To calculate the modified convexity of the bond, follow these steps:

Formula for Modified Convexity:

$\text{Modified Convexity} = \frac{\sum \left( \frac{t(t+1) \cdot PV}{(1+y)^2} \right)}{P \cdot (1+y)^2}$ Where:

• $t$: Time period in years
• $y$: Annual effective interest rate
• $PV$: Present Value of the cash flows
• $P$: Price of the bond (sum of present values of all cash flows)

Step 1: Identify Bond Cash Flows

• Coupons: $C = 300$ paid annually for 3 years
• Face Value: $F = 5000$, paid at maturity (year 3)

Step 2: Calculate Present Values of Cash Flows

Using the formula: $PV_t = \frac{\text{Cash Flow at Time } t}{(1+y)^t}$ Where $y = 7.75\% = 0.0775$.

For $t = 1, 2, 3$:

• Year 1: $PV_1 = \frac{300}{(1 + 0.0775)^1} = \frac{300}{1.0775} = 278.34$
• Year 2: $PV_2 = \frac{300}{(1 + 0.0775)^2} = \frac{300}{1.1610} = 258.39$
• Year 3 (coupon + face value): $PV_3 = \frac{5300}{(1 + 0.0775)^3} = \frac{5300}{1.2494} = 4241.63$

Total Bond Price $P$: $P = PV_1 + PV_2 + PV_3 = 278.34 + 258.39 + 4241.63 = 4778.36$

Step 3: Calculate Time-Weighted Present Value

For each time $t$, calculate $t(t+1) \cdot PV_t$:

• Year 1: $1(1+1) \cdot PV_1 = 2 \cdot 278.34 = 556.68$
• Year 2: $2(2+1) \cdot PV_2 = 6 \cdot 258.39 = 1550.34$
• Year 3: $3(3+1) \cdot PV_3 = 12 \cdot 4241.63 = 50899.56$

Sum of Weighted PVs: $\sum \text{Weighted PVs} = 556.68 + 1550.34 + 50899.56 = 52906.58$

Step 4: Apply Convexity Formula

$\text{Modified Convexity} = \frac{\sum \text{Weighted PVs}}{P \cdot (1+y)^2}$ $\text{Modified Convexity} = \frac{52906.58}{4778.36 \cdot (1.0775)^2} = \frac{52906.58}{5533.83} = 9.56$

Final Answer:

$\text{Modified Convexity} = \boxed{9.56}$

Would you like a deeper breakdown or clarification on any steps?

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Related Questions:

1. How is modified convexity different from Macaulay duration?
2. What are the practical applications of bond convexity in finance?
3. How would the modified convexity change with a higher interest rate?
4. Can convexity be negative, and under what circumstances?
5. How does convexity help in managing interest rate risk?

Tip:

Modified convexity is crucial for understanding how the price of a bond reacts to interest rate changes, especially for large fluctuations.