You have decided to invest in two bonds. Bond X is an n-year bonod with semiannual coupons, while bond Y is zero-coupon bond, which is re- deemable in 2 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 r/i 1.03125.
b. 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?

The price of the bond X is 1055.09.

Explanation:

The price of Bond X can be determined as follows:

Let:

R = Present value of the Redemption amount of X and Y

i = yield of both Bond X  and Bond Y

For Bond X, we have:

381.5 = R / (1+i)n ………………… (1)

For Bond Y, we have:

647.80. = R/(1+i)(n/2) ……………….. (2)

Use equation (2) to divide equation (1), we have:

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

0.588916332201297= 1/ (1+i)(n/2)

(1+i)(n/2) = 1 / 0.588916332201297

(1+i)(n/2) = 1.69803407601573

647.80 = R / 1.69803407601573

R = 647.80 * 1.69803407601573 = 1,099.99, or 1,100

Now, also let:

r = coupon rate of Bond X

P= present value of Cash flows of Bond X  

Therefore, we have:

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

It can be observed that we have 2n terms indicating present value of 2n semiannual coupon payments and last term is present value of Redemption amount.

Applying GP formula, we have:

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

Looking at equation (3), we it can be observed that we can have:

(1+i)-(n/2) = 381.5/647.8 = 0.588916

Therefore, we have:

(1+i)-n = 0.5889162= 0.346822

By employing the Binomial approximation, we have:

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

This indicates that:

(1+i)(1/2)-1 = i/2

If we substitute this into the price equation, we will have

P = 1000 *(r/2) * (1-0.346822) / ( i/2) + 1100 * 0.346822 = 1000 * (r/i) * 0.653178 + 381.504

Since r/i = 1.03125, we have:

P = 653.178 * 1.03125 + 381.504 = 1055.09

Therefore, the price of the bond X is 1055.09.