Question
Bond A is zero-coupon bond paying $100 one year from now. Bond B is a zero-coupon bond paying $100 two years from now. Bond C is a 10% coupon bond that pays $10 one year from now and $10 plus the $100 principal two years from now. The yield to maturity on bond A is 10%, and the price of bond B is $84.18. Assuming annual compounding, what is the yield to maturity on Bond B?
*Make sure to input all percentage answers as numeric values without symbols, and use four decimal places of precision. For example, if the answer is 6%, then enter 0.0600.
*Make sure to input all percentage answers as numeric values without symbols, and use four decimal places of precision. For example, if the answer is 6%, then enter 0.0600.
Answers
GPT-4o mini
To find the yield to maturity (YTM) of Bond B, we will use the formula for the price of a zero-coupon bond. The price of a zero-coupon bond can be expressed as:
\[
P = \frac{F}{(1 + r)^n}
\]
Where:
- \(P\) is the price of the bond,
- \(F\) is the face value of the bond,
- \(r\) is the yield to maturity (YTM),
- \(n\) is the number of years until maturity.
For Bond B:
- \(P = 84.18\)
- \(F = 100\)
- \(n = 2\)
Plugging these values into the formula, we have:
\[
84.18 = \frac{100}{(1 + r)^2}
\]
To isolate \(r\), first multiply both sides by \((1 + r)^2\):
\[
84.18(1 + r)^2 = 100
\]
Next, divide both sides by 84.18:
\[
(1 + r)^2 = \frac{100}{84.18}
\]
Calculating the right-hand side:
\[
(1 + r)^2 \approx 1.1865
\]
Now take the square root of both sides:
\[
1 + r = \sqrt{1.1865}
\]
Calculating the square root gives:
\[
1 + r \approx 1.090
\]
Subtracting 1 from both sides:
\[
r \approx 0.090
\]
Thus, the yield to maturity (YTM) of Bond B is:
\[
r \approx 0.0900
\]
Finally, to express the yield to maturity for Bond B in the required format:
\[
\boxed{0.0900}
\]
\[
P = \frac{F}{(1 + r)^n}
\]
Where:
- \(P\) is the price of the bond,
- \(F\) is the face value of the bond,
- \(r\) is the yield to maturity (YTM),
- \(n\) is the number of years until maturity.
For Bond B:
- \(P = 84.18\)
- \(F = 100\)
- \(n = 2\)
Plugging these values into the formula, we have:
\[
84.18 = \frac{100}{(1 + r)^2}
\]
To isolate \(r\), first multiply both sides by \((1 + r)^2\):
\[
84.18(1 + r)^2 = 100
\]
Next, divide both sides by 84.18:
\[
(1 + r)^2 = \frac{100}{84.18}
\]
Calculating the right-hand side:
\[
(1 + r)^2 \approx 1.1865
\]
Now take the square root of both sides:
\[
1 + r = \sqrt{1.1865}
\]
Calculating the square root gives:
\[
1 + r \approx 1.090
\]
Subtracting 1 from both sides:
\[
r \approx 0.090
\]
Thus, the yield to maturity (YTM) of Bond B is:
\[
r \approx 0.0900
\]
Finally, to express the yield to maturity for Bond B in the required format:
\[
\boxed{0.0900}
\]