Question

Determine whether the following statement is true for false:
Given two bonds with the same price, face value, expiration date and yield, their coupons
payments must be identical.
Justify your answer by providing either a proof if true, or a counterexample if false

Answer 1

The value of an assets/security is the present/discounted value of all future cash flows (returns) associated with its over the relevant/specified period.

V = A1/(1+k)^1 + A2/(1+k)^2 + ..............+An/(1+k)^n

Where V = Value of the assets/security at time zero

At = Cash flow streams expected at the end of year t

k = appropriate required/capitalization/discount rate

so from the above formula there are three variables required to determine the price of a bond.

1. Face value of bond

2. discounted rate

3. coupon rate

Suppose A bond currently selling for $ 10800 assuming (1) coupon rate of interest 10% (2) par value $ 10000 (3

) Yield to maturity, 10 years (4) Annual Interest payment

Yield to maturity = ?

YTM = 1000*PVIFA(9%,10 years) + 10000*PVIF(9%,10 years)

= 1000*6.418 + 10000*0.422

= 6418 + 4220

= 10638

we try a lower rate of discount as value come 10638 which is lower than 10800

using 8%

1000*6.710 + 10000*0.463

= 6710 + 4630

= 11340

now by extrapolation

Difference between the bond values at 8 and 9 percent = 11340 - 10638 = 702

Difference between the desired value and the value with the lower = 11340 - 10800 = 540

now 540/702 = 0.77

now the YTM = 8.77 or can be calculated by MS Excel also by YTM formula

now Come to main question

if YTM of second bond is same, Market price, face value is also same and Maturity period is same then the coupon rate would also be the same as

10800 = x*PVIFA(8.77%,10) + x*PVIF(8.77%,10)

10800 = x*6.483 + 10000*0.431

10800 = 6.9146x + 4314.284

6.9146x = 10800 - 4314.284

6.9146x = 6485.716

x = 6485.716/6.9146

= 1000.4

= 1000 (round off error excused)

x = 1000

Coupon amount = $ 1000 now the face value of bond = 10000

Coupon rate = 1000/10000

= 10% which is equal to first bond.

hence proved their coupon rate must be identical.

Please check with your answer and let me know.