A portfolio has an expected annual return of 10% and a standard deviation of 16.75%. What is the 1% (analytical) VAR of $100,000 in this portfolio?

Question

BlackTom AI · Accepted Answer

We are given a portfolio with an expected annual return (mu) of 10% and a standard deviation (sigma) of 16.75% on a value of $100,000. The task asks for the 1% analytical Value at Risk (VaR).

First, recall a common analytical (parametric) VaR formula for a one-year horizon under normality: VaR at 1% = V × (mu − z_{0.99} × sigma), where z_{0.99} is the 99th percentile of the standard normal distribution (negative tail for losses is captured by the subtraction).

Key constants: z_{0.99} ≈ 2.3263. Convert percentages to decimals: mu = 0.10, sigma = 0.1675. V = $100,000.

Proceed option by option:

Option A: $29,027.
- To assess, compute the VaR using the standard formula: VaR = 100,000 × (0.10 − 2.3263 × 0.1675).
- Compute the product: 2.3263 × 0.1675 ≈ 0.3896. Subtract from mu: 0.10 − 0.3896 ≈ −0.2896.
- Multiply by 100,000: 100,000 × (−0.2896) ≈ −$28,960. The magnitude of loss is about $28,960, which is close to $29,000. Since VaR is a loss amount, the positive figure often quoted is the magnitude, so Option A is in the ballpark of the standard calculation. Small rounding differences can account for the $67 discrepancy.

Option B: $39,280.
- This value aligns with the simple “volatility-based” component V × z_{0.99} × sigma, i.e., ignoring the expected return mu: 100,000 × 2.3263 × 0.1675 ≈ 100,000 × 0.3896 ≈ $38,960. This rounds to about $38,960–$39,000, which is higher than the 1% VaR that also accounts for the expected return (mu). If one uses only the volatility term, you obtain a VaR around $39k, which is larger because it does not subtract mu. Depending on rounding, $39,280 could be a rounded variant of this approach. This makes Option B consistent with a volatility-only VaR calculation, not the mu-adjusted VaR.

Option C: $17,637.
- This value does not emerge from the standard 1% VaR calculation with the given mu and sigma. The magnitude is far smaller than both the mu-adjusted VaR (≈ $28.9k) and the volatility-only VaR (≈ $39k). If someone used a different horizon, a different z-score, or a substantially different interpretation (e.g., applying VaR to a subset or misapplying the formula), they might land on a number this low, but it would not match the conventional analytical 1% VaR for these inputs.

Summary of reasoning for each option:
- Option A (≈ $29k) matches the mu-adjusted VaR calculation, with the loss magnitude close to the computed value when mu is included. It aligns with the standard formula VaR = V × (mu − z × sigma) and is plausible given rounding.
- Option B (≈ $39k) corresponds to using only the volatility term VaR = V × z × sigma, effectively ignoring mu. This gives a larger VaR and is a valid variant in some contexts but not the mu-adjusted version.
- Option C (≈ $17.6k) is inconsistent with the standard 1% VaR for these parameters and would arise from an atypical calculation or misapplication of the formula.

A portfolio has an expected annual return of 10% and a standard deviation of 16.75%. What is the 1% (analytical) VAR of $100,000 in this portfolio?单项选择题

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