Business sciences

The financial transaction. Classification of financial transactions. The capital market. Characteristics of the ideal market and real markets. Elementary financial transactions. The principle of financial equivalence. Investment and discount operations. Interest and discount. The value function. Periodic interest rate and discount rate. Fundamental relationships between financial quantities. Spot and forward contracts. Properties and terminological aspects. Outline of the spot structure. Outline of the forward structure. Relationship between spot and forward transactions. Principle of absence of arbitrage (Theorem of implicit prices). Periodic rates and rates per unit period. Periodic rate and effective interest rate per unit period. Average interest rate per unit period. Equivalent rates. Intertemporal financial laws and financial regimes. Compound capitalization scheme. Simple capitalization scheme. Trade discount scheme. Comparisons. Convertible interest rate. Instantaneous intensity of interest or Interest force. Nominal discount rate convertible to times in unit period. Generalization to two-variable financial laws. Translatability and separability: Cantelli theorem. Annuities. Periodic annuities. Present value and capital values of financial annuities (with periodic rates, spot rates, term rates, average rates). Values of annuities (all cases). Inverse problems for annuities. Internal rate of return (IRR): definition, existence and uniqueness. Numerical methods for the estimation of the IRR: iterative method, interpolation method, bisection method, Newton's method. Time and variability indices. Average financial maturity. Arithmetic mean maturity. Duration. Flat yield curve duration. Duration of a ZCB. Duration of a CB. Duration of a par bond. Duration of an irredeemable title. Rate risk: definition and examples. Measure of interest rate risk for ZCB securities. Measure of interest rate risk for coupon securities. Modified duration: definition. Convexity: definition and properties. Analysis of the adjustment provided by convexity to rate risk measurement. Basics of financial immunization. Additive shift hypotheses: definition, observations and graphical analysis. Asset Liability Management. Post-shift financial immunization. Portfolio of immunizing assets: budget constraint, duration matching and portfolio convexity. Fisher and Weil theorem. Examples. Redington theorem. Examples. Loans and mortgages. Generalities. Amortization plan. Typical schemes. Uniform amortization (Italian). Progressive amortization (French). Amortization at two rates (American). Depreciation at anticipated interest (German). Introduction to the valuation of the main classes derivative securities (forwards, futures, swaps, options). Principle of risk-neutral valuation (outline).

Calculus

Theory Sergio Bianchi, Matematica Finanziaria, materiale didattico elaborato per il corso 2020, https://web.uniroma1.it/memotef/node/7495 (download gratuito) Arsen Palestini, Matematica Finanziaria dispense 2017, https://web.uniroma1.it/memotef/node/6239 (download gratuito) Exercises M. Frezza, Esercizi di Matematica Finanziaria svolti e commentati, McGraw Hill, 2019

In-person lecture is conducted in the classroom using traditional methods (blackboard) and slides. At the end of each lesson, the material presented is uploaded on GClassroom. The distance learning lesson is conducted in synchronous mode. The video of the lesson is made available on GClassroom and can be viewed in asynchronous mode.

Classes are scheduled to be attended in presence

The written test consists of 4 exercises, possibly divided into sub-questions. For each question is reported the maximum score given to the question itself. The written test lasts 2 hours. In case of a positive result in the written test, the student can take the oral test. At the discretion of the instructor, the student may be summoned for an oral examination if it is deemed necessary to further assess the written test. Any student who is caught copying from electronic, computer or telematic instruments will be immediately removed, his or her test (both written and theory) will be cancelled and a failing grade will be recorded. During the theory test, the student may not carry any book, paper or note or any computer, telematic or electronic instrument. The presence of any of the materials listed above will result in the annulment of the entire examination (including the written test, whatever the mark obtained) and the recording of the rejection.

In-person lecture is conducted in the classroom using traditional methods (blackboard) and slides. At the end of each lesson, the material presented is uploaded on GClassroom. The distance learning lesson is conducted in synchronous mode. The video of the lesson is made available on GClassroom and can be viewed in asynchronous mode.

Definition and classification of financial operations. Basic financial operations. Principle of financial equivalence. Assumptions of the ideal market. Characteristics of real financial markets. Schemes of investment and anticipation financial operations. Assumptions regarding price functions. Capitalization factor. Discounting factor. Interest. Effective interest rate. Discount. Effective discount rate. Spot and forward financial operations. Properties of value functions in the context of forward schemes. Relationship between spot and forward operations (in the case of two periods). No-arbitrage condition. Market structure based on interest rates. Financial regimes. Financial regime of compound capitalization (periodic rate, variable rates, average rate). Equivalent rates. Nominal interest rate convertible. Instantaneous interest intensity. Nominal discount rate convertible. Instantaneous discount intensity. Interest force for financial laws of one variable. Interest force for financial laws of two variables. Divisibility. Uniformity. Financial regime of simple capitalization. Financial regime of commercial discounting. Complex financial operations. Annuities. Valuation of an annuity. Internal rate of return. Classification of annuities. Present values and amounts for immediate, deferred, advanced, postponed, whole, and fractional annuities. Present values for perpetuities. Present values and amounts for continuous annuities. Mixed capitalization (overview). Determining the number of installments, the installment amount, and the interest rate. Rate search problem: iterative method, interpolation method, and successive approximation method. Temporal and variability indices. Maturity. Financial average maturity. Arithmetic average maturity. Duration: definition, properties, and financial interpretation. Duration of fixed-coupon securities. Convexity: definition and classification. Bond loans. Duration of: bonds (TCN, TCF), bond portfolios (with examples), immediate deferred annuity with constant installments and properties. Convexity. Capital formation and amortizations. Issues related to capital formation. Loan contract (generalities). Residual debt. Settled debt. Capital portion. Interest portion. Installment. Amortization: classification. Amortization with periodic interest payments and final capital repayment. Progressive amortization. Uniform amortization. Amortization with two rates. Amortization with prepaid interest.

In order to successfully engage with the Financial Mathematics course, students should have a fundamental understanding of calculus, particularly in areas such as algebra, limits, derivatives, integrals, and basic concepts of statistics and probability. Familiarity with the principles of economics, as well as the fundamental concepts of banking and financial operations, is also recommended. Knowledge of key calculation tools, such as scientific calculators or spreadsheet software, is advisable to facilitate the practical application of the methodologies covered throughout the course

• Teaching material developed for the course (by Prof. Sergio Bianchi) • M. Frezza, Esercizi di Matematica Finanziaria svolti e commentati, McGraw Hill, 2019 • Past exams

Attendance to the course is strongly recommended

A comprehensive written exam aimed at assessing students' understanding of fundamental mathematical concepts and their application to economic and financial problems. The exam may include both theoretical questions and practical problem-solving exercises.

The course will be delivered through a combination of the following teaching methods: - Lectures Formal classes where the instructor presents key mathematical concepts, theories, and techniques relevant to economics and finance. Lectures are designed to build foundational understanding and provide structured explanations of the course material. - Interactive Exercises and In-Class Activities During selected lectures or tutorials, students may engage in guided exercises or collaborative problem-solving tasks to reinforce comprehension and encourage active participation. - Assignments and Homework Regular problem sets will be assigned to help students consolidate their understanding and apply the techniques learned in class to practical problems.