The trader is investing $1,000,000. She borrowed $800,000 from her employer and invested $200,000 of her own money. Her employer requires that the trader's one-week portfolio VaR to not exceed $100,000 with a 99% degree of confidence. She invest in a three-stock portfolio as described in the below table: [table]
i | w_i | σ_(1,i) | σ_(2,i) | σ_(3,i) 1 | 0.3 | 0.04 | | 2 | 0.1 | 0.01 | 0.16 | 3 | 0.6 | 0.02 | 0.04 | 0.36
[/table] Calculate Value at Risk of this portfolio. Use z = 2.326. All the variances and covariances in the table are annualized.[Fill in the blank]

Question

The trader is investing $1,000,000. She borrowed $800,000 from her employer and invested $200,000 of her own money. Her employer requires that the trader's one-week portfolio VaR to not exceed $100,000 with a 99% degree of confidence. She invest in a three-stock portfolio as described in the below table: [table]
i | w_i | σ_(1,i) | σ_(2,i) | σ_(3,i) 1 | 0.3 | 0.04 |  |  2 | 0.1 | 0.01 | 0.16 |  3 | 0.6 | 0.02 | 0.04 | 0.36
[/table] Calculate Value at Risk of this portfolio. Use z = 2.326. All the variances and covariances in the table are annualized.[Fill in the blank]

BlackTom AI · Accepted Answer

Portfolio VaR uses the portfolio standard deviation and the z‑score: VaR = z * V * σ_p(horizon). Portfolio variance is σ_p^2 = w' Σ w where Σ contains annual variances/covariances; convert annual volatility to one‑week by multiplying by sqrt(1/52).

1) Relevant concepts:
- Portfolio variance: σ_p^2 = Σ_i Σ_j w_i w_j cov(i,j).
- Convert annual variance to one‑week variance by multiplying by 1/52 (or volatility by sqrt(1/52)).
- One‑week VaR (99%): VaR = z * V * σ_week where z = 2.326 and V = $1,000,000.

2) Step‑by‑step calculation (showing all intermediate steps):
Weights w = [0.3, 0.1, 0.6].
Annual covariance/variance matrix entries (from table):
var1 = 0.04, var2 = 0.16, var3 = 0.36;
cov12 = 0.01, cov13 = 0.02, cov23 = 0.04.

Compute annual portfolio variance:
w1^2·var1 = 0.3^2 * 0.04 = 0.09 * 0.04 = 0.0036
w2^2·var2 = 0.1^2 * 0.16 = 0.01 * 0.16 = 0.0016
w3^2·var3 = 0.6^2 * 0.36 = 0.36 * 0.36 = 0.1296
Cross terms: 2*(w1 w2 cov12 + w1 w3 cov13 + w2 w3 cov23)
= 2*(0.3*0.1*0.01 + 0.3*0.6*0.02 + 0.1*0.6*0.04)
= 2*(0.0003 + 0.0036 + 0.0024) = 2*(0.0063) = 0.0126

Sum: σ_annual^2 = 0.0036 + 0.0016 + 0.1296 + 0.0126 = 0.1474

Annual σ = sqrt(0.1474) = 0.3840 (≈38.40%)

One‑week σ = 0.3840 * sqrt(1/52) = 0.3840 / sqrt(52) ≈ 0.3840 / 7.211 = 0.05323 (≈5.323%)

One‑week VaR (99%): VaR = 2.326 * $1,000,000 * 0.05323
= 2.326 * 53,229.6 ≈ $123,839

3) Why this answer is correct:
The calculation used the correct portfolio variance formula, included all covariances, converted annual volatility to the one‑week horizon, and applied the 99% z‑score. The numeric result is ≈ $123,839, matching option (a).

4) Why the other options are incorrect (briefly):
- b. $83,536: too small — likely due to wrong z or forgetting to scale by portfolio value or mis‑scaling time.
- c. $147,400: does not match correct time scaling/variance aggregation (likely arithmetic or scaling error).
- d. $383,927: far too large — probably misapplied z or used annual VaR without converting to weekly.
- e. $100,000: employer limit, not the calculated VaR.

Conclusion: The computed one‑week 99% VaR is approximately $123,839, so the correct choice is (a) $123,839 (matches provided answer).

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