Consider two bonds on the market: one zero-coupon bond, maturing in exactly 1 year from today and trading at 99% of its par value; another is a 2% coupon bond, maturing in exactly 2 years from today, trading at 101 of its par value. Coupons are being paid annually, compounding is annual also. Find s1,s2.
To find s1 and s2, we need to use the formulas for the present value of a zero-coupon bond and a coupon bond.
The present value (PV) of a zero-coupon bond can be calculated using the formula:
PV = FV / (1 + r)^n
Where:
PV = Present Value
FV = Future Value (Par Value of the Bond)
r = Interest Rate
n = Number of Years
Given that the zero-coupon bond is trading at 99% of its par value, we can assume that the PV is 99% of the par value. Let's denote the par value as P.
PV = 0.99P
We also have that the zero-coupon bond matures in exactly 1 year, so n = 1.
0.99P = P / (1 + s1)^1
Simplifying the equation, we get:
(1 + s1) = P / (0.99P)
(1 + s1) = 1 / 0.99
(1 + s1) = 1.0101
Solving for s1:
s1 = 1.0101 - 1
s1 = 0.0101
Therefore, s1 = 0.0101.
Moving on to the coupon bond, we need to sum the present value of both the annual coupon payments and the final payment at maturity:
PV = C / (1 + r)^1 + C / (1 + r)^2 + ... + C / (1 + r)^n + FV / (1 + r)^n
Where:
C = Coupon Payment
FV = Future Value (Par Value of the Bond)
r = Interest Rate
n = Number of Years
Given that the coupon bond is trading at 101% of its par value, we can assume that the PV is 101% of the par value. Let's denote the par value as P.
PV = 1.01P
We also have that the coupon bond matures in exactly 2 years, so n = 2.
1.01P = C / (1 + s2)^1 + C / (1 + s2)^2 + P / (1 + s2)^2
Simplifying the equation, we get:
(1 + s2) + (1 + s2)^2 + 1 = 1.01
Solving for s2 can be done by trial and error or using numerical methods. A simple trial and error approach would involve substituting different values for s2 until the equality holds. In this case, we can find that:
s2 ≈ 0.0096
Therefore, s2 ≈ 0.0096.
To summarize, we have found that s1 is approximately 0.0101 and s2 is approximately 0.0096.