Applied models: illustrative numbers, real mechanics

The individual modules each show one idea. These six cases put the same building blocks (Poisson counts, Log-Normal claim sizes, compounding growth, correlation and search over options) to work on six different business questions. Each case has sliders, and every figure, chart and finding recomputes live.

Motor pricing builds a premium from a Poisson claim rate and a Log-Normal claim size: at the defaults, 0.08 claims a year × a $4,200 average claim gives a $336 pure premium, loaded by 34% to about $450. The investment reserve reads the spread of ten-year outcomes from the closed-form log-normal distribution of geometric Brownian motion. Hospital surge planning reads capacity off a Poisson tail: with 42 admissions a day and 60 beds, demand exceeds capacity on about 0.35% of days, roughly once a year. The efficient frontier is the two-asset Markowitz problem from the CFA curriculum. The loan portfolio compares expected loss (PD × LGD × EAD) with Basel IRB capital from the Vasicek one-factor model at 99.9%: for 500 loans of $200,000 with a 2% PD, 45% LGD and asset correlation ρ = 0.15, expected loss is $0.9 million but capital is about $7.0 million, around seven times what independent defaults would require. Fleet routing compares a naive route, a nearest-neighbour heuristic and the brute-force optimum (62.9, 48.1 and 43.7 km for the default stops) and shows why exhaustive search stops being feasible.

All six are illustrative: parameters are chosen to be realistic for teaching, not drawn from any client engagement. Cases D and E follow CFA and FRM material in simplified form; the credit case, for example, omits the IRB maturity adjustment, which is equivalent to assuming one-year loans.