Time series: interview questions and key results

AR(1), mean reversion, random walks. The results to know for quant trading interviews, then a worked example of each of the 5 question types. Free, no sign-up.

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Key results

AR(1): Var = σ_ε²/(1−φ²), ACF ρ(k) = φ^k.
Half-life = ln(½)/ln φ.
Random walk: variance grows linearly with time.

Worked examples (5)

Try each one before opening the solution. In the app, every type generates fresh numbers each time.

AR(1) variance

X_t = 0.9·X_t−1 + ε_t, where ε_t is white noise with variance 4. What is the stationary variance of X?

Answer and worked solution

Answer: 400/19 ≈ 21.053

Var(X) = φ²Var(X) + σ_ε² ⇒ Var(X) = σ_ε²/(1 − φ²) = 4/0.19 ≈ 21.053.

Mean-reversion half-life

A spread follows an AR(1) with daily coefficient φ = 0.98. What is its half-life of mean reversion in days?

Answer and worked solution

Answer: 34.31

Deviations decay like φ^h. Setting φ^h = ½ gives h = ln(½)/ln(φ) = 34.31 days. This is a key number for sizing and holding stat-arb trades.

AR(1) forecast

X_t − 4 = 0.9(X_t−1 − 4) + ε_t. Today X = 12. What is the forecast E[X_t+4]?

Answer and worked solution

Answer: 9.2488

E[X_t+h] = μ + φ^h(X_t − μ) = 4 + 0.9⁴·(8) = 9.2488.

AR(1) autocorrelation

For a stationary AR(1) with φ = -0.6, what is the autocorrelation at lag 4?

Answer and worked solution

Answer: 0.1296

ρ(k) = φ^k = -0.6⁴ ≈ 0.1296. The ACF decays geometrically while the PACF cuts off after lag 1, which is how you identify an AR(1).

Random-walk scaling

A price follows a random walk with daily change standard deviation $1.5. What is the standard deviation of the change over 25 days?

Answer and worked solution

Answer: 7.5

Independent increments mean variance scales with n, so σ₂₅ = 1.5·√25 = 7.5.

Time series: interview questions and key results

Key results

Worked examples (5)

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