The School of Numbers & Logic · mathematics, pure and applied
Applied & Financial Math
Interest, optimization, games, and models — the working mathematics behind loans, schedules, strategies, and forecasts.
Compound interest, annuities, and amortization — enough to read a loan document as an equation rather than a promise.
Syllabus · 4 units · ~16 hours
Unit I — Interest
Simple interest and where it still appears · Compound interest and the compounding period · APR versus APY: the same loan, two numbers · Continuous compounding and the number e
Unit II — Time Value
Present value and discounting · Comparing money across time · Net present value of a simple project
Unit III — Annuities & Loans
Ordinary annuities and their formulas · Amortization: how a mortgage payment splits · Total interest over a loan's life · Extra payments and what they buy
Unit IV — The Fine Print
Inflation and real versus nominal returns · The rule of 72 · Credit card minimums as slow-motion arithmetic · Reading an amortization table critically
The quantitative core of fixed income — valuing cash flow streams, pricing bonds, and measuring their sensitivity to rates.
Syllabus · 4 units · ~38 hours
Unit I — The Time Value of Money, Formalized
Accumulation functions and effective rates · Nominal rates and compounding conventions · Force of interest · Equations of value
Unit II — Annuities & Perpetuities
Annuities immediate and due · Deferred and growing annuities · Perpetuities and their uses in valuation
Unit III — Loans & Bonds
Loan schedules: amortization and sinking funds · Bond pricing between coupon dates · Premium, discount, and book value · Callable bonds in outline
Unit IV — Yield & Sensitivity
Yield to maturity and the yield curve · Spot rates and forward rates · Duration and convexity · Immunizing a portfolio against rate moves
Finding the best feasible answer — modeling constraints, the simplex method, and the shadow prices hiding in every optimum.
Syllabus · 4 units · ~30 hours
Unit I — Modeling Decisions
Decision variables, objective, constraints · Formulating production, diet, and blending problems · Feasible regions in two variables · Solving graphically at the corner points
Unit II — The Simplex Method
Standard form and slack variables · Pivoting from corner to corner · Degeneracy and unboundedness · Reading the final tableau
Unit III — Duality
The dual problem and what it means · Shadow prices: the value of one more unit · Sensitivity analysis for changing data
Unit IV — Beyond Linear
Integer programming and when rounding fails · Transportation and assignment problems · Scheduling as optimization · A taste of nonlinear and convex problems
Rational play among rational opponents — equilibrium, bluffing, and why cooperation survives in a self-interested world.
Syllabus · 4 units · ~20 hours
Unit I — Games in Normal Form
Players, strategies, payoffs · Dominant and dominated strategies · The prisoner's dilemma, taken seriously
Unit II — Equilibrium
Best responses and Nash equilibrium · Multiple equilibria and coordination · Mixed strategies: when to randomize · Zero-sum games and the minimax theorem
Unit III — Games in Sequence
Game trees and backward induction · Credible and incredible threats · First-mover advantage, and its absence
Unit IV — Repetition & the Real World
Repeated games and the shadow of the future · Tit-for-tat and the evolution of cooperation · Auctions in outline · Applications: pricing, treaties, and traffic
Turning messy situations into tractable mathematics — building, testing, and honestly criticizing models of the real world.
Syllabus · 5 units · ~34 hours
Unit I — The Modeling Cycle
From situation to assumptions to equations · Choosing what to ignore · Validation: comparing a model against reality · Iterating without falling in love with the model
Unit II — Scaling Arguments
Proportionality and dimensional analysis · Why elephants have thick legs: allometry · Order-of-magnitude modeling
Unit III — Dynamic Models
Exponential and logistic growth · Equilibria and stability in difference equations · SIR models of epidemic spread · Chaos in the logistic map: a cautionary tale
Unit IV — Fitting & Judging
Fitting a model to data by least squares · Residuals as the model's confession · Overfitting: the model that memorized
Unit V — Case Studies
Queues: why the other line moves faster · Traffic flow · Fisheries and sustainable harvest · Writing up a model so others can attack it fairly