Question

Please post with mathematical formulas, not an excel sheet

5. You have decided to invest in two bonds. Bond X is an n-year bond with semiannual coupons, while bond Y is zero-coupon bond, which is re- deemable in years. The desired yield rate is the same for both bonds. You also have the following information: Bond X • Par value is 1000. • The ratio of the semi-annual bond rate to the desired semi-annual yield rate, that is is 1.03125. • The present value of the redemption amount is 381.50. Bond Y • Redemption value is the same as that of bond X. • Price to yield is 647.80. What is the price of bond X?

Answer 1

For Bond X,

As the present value of the Redemption amount (R) is 381.5,, we get the following equation

381.5 = R / (1+i)ⁿ, ( eq. 1) where i is the yield of Bond X as well as that of bond Y

For Bond Y, the redemption amount is the same i.e. R , so we get the equation as

647.80. = R/(1+i)^(n/2) , (eq. 2)

Dividing equation 1 and 2, we get

381.5/647.8 = 1/ (1+i)^(n/2)   (eq. 3)

Substituting the above in equation 2, we get

647.8 = R* 381.5/647.8 which gives

R = 647.8*647.8/381.5 = 1099. 986 or apx. 1100

Taking R as 1100 , we now get the price(P) for Bond X as

P= present value of Cash flows of Bond X  

P = 1000* (r/2)/ (1+i)^0.5 + 1000*(r/2)/ (1+i)¹+....+1000 * (r/2) /(1+i)ⁿ+ 1100/(1+i)ⁿ

where r= coupon rate of Bond X,

Here, there are 2n terms representing present value of 2n semiannual coupon payments and last term is present value of Redemption amount

By applying GP formula

P = 1000 *(r/2) * (1- (1+i)^-n) / ((1+i)^(1/2) -1) + 1100 * (1+i)^-n

From equation 3, we know. (1+i)^-(n/2) = 381.5/647.8 = 0.588916

So, (1+i)^-n = 0.588916²= 0.346822

And Using the Binomial approximation ,

(1+i)^(1/2)= 1+i/2+.... very small terms = 1+i/2

So, (1+i)^(1/2)-1 = i/2

Substituting in the price equation we get

P = 1000 *(r/2) * (1-0.346822) / ( i/2) + 1100 * 0.346822

= 1000 * (r/i) * 0.653178 + 381.504

= 653.178 * 1.03125 + 381.504

= 1055.09 or appx. 1055

So, the price of the bond is 1055