Financial Mathematics - Theory of Interest & Cashflow Models

Use timeline diagrams to map cash flows of financial instruments, highlighting timing, magnitude, and probability distribution across bonds, annuities, equities, and insurance products.

Learn zero coupon bonds, where a single future payout is discounted to present value with price = 100(1+i)^(-n); grasp risk-free rate, risk premium, and discounting pitfalls.

Explore fixed interest securities that pay regular coupons and redemption at maturity, priced by discounting cash flows. Understand how duration and interest rates influence valuation.

Understand index linked securities, where coupons and redemption payments vary with an index such as the consumer price index or an equity index. This hedges inflation risk with index changes.

Explore how annuities manage cash flow for retirement, distinguishing annuity certain from contingent annuities, and price them using survival probabilities and discounting.

Explore equity valuation through growing dividends and price, showing how perpetual dividends are discounted at a given interest rate using actuarial notation.

Explore how common insurance products work, including whole life, term life, endowments, contingent annuities, general insurance, and medical aids, with premiums, benefits, and customization.

Explore the mathematics of interest with a 100 grand, 10 percent example, comparing simple and compound interest. See why compound interest grows on the total amount, not just the principal.

Explore the power of interest rates with the Manhattan island example, and compare simple versus compound rates, highlighting how 5.5% can grow a small sum into enormous value.

Explore how discount rates relate to interest rates in converting future values to present values and vice versa, using (1+i)^-1 = 1 - D with a 100 grand example.

Explore the discount factor v, present value of future cash flows, and derive annuity formulas for immediate and due payments, introducing actuarial notations a-angle-n and ä-angle-n.

Explore advanced aspects of the theory of interest by examining money rates, real rates, and nominal rates. Analyze the force of interest as a function of time.

Compare money rates and real rates to see how inflation erodes wealth. Learn why real rates matter for wealth growth using Japan's 1% and South Africa's 7% example.

Compare nominal rates to effective rates, where six percent per annum is convertible quarterly. See how quarterly compounding yields the effective rate and how I_p and P convert between forms.

Investigate the force of interest by taking the limit as compounding frequency approaches infinity, linking nominal and effective rates via 1+i = e^{Δ}, with Δ = log(1+i).

Explore a quick recap of integration as the area under a curve and the inverse of differentiation, with examples like x^2 to x^3 and e^x to e^x.

The force of interest is defined as a function of time, delta(t), the instantaneous change of the fund divided by the fund. Discounting uses e^{-∫_0^t delta(s) ds}.

Explore how a time-varying force of interest, defined as 0.08 for 0-5 years and 0.13-0.01t thereafter, accumulates a $500 investment via integration, yielding an equivalent effective rate.

Explore actuarial notation for present, accumulated, and future values using discounting factors and annuity types. Visualize concepts with timeline methods to master force of interest and increasing or decreasing annuities.

Define discounting factor V to compute present values; a cash flow X one year equals X times V, and two years equals X times V^2, linking i and D.

Explore how the present value of an annuity certain is derived from the zero-coupon bond idea, and learn the actuarial notation a-angle-n with the formula pv = (1 - v^n)/i.

Explains the present value of an annuity in advance, built from zero coupon bonds and adjusted by one minus d, with PD = (1 − v^n)/d.

Explains actuarial notation for the present value of a deferred annuity, showing how deferral period m shifts payments to end of year and derives the A-angle-n with deferment formula.

Learn to compute the present value of annuities payable with P payments per year under a nominal rate i_p, using monthly payments (P=12) and a familiar annuity formula.

Compute the present value of a continuously paid annuity by integrating v^t from 0 to n, then obtain ä_n by 1/δ from the ordinary annuity value using the orange book.

Using actuarial notation, this lecture derives the present value of a perpetuity, showing that an infinite annuity paying one dollar at year-end equals one over the interest rate.

Learn to calculate the present value of an increasing annuity and express it with actuarial notation, including annuity in advance and force of interest scenarios.

Explore equations of value by equating the present value of income to the present value of outgo, using cash flow parameters and linear interpolation to isolate the interest rate.

Explore linear interpolation to estimate interest rates in a discounted cash flow model, using an annuity example and the equation of value to relate present values and rates.

Explore loan schedules that split payments into interest and capital under fixed rates and regular repayments, and compare APR with flat rate using prospective and retrospective methods.

Evaluate projects using net present value, internal rate of return, and discounted payback period, considering capital needs, duration, and liquidity risks to compare opportunities.

Examine the term structure of interest rates in the discounted cash flow framework, covering spot and forward rates, theories, yields to maturity, duration, convexity, immunization, and stochastic models.

Explore spot rates and how zero coupon bonds establish the yield curve, showing how price, redemption, and duration reveal term-dependent interest rates.

Explore forward rates and their notation, connect them to spot rates, and learn to chain forward rates to derive spot rates, including continuous forward rates and practical timeline diagrams.

Explore theories on the term structure of interest rates using the liar framework—liquidity preference, inflation risk, market segmentation, and expectation theory—to explain yield curves and self-fulfilling price-rate expectations.

Calculate yield to maturity for a fixed-rate bond using spot rates, coupons, and redemption at par; learn the equation of value linking price and cash flows.

Examine par yields and their link to spot rates, showing how a bond's price, coupon payments, and redemption at par determine the par yield and the coupon bias.

Immunization is an interest rate risk management technique that uses a flat yield curve to balance assets and liabilities, via effective duration and convexity.

Explore stochastic interest rate models that use distributions to estimate expected values, variance, and percentiles of future rates. Compare fixed and varying rate models and spot versus forward rates.

Explains the lognormal distribution for interest rates, showing that the log of one plus the rate is normal, and that accumulating lognormals enables range, confidence intervals, and value-at-risk-like loss analysis.

Explore single factor interest rate models, including Vasicek and Cox-Ingersoll-Ross, featuring a random walk with drift, mean reversion, time-varying volatility, and market-calibrated forward rates; note limitations for complex derivatives.