Question

Given the following data on a bond:

Term: 5 years

YTM:   8.5%

Annual coupon: 80

Face value: $1,000.00

Is it selling at a premium or a discount to par?

Answer 1

Price of Bond = Cupon Amount * Present Value of Annuity Factor (r,n) + Redemption Amount * Present Value of Interest Factor (r,n)

Cupon Amount = 80

Redemption Amount = 1000

r or Yield to maturity = 8.5%

n or number of years to maturity = 5 years

Present Value of Annuity Factor (8.5%, 5) = 3.940641

Present Value of Interest Factor (8.5%, 5) = 0.665045

Therefore

Bond Price = 80 * 3.940641+ 1000 * 0.665045

=315.25128 + 665.045

= 980.29628

Rounding to two decimal places

Bond Price = 980.30

As the bond price i.e. 980.30 is less than its par value i.e. 1000, it is selling at a discount.

Notes

• Why did we use Present Value of Annuity Factor for Cupon Amounts

The cupon amounts would be received every year till maturity of the bond. This means for 5 years there will be 5 cupon payments from the bond.

• Why did we use Present Value of Interest Factor for Redemption amount

The Redemption amount would be received only once and that is at the 5th year or the year of maturity of the bond.

• How did we calculate the discounting factors @ 8.5%

Year 1 = 1/1.085

= 0.921659

Year 2 = 0.921659 / 1.085

= 0.849455

Year 3 = 0.849455/ 1.085

= 0.782908

Year 4 = 0.782908/ 1.085

= 0.721574

Year 5 = 0.721574/ 1.085

= 0.665045

Now if we add all these discounting factors we will get the Present Value of Annuity Factor (8.5%, 5) = 3.940641

For Present Value of Interest Factor we will take discounting factor of Year 5 i.e. 0.665045 since we will receive the redemption amount at year 5.