ETF5952 · Quantitative Methods for Risk Analysis
Estimating Return Distributions: Monte Carlo, Parametric and Non-parametric
This chapter of Monash University's ETF5952 Quantitative Methods for Risk Analysis tackles the problem underneath every risk number: the next return RT+1 is drawn from an unknown distribution F, and expected utility, VaR and Expected Shortfall are all expectations under F. You estimate F three ways — Monte Carlo simulation (draw many returns, map to wealth, average the quantity), parametric models (assume a Normal or Student-t and estimate μ, σ), and non-parametric methods (use the empirical distribution directly). These are the hand skills examined in the closed-book final and rehearsed in the R workshops.
What this chapter covers
- 01The estimation problem: R_{T+1} ~ unknown F, and every risk measure is an expectation under F
- 02Three routes to F: Monte Carlo simulation, parametric (Normal / Student-t), non-parametric (empirical)
- 03Monte Carlo for expected utility: draw N returns, convert to wealth, average U, choose the higher E[U]
- 04Simulate returns then map to wealth: P = P_T(1+r) for simple returns, P = P_T·e^r for log returns
- 05Parametric location-scale model: R = μ + σZ, quantile q_p = μ + σ z_p, VaR(p) = −(μ + σ z_p)
- 06Normal quantiles z_0.05 = −1.645, z_0.01 = −2.326; the 1% VaR always exceeds the 5% VaR
- 07Student-t: standardise the raw t-quantile by √(ν/(ν−2)); heavier tails give a larger deep-tail VaR
- 08Non-parametric empirical distribution: historical-simulation quantile r₍⌈pT⌉₎ and bootstrap resampling
- 09Why simulate: no closed form for E[U], error shrinks like 1/√N, and a risk-averse investor ranks on E[U] not mean return
Monte Carlo expected-utility choice between two funds
- +1Map Fund X returns to wealth: P = 100(1 + r) gives {121, 100, 144}. Apply U(P) = √P: {11, 10, 12}.
- +1Estimate E[U] for X by averaging the utilities: (11 + 10 + 12) / 3 = 33 / 3 = 11.00.
- +1Map Fund Y returns to wealth: {169, 81, 100}, so √P = {13, 9, 10}.
- +1Estimate E[U] for Y: (13 + 9 + 10) / 3 = 32 / 3 = 10.667.
- +1Compare: E[U_X] = 11.00 > E[U_Y] = 10.667, so choose Fund X — even though Y has a higher single upside draw, its bigger spread lowers expected utility for the concave (risk-averse) investor.
Key terms
- Monte Carlo simulation
- Estimating an expectation by generating a large sample: draw N returns r^(s) from an estimated distribution F̂, map each to wealth, and average the quantity of interest, e.g. E[U(P_{T+1})] ≈ (1/N)ΣU(P^(s)). Used whenever there is no closed form.
- Expected utility
- The average utility of end wealth, E[U(P_{T+1})], which a risk-averse (concave-U) investor maximises. Ranking on E[U] — not on mean return — accounts for the penalty a concave utility places on volatility.
- Parametric model
- An estimate of F that assumes a distribution family (Normal or Student-t) and estimates its few parameters (μ, σ, and ν for the t). Quantiles then come from a formula rather than from the raw data.
- Location-scale model
- Writing the return as R = μ + σZ with Z standardised (mean 0, variance 1). Its p-quantile is q_p = μ + σ z_p, giving parametric VaR(p) = −(μ + σ z_p); only z_p changes between the Normal and the Student-t.
- Empirical distribution (historical simulation)
- A non-parametric estimate that treats the observed returns as the distribution: sort them and read the ⌈pT⌉-th worst for the p-quantile (round pT up). No shape is assumed; the bootstrap resamples the same data with replacement.
- Student-t distribution
- A symmetric family with heavier tails than the Normal (kurtosis > 3), governed by degrees of freedom ν. Standardised to unit variance (divide the raw quantile by √(ν/(ν−2))), its deep-tail quantile is more negative than the Normal's, so VaR and ES are larger; t → Normal as ν → ∞.
- Certainty equivalent (x_CE)
- The guaranteed wealth giving the same utility as a gamble: U(x_CE) = E[U(P)], so x_CE = U⁻¹(E[U]). For a risk-averse investor x_CE < E[wealth]; the gap is the risk premium.
Estimating Return Distributions: Monte Carlo, Parametric and Non-parametric FAQ
When should I use Monte Carlo instead of a parametric formula?
Use the parametric formula when a closed form exists — for example VaR(p) = −(μ + σ z_p) under a Normal or standardised Student-t. Reach for Monte Carlo when there is no closed form: any non-linear utility U(P), an option-like payoff, or a multi-asset portfolio. You draw N returns from your estimated distribution, convert to wealth, and average the quantity; the same simulated sample also gives VaR (sort the losses) and ES (average the worst tail). The trade-off is simulation error, which shrinks only like 1/√N, so use a large N.
What is the difference between the parametric and non-parametric routes?
Both estimate the same distribution F, but with different assumptions. The parametric route assumes a family (Normal or Student-t), estimates μ and σ, and reads quantiles off R = μ + σZ — clean, but wrong if the true tails are fatter than assumed. The non-parametric route assumes no shape: historical simulation reads the quantile straight off the sorted returns (the ⌈pT⌉-th worst), and the bootstrap resamples them. It captures the real tail but needs a long window and cannot see a loss worse than any yet observed.
Can AI help me with estimating return distributions in ETF5952?
Yes — Sia can explain the method step by step: how to set up a Monte Carlo run for expected utility (simulate returns, map to wealth, average U, choose the higher E[U]), how to standardise a Student-t before reading its quantile, and how to round ⌈pT⌉ up for a historical-simulation estimate. It works through practice problems with you and checks your sign, quantile and units, but it does not complete graded assessments for you or promise any particular mark — always follow Monash's academic-integrity rules and confirm details on Moodle.
Exam move
Anchor everything on one idea: F is unknown, and each route is a different way to estimate it. Drill the Monte Carlo mechanic until it is automatic — simulate returns, convert to wealth, average the utility, choose the higher E[U] — and never average returns before applying a concave U (that is the Jensen trap). For the parametric route, memorise the location-scale formula VaR(p) = −(μ + σ z_p) with z_0.05 = −1.645 and z_0.01 = −2.326, and remember to standardise a Student-t by dividing its raw quantile by √(ν/(ν−2)). For the non-parametric route, practise reading the ⌈pT⌉-th worst sorted return (round pT up) and the bootstrap. The final exam is worth 40%, is closed book and centrally scheduled in the Monash Semester-1 period (~June 2027 — confirm the exact date and length on Moodle); because the duration is not published in the unit materials, budget your time in proportion to the marks and bank the one-line definitional marks before the arithmetic. Assignment 2 is set in the style of the final, so treat it as your best mock.
Working through Estimating Return Distributions: Monte Carlo, Parametric and Non-parametric in ETF5952? Sia is AskSia’s AI Finance tutor — ask any ETF5952 Estimating Return Distributions: Monte Carlo, Parametric and Non-parametric question and get a clear, step-by-step explanation grounded in how ETF5952 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
