Answer: 5.80%

## Explanation This question involves the Ornstein-Uhlenbeck process for mean reversion in the Gauss+ model. The process is given by: $$dx_t = \kappa(\theta - x_t)dt + \sigma dW_t$$ Where: - $x_t$ = current factor level (3.0%) - $\kappa$ = speed of mean reversion (0.04) - $\theta$ = long-term mean (what we're solving for) - $\sigma$ = volatility (120 bps = 1.20%) - $dW_t$ = Wiener process increment (0.24) - $dx_t$ = change in factor over period (0.40%) - $dt$ = time period (6 months = 0.5 years) Plugging into the formula: $$0.40\% = 0.04(\theta - 3.0\%) \times 0.5 + 1.20\% \times 0.24$$ Solving step by step: 1. Calculate the diffusion term: $1.20\% \times 0.24 = 0.288\%$ 2. Rearrange the equation: $0.40\% - 0.288\% = 0.04(\theta - 3.0\%) \times 0.5$ 3. $0.112\% = 0.02(\theta - 3.0\%)$ 4. $\theta - 3.0\% = \frac{0.112\%}{0.02} = 5.60\%$ 5. $\theta = 3.0\% + 5.60\% = 8.60\%$ Wait, this gives 8.60%, which is option C, but the correct answer should be B (5.80%). Let me recalculate more carefully: Actually, the correct calculation should be: $$dx_t = \kappa(\theta - x_t)dt + \sigma dW_t$$ $$0.0040 = 0.04(\theta - 0.03) \times 0.5 + 0.012 \times 0.24$$ $$0.0040 = 0.02(\theta - 0.03) + 0.00288$$ $$0.0040 - 0.00288 = 0.02(\theta - 0.03)$$ $$0.00112 = 0.02(\theta - 0.03)$$ $$\theta - 0.03 = \frac{0.00112}{0.02} = 0.056$$ $$\theta = 0.03 + 0.056 = 0.086 = 8.60\%$$ This confirms my initial calculation of 8.60%. However, the correct answer is listed as B (5.80%). There might be an error in the question or answer key. Based on the mathematical derivation, the correct answer should be **8.60%** (option C).

Author: LeetQuiz .