Explanation

Convexity measures the curvature of the price-yield relationship of a bond. The approximate convexity formula is:

$\text{Convexity} \approx \frac{P_+ + P_- - 2P_0}{P_0 \times (\Delta y)^2}$

Where:

• $P_+$ = price when yield increases by $\Delta y$
• $P_-$ = price when yield decreases by $\Delta y$
• $P_0$ = initial price
• $\Delta y$ = change in yield (in decimal form)

From the table:

• Initial yield = 1.50% with price = 97.0 (this is $P_0$)
• When yield decreases to 1.35% (Δy = -0.15%), price = 97.5 (this is $P_-$)
• When yield increases to 1.65% (Δy = +0.15%), price = 96.8 (this is $P_+$)

Note: The table shows 1.35% twice with price 97.5, but we only need one instance.

Step 1: Calculate the numerator $P_+ + P_- - 2P_0 = 96.8 + 97.5 - 2 \times 97.0 = 194.3 - 194.0 = 0.3$

Step 2: Calculate the denominator $P_0 \times (\Delta y)^2 = 97.0 \times (0.0015)^2 = 97.0 \times 0.00000225 = 0.00021825$

Step 3: Calculate convexity $\text{Convexity} \approx \frac{0.3}{0.00021825} \approx 1,374.57$

Step 4: Adjust for convention In bond convexity calculations, the result is often divided by 100 to express it in a more manageable form. However, looking at the options:

• Option A: 2 (too small)
• Option B: 687 (half of our calculation)
• Option C: 1.375 (our calculation divided by 1000)

Actually, the standard convexity formula gives a large number. If we use the formula: $\text{Convexity} = \frac{P_+ + P_- - 2P_0}{P_0 \times (\Delta y)^2}$ With Δy = 0.0015 (0.15% in decimal): $\text{Convexity} = \frac{0.3}{97.0 \times (0.0015)^2} = \frac{0.3}{97.0 \times 0.00000225} = \frac{0.3}{0.00021825} \approx 1,374.57$

This is approximately 1,375, which is close to option C (1.375) if we consider it might be expressed differently. However, 1,374.57 is much closer to 1,375 than to 687.

Important note: There might be a scaling factor. Sometimes convexity is divided by 100. If we divide 1,374.57 by 100, we get 13.7457, which doesn't match any option. If we divide by 2, we get 687.285, which matches option B.

Given that:

1. The calculation yields approximately 1,375
2. Option C is 1.375 (which is exactly 1,375 divided by 1,000)
3. Option B is 687 (which is approximately half of 1,375)

The most likely correct answer is B. 687. because convexity is often scaled or there might be a different convention being used. The exact calculation gives 1,374.57, and 687 is approximately half of that, which could result from using a different formula or scaling factor.

Final Answer: B