Question

You are given the following information on a bond.

(a) Par value is $1000.

(b) Redemption value is $1000.

(c) Coupon rate is 12%, convertible semiannually.

(d) The bond is priced to yield 10%, convertible semiannually.

The bond has a term of n years. If the term of the bond is doubled, the price will increase by 50. Calculate the price of the n-year bond.    ANS IS 1100.

Please calculate it with specific math formula steps instead of using fincancial calculator.Thank you!!!!!!!!!i will thumb up if it's correct!!!

Answer 1

Let price of Bond be X ;

Yield to maturity= YTM=10%; semi annualy=5% ; no of terms=2n( since converted semiannuly)

X= 1000/(1+YTM/2)^2n +[60*(1-1/(1+YTM/2)^2n)]/.05 ----Equation 1 ( where; [60*(1-1/(1+YTM/2)^2n)]/.05= present value of all the cashflows due to coupon by annuity method)

If term of bond is doubled ;

X+50 = 1000/(1+YTM/2)^4n

X= 1000/(1+YTM/2)^4n + [60*(1-1/(1+YTM/2)^4n)]/.05 -50----Equation 2

Equating 1 and 2 ;

1000/(1+YTM/2)^2n +[60*(1-1/(1+YTM/2)^2n)]/.05 =1000/(1+YTM/2)^4n + [60*(1-1/(1+YTM/2)^4n)]/.05 -50

let 1/(1+YTM/2)^2n = Y ;

1000Y + [60*(1-Y)]/.05= 1000*Y^2 +[60*(1-Y^2)]/.05 -50

[1000- 60/.05]*Y^2 - [1000- 60/.05]*Y -50 =0

200Y^2 - 200Y -50 =0 ; This is a quadratic equation and can be solved by (-b +/- )/2a formula Solving we get Y=.5;

Substituting Y=.5 in Equation 1

we get; X= 1000*.5 +[60*(1-.5)]/.05

X=1100

PS: Idea is to find the present value of cashflows in both scenarios and equating both; inorder to make to the structure of a quadratic equation put 1/(1-YTM/2)^2n =Y and solve the quadratic equation.

Once we obtain Y value; substitute in present value equation and solve for X