Magister Scientiae - MSc (Mathematics and Applied Mathematics)

Permanent URI for this collectionhttps://hdl.handle.net/10566/19488

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    Modelling the Basel III capital adequacy and net stable funding ratios for a commercial bank following an optimal investment strategy
    (University of the Western Cape, 2024) Hercules, Richard Stephen
    Commercial banks have a significant impact on a country’s economy as they raise capital, create liquidity by converting their customers’ deposits into loans, and deliver essential services such as loans, certificates of deposits and savings accounts to their clients. As a result, commercial banks are heavily regulated in most countries. The Basel Committee on Banking Supervision (BCBS) introduced an international set of capital standards, known as the Basel Accords, in an attempt to improve the regulation of internationally active banks. These accords resulted from a series of international banking regulatory meetings that established capital and risk manage-ment measurements for internationally active banks. Under the accords, banks are required to maintain a minimum level of capital as a bu˙er to protect their depositors, and the financial market, in the event of severe losses caused by financial risk. The latest of these accords, that is, the Basel III Accord, consists of three key pillars. These are firstly, minimum capital require-ments, secondly, supervisory review, and lastly, market discipline. In this regard, the BCBS introduced, respectively, the Capital Adequacy Ratio (CAR) and the Net Stable Funding Ratio (NSFR). The purpose of the CAR is to determine whether or not an absolute amount of a bank’s capital is adequate when compared to its absolute risk. The purpose of the NSFR, on the other hand, is to determine whether the bank has enough stable funding to cover its long term assets. Furthermore, government regulators aim to maintain the confidence and trust of the general public through the use of a deposit insurance scheme (DIS). In the event of a bank failure, deposit insurance (DI) has the e˙ect of reducing the probability of mass deposit with-drawals. An insuring agent is tasked with estimating a fairly priced premium for DI coverage.
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    Persistent homology and the application of topological data analysis to astronomical data
    (University of the Western Cape, 2024) Randall, Jessica
    This thesis addresses a key challenge in topology: determining whether spaces are homeomorphic, which requires establishing a continuous, invertible mapping. Traditional methods based on basic invariants like connectivity and compactness often fall short for complex classifications. To address this, we draw on advanced concepts from algebraic topology, particularly the fundamental and homology groups introduced by Henri Poincar ́e and Enrico Betti, as well as Betti numbers that quantify the dimensions of “ holes ” in spaces. We analyze spatial structures through homology by focusing on missing elements. The p-th homology group Hp enables comparisons between the p-th cycle group Zp and the p-th boundary group Bp, allowing us to isolate sig- nificant topological features.
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    A mathematical model for managing equity-linked pensions
    (University of the Western Cape, 2006) Julie, Elmerie
    Pension fund companies manage and invest large amounts of money on behalf of their members. In return for their contributions, members expect a benefit at termination of their contract. Due to the volatile nature of returns that pension funds attain, pension companies started attaching a minimum guaranteed amount to member's benefits. In this mini-thesis we look at the pioneering work of Brennan and Schwartz [10] for pricing these minimum guarantees. The model they developed prices these minimum guarantees using option pricing theory. We also look at the model proposed by Deelstra et al. [13] which prices minimum guarantees in stochastic financial setting. We conclude this mini-thesis with new contributions where we look at simple alternative ways of pricing minimum guarantees. We conclude this mini-thesis with an approach, related to the work of Brenan and Schwartz [10], whereby the member's benefit is maximised for a given minimum guaranteed amount, which comprises of multi-period guarantees. We formulate a method to find the optimal stream of these multi-period guarantees.