A European government bond maturing in 5 years has a fixed coupon rate of 2% pa, paid annually.The bond's yield-to-maturity (YTM) is currently 0.5% pa, given as an annualised percentage rate (APR) compounding annually.The face value of the bond is $1,000. Calculate the bond price. All answer options below are rounded to 6 decimal points.Question 2Select one:a.$1,073.887995b.$1,025.251253c.$1,020d.$994.187674e.$929.298107
Question
A European government bond maturing in 5 years has a fixed coupon rate of 2% pa, paid annually.The bond's yield-to-maturity (YTM) is currently 0.5% pa, given as an annualised percentage rate (APR) compounding annually.The face value of the bond is 1,073.887995b.1,020d.929.298107
Solution
To calculate the price of the bond, we need to calculate the present value of the future cash flows, which are the annual coupon payments and the face value at maturity.
Step 1: Calculate the present value of the coupon payments.
The bond pays an annual coupon of 2% on a face value of 20.
As these are annual payments, we discount them at the yield-to-maturity rate of 0.5% to get their present value.
The formula for the present value of an annuity is:
PV = C * [(1 - (1 + r)^-n) / r]
where: C = annual coupon payment = $20 r = yield-to-maturity = 0.5% = 0.005 (in decimal form) n = number of years to maturity = 5
Substituting these values into the formula, we get:
PV = 20 * [(1 - (1 + 0.005)^-5) / 0.005] = $99.502487
Step 2: Calculate the present value of the face value.
The face value of the bond is $1,000, which will be returned to the bondholder at maturity.
We discount this at the yield-to-maturity rate to get its present value.
The formula for the present value of a single sum is:
PV = FV / (1 + r)^n
where: FV = face value = $1,000 r = yield-to-maturity = 0.5% = 0.005 (in decimal form) n = number of years to maturity = 5
Substituting these values into the formula, we get:
PV = 1000 / (1 + 0.005)^5 = $974.385508
Step 3: Add the present values calculated in steps 1 and 2 to get the bond price.
Bond price = PV of coupon payments + PV of face value = 974.385508 = $1,073.887995
So, the bond price is approximately $1,073.887995, which corresponds to option a.
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