Sumitomo Bank's risk manager has estimated that the DEARs of two of its major assets in its trading portfolio, foreign exchange and bonds, are -$300,000 and -$500,000, respectively. What is the 10-day VaR of Sumitomo's trading portfolio if the correlation among assets is assumed to be 0.8?

Question

BlackTom AI · Accepted Answer

Using the variance–covariance (delta–normal) method, treat the given DEARs as the one‑day VaR (absolute daily volatilities) for each asset: |DEAR_fx| = 300,000 and |DEAR_bonds| = 500,000. Portfolio one‑day variance = σ1^2 + σ2^2 + 2ρσ1σ2, then scale to 10 days by multiplying the standard deviation by sqrt(10).

1) Relevant concepts
- DEAR (daily expected adverse excursion) here is the one‑day VaR magnitude (use absolute values as σ).
- For two correlated assets, portfolio variance includes the covariance term 2ρσ1σ2.
- Time scaling for VaR under iid returns: VaR_T = VaR_1day * sqrt(T).

2) Step‑by‑step reasoning and calculations
- σ1 = 300,000; σ2 = 500,000; ρ = 0.8.
- Compute variances and covariance term:
  σ1^2 = (300,000)^2 = 90,000,000,000
  σ2^2 = (500,000)^2 = 250,000,000,000
  2ρσ1σ2 = 2 * 0.8 * 300,000 * 500,000
          = 1.6 * 150,000,000,000
          = 240,000,000,000
- Portfolio one‑day variance = 90,000,000,000 + 250,000,000,000 + 240,000,000,000 = 580,000,000,000.
- One‑day portfolio standard deviation (VaR magnitude) = sqrt(580,000,000,000) ≈ 761,577.3.
- Scale to 10 days: VaR_10day = 761,577.3 * sqrt(10) = 761,577.3 * 3.16227766 ≈ 2,408,317.7.

3) Intermediate steps shown above for each arithmetic operation.

4) Why this answer is correct
The variance–covariance formula properly combines individual risks and the positive correlation (0.8) increases portfolio variance via the covariance term. Time scaling by sqrt(10) converts the one‑day VaR to a 10‑day horizon under the usual iid assumption. The computed value rounds to -$2,408,318 (negative sign denotes potential loss).

5) Why other options are incorrect
- Option d (-$761,577) is the one‑day portfolio VaR, not 10‑day.
- Other numeric choices reflect incorrect combinations (e.g., ignoring covariance, incorrect scaling, or arithmetic errors) and therefore do not match the correct variance‑covariance and time‑scaling result.

6) Conclusion
The 10‑day VaR is -$2,408,318, which matches option a.

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