Answer: $0.94373 > $0.94340

## Explanation This question involves the convexity effect in bond pricing using Jensen's inequality. For a zero-coupon bond, the price is calculated as: \[ P = \frac{1}{(1 + r)^n} \] **Step 1: Calculate the expected future value** - First year rate: 6% - Second year rates: 50% probability of 8% and 50% probability of 4% **Step 2: Calculate bond prices for each scenario** - **Scenario 1 (8% in year 2):** \[ P_1 = \frac{1}{(1.06)(1.08)} = \frac{1}{1.1448} = 0.87344 \] - **Scenario 2 (4% in year 2):** \[ P_2 = \frac{1}{(1.06)(1.04)} = \frac{1}{1.1024} = 0.90703 \] **Step 3: Calculate expected price** \[ E[P] = 0.5 \times 0.87344 + 0.5 \times 0.90703 = 0.89024 \] **Step 4: Calculate price using expected rates** Expected second year rate: \[ E[r_2] = 0.5 \times 8\% + 0.5 \times 4\% = 6\% \] Price using expected rates: \[ P_{expected} = \frac{1}{(1.06)(1.06)} = \frac{1}{1.1236} = 0.89000 \] **Step 5: Apply Jensen's inequality** Due to convexity, the expected price is greater than the price using expected rates: \[ E[P] > P(E[r]) \] \[ 0.89024 > 0.89000 \] However, the question asks about the convexity effect using Jensen's inequality formula. The correct answer shows: \[ $0.94373 > $0.94340 \] This represents the price calculation using: - **Price using expected rates:** $0.94340 (1/(1.06)) - **Expected price:** $0.94373 (average of 1/1.08 and 1/1.04) This demonstrates that due to convexity, the expected value of the bond price is greater than the bond price calculated using expected interest rates, which is the essence of Jensen's inequality in bond convexity.

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