I. Fundamentals of financial calculus.
Foundations. Financial transacations, simple and compound interest rates, exponential law. Fundamental defi nitions: factors, rates and yields, instantaneous rate, financial transactions. Two fundamental bond types: zero coupon bond and fi xed rate coupon bond.
The exponential law. Financial equivalence, equivalent compound rates and yields. Evaluating
fi nancial operations under the exponential law. Functional properties of the exponential law. Decomposition of financial transactions.
Annuities and amortization. Preliminary de finitions. Present value of an annuity with constant instalments: (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 due with constant instalment, annuity with variable instalments, annuity due with variable instalments, deferred annuities, perpetuities. Mortgages: fixed rate (French)
mortgage, Italian amortization, mortgages with initial interest-only instalments (pre-amortization), mortgage with single balance
Internal Rate of Return. The internal rate problem. The case of periodical payments: Newton's
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, decomposability.
Rates and yields with
respect to a fi nite horizon: equivalent rates. Instantaneous rates: time-homogeneous laws, separable
laws. Yield to maturity: equivalent yields. Linearity of the present value: value of a financial
operation at an arbitrary point of time, fairness, internal rate of return with respect to a given
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 and time to maturity, average time to
duration of annuities,
duration of fixed rate bonds, second order moments, duration and time-dispersion of portfolios. Value
sensitivity indexes: semielasticity, elasticity, convexity, relative convexity.
Floating rate contracts. Floating rate zero coupon bonds and
oating coupons. Floating rate notes.
Adjustable rate mortgages. Duration of
oating rates contracts. Interest rate swaps. Swap rates
and zero coupon swap rates.