Question

1. Derive the formula for pricing a coupon bond that pays a coupon at the end of each period. What will the formula be if the coupons are paid at the beginning of each period (i.e. you receive a coupon immediately after you acquired the bond and only receive the face value when it matures).

Answer 1

The price of a bond is the PV of the expected cash flows from the

bond if it is held till maturity, the discount rate being the market

rate of interest (market rate may be different from the coupon rate)

The expected cash flows from the bond are:

1) The maturity value of the bond, which may be equal to the face

value of the bond, and the

2) The periodic interest payment stream which constitutes an

annuity. The annuity may be an ordinary annuity or an annuity due.

The formulae for finding out the price of bond would then be:

the sum of:

1) PV of the MV given by MV/(1+r)^n and

2) PV of the periodic interest payments given by:

*In the case where the interest payments are made

at the end of each period (ordinary annuity):

= I*((1+r)^n-1))/((r*(1+r)^n)). [This is the formula for finding

the PV of an ordinary annuity]

*In the case where the interest payments are made

at the end of beginning of each period (annuity due):

= I*r*((1+r)^n-1))/((r*(1+r)^n)).

[This is the formula for finding PV of an annuity due]

Where,

MV = market value

n = number of periods (can be year or part of the year)

r = market rate of interest (per year or part of the year)

I = interest in $ for a period, which is face value *

coupon rate for a year or part of a year as the case may

Hence, we may write the formula for the price of a bond

as:

1) Where the coupons are paid at the end of each

period,

Price of the bond = MV/(1+r)^n+= I*((1+r)^n-1))/((r*(1+r)^n))

2) Where the coupons are paid at the beginning of each

period,

Price of the bond = MV/(1+r)^n+= I*r*((1+r)^n-1))/((r*(1+r)^n))