CM1 Statistics: Returns, Dependence and Simple Stochastic Models
The Statistics domain tests the descriptive and probabilistic tools CM1 applies to investment returns and mortality: moments, dependence between variables, return distributions, and simple stochastic models used to represent interest rates and decrements.
Summarising data and dependence
- Sample variance divides by n − 1; population variance divides by n. Covariance σ_xy = Σ(x−x̄)(y−ȳ)/(n−1); correlation ρ = σ_xy/(σ_x σ_y), and ρ = 0 does not imply independence.
- Two-asset portfolio: E = w μ_1 + (1−w) μ_2; Var = w²σ_1² + (1−w)²σ_2² + 2w(1−w)ρσ_1σ_2.
Return and rate models
- If 1 + R is lognormal with log-parameters μ and σ, then E[1+R] = exp(μ + σ²/2) and the median is exp(μ); the σ²/2 term is essential.
- Log returns are additive: n independent years give a sum distributed N(nμ, nσ²), which is why long-horizon accumulation reflects the geometric, not the arithmetic, average return.
- The AR(1) model X_t = α + βX_(t−1) + ε_t has mean α/(1−β) and variance σ²/(1−β²) only when |β| < 1.
- With d deaths over central exposure E under a constant force, μ̂ = d/E with approximate variance d/E².
Worked example
Two assets with w = 0.6, σ_1 = 10%, σ_2 = 20%, ρ = 0.25. Variance = 0.36(0.01) + 0.16(0.04) + 2(0.6)(0.4)(0.25)(0.10)(0.20) = 0.0036 + 0.0064 + 0.0024 = 0.0124, so σ = 11.1%. The covariance term is about 19% of total variance; omitting it would understate portfolio risk.
Traps that cost marks
- Projecting long-term accumulation with the arithmetic mean return.
- Dropping the σ²/2 term when converting log parameters to expected returns.
- Reading a high R² from a returns regression as evidence of a stable relationship.
- Quoting μ̂ = d/E without its standard error, so the sampling uncertainty disappears.