the corner grocer has a 7-year, 6 percent annual coupon bond outstanding with a $1,000 par value. the bond has a yield to maturity of 5.5 percent. Which one of the following statements is correct if the market yield suddenly increases to 6.5 percent?
The bond price will increase by $57.14.
The bond price will increase by 5.29 percent.
The bond price will decrease by $53.62.
The bond price will decrease by 5.43 percent.
The bond price will decrease by 5.06 percent.
To determine the impact of a change in market yield on the bond price, you need to understand the inverse relationship between bond prices and yields. As market yields increase, bond prices generally decrease, and vice versa.
To calculate the bond's price, you can use the present value formula, which consists of calculating the present value of each future cash flow (coupon payments and the face value).
In this case, the bond has a 7-year maturity, with an annual coupon rate of 6 percent and a $1,000 par value. The yield to maturity (YTM) is given as 5.5 percent. Now, let's calculate the bond price using these parameters.
1. Calculate the present value of each coupon payment:
- The bond has an annual coupon payment of 6% * $1,000 = $60.
- Since the coupon payments are received annually for 7 years, calculate the present value of each cash flow using the YTM of 5.5 percent:
Year 1: $60 / (1 + 0.055)^1 = $56.82
Year 2: $60 / (1 + 0.055)^2 = $53.97
Year 3: $60 / (1 + 0.055)^3 = $51.25
Year 4: $60 / (1 + 0.055)^4 = $48.67
Year 5: $60 / (1 + 0.055)^5 = $46.21
Year 6: $60 / (1 + 0.055)^6 = $43.87
Year 7: $60 / (1 + 0.055)^7 + $1,000 / (1 + 0.055)^7 = $ 44.43 + $799.21 = $843.64
2. Calculate the present value of the face value (the value received at maturity):
- The face value is $1,000.
- Calculate the present value using the YTM:
$1,000 / (1 + 0.055)^7 = $774.99
3. Calculate the bond price by summing the present values of all cash flows:
Bond price = Sum of present values of coupon payments + Present value of face value
Bond price = $56.82 + $53.97 + $51.25 + $48.67 + $46.21 + $43.87 + $843.64 + $774.99 = $1,319.42
Now, let's consider a scenario where the market yield suddenly increases to 6.5 percent. We need to recalculate the bond price using the new yield.
4. Calculate the present value of each coupon payment using the new yield of 6.5 percent:
Year 1: $60 / (1 + 0.065)^1 = $56.50
Year 2: $60 / (1 + 0.065)^2 = $53.23
Year 3: $60 / (1 + 0.065)^3 = $50.20
Year 4: $60 / (1 + 0.065)^4 = $47.41
Year 5: $60 / (1 + 0.065)^5 = $44.85
Year 6: $60 / (1 + 0.065)^6 = $42.51
Year 7: $60 / (1 + 0.065)^7 + $1,000 / (1 + 0.065)^7 = $42.07 + $790.08 = $832.15
5. Calculate the present value of the face value using the new yield:
$1,000 / (1 + 0.065)^7 = $711.78
6. Calculate the new bond price by summing the present values of all cash flows:
Bond price = Sum of present values of coupon payments + Present value of face value
Bond price = $56.50 + $53.23 + $50.20 + $47.41 + $44.85 + $42.51 + $832.15 + $711.78 = $1,338.63
Comparing the old and new bond prices, we can see that the bond price increased from $1,319.42 to $1,338.63 when the market yield increased from 5.5 percent to 6.5 percent. This means that the bond price increased by $1,338.63 - $1,319.42 = $19.21.
Therefore, the correct statement would be "The bond price will increase by $19.21."