ETF5952 · Quantitative Methods for Risk Analysis
Conditional Mean, Volatility and Time-Varying Risk
This is the time-varying-risk core of ETF5952 Quantitative Methods for Risk Analysis at Monash University: the topic that stops treating risk as a fixed number and makes it move with the market. You condition on the information set I_t = {r_1,...,r_t}, work with the conditional distribution of next period's return, and separate the roughly constant conditional mean from the strongly time-varying conditional variance that produces volatility clustering. From a rolling-window estimate of today's volatility you compute the conditional Value-at-Risk (VaR) as a tail quantile and the conditional Expected Shortfall (ES) as the average loss beyond it, using either the parametric-normal formula or non-parametric historical simulation - the workshop-and-exam skill that runs setup → method → a computed risk number.
What this chapter covers
- 011. Information set - I_t = {r_1,...,r_t} collects everything known at time t and GROWS over time; every conditional quantity is an expectation given I_t
- 022. Conditional vs unconditional distribution - the unconditional distribution averages over all time; the conditional distribution F_{t+1|t} uses today's state, so it widens in a storm
- 033. Conditional mean, variance, volatility - E_t(R_{t+1}), Var_t(R_{t+1}) and σ_{t+1|t} = √Var_t; the mean is ~flat while the variance moves
- 044. Volatility clustering - large moves bunch with large, calm with calm; the return model R_t = μ + σ_t Z_t puts all the time-variation in σ_t
- 055. Rolling-window (moving-average) volatility - σ-hat²_{t+1|t} = (1/m)Σ(r_{t-i} - μ-hat)²; short m is reactive, long m is smooth
- 066. Conditional quantile and VaR - q_p solves Pr(R_{t+1} ≤ q_p | I_t) = p, and VaR(p) = -q_p is a positive loss; normal case VaR = -(μ + σ z_p)
- 077. Conditional Expected Shortfall - ES(p) = -E_t[R_{t+1} | R_{t+1} ≤ q_p], the mean loss in the worst-p tail; ES ≥ VaR always
- 088. Historical simulation (non-parametric) - sort the window, empirical quantile q-hat_p = r_(⌈pm⌉), VaR_HS = -q-hat_p, ES_HS = mean of the ⌈pm⌉ worst
- 099. Square-root-of-time scaling - under flat volatility, h-period volatility = √h·σ (variance scales with h), an approximation once volatility clusters
- 1010. Time-varying dynamics - as the window rolls and I_t updates, a fresh shock raises VaR and ES and a run of calm lowers them, unlike a static unconditional VaR
Rolling-window volatility to a conditional normal VaR and ES
- +1Sum the squared returns (with μ = 0 the deviations are just the returns): 0.012² + 0.008² + 0.020² + 0.004² + 0.010² = 0.000144 + 0.000064 + 0.000400 + 0.000016 + 0.000100 = 0.000724.
- +1Rolling-window variance. Divide by the window length m = 5: σ-hat²_{t+1|t} = 0.000724 / 5 = 0.0001448.
- +1Conditional volatility. Take the square root: σ-hat_{t+1|t} = √0.0001448 = 0.01203, i.e. 1.203% per day.
- +1Conditional 5% quantile (Normal). With z_0.05 = -1.645, q_0.05 = μ + σ·z_0.05 = 0 + 0.01203×(-1.645) = -0.01979 (a bad, negative return).
- +1Conditional VaR. VaR(0.05) = -q_0.05 = 0.01979 = 1.98%. The minus sign turns the negative tail return into a positive loss magnitude.
- +1Dollar VaR. Multiply by portfolio value: 0.01979 × $2,000,000 = $39,590. Reading: a 5% chance of losing more than about $39,590 over one day.
- +1Conditional ES (Normal). ES(0.05) = -μ + σ·φ(z_0.05)/p = 0 + 0.01203 × (0.1031/0.05) = 0.01203 × 2.062 = 0.02481 = 2.48%; in dollars 0.02481 × $2,000,000 = $49,624.
- +1Sanity check. ES (2.48%, $49,624) > VaR (1.98%, $39,590), as it must be - ES averages the losses beyond the VaR threshold, so it can never be smaller.
Key terms
- Information set (I_t)
- The collection of everything known at time t, I_t = {r_1, r_2, ..., r_t}, which grows as new returns arrive. Every 'conditional' quantity in this topic - conditional mean, variance, volatility, VaR and ES - is an expectation taken GIVEN I_t, i.e. using today's information to describe next period's return R_{t+1}.
- Conditional volatility (σ_{t+1|t})
- The standard deviation of next period's return given today's information, σ_{t+1|t} = √Var_t(R_{t+1}). Unlike a single unconditional volatility it changes over time, rising in turbulent periods and falling in calm ones. It is the input that drives every conditional VaR and ES figure.
- Volatility clustering
- The empirical regularity that large price moves tend to be followed by large moves and calm by calm, so the conditional variance is strongly time-varying even though the conditional mean is roughly constant. It is why risk must be modelled as time-varying rather than fixed, and it motivates rolling-window, EWMA and GARCH volatility estimates.
- Rolling-window (moving-average) volatility
- A non-parametric estimate of the conditional variance from the last m returns with equal weights: σ-hat²_{t+1|t} = (1/m)Σ_{i=0}^{m-1}(r_{t-i} - μ-hat)². A short window reacts quickly to new shocks but is noisy; a long window is smooth but slow, and old extreme returns leave the estimate abruptly when they drop out of the window.
- Conditional Value-at-Risk (VaR)
- The loss threshold that the worst-p tail of the conditional return distribution just reaches: VaR_{t+1|t}(p) = -q_p, where the p-quantile q_p solves Pr(R_{t+1} ≤ q_p | I_t) = p. It is reported as a positive loss; for a conditional normal, VaR(p) = -(μ + σ_{t+1|t} z_p) with z_0.05 = -1.645.
- Conditional Expected Shortfall (ES)
- The AVERAGE loss given that the loss lands in the worst-p tail: ES_{t+1|t}(p) = -E_t[R_{t+1} | R_{t+1} ≤ q_p]. Because it averages losses beyond the VaR threshold, ES ≥ VaR always. For a conditional normal, ES(p) = -μ + σ_{t+1|t}·φ(z_p)/p, where φ is the standard-normal density.
- Conditional quantile (q_p)
- The value q_p with Pr(R_{t+1} ≤ q_p | I_t) = p. For downside risk p is small (e.g. 0.05 or 0.01), so q_p is a negative return, and q_0.01 < q_0.05 because the 1% tail sits inside the 5% tail. VaR and ES are both built directly from this quantile.
- Historical simulation (HS)
- A non-parametric way to get VaR and ES with no distributional assumption: sort the window of returns r_(1) ≤ ... ≤ r_(m), read the empirical quantile q-hat_p = r_(⌈pm⌉) (rounding up), then VaR_HS(p) = -q-hat_p and ES_HS(p) = minus the average of the ⌈pm⌉ worst returns. It captures fat tails automatically but is limited by the observed window.
Conditional Mean, Volatility and Time-Varying Risk FAQ
What is the difference between conditional and unconditional VaR, and why does it matter?
An unconditional VaR treats the return distribution as fixed and averages over all time, so it produces a single risk number that sits still through calm and crisis alike. A conditional VaR is computed from the conditional distribution given the information set I_t, so it uses today's estimated volatility: when a fresh shock enters the rolling window the conditional volatility jumps and the VaR rises, and when markets calm the shock eventually drops out and the VaR falls. Because financial volatility clusters, the unconditional number badly understates risk during a turbulent period and overstates it afterwards, which is exactly why ETF5952 emphasises the time-varying, conditional version.
Why is Expected Shortfall always at least as large as VaR at the same coverage?
VaR(p) is only the THRESHOLD of the worst-p tail - the p-quantile loss - whereas ES(p) is the AVERAGE loss across everything inside that tail, i.e. the losses that are at least as bad as VaR. Averaging a set of numbers that are all ≥ the threshold can never give something below the threshold, so ES(p) ≥ VaR(p) by construction. Intuitively, VaR answers 'how bad is the tail edge?' and ES answers 'given we are past that edge, how bad is it on average?', which is why regulators favour ES for capturing the size of extreme losses. If a calculation ever shows ES below VaR, it signals an arithmetic or sign error.
Can AI help me with conditional VaR, ES and time-varying volatility in ETF5952?
Yes, as a study aid for the method. You can ask Sia to explain the ideas step by step - how the information set makes a quantity conditional, why VaR carries a minus sign, how the rolling-window volatility feeds the normal VaR and ES formulas, why ES is always at least VaR, and how the square-root-of-time rule is derived - and to generate fresh practice problems with new numbers so you rehearse the technique rather than memorise an answer. Sia explains and coaches the working; it does not sit your quizzes, assignments or the closed-book final exam for you, and you should always check formulas and assumptions against your own Moodle materials.
Exam move
Treat this topic as a pipeline and drill it end to end, because the exam and the at-home Assignment 2 (written in the style of the final) run the same route: setup, then a computed risk number. From a short window of returns, compute the rolling-window variance by averaging the squared deviations and dividing by m (not m-1), take the square root for the conditional volatility, then turn it into a 5% and a 1% VaR with VaR = -(μ + σ z_p) and z_0.05 = -1.645, z_0.01 = -2.326, and into an ES with the normal formula ES = -μ + σ·φ(z_p)/p. Practise the non-parametric route in parallel: sort a window, round ⌈pm⌉ UP, read the VaR off the ⌈pm⌉-th worst return and average the worst ⌈pm⌉ for the ES. Make the direction words reflexive - VaR is a positive loss equal to -q_p, ES ≥ VaR always, volatility scales with √h while variance scales with h, conditional (given I_t) is not the same as unconditional - because a right number attached to the wrong rule earns nothing. Always finish by converting to dollars with the portfolio value and sanity-checking that ES exceeds VaR. The final examination is closed book and worth 40% of the unit; its duration is not published in the unit materials, so budget your time in proportion to the marks on each question rather than assuming a length, and confirm the exact date and duration on Moodle or the Monash exam timetable.
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