60
Mathematics for Economics, Actuarial Studies and Finance
REGGIO DI CALABRIA
Overview
Date/time interval
Syllabus
Course Objectives
Methodologically, the course aims to make students aware of the assumptions underlying each model and of the consequences of their violation. The module on digital finance offers a natural testing ground in this respect: crypto-asset markets systematically violate the normality, liquidity and integration conditions assumed by classical theory, and thereby compel a critical use of the instruments acquired in the preceding modules.
The individual project work pursues a distinct and complementary objective: autonomy in framing a financial problem, in the reasoned selection of instruments, in the handling of empirical data, and in the scientific communication of results.
Expected learning outcomes (Dublin Descriptors)
Knowledge and understanding. Students know financial laws and their axiomatic properties, the structure of investment appraisal criteria, the formal framework of mean-variance theory, capital market equilibrium models, and the economic architecture of crypto-asset markets.
Applying knowledge and understanding. Students can construct amortisation schedules, compute sensitivity measures, reconstruct a yield curve, solve constrained and unconstrained portfolio optimisation problems, handle digital asset return series, and implement such procedures in a computational environment.
Making judgements. Students assess the adequacy of a model to its applied context, recognise the effects of estimation error on optimal solutions, distinguish robust results from procedural artefacts, and identify the limits of extending classical instruments to unregulated or illiquid markets.
Communication skills. Students present the results of a quantitative analysis in structured written form and in oral presentation, with appropriate terminology and effective graphical representation.
Learning skills. Students can independently access the scientific and professional literature of the field and update their methodological toolkit in a domain subject to rapid technological and regulatory change.
Course Prerequisites
Teaching Methods
Assessment Methods
Written exam (weighted at 60% of the final grade, based on a scale of 30). Format: Numerical exercises and theoretical questions; duration: 120 minutes.
Project work (30%). Format: Evaluation of the submitted work based on the Module D rubric.
Oral discussion (10%). Format: Presentation of the project work and questions on the course syllabus.
Submission of the project work is a prerequisite for admission to the final assessment. Grades are awarded on a scale of 30; distinction (*cum laude*) is reserved for projects demonstrating originality in their formulation or in the treatment of data.
Texts
Lecture notes, handouts, datasets and notebooks provided by the instructor during the course, which constitute the primary reference for the syllabus and for exam preparation.
Ferrara M., Intelligenza Artificiale Affidabile. Verso un framework unificato tra geometria, teoria dei giochi e quantum computing, 2nd edition, EGEA – Università Bocconi Editore, Milan, with a preface by Paolo Benanti (reference for the methodological framework of the course and for the critical and responsible use of advanced computational methods in financial decision-making).
Recommended textbooks
Luciano E., Peccati L., D'Amico M., Calcolo finanziario. Temi di base e temi moderni, 2nd edition, EGEA, Milan, 2018, I Manuali series, XII-461 pp., EAN 9788823822726. The volume includes digital content available through the MyBook platform, notably spreadsheets with worked solutions to the examples of all ten chapters (main reference for Module A).
Castellani G., De Felice M., Moriconi F., Manuale di finanza. I. Tassi d'interesse. Mutui e obbligazioni, il Mulino, Bologna (Module A).
Castellani G., De Felice M., Moriconi F., Manuale di finanza. II. Teoria del portafoglio e del mercato azionario, il Mulino, Bologna (Module B).
Cesari R., Introduzione alla finanza matematica. Mercati azionari, rischi e portafogli, McGraw-Hill, Milan.
Elton E.J., Gruber M.J., Brown S.J., Goetzmann W.N., Modern Portfolio Theory and Investment Analysis, Wiley.
Luenberger D.G., Investment Science, Oxford University Press.
Readings for Module C
Schär F., Decentralized Finance: On Blockchain- and Smart Contract-Based Financial Markets, Federal Reserve Bank of St. Louis Review, 103(2), 2021, pp. 153–174.
Makarov I., Schoar A., Trading and Arbitrage in Cryptocurrency Markets, Journal of Financial Economics, 135(2), 2020, pp. 293–319.
Liu Y., Tsyvinski A., Risks and Returns of Cryptocurrency, The Review of Financial Studies, 34(6), 2021, pp. 2689–2727.
Liu Y., Tsyvinski A., Wu X., Common Risk Factors in Cryptocurrency, The Journal of Finance, 77(2), 2022, pp. 1133–1177.
Regulation (EU) 2023/1114 of the European Parliament and of the Council of 31 May 2023 on markets in crypto-assets (MiCA), OJ L 150, 9.6.2023.
Optional further reading
Meucci A., Risk and Asset Allocation, Springer.
Fabozzi F.J., Kolm P.N., Pachamanova D., Focardi S.M., Robust Portfolio Optimization and Management, Wiley.
Narayanan A., Bonneau J., Felten E., Miller A., Goldfeder S., Bitcoin and Cryptocurrency Technologies, Princeton University Press.
Contents
Module A — Financial calculus (12 hours)
A.1 Foundations of financial laws. Financial operations and preference criteria. Accumulation and discount functions; simple, compound and continuous regimes. Decomposability and time-uniformity axioms; characterisation of the exponential law. Equivalent rates, nominal convertible rates, force of interest.
A.2 Annuities and repayment schedules. Valuation of certain annuities, temporary and perpetual, in advance and in arrears. Constant-instalment, constant-principal and bullet amortisation; construction and interpretation of the amortisation schedule. Floating-rate operations.
A.3 Investment appraisal criteria. Net present value and its functional structure. Internal rate of return: existence, uniqueness, Norström condition. Comparison of criteria and cases of ranking conflict; modified rate of return; profitability index. Capital rationing constraints.
A.4 Term structure and bond valuation. Spot and forward prices, spot and forward rates, discount function. Bootstrapping of the yield curve. Clean and dirty price, yield to maturity.
A.5 Sensitivity measures and immunisation. Macaulay and modified duration; convexity. Price-change approximation and the limits of the second-order expansion. Fisher–Weil theorem, simple immunisation and cash-flow matching; interest rate risk management strategies.
Module B — Mathematical portfolio theory (12 hours)
B.1 Returns and risk. Simple and logarithmic returns; temporal and cross-sectional aggregation. Sample moments, covariance matrix estimation and conditioning issues. Stylised facts of financial time series.
B.2 The Markowitz problem. Mean-variance formulation as a constrained quadratic programme. Portfolio frontier and efficient frontier without a riskless asset; analytical solution via Lagrange multipliers. Global minimum-variance portfolio. Two-fund separation theorem.
B.3 Riskless asset and equilibrium models. Frontier with a riskless asset, capital market line, tangency portfolio. Sharpe ratio maximisation. Capital Asset Pricing Model: derivation, security market line, interpretation of beta. Factor models and Arbitrage Pricing Theory; decomposition of systematic and idiosyncratic risk.
B.4 Expected utility and choice criteria. Utility functions and risk aversion; Arrow–Pratt measures. Consistency between the mean-variance criterion and expected utility. Stochastic dominance.
B.5 Risk measures and estimation issues. Value at Risk and Expected Shortfall; coherence axioms. Fragility of the Markowitz solution with respect to estimation error; the equally weighted portfolio as a benchmark. Constraints, regularisation and shrinkage; introduction to the Black–Litterman model, risk parity and robust formulations of portfolio optimisation.
Module C — Digital finance and crypto-assets (8 hours)
C.1 Architecture of digital markets. Distributed ledgers and block structure; proof-of-work and proof-of-stake consensus mechanisms and their underlying economic incentive logic. Smart contracts and the automation of contractual execution. Transaction costs, latency and settlement finality.
C.2 Taxonomy of crypto-assets. Payment tokens, utility tokens, asset-referenced tokens and e-money tokens. Collateralised and algorithmic stablecoins: pegging mechanisms and conditions for peg failure. Non-fungible tokens. Central bank digital currencies and the digital euro project.
C.3 Statistical properties of crypto returns. Realised volatility, heavy tails and skewness; inadequacy of the Gaussian hypothesis. Time-series momentum and investor attention effects. Correlation structure with traditional asset classes and its instability under stress.
C.4 Market fragmentation and limits to arbitrage. Price deviations across trading venues, the role of capital controls and of operational constraints on the movement of arbitrage capital. Perpetual futures and funding rates; basis and no-arbitrage relations.
C.5 Valuation and risk factors. Valuation difficulties in the absence of contractual cash flows; adoption and network models. Risk factors specific to crypto markets and their explanatory power for expected returns.
C.6 Crypto-assets in the optimal portfolio. Inclusion of a highly volatile asset in the mean-variance problem: effects on the efficient frontier and on optimal weights. Risk measures under heavy tails; maximum drawdown. Liquidity, custody and counterparty risk as additional constraints of the allocation problem.
C.7 Decentralised finance and the regulatory framework. Automated market makers and the constant product formula; impermanent loss and its analytical derivation. Overcollateralised lending protocols, liquidation thresholds, price oracles. Regulation (EU) 2023/1114 (MiCA) and its operational implications for issuers and service providers.
Module D — Individual project work (1 ECTS)
Each student develops an individual project on a topic covered in class, agreed with the instructor by the fourth week of the course. The work must comprise problem formulation, a methodological section, a computational implementation on real data, and a critical discussion of the results.
Formal requirements: report of 8–12 pages (excluding appendices), reproducible code in Python or R, documented data and sources, essential bibliography. Ten-minute oral presentation supported by slides.
Indicative topics
Financial calculus
- Immunisation of a bond position: duration matching versus cash-flow matching
- Bootstrapping the term structure from market data and valuation of an illiquid security
- Valuation of a complex amortisation plan and sensitivity analysis with respect to the interest rate
- Comparison of NPV and IRR criteria for projects with non-conventional cash flows
Portfolio theory
- Construction of the efficient frontier on a basket of listed securities and stability analysis of the solutions across estimation windows
- Empirical test of the CAPM on a European equity market: beta estimation and security market line testing
- Out-of-sample comparison of minimum-variance, tangency and equally weighted portfolios
- Estimation and backtesting of VaR and Expected Shortfall with parametric, historical and Monte Carlo approaches
- Effects of short-selling and concentration constraints on the optimal portfolio composition
- Application of the Black–Litterman model with student-formulated views
Digital finance and crypto-assets
- Effect of adding a Bitcoin allocation to an equity-bond portfolio: shift of the efficient frontier and cost in terms of tail risk
- Measurement of price deviations across two or more trading venues and estimation of implicit arbitrage costs
- Analysis of stablecoin peg stability and construction of a stress indicator
- Funding rate dynamics of perpetual futures and its relation to the forward premium
- Simulation of impermanent loss in a constant product market maker as a function of underlying volatility
- Dynamic correlation between crypto-assets and equity indices: rolling-window estimation and interpretation of breaks