Returns, risk & sizing: interview questions and key results

Sharpe, vol, VaR, Kelly. The results to know for quant trading interviews, then a worked example of each of the 7 question types. Free, no sign-up.

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Key results

The mean scales with T, volatility with √T; √252 ≈ 15.87.
Kelly: f* = p − q/b.
Geometric growth ≈ μ − σ²/2.
Normal VaR₉₅ = 1.645σ.

Worked examples (7)

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

Annualised Sharpe

A strategy's daily returns have mean 0.06% and standard deviation 1.9%. What is its annualised Sharpe ratio? (252 trading days, zero risk-free rate)

Answer and worked solution

Answer: 0.5013

The mean scales with T and σ with √T, so SR_annual = (μ/σ)·√252 = (0.06/1.9)·15.87 ≈ 0.5013.

Two-asset portfolio vol

Portfolio: 30% in asset A (vol 20%), 70% in asset B (vol 25%), correlation 0.6. What is the portfolio volatility? (decimal or %)

Answer and worked solution

Answer: 0.21639

σ_p² = w²σ_A² + (1−w)²σ_B² + 2w(1−w)ρσ_Aσ_B = 0.046825, so σ_p ≈ 0.21639 (21.64%).

Kelly fraction

A bet wins with probability 0.6 and pays 1:1 (you lose your stake otherwise). What fraction of bankroll does the Kelly criterion stake?

Answer and worked solution

Answer: 0.2

f* = p − q/b = 0.6 − 0.4/1 = 0.2. Traders often use "half Kelly": edge estimates are noisy, and over-betting is much worse than under-betting.

Compounding returns

A position returns -4%, 9%, -8% over three consecutive months. What is the total return? (decimal or %)

Answer and worked solution

Answer: -0.037312

Returns compound: (0.96)·(1.09)·(0.92) − 1 ≈ -0.037312 (-3.731%), not the simple sum -3%. Log returns do add: Σ ln(1+r_i) = -0.038026.

Annualising volatility

A stock has daily volatility 1.6%. What is its annualised volatility? (252 trading days; decimal or %)

Answer and worked solution

Answer: 0.25399

σ_annual = σ_daily·√252 ≈ 1.6% × 15.87 = 25.4%. Handy rule: divide annual vol by 16 to get daily.

Normal VaR

You hold a $10m position with daily volatility 2%. Assuming normal returns with zero mean, what is the 1-day 95% VaR in dollars?

Answer and worked solution

Answer: 329000

VaR₉₅ = 1.645·σ·V = 1.645 × 0.02 × 10,000,000 ≈ $329,000. Real returns have fat tails, so normal VaR understates extreme losses.

Volatility drag

An asset has expected arithmetic return 8% per year and volatility 20%. Approximately what is its expected geometric (compound) growth rate? (decimal or %)

Answer and worked solution

Answer: 0.06

g ≈ μ − σ²/2 = 0.08 − 0.02 = 0.06. This is "volatility drag": high vol eats compound growth even when the average return is positive. You can see it in the Stats lab.

Returns, risk & sizing: interview questions and key results

Key results

Worked examples (7)

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