Corporate Bond Dollar Duration Calculation

Explanation

To calculate the dollar duration of a bond position, we need to use the following formula:

Dollar Duration = Macaulay Duration * Portfolio Value * (1 + Yield)

In this case, we are given the following information:

Portfolio Value: $12 million Percentage change in bond price for a 50-basis point increase in rates: 4%

First, let's calculate the Macaulay Duration. The Macaulay Duration is a measure of the weighted average time until the cash flows from the bond are received. It is calculated as the present value of each cash flow multiplied by the time of receipt, divided by the bond's current price.

Since we are not given any specific cash flow schedule, we'll assume the bond pays a single coupon payment annually and the principal at maturity.

Let's assume the bond has a par value of $1,000 and a coupon rate of 5% (which means it pays $50 as a coupon each year). We'll also assume a yield to maturity of Y%.

Using the formula for the present value of a bond's cash flows:

Price = (Coupon Payment / (1 + Y)^1) + (Coupon Payment / (1 + Y)^2) + ... + (Coupon Payment / (1 + Y)^n) + (Principal / (1 + Y)^n)

where n is the number of periods until maturity.

Since the bond pays a single coupon payment each year and the principal at maturity, the price formula simplifies to:

Price = Coupon Payment / (1 + Y) + Coupon Payment / (1 + Y)^2 + ... + Coupon Payment / (1 + Y)^(n-1) + (Coupon Payment + Principal) / (1 + Y)^n

Using the given information, we know that a 50-basis point increase in rates (0.50%) causes a 4% decline in the bond price. Therefore, we can say:

0.04 = 0.005 / (1 + 0.005Y) + 0.005 / (1 + 0.005Y)^2 + ... + 0.005 / (1 + 0.005Y)^(n-1) + (0.005 + 1) / (1 + 0.005Y)^n

Simplifying the equation, we have:

0.04 = 0.005 * [1 / (1 + 0.005Y) + 1 / (1 + 0.005Y)^2 + ... + 1 / (1 + 0.005Y)^(n-1)] + 1 / (1 + 0.005Y)^n

Next, we need to solve this equation to find the yield to maturity (Y). This requires using numerical methods or financial calculators/software to approximate the yield. Once we find the yield, we can calculate the Macaulay Duration using the cash flow schedule.

Finally, we can use the calculated Macaulay Duration, the portfolio value ($12 million), and the yield (found earlier) to calculate the dollar duration using the formula mentioned earlier:

Dollar Duration = Macaulay Duration * Portfolio Value * (1 + Yield)

The answer will be the closest option provided: A. $96,000, B. $480,000, or C. $960,000.

Please note that the calculation requires additional information, such as the bond's cash flow schedule and the yield to maturity. Without this information, we cannot provide an exact answer.