Financial Mathematics
A.Y. 2016-17
Instructor:
Dr. Luca Regis
email: [email protected]
homepage: docenti.unisi.it/lucaregis/
Office: Room 216, 2nd Floor Office Hours: Wed, 11-13 Tel: +39 0577 232785
Class Schedule: Tue 18-19.30 Aula 11; Wed 14-16 Aula 5; Thu 8.30-10 Aula 11.
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).
Course Material
Teaching material will be made available in the form of slides and lecture notes. Students who can speak Italian can 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 operations, simple and compound interest rates, exponential law. Funda- mental definitions: factors, rates e yields, instantaneous rate, financial operations. Two fundamen- tal 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) 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
1
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 operations 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.
Term structure dynamics (optional). Term structure evolution under certainty assumption. Ex- pectation assumptions: pure expectations assumption, liquidity preference assumption, market segmentation assumption, preferred habitat assumption.
Exam
The exam is written only. A partial is scheduled for mid April.
2
