ETF5952 · Quantitative Methods for Risk Analysis
Risk, Uncertainty and Random Variables
This is the opening topic of ETF5952 Quantitative Methods for Risk Analysis at Monash University, and it lays the probability foundation the rest of the unit is built on. The core idea is that a future price or return is not a fixed number but a random variable whose spread of possible outcomes is exactly what we mean by risk. You learn to distinguish risk (uncertainty you can attach probabilities to) from uncertainty proper, to separate a random variable X from its realisation x, to describe a distribution three ways (pmf, pdf, CDF), to read off its four moments (mean, variance, skewness, kurtosis) and estimate them from data, and to define simple versus log returns. Master these and Value-at-Risk, Expected Shortfall, volatility models and backtesting all become applications of the same objects.
What this chapter covers
- 011. Risk vs uncertainty - risk is uncertainty you can describe with probabilities; the unit models risk as a random variable
- 022. States of the world - the future lands in one of several states, each with a probability; a random variable assigns a number to each
- 033. Random variable X vs realisation x - capital letters are the (random) model of the future, lowercase are the observed data
- 044. pmf, pdf and CDF - discrete mass p(x)=Pr(X=x), continuous density with Pr(a≤X<b) as area, and F(x)=Pr(X≤x)
- 055. Mean and variance - μ=E[X] locates the distribution; Var(X)=E[(X-μ)²] and volatility σ measure its spread
- 066. Skewness and kurtosis - the 3rd and 4th moments: negative skew = heavier downside tail, kurtosis > 3 = fatter-than-normal tails
- 077. Sample estimators - the sample mean x̄ and the sample variance s² (divide by T-1, not T) estimate μ and σ²
- 088. Prices and returns as random variables - simple return R=(P_t-P_{t-1})/P_{t-1} vs log return r=ln(P_t/P_{t-1}), and why log returns add across time
Expected return, variance and volatility over three states of the world
- +1Check the pmf. The three states are mutually exclusive and their probabilities sum to 0.25 + 0.50 + 0.25 = 1, with each non-negative, so this is a valid probability mass function over the three states of the world.
- +1Expected return (probability-weighted average). E[R] = 0.25(0.20) + 0.50(0.05) + 0.25(-0.15) = 0.05 + 0.025 - 0.0375 = 0.0375, i.e. an expected return of 3.75%.
- +1Variance = probability-weighted squared deviations about μ = 0.0375. The deviations are 0.1625, 0.0125 and -0.1875, so Var(R) = 0.25(0.1625)² + 0.50(0.0125)² + 0.25(0.1875)² = 0.006602 + 0.000078 + 0.008789 = 0.015469 (in return-squared units).
- +1Volatility. σ = √0.015469 = 0.1244, i.e. a standard deviation of 12.44%, which puts the risk back in the same units as the return.
Key terms
- Risk vs uncertainty
- Risk is uncertainty that can be described with probabilities, so it can be modelled as a random variable with a distribution. Uncertainty proper refers to situations where the probabilities are not well defined. The unit focuses on risk, and risk is always defined relative to the information available at a given time.
- State of the world
- One of several mutually exclusive scenarios the future can land in (for example boom, normal or recession), each carrying a probability. A random variable assigns a number to every state, so the distribution of the random variable is just the list of state values with their probabilities.
- Random variable vs realisation
- A random variable (capital letter, e.g. X, P_t, R_t) is the pre-outcome model of an uncertain quantity and has a distribution, an expectation and a variance. A realisation (lowercase, e.g. x, r_t) is the single value it takes once the state is revealed - just a number, with no distribution of its own.
- pmf, pdf and CDF
- The probability mass function p(x)=Pr(X=x) gives probabilities to discrete outcomes (summing to 1). The probability density function f(x) describes a continuous variable, with Pr(a≤X<b) equal to the area under f and Pr(X=x)=0. The cumulative distribution function F(x)=Pr(X≤x) works for both, is non-decreasing and runs from 0 to 1.
- Moments (mean, variance, skewness, kurtosis)
- Four summaries of a distribution: the mean μ=E[X] (centre), the variance Var(X)=E[(X-μ)²] with volatility σ (spread), the skewness E[(X-μ)³]/σ³ (asymmetry - negative means a heavier downside tail), and the kurtosis E[(X-μ)⁴]/σ⁴ (tail-heaviness - 3 for a normal, above 3 for the fat tails typical of returns).
- Sample mean and sample variance
- Estimators computed from observed data {x_1,...,x_T}: the sample mean x̄ = (1/T)Σx_t estimates E[X], and the sample variance s² = (1/(T-1))Σ(x_t - x̄)² estimates Var(X). The divisor is T-1 (not T) to make the variance estimator unbiased; s = √s² is the sample volatility.
- Simple vs log return
- The simple (net) return is R_t = (P_t - P_{t-1})/P_{t-1}, so the gross return is 1 + R_t. The log (continuously compounded) return is r_t = ln(P_t/P_{t-1}) = ln(1 + R_t). Log returns are additive across time - the multi-period log return is the sum of the single-period log returns - which is why the unit uses them for multi-period risk.
- i.i.d. returns
- The baseline assumption that successive returns are independent and identically distributed draws from a single distribution F. This is the setting in which the pmf/pdf/CDF and the moments describe risk; later topics relax it to allow volatility to change over time via conditional distributions and GARCH-type models.
Risk, Uncertainty and Random Variables FAQ
What is the difference between a random variable and its realisation, and why does it matter?
A random variable (written with a capital letter, like R for a return) is the model of an uncertain future quantity: it has a whole distribution of possible values with probabilities, and therefore an expectation and a variance. A realisation (lowercase, like the return you actually observed yesterday) is the single number it turned out to be, with no distribution of its own. It matters because risk - variance, expected loss, VaR - is a property of the random variable, which you then estimate from a sample of past realisations. Speaking of 'the variance of yesterday's return' mixes the two up and is a common exam error.
Why does the sample variance divide by T - 1 instead of T?
Dividing by T - 1 (the number of observations minus one) makes the sample variance an unbiased estimator of the true population variance: because the deviations are measured about the estimated sample mean rather than the unknown true mean, one degree of freedom is 'used up', and dividing by T would systematically underestimate the variance, especially in small samples. In ETF5952 the sample variance is always s² = (1/(T-1))Σ(x_t - x̄)², and dividing by T is a routine mark-losing slip.
Can AI help me with random variables and moments in ETF5952?
Yes, as a study aid for the method. You can ask Sia to explain the concepts step by step - how to set up an expected value over states of the world, why the sample variance uses T - 1, how to read the sign of skewness, or how simple and log returns differ and aggregate - and to walk through a practice problem so you can reproduce the working yourself. Sia explains and coaches the technique; it does not sit your quizzes, assignments or the closed-book final exam for you, and you should always check definitions and formulas against your own Moodle materials.
Exam move
Treat this chapter as vocabulary you must be able to write down without hesitation, because the marks here are the cheapest in the unit and everything later reuses these objects. First, be able to state the risk-versus-uncertainty distinction and the random-variable-versus-realisation split in a sentence each. Then drill the small calculations by hand: an expected value and variance over a handful of states, a sample mean and sample variance from a short return series (always dividing by T - 1), and a simple and log return from two prices. Practise saying the direction words out loud - negative skew means a heavier downside tail, kurtosis above 3 means fatter-than-normal tails and more frequent extreme losses - because a correct number with the wrong direction earns nothing. Finally, connect the pieces forward: the CDF and its inverse (quantiles) become Value-at-Risk, and log-return additivity becomes the square-root-of-time rule, so time spent here pays off across the whole unit. The final exam is closed book and worth 40% of the unit; the exam duration is not published in the unit materials, so plan your time in proportion to the marks on each question and confirm the exact date and length on Moodle or the Monash exam timetable.
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