Answer: CNY 3.03 million

The question tests your understanding of how to calculate the **standard deviation (unexpected loss)** of credit losses for a portfolio of loans under the assumption of a binomial loss distribution for each individual loan. This is a core topic in the FRM Part 1 Valuation and Risk Models reading on measuring credit risk. ### Key Concepts For a single loan, the loss is either: - 0 (with probability 1 – PD), or - Exposure × (1 – Recovery rate) = LGD amount (with probability PD). This is a **Bernoulli (binomial with n=1)** random variable. The standard deviation of loss for loan *i* (denoted σᵢ) is therefore: **σᵢ = [Lᵢ × (1 – Rᵢ)] × √[PDᵢ × (1 – PDᵢ)]** Where: - Lᵢ = amount borrowed (exposure), - Rᵢ = recovery rate, - PDᵢ = probability of default. For a portfolio, the **variance of portfolio losses** accounts for the individual variances plus the covariances driven by default correlations (ρᵢⱼ): **σₚ² = Σ Σ ρᵢⱼ × σᵢ × σⱼ** (for all pairs i, j; when i = j, ρ = 1) The portfolio standard deviation is then **σₚ = √(σₚ²)**. Here, both loans have the same PD = 2% (0.02), so √[PD(1–PD)] = √[0.02 × 0.98] = √0.0196 ≈ 0.14. ### Step-by-Step Calculation (Correct Approach) 1. **Loss given default (LGD) amount for each loan**: - Loan 1: 15 million × (1 – 0.40) = 15 × 0.60 = **9 million** - Loan 2: 20 million × (1 – 0.25) = 20 × 0.75 = **15 million** 2. **Individual standard deviations (σ₁ and σ₂)**: - σ₁ = 9 million × √(0.02 × 0.98) = 9 × 0.14 = **1.26 million** - σ₂ = 15 million × √(0.02 × 0.98) = 15 × 0.14 = **2.10 million** 3. **Portfolio variance**: - σₚ² = σ₁² + σ₂ - + 2 × ρ × σ₁ × σ₂ - = (1.26)² + (2.10)² + 2 × 0.6 × 1.26 × 2.10 - = 1.5876 + 4.41 + 3.1752 = **9.1728** 4. **Portfolio standard deviation**: - σₚ = √9.1728 ≈ **3.0287 million** (rounds to **3.03 million**) ### Explanation for Each Option **A: CNY 1.38 million** — Incorrect. This likely comes from an error in the individual σ calculation, such as mistakenly using √[PD × (1–PD)] multiplied by something other than the full LGD amount (e.g., using exposure without proper LGD adjustment or an incorrect variance formula like √[PD(1–PD) × LGD²] without the correct scaling). It understates the risk significantly and ignores proper portfolio aggregation. **B: CNY 1.59 million** — Incorrect. This appears to be an intermediate miscalculation, possibly ρ × σ₁ × σ₂ (0.6 × 1.26 × 2.1 ≈ 1.5876, rounded up) or a partial covariance term treated as the full portfolio SD. It ignores the individual variances (σ₁² and σ₂²) and the full double-counting of covariance in the portfolio variance formula. **C: CNY 3.03 million** — **Correct**. This matches the full calculation shown above: proper individual σᵢ using LGD amounts, then full portfolio variance including variances + twice the covariance term, followed by the square root. It correctly applies the binomial assumption and the two-asset portfolio volatility formula with the given default correlation of 0.6. **D: CNY 3.36 million** — Incorrect. This is a common error of simply **adding** the individual standard deviations (1.26 + 2.10 = 3.36) without any diversification adjustment. It ignores correlation entirely and treats the risks as perfectly additive (which would only be true if ρ = +1 and you were adding absolute losses, not standard deviations properly). In reality, correlation of 0.6 provides some (but not complete) diversification benefit, so the portfolio SD is less than the simple sum. ### Reference Answer The correct answer is **C: CNY 3.03 million**. This question emphasizes that while expected loss is simply additive (ELₚ = EL₁ + EL₂), unexpected loss (standard deviation) depends on correlations and requires the full variance-covariance approach. Mastering the formula for σᵢ and the portfolio variance summation is essential for FRM Part 1 credit risk questions involving small portfolios. Practice varying the correlation (e.g., ρ=0 vs. ρ=1) to see the diversification impact clearly.