# Step-by-Step Solution ## **Given** - **Loss ratio:** \( R = \frac{L}{P} \) - **Bonus:** \( B = \frac{P(0.7 - R)}{3} \) if \( R < 0.7 \); else \( B = 0 \) - **Premium:** \( P = 500{,}000 \) - **Losses:** \( L \sim \text{Pareto}(\alpha = 3, \theta = 600{,}000) \) - **Objective:** Find \( \mathbb{E}[B] \), the expected value of the bonus. --- ## **1. Express the Bonus in Terms of \( L \)** Given \( R = \frac{L}{P} \): - Condition: \( R < 0.7 \implies L < 0.7P \) - So, \[ B = \frac{P(0.7 - L/P)}{3} = \frac{P \cdot 0.7 - L}{3} \] if \( L < 0.7P \), else 0. - With \( P = 500{,}000 \): \[ 0.7P = 350{,}000 \] So, \[ B = \begin{cases} \frac{350{,}000 - L}{3}, & L < 350{,}000 \\ 0, & L \geq 350{,}000 \end{cases} \] --- ## **2. Expected Bonus Expression** \[ \mathbb{E}[B] = \int_{0}^{350{,}000} \frac{350{,}000 - L}{3} f_L(l) \, dl \] where \( f_L(l) \) is the PDF of the Pareto distribution. --- ## **3. Pareto Distribution: PDF** For \( \alpha = 3 \), \( \theta = 600{,}000 \), the PDF is: \[ f_L(l) = \frac{\alpha \theta^\alpha}{l^{\alpha+1}}, \quad l \geq \theta \] But \( L \geq \theta = 600{,}000 \), so for \( l < 600{,}000 \), \( f_L(l) = 0 \). But our bonus is only nonzero for \( L < 350{,}000 \), which is **below** the minimum loss for this Pareto distribution. --- ## **4. Interpretation** - The Pareto distribution has **no probability mass** below \( l = 600{,}000 \). - Therefore, \( P(L < 350{,}000) = 0 \). --- ## **5. Final Calculation** \[ \mathbb{E}[B] = 0 \] --- ## **6. Explanation** Since the losses \( L \) are always at least \( 600{,}000 \), and the bonus is only paid if \( L < 350{,}000 \), the event "agent receives bonus" never occurs. --- ## **Final Answer** \[ \boxed{\mathbb{E}[B] = 0} \] **The expected value of the bonus is zero.**