BERGAMO
Overview
Date/time interval
Syllabus
Course Objectives
At the end of the course, students will know and understand the main principles of financial mathematics, including the time value of money, accumulation and discounting, equivalent interest rates, annuities, loan amortisation, investment appraisal criteria, basic bond valuation and elementary actuarial valuation.
Students will be able to apply these tools to standard financial problems. In particular, they will be able to compute present and future values of cash flows, determine equivalent rates, value annuities and simple bonds, construct and interpret the amortisation schedule of a constant-annuity loan, distinguish the interest and principal components of each instalment, evaluate basic investment projects through net present value and internal rate of return, and compute elementary actuarial present values and pure premiums using mortality tables.
The course favours understanding and interpretation over the mechanical use of formulas.
Course Prerequisites
Students are expected to be familiar with basic algebra, elementary functions, logarithms and the basic notions of calculus usually covered in a first-year mathematics course. No previous knowledge of finance is required.
Compulsory prerequisites required (Propedeuticità) are published on the web site: https://lt-ea.unibg.it/it/node/106.
Since sequences and series may not have been covered in detail in previous courses, the course begins with a short application-oriented refresher on sequences, finite sums and geometric progressions, which are essential for understanding compound interest, annuities, loan amortisation and actuarial present values.
Teaching Methods
Students will regularly solve exercises on interest rates, annuities, loans, investment projects, bonds and elementary actuarial valuation. A standard calculator is sufficient. Simple spreadsheet examples may be used in class to illustrate amortisation schedules or cash-flow calculations, but spreadsheet skills are not a prerequisite and are not directly assessed.
Assessment Methods
The assessment consists of a written examination. The written exam includes numerical exercises and short theoretical or interpretative questions. The exercises are designed to assess the student’s ability to apply the quantitative tools covered in the course; the theoretical questions assess understanding of definitions, assumptions and interpretation of results. The final grade is expressed out of 30. The grade depends on correctness of calculations, appropriate use of formulas, clarity of reasoning, interpretation of financial quantities and ability to connect the numerical answer with the underlying financial problem.
The lecturer reserves the right to require an additional oral interview, including after the written examination has taken place, whenever this is deemed necessary in order to verify the student’s effective acquisition of the knowledge and skills set out in the course learning objectives and to ensure the appropriate attribution of the final grade.
Unless otherwise specified at the beginning of the course, attending and non-attending students take the same exam and are assessed according to the same criteria.
Contents
Mathematical refresher: sequences and finite sums
Sequences and recursive definitions. Arithmetic and geometric sequences. Sigma notation. Finite geometric sums. Powers and logarithms. Exponential growth and decay. Applications to repeated compounding and repeated payments.
Foundations of financial mathematics
Time value of money. Financial operations and cash flows. Accumulation and discounting. Simple interest, compound interest and continuous compounding. Interest rates, discount rates and force of interest. Equivalent rates and nominal rates convertible several times per year.
Valuation of cash-flow streams and annuities
Present value and accumulated value of a sequence of cash flows. Fair financial operations. Annuities-immediate and annuities-due. Temporary annuities, deferred annuities and perpetuities. Accumulation of a target capital through periodic payments.
Loans and amortisation
Loan repayment. Constant-annuity loans. Outstanding balance. Interest component and principal component of each instalment. Construction and interpretation of the amortisation schedule. Basic comparison with other repayment structures.
Investment project appraisal
Financial projects and cash-flow representation. Net present value. Internal rate of return. Comparison of projects. Interpretation, advantages and limitations of NPV and IRR.
Introduction to bonds and interest-rate applications
Zero-coupon and coupon bonds. Bond price as present value of promised cash flows. Yield to maturity. Relationship between price, coupon rate and yield. Basic interpretation of interest-rate sensitivity.
Elementary actuarial mathematics
Basic probability notions. Survival and death probabilities. Mortality tables. Expected present value of life-contingent payments. Pure endowment, term life insurance and whole-life insurance in simple settings. Single pure premiums. Brief introduction to periodic premiums, loaded premiums and the conceptual meaning of actuarial reserves.