IAI Actuarial Core Principles · Economic Modelling

Measures of investment risk: formula sheet

Full chapter guide

Key formulas

Expected utility: E[U(W)] = Σ p_i × U(w_i); For a continuous outcome use the integral of U(w) f(w) dw. Choose the option with the higher value.
Non-satiation: U'(w) > 0; More wealth is always preferred.
Risk aversion, neutrality, seeking: U''(w) < 0, = 0, > 0; Concave, linear, convex respectively.
Jensen's inequality: E[U(W)] ≤ U(E[W]) for concave U; A risk-averse investor prefers the certain expected wealth to the gamble.
Absolute risk aversion: A(w) = −U''(w) ÷ U'(w); Unchanged by a positive linear transformation of U.
Relative risk aversion: R(w) = w × A(w) = −w U''(w) ÷ U'(w); Measures aversion to proportional wealth risk.
Certainty equivalent: U(CE) = E[U(W)]; CE is the certain wealth giving the same utility as the gamble. Solve by inverting U.
Exponential utility: U(w) = −e^(−aw); A(w) = a; R(w) = a w; Constant absolute, increasing relative risk aversion.
Power utility: U(w) = w^(1−γ) ÷ (1−γ), γ > 0, γ ≠ 1; A(w) = γ ÷ w; R(w) = γ; Constant relative, decreasing absolute risk aversion.
Logarithmic utility: U(w) = ln w; A(w) = 1 ÷ w; R(w) = 1; Special case of power utility with γ = 1.
Quadratic utility: U(w) = w − b w²; A(w) = 2b ÷ (1 − 2b w); b > 0, valid only for w < 1 ÷ (2b). Absolute risk aversion increases with wealth.
Expected return (discrete): μ = E[R] = Σ pᵢ rᵢ; pᵢ are probabilities that sum to 1.
Variance: Var(R) = σ² = E[(R − μ)²] = E[R²] − μ²; The second form is usually faster to compute.
Standard deviation: σ = √Var(R); Same units as the return.
Semi-variance (below the mean): SV = Σ pᵢ [min(rᵢ − μ, 0)]²; Outcomes at or above the target contribute zero. State the target used.
Sample variance: s² = Σ (rᵢ − r̄)² ÷ (n − 1); Use when estimating from observed data. Divide by n if the question says to treat the data as the whole population.
Two-asset portfolio variance: σp² = w₁²σ₁² + w₂²σ₂² + 2 w₁ w₂ ρ σ₁ σ₂; w₁ + w₂ = 1 for a fully invested portfolio. ρ is the correlation; Cov = ρσ₁σ₂.
Scaling a return: Var(aR + b) = a² Var(R); Adding a constant does not change variance.
Shortfall probability: P(X < L); L is the target. For a continuous X, P(X < L) = P(X ≤ L). For a normal X, use Φ((L − μ) ÷ σ).
Shortfall (one outcome): (L − X)+ = max(L − X, 0); Zero when the target is met.
Expected shortfall below target: E[(L − X)+] = Σ over x < L of (L − x) × P(X = x); For discrete X. For continuous X, integrate (L − x) f(x) from −∞ to L. Includes zeros when the target is met, so it is not conditional on a shortfall.
Conditional expected shortfall: E[L − X | X < L] = E[(L − X)+] ÷ P(X < L); Average size of the shortfall given that one occurs.
Below-target semi-variance: E[((L − X)+)²] = Σ over x < L of (L − x)² × P(X = x); Measured about the target L, not about the mean. The square root is the target semi-deviation.
Normal shortfall probability: P(X < L) = Φ((L − μ) ÷ σ); Valid only if X is normally distributed with mean μ and standard deviation σ.
Value at Risk: VaR_α = smallest x such that P(L ≤ x) ≥ α; L is the loss. For a continuous distribution, P(L ≤ VaR_α) = α. Always state α and the time horizon.
VaR for a normal loss: VaR_α = μ + z_α × σ; L ~ N(μ, σ²). z_α is the standard normal value with Φ(z_α) = α. Common values: z = 1.645 (95%), z = 2.326 (99%).
Tail Value at Risk: TVaR_α = (1 ÷ (1 − α)) × ∫ from α to 1 of VaR_u du; Average loss in the worst (1 − α) of probability (expected shortfall). For a continuous loss this equals E[L | L ≥ VaR_α]. For a discrete loss, E[L | L ≥ VaR_α] can differ from it (see the worked example).
TVaR for a normal loss: TVaR_α = μ + σ × φ(z_α) ÷ (1 − α); φ is the standard normal density, φ(z) = e^(−z²/2) ÷ √(2π). Applies to a normal loss.
Scaling over time: σ over n periods = σ × √n, mean over n periods = n × μ; Valid only if period returns are independent and identically distributed with the same variance.
Coherence conditions: Monotonicity; ρ(X + c) = ρ(X) + c; ρ(kX) = kρ(X) for k > 0; ρ(X + Y) ≤ ρ(X) + ρ(Y); All four must hold. VaR fails subadditivity in general. TVaR defined as expected shortfall satisfies all four. The E[L | L ≥ VaR] form is coherent for continuous losses but can fail subadditivity for discrete losses.
Active return: a_t = R_P,t − R_B,t; Portfolio return minus benchmark return in period t.
Mean active return: ā = (1/n) Σ a_t; Average of the active returns over n periods.
Tracking error (sample, from data): TE = √[ Σ (a_t − ā)² ÷ (n − 1) ]; Use n − 1 for a sample unless the question says to use n. State your choice.
Tracking error from variances: TE² = σ_P² + σ_B² − 2 ρ σ_P σ_B; ρ is the correlation between portfolio and benchmark returns.
Tracking error from active weights: TE² = Σ Σ (w_i − b_i)(w_j − b_j) σ_ij; w_i is the portfolio weight and b_i the benchmark weight. The difference is the active weight.
Information ratio: IR = ā ÷ TE; Mean active return per unit of tracking error. Use the same time period for both.
Matching (mismatch) risk: Var(A − L) = σ_A² + σ_L² − 2 ρ σ_A σ_L; A and L are asset and liability values or returns on a consistent basis.
Annualising: TE_annual = TE_monthly × √12; Valid when active returns are independent across periods.
Beta: β_i = Cov(R_i, R_m) ÷ Var(R_m) = ρ_im × σ_i ÷ σ_m; Measures systematic risk relative to the market. ρ_im is the correlation between asset i and the market.
Beta of a portfolio: β_p = Σ w_i β_i; Beta is a weighted average of the asset betas, using portfolio weights that sum to 1.
Systematic and specific variance (single-factor model): σ_i² = β_i² σ_m² + σ_ei²; Assumes the residual risk is uncorrelated with the market. The first term is systematic, the second is specific.
Approximate price change using modified duration: ΔP ÷ P ≈ −D_mod × Δy; Valid for small changes in yield. D_mod is modified duration, with yield y.
Macaulay duration and modified duration: D_mod = D_mac ÷ (1 + y); Applies when y is an effective annual yield and cash flows are annual.
Surplus: S = A − L; Risk for a liability-driven investor is the variability of S, or of the funding ratio A ÷ L.

Quick revision

Risk-averse investors have a concave utility function, so U″(w) < 0.
Variance of returns measures spread around the mean and treats gains and losses alike.
Standard deviation is the square root of variance and is in the same units as returns.
Shortfall probability is P(return < target), and you must state the target clearly.
Semi-variance looks only at outcomes below a chosen level, so it focuses on downside.
VaR at level α is a quantile of the loss distribution over a stated time horizon.
Always state the confidence level and horizon when quoting VaR.
VaR says nothing about how large losses are beyond the VaR point.
TVaR is the expected loss given that the loss exceeds VaR, so it is at least as large as VaR.
Tracking error is the standard deviation of the difference between portfolio and benchmark returns.
Choose a risk measure to match the question being asked and the data available.
Always interpret your answer in words, not just give a number.

Common mistakes

Choosing the option with the highest expected wealth. Fix: Compute E[U(W)] for each option. A risk-averse investor can prefer a lower-mean, safer option.
Defining A(w) as −U''(w) without dividing by U'(w). Fix: Always use A(w) = −U''(w) ÷ U'(w). Dividing removes the effect of rescaling U.
Reporting variance when the question asks for standard deviation, or the reverse. Fix: Underline the required measure in the question and finish with the square root if σ is asked.
Forgetting to square the weights in portfolio variance. Fix: Write σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂ before substituting.
Measuring the semi-variance about the mean instead of the target. Fix: Read the definition. Below-target semi-variance uses L, and the shortfall is (L − x).
Including outcomes equal to or above the target in the sums. Fix: Mark each row as shortfall or no shortfall first. Zero-shortfall rows add nothing.
Treating a return as a loss and getting the sign wrong. Fix: Write L = −(return) × value first. For a normal return R, the loss is −R, and VaR at 95% is −μ_R + 1.645σ_R times the value.
Forgetting to scale mean and standard deviation to the horizon. Fix: Multiply the mean by n and the standard deviation by √n before using z. State the independence assumption.
Using portfolio standard deviation instead of the standard deviation of active returns. Fix: Always form the difference series first. Tracking error is the standard deviation of that series.
Dividing by n instead of n − 1 without saying so. Fix: Treat the data as a sample and use n − 1, unless the question says otherwise. Write down your choice.

Exam tips

Show U' and U'' every time. Marks go for the derivatives and signs even if the final number slips.
State what each risk aversion measure means in words: A relates to rupee amounts at risk, R to proportions of wealth.
Be ready to discuss the realism of each utility function, for example quadratic increasing A and log utility R = 1.
In certainty equivalent questions, invert U carefully and give the risk premium as E[W] − CE with units.
Link utility to portfolio choice: a more risk-averse investor chooses a point on the efficient frontier with lower risk.
Show the formula, the substitution and the result. Method marks are awarded even if arithmetic slips.
State units clearly and stay in one unit (decimals or per cent) throughout.
For discussion questions, give both a reason variance is used (tractability, diversification, normality) and at least two limitations.