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?

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BlackTom AI · Accepted Answer

We start by identifying the components of the problem: the portfolio value V0 = $100,000, the expected annual return mu = 10% (0.10), and the standard deviation sigma = 16.75% (0.1675). The 1% analytical VaR uses the 1st percentile of the portfolio's return distribution, assuming normality.

Option C: $17,637.
To reach this figure, one would either misplace the tail (e.g., using a 1% right tail loss instead of left tail) or apply an incorrect z-score. The correct 1% left-tail z-score for a standard normal is about -2.326, not a smaller magnitude. Using r1% = mu + z*sigma with z = -2.326 and sigma = 0.1675 gives r1% ≈ 0.10 + (-2.326)(0.1675) ≈ 0.10 - 0.389 ≈ -0.289. The resulting 1% VaR in dollars would be V0 * (-r1%) ≈ 100,000 * 0.289 ≈ $28,900, not $17,637. So this option underestimates the potential loss and reflects a miscalculation of tail risk.

Option B: $39,280.
This value is larger than the standard 1% VaR computed from the same inputs, implying an overestimation of downside or an using a more extreme tail (e.g., using z for a 0.5% or misapplying the sign). If you plug in the correct 1% left-tail z of -2.326 with the given mu and sigma, you do not arrive at 39,280; the math yields around $28,900. Therefore, this choice reflects a misapplication of the z-score or an incorrect interpretation of the distribution.

Option A: $29,027.
This option is closest to the calculation using the standard normal 1% left tail. Computing precisely: r1% = 0.10 + (-2.3263)(0.1675) ≈ 0.10 - 0.3896 ≈ -0.2896. The 1% VaR in dollars is V0 * (-r1%) ≈ 100,000 * 0.2896 ≈ $28,960. Rounding depending on the precision used for z and sigma can yield about $28,960 to $29,030; $29,027 is a very reasonable rounded figure. This aligns with the typical VaR formula VaR_{1%} = V0 * [-(mu + z_{0.01} * sigma)], using z_{0.01} ≈ -2.326.

Overall, the reasoning shows that the 1% VaR involves subtracting a substantial negative return (due to the large negative z-score) from the expected return, and converting that into a dollar loss on the initial portfolio value. The other options arise from misplacing the tail, misapplying the sign, or using an incorrect multiplier, leading to values that are either too small or too large relative to the standard 1% VaR calculation.

BU.232.750.81.FA25 Final Exam Fall 2025- Requires Respondus LockDown Browser

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?

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