Financial Mathematics
A.Y. 2018-19
Instructor:
Dr. Luca Regis
email: luca.regis@unisi.it
homepage: docenti.unisi.it/lucaregis/
Office: Room 216, 2nd Floor Office Hours: Wed, 14-16 Tel: +39 0577 232785
Class Schedule: Tue 16-18 Aula Chiostro; Wed 8.30-10 Aula 11; Thu 8.30-10 Aula 11.
Midterm exam: Thu, November 8th, 8.30-10 Room Caparrelli A.
Course Objectives.
The course covers the fundamental principles of financial mathematics. Students are expected to be familiar with financial laws, interest rates and valuation under certainty by the end of the course.
Prerequisites: Principles of Mathematics (formal), Statistics, Political Economy (recommended).
Required basic mathematical knowledge: solving first order and second order equations, properties of logarithms, exponential functions, linear systems, fundamental derivatives and integral calculations.
Course Material
Lecture notes, past exams, exercises, are available on the website, and are updated every week during the course. They are a comprehensive reference for all the material taught in the course.
Students who can read Italian can also refer to the textbook:
G. Castellani, M. De Felice, F. Moriconi, Manuale di Finanza. I. Tassi dinteresse. Mutui e obbligazioni, il Mulino, Bologna 2005.
Course Program
I. Fundamentals of financial calculus
Foundations. Financial transactions, simple and compound interest rates, linear and exponential law. Fundamental definitions: factors, rates and yields, instantaneous rate, financial operations.
Two fundamental bond types: zero coupon bond and fixed rate coupon bond.
The exponential law. Financial equivalence, equivalent compound rates and yields. Evaluating financial operations under the exponential law. Functional properties of the exponential law. De- composition of financial operations.
Annuities and amortization. Preliminary definitions. Present value of an annuity with constant instalment: (immediate) annuity with duration m, perpetuity, (immediate) annuity with duration m and payments in advance, perpetuity with payments in advance, deferred annuities. Fractional annuities. Dynamics of annuities: annuity with constant instalment, annuity with constant instal- ment and payments in advance, annuity with variable instalments, annuity with variable instalments and payments in advance, annuity with constant balance share. Mortgages: fixed rate (French)
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mortgage, fixed rate mortgage with payments in advance (German mortgage), mortgage with con- stant balance share, mortgages with initial interest-only instalments, mortgage with single balance repayment, mortgages with fractional instalment.
Internal Rate of Return. The internal rate problem. The case of periodical payments: Newtons method. Non-periodical payments.
Theory of financial equivalence. The value function in a spot contract. The value function in a forward contract: time-uniformity property. Discount and capitalisation factors: time-homogeneity property, the spot-forward consistency assumption, the separation property. Rates and yields with respect to a finite horizon: equivalent rates. Instantaneous rate: time-homogeneous laws, separable laws. Yield to maturity: equivalent yields. Linear and hyperbolic laws: the linear law (rational discount), the hyperbolic law (commercial law). Linearity of present value: value of a financial operation in an arbitrary point of time, fairness, internal rate of return with respect to a given value.
II. Financial contracts and market structure.
Value function and market prices. Market assumptions: frictionless, competitive, no arbitrage. Unit zero coupon bonds. General zero coupon bonds. Portfolios of zero coupon bonds with different maturities. Forward contracts. Implied rates.
The term structure of interest rates. Spot term structures. Implied term structures. Term structure with respect to a term set: discrete sets, continuous terms, discrete sets with continuous underlying model. Internal rate of return and par yield of fixed rate bonds and mortgages.
Time and value sensitivity indeces. Time indexes: maturity e time to maturity, mean time to maturity, duration, flat yield curve duration, flat yield curve duration of annuities, flat yield curve duration of fixed rate bonds, second order moments, duration e time-dispersion of portfolios. Value sensitivity indexes: semielasticity, elasticity, convexity, relative convexity.
Floating rate contracts. Floating rate zero coupon bonds and floating coupons. Floating rate notes.
Adjustable rate mortgages. Duration of floating rates contracts. Interest rate swaps. Swap rates and zero coupon swap rates.
Exam
The exam is written only. A midterm exam is scheduled during the class break week.
The exam consists of two parts:
1. A theory part, tested by means of 10 multiple choice questions. Each correct answer earns 3 points, unanswered questions give 0 points, incorrect answers earn -1 points. A minimum of 5 correct answers are necessary to pass the exam.
2. Exercises: 5 exercises, on the topics covered in class.
Both parts are graded separately, out of 30. The final mark is obtained as a weighted average of the two marks, with the theory part accounting for 40% of the final mark and the exercise for the remaining 60%.
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