Compare your answers to Questions 2 and 3. Explain the source of any difference. Which is more

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Question:

Compare your answers to Questions 2 and 3. Explain the source of any difference. Which is more correct?

Question 2:

Compute the duration of this bond and use it to estimate the new value of the bond if rates were to suddenly decline by 0.80%.

To compute the duration of the bond, we can use the formula:

Duration = [C1 x (1 + r)^-1 x t1 + C2 x (1 + r)^-2 x t2 + ... + Cn x (1 + r)^-n x tn + Fn x (1 + r)^-n x n]/[P x r]

where C is the coupon payment, t is the time to the cash flow (in years), F is the face value, P is the price, and r is the yield to maturity.

For this bond, we have:

Coupon payment (C) = $25 (semi-annual payments of $12.50)
Time to cash flow (t) = 0.5 years (semi-annual payments)
Face value (F) = $1000
Price (P) = $963.828
Yield to maturity (r) = 2.622% (computed in previous question)

Using these values, we can calculate the duration of the bond:

Duration = [$12.50 x (1 + 0.02622/2)^-1 x 0.5 + $12.50 x (1 + 0.02622/2)^-2 x 1 + ... + $12.50 x (1 + 0.02622/2)^-20 x 10 + $1000 x (1 + 0.02622/2)^-20 x 10]/[$963.828 x 0.02622/2]

Duration = 7.583 years

To estimate the new value of the bond if rates were to suddenly decline by 0.80%, we can use the modified duration formula:

Change in bond price = - Duration x Change in yield x Bond price

Change in yield = -0.008 (0.80% decrease)

Change in bond price = -7.583 x (-0.008) x $963.828

Change in bond price = $61.90 (rounded)

The new value of the bond would be $963.828 + $61.90 = $1,025.73 (rounded).

Question 3:

Calculate the bond's value directly (using the present value approach) assuming that rates declined 0.80% from the yield to maturity you estimated in the first question.

To calculate the bond's value directly using the present value approach, we need to adjust the discount rate used in the calculation to reflect the change in interest rates. In this case, we assume that rates have declined by 0.80% from the yield to maturity estimated in the first question, which was 2.622%.

Explanation:

Step 1:

Therefore, we will use a discount rate of 1.822% (2.622% - 0.80%) in the calculation.

We can use the same formula we used in the first question to calculate the present value of the bond's cash flows using the new discount rate:

PV = $12.50 x [1 - (1 + 1.822%/2)^(-20)]/(1.822%/2) + $1000/(1+1.822%/2)^20

Solving for PV, we get:

PV = $1024.19.

Step 2:

Therefore, the bond's value using the present value approach with the new discount rate is $1024.19. The increase in bond value from the first question's value of $963.828 to the new value of $1024.19 is due to the decrease in the discount rate. As interest rates decline, the present value of future cash flows increases, resulting in a higher bond price. This concept is known as the inverse relationship between bond prices and interest rates. As interest rates rise, bond prices fall, and vice versa.

Overall, the present value approach is a useful method for valuing bonds because it considers the time value of money and accounts for the present value of all future cash flows. It allows investors to compare the value of different bonds and make informed investment decisions based on their risk tolerance and investment objectives.