ETF5952 · Quantitative Methods for Risk Analysis
Financial Risk and Returns
This is the measurement layer of ETF5952 Quantitative Methods for Risk Analysis at Monash University: the topic that turns a price series into the returns on which every later risk measure is computed. You define the simple (net) return R_t = (P_t - P_{t-1})/P_{t-1} and the log (continuously compounded) return r_t = ln(P_t/P_{t-1}), see why the unit prefers log returns because they are additive across time, and learn how returns at different frequencies aggregate over a horizon h. Adding the i.i.d. assumption gives the square-root-of-time rule - mean scales with h, variance with h, volatility with √h - which is the plumbing under the Value-at-Risk and Expected-Shortfall calculations that follow.
What this chapter covers
- 011. Simple vs log return - simple R_t = (P_t - P_{t-1})/P_{t-1}; log r_t = ln(P_t/P_{t-1}); both are dimensionless fractions of the base price
- 022. Gross return and the link - gross return = P_t/P_{t-1} = 1 + R_t, so r_t = ln(1 + R_t) and R_t = e^(r_t) - 1: two encodings of one price move
- 033. Log returns are always ≤ simple returns - they agree for small moves and diverge as moves grow (a gain: r < R; a loss: |r| > |R|)
- 044. Rebuilding the price - roll forward with P_{t+1} = P_t(1 + R_{t+1}) or P_{t+1} = P_t·e^(r_{t+1})
- 055. Log-return additivity - the k-period log return is the SUM of the one-period log returns; simple returns instead COMPOUND (multiply the gross returns)
- 066. Return frequencies - returns are measured daily, weekly, monthly or annually, and must be aggregated to move between frequencies
- 077. Time aggregation under i.i.d. - if returns are independent and identically distributed, the h-period mean is h·μ and variance is h·σ²
- 088. Square-root-of-time rule - volatility scales as √h·σ (variance adds, so volatility grows with the square root), valid only under flat volatility
- 099. The i.i.d. assumption and its limits - real returns show volatility clustering, so √h is a first-order approximation the later EWMA/GARCH models refine
Two-day log-return additivity and the log-vs-simple gap
- +1Daily log returns. r_1 = ln(P_1/P_0) = ln(25.50/25.00) = ln(1.02) = 0.019803, i.e. +1.980%. r_2 = ln(P_2/P_1) = ln(24.75/25.50) = ln(0.970588) = -0.029853, i.e. -2.985%. Each is the log of the gross return for that day.
- +1Two-day log return by adding. Log returns are additive across time, so r_{0,2} = r_1 + r_2 = 0.019803 + (-0.029853) = -0.010050, i.e. -1.005%. The intermediate price P_1 cancels in the sum.
- +1Verify directly. Computing the two-day log return in one step gives ln(P_2/P_0) = ln(24.75/25.00) = ln(0.99) = -0.010050 = -1.005%, identical to the sum - which is exactly what additivity guarantees.
- +1Two-day simple return. R_{0,2} = (P_2 - P_0)/P_0 = (24.75 - 25.00)/25.00 = -0.010 = -1.000%. Note this is NOT the sum of the daily simple returns (+2.000% and -2.941% sum to -0.941%): simple returns compound as (1 + R_1)(1 + R_2) - 1, they do not add. That is why the unit uses log returns for multi-period risk.
Key terms
- Simple (net) return
- The proportional price change over one period, R_t = (P_t - P_{t-1})/P_{t-1} = P_t/P_{t-1} - 1. It is dimensionless (a fraction of the starting price, usually quoted as a percentage). Multi-period simple returns COMPOUND: the k-period gross return is the product (1 + R_{t+1})...(1 + R_{t+k}), so simple returns are multiplied, not added.
- Log (continuously compounded) return
- The log of the gross return, r_t = ln(P_t/P_{t-1}) = ln(P_t) - ln(P_{t-1}) = ln(1 + R_t). The unit's default return: because logarithms telescope, log returns are additive across time, which makes multi-period means, variances and the square-root-of-time rule work cleanly.
- Gross return
- The price ratio P_t/P_{t-1} = 1 + R_t = e^(r_t). It is always positive (prices cannot go below zero). The two return definitions are linked through it: r_t = ln(1 + R_t) and, inverting, R_t = e^(r_t) - 1, so a log and a simple return describe the same price move in different units.
- Log-return additivity
- The property that the k-period log return equals the sum of the one-period log returns: ln(P_{t+k}/P_t) = r_{t+1} + ... + r_{t+k}, because every intermediate log-price cancels. Simple returns do not have this property (they compound), which is the main practical reason the unit works in log space.
- Return frequency
- The interval over which a return is measured - daily, weekly, monthly or annual. Risk figures are frequency-specific, so a daily volatility must be aggregated before it can be compared with a monthly or annual one. Converting between frequencies uses the same time-aggregation rules as horizon scaling.
- i.i.d. returns
- The baseline assumption that successive returns are independent and identically distributed: every period shares the same mean and variance (identical), and today's return carries no information about tomorrow's (independent, hence serially uncorrelated). Under i.i.d. the covariance terms vanish when returns are added, so means and variances of multi-period returns simply add.
- Time aggregation (mean and variance)
- Under i.i.d. returns with per-period mean μ and variance σ², the h-period mean is μ_h = h·μ and the h-period variance is σ²_h = h·σ² (variances add because returns are uncorrelated). The same rules annualise a rate using N periods per year: μ_ann = N·μ, σ²_ann = N·σ².
- Square-root-of-time rule
- Because variance scales with the horizon h, volatility - its square root - scales with √h: σ_h = √h·σ. So a 10-day volatility is √10 ≈ 3.16 daily volatilities, not 10. The rule is exact only under i.i.d. / flat volatility; volatility clustering (captured later by EWMA and GARCH) means it is a first-order approximation, not a law.
Financial Risk and Returns FAQ
What is the difference between a simple return and a log return, and when should I use each?
A simple (net) return R_t = (P_t - P_{t-1})/P_{t-1} is the ordinary percentage price change and is the natural way to describe the return on a portfolio at a single point in time or across assets held together. A log return r_t = ln(P_t/P_{t-1}) = ln(1 + R_t) is the continuously compounded version and is the one to reach for whenever you aggregate over TIME, because log returns add across periods while simple returns compound. The two agree for small moves and diverge as moves get larger. In ETF5952 the working default is the log return, but read each question carefully - a dollar-P&L or single-period portfolio question may want the simple return.
Why does volatility scale with the square root of time rather than with time itself?
Because it is the VARIANCE, not the volatility, that adds up over independent periods. If daily returns are i.i.d. with variance σ², then the variance over h days is h·σ² (the covariance terms are zero because returns are uncorrelated). Volatility is the square root of variance, so the h-day volatility is √(h·σ²) = √h·σ. Scaling volatility linearly by h instead of √h badly overstates risk - for a 10-day horizon it would inflate a figure that should only grow by √10 ≈ 3.16 by a factor of 10. The rule holds only while volatility is roughly constant; once volatility clusters, it is just an approximation.
Can AI help me with simple vs log returns and time aggregation in ETF5952?
Yes, as a study aid for the method. You can ask Sia to explain the ideas step by step - how log-return additivity follows from the logarithm cancelling intermediate prices, why the sum of daily simple returns is not the multi-period simple return, how to annualise a daily mean and volatility, and how the square-root-of-time rule drops out of the i.i.d. assumption - 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 formulas and assumptions against your own Moodle materials.
Exam move
Make the two return definitions and the three scaling rules automatic, because this topic is the setup step inside almost every later Value-at-Risk, simulation and volatility question, so errors here propagate. Drill the mechanics by hand: from two prices compute both a simple and a log return; from a short string of daily log returns add them to a multi-period return and rebuild the final price with P_0·e^(sum); and annualise a daily mean and volatility using μ_ann = N·μ and σ_ann = √N·σ. Rehearse the direction words until they are reflexive - log returns ADD while simple returns COMPOUND, variance scales with h while volatility scales with √h - because a correct number attached to the wrong rule earns nothing. Watch the recurring traps: adding simple returns to aggregate over time, scaling volatility linearly in h, and forgetting that the square-root-of-time rule assumes i.i.d. / flat volatility. The final examination 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 rather than assuming a length, and confirm the exact date and duration on Moodle or the Monash exam timetable. Assignment 2 is written in the style of the final exam, so its problems are the best rehearsal for how this material is tested.
Working through Financial Risk and Returns in ETF5952? Sia is AskSia’s AI Finance tutor — ask any ETF5952 Financial Risk and Returns 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.
