MATH 102 · Year 1 · Semester 2 · 4 credits · Mathematics & Statistics

Calculus II & Multivariable Calculus

From infinite cash-flow streams to continuous risk surfaces: mastering the multidimensional calculus behind joint probability and portfolio optimization.

Full practice exam · 30 Q · 120 min

Season 1 · 8 episodes

The Solvency Surface

When a catastrophic climate and longevity shock threatens Aurelia Re with regulatory liquidation, junior quant Maya Lin must rebuild the syndicate's multivariable valuation models from first principles to prove enterprise solvency.

Protagonist · Maya Lin, Junior Quantitative Risk Analyst at Aurelia Reinsurance Syndicate

Setting · Aurelia Re Trading Floor & Solvency Audit War Room, London

Stakes · A multi-billion dollar capital deficit and imminent regulatory shutdown under Solvency II audit if joint tail risks, portfolio boundaries, and cash flow fields are miscalculated.

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Episodes

Season finale: Maya deploys a conservative multi-peril vector potential model and joint density integration engine before the regulatory audit panel, proving enterprise solvency and exposing an unhedged cross-peril arbitrage.

From the lecturer: Welcome to MATH 102. Multivariable calculus is the native language of actuarial science and modern risk analytics. In MATH 101, you evaluated risks along a single axis; here, you will learn to navigate multidimensional probability surfaces, find optimal asset allocations under binding regulatory constraints, and evaluate infinite series representing life annuities. Keep two notebooks open beside you: an Analytical Derivation Journal for working out boundary conditions and Hessian matrices line by line, and a Geometric Sketchbook for drawing cross-sections of integration regions before setting up double integrals. When evaluating joint probability densities for SOA Exam P, ninety percent of student errors stem not from integration rules, but from incorrectly sketching non-rectangular support regions. Pause the videos whenever a prompt appears, sketch the domain, derive the Jacobian or critical points yourself, and then resume. We will treat every theorem with strict mathematical rigor while constantly linking it to real-world capital optimization, joint survival distributions, and machine learning gradient descent.

Cold open

At two in the morning, the risk engine at Aurelia Re triggers a red alert. An automated longevity swap evaluation has returned an infinite liability, freezing forty million dollars in daily collateral. Maya Lin stares at an unverified infinite series in the legacy discounting codebase.

preparing lecture…

Transcript

Define sequences and infinite series as limits of partial sums in financial and actuarial contexts
Determine the convergence or divergence of infinite series using integral, comparison, ratio, and alternating series tests
Calculate exact sums for geometric, telescoping, and arithmetico-geometric series used in pricing perpetuities and discrete probability models
Differentiate between absolute and conditional convergence and evaluate their implications for cash flow reordering