Final answer:
The current price, Macaulay duration, and modified convexity of a bond are calculated using the coupon payments, face value, and effective yield rate.
Step-by-step explanation:
To determine the current price, Macaulay duration, and modified convexity of a 10-year bond with a face value of $1000, an annual coupon rate of 6%, and an annual effective yield rate (i) of 4%, we need to calculate each component step by step:
- Current Price: The bond pays a monthly coupon, which means the coupon payment is $1000 x 0.06 / 12 = $5 per month. To get the present value of these coupon payments, we would discount each monthly payment back to the present using the annual effective yield rate. Additionally, we would discount the face value of the bond back to the present. The sum of these present values is the current bond price.
- Macaulay Duration: Macaulay duration measures the weighted average time to receive the bond's cash flows. We calculate this by taking the present value of each cash flow multiplied by the time it is received and then dividing by the current price of the bond.
- Modified Convexity: This measures the sensitivity of the bond's duration to changes in yield rates. It is calculated by taking the second derivative of the price function with respect to yield, and then normalizing it by the current bond price.
For the second part of the question, if the annual effective yield rate is raised to 4.8%, we use second order approximation, which involves both the duration and convexity of the bond:
- The price change due to the duration is calculated by multiplying the duration by the change in yield (in decimal form).
- The price change due to the convexity is calculated by multiplying one-half of the modified convexity by the square of the change in yield (in decimal form).
The estimated new price of the bond is the current price adjusted for these two factors.