The worked solutions are included for this question but I need an explanation for the worked solution.

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The worked solutions are included for this question but I need an explanation for the worked solution.

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Why must the equation of step 2 include the 1/1.0246956^20?

Bond “ABC” is a 10-year, $1,000 par value bond which pays a 6% coupon with quarterly payments during its first five years (you receive $15 a quarter for the first 20 quarters). During the remaining five years the security has an 8% quarterly coupon (you receive $20 a quarter for the second 20 quarters). At the end of 10 years (40 quarters) you will also receive the par value. 

Bond “DEF” is another 10-year bond issued by the same company, and it has a 10% semiannual coupon. This bond is selling at its par value $1,000 and has the same risk as the bond “ABC”. Given this information, what should be the price of the bond “ABC”? 

Worked Solution:

Step 1: Find the periodic interest rate on “ABC”.  

Since the securities are of equal risk and maturity, they must have the same effective annual rate. Since “DEF” is a 10-year bond is selling at par, its nominal yield is 10%, the same as its coupon rate.      DEF’s effective annual rate is (1 + 0.10/2)2 – 1 = 10.25%.   Since “ABC” has quarterly payments, its periodic rate = (1.1025)0.25 – 1 = 2.4695%  

Step 2: Price of ABC Price of “ABC” = PV of Annuity (20 payments of $15) + PV of Annuity (next 19 payments of $20) + PV of (last $20 + $1,000) = 15/0.024695 (1 − 1/1.024695^20 ) + 1/1.024695^20 { 20/0.024695 (1 − 1/1.024695^19 )} + 1020/1.024695^40  

= 234.51 + 184.42 + 384.42 = $ 803.36  

The final answer is $803.36

Why must the equation of step 2 include the 1/1.0246956^20?

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