilised the one-factor Hull-White model to simulate various interest rate scenarios, while an interest rate-dependent logistic prepayment model provided monthly prepayment rates based on the mortgagors’ refinancing incentives. Ten different hedging techniques were explored, including the internal funding, a static and dynamic notional hedge, and a static and dynamic value hedge. Additionally, a calibrated receiver swaption was included in each of these five hedging approaches. Subsequently, each of these hedging approaches was assessed for its margin stability, measured by the variance of the net interest margin, and its value stability, evaluated through the variance of the net present value, the average basis point value, and the NPV-at-Risk in ±200 basis point shocked interest rate scenarios.

The analysis indicated that relying solely on internal funding performs poorly in terms of both margin and value stability. Dynamic hedges were found to generally outperform their static counterparts, due to their ability to respond to market changes. Furthermore, the notional hedge demonstrates superior margin stability, while the value hedge exhibits the best value stability. Additionally, the analysis revealed that the incorporation of a receiver swaption significantly improves the NPV-at-Risk but has limited impact on the other risk metrics. Based on the conducted research, it is concluded that for a bank aiming for both value and margin stability, the most effective hedge strategy is the dynamic value hedge without the utilisation of a swaption. However, it should be noted that the ultimate choice for a hedging strategy depends heavily on the risk appetite of each bank. If a bank prioritises attaining margin stability, the recommended choice would be the dynamic notional hedge without the incorporation of the receiver swaption. On the other hand, for a bank that prefers value stability over margin stability, the dynamic value hedge without the inclusion of a swaption should be considered. Moreover, the final decision may also be influenced by mandatory requirements imposed by financial regulators, such as the European Central Bank.

We live in an online world: we date online, we do business online and we communicate online. To make sure this happens securely, almost all transferred data is encrypted by cryptosystems. This paper focuses on one of these: the knapsack cryptosystem of Merkle and Hellman, which relies on the hardness of solving the knapsack problem (1978). In general it works as follows.
Bob, who wants to communicate with Alice, encrypts his message with a public key and sends this to her. If a third party now intercepts it, he must solve an instance of the knapsack problem, which is NP-hard in general. However, this becomes computationally infeasible if the number of items is large, and therefore Bob’s message is safe. Alice has access to a private key, which she uses to transform the hard knapsack problem into an easier one: one where the vector of weights is superincreasing. That is, each component of the vector is larger than the sum of all previous components. In this case she can solve the problem efficiently and read what Bob sent her.
This method seems to be reliable at first sight. However, a few years after it was published, cryptographer Shamir proposed an algorithm that breaks the system (1984). The algorithm finds a pair of numbers by solving two systems of inequalities. Then a third party can use this pair to transform the hard knapsack problem into one he can solve, which may be different from the one Alice finds. However, he will find the same solution, and therefore he can also read Bob’s message.
This algorithm can be implemented as a computer program and in this paper we used Python. The first system of inequalities is written as a optimization problem and since there is fixed number of unknowns, we can solve it in polynomial time using Lenstra’s integer programming algorithm (1983). In this paper we used the Gurobi optimizer to solve it, as Lenstra’s algorithm is hard to use in practice. The second system of inequalities is solved by comparing lower and upper bound, which can also be carried out in polynomial time. Nevertheless, the total algorithm finishes in polynomial time only with a certain high probability since we made some probabilistic assumptions. This implies that there is a small probability of failure. However, if it finds a solution, then the algorithm is fast, and therefore it is still valuable to use in real life.