Question

A $1,000 bond with a coupon rate of 6.2% paid semiannually has eight years to maturity and a yield to maturity (YTM) of 8.30%. If the YTM increases to 8.70%, what will happen to the price of the bond? A. The price of the bond will fall by $18.93 B. The price of the bond will rise by $15.77 C. The price of the bond will fall by $20.96 D. The price of the bond will not change

Answer 1

Formula for bond price is:

Bond Price = C x [1-{1/ (1+r) ⁿ}]/r + M/(1+r)ⁿ

Where,

M = Face Value= $ 1,000

C = Coupon Amount = Face value x coupon rate

                                    = $ 1,000 x 6.2 %/2 = $ 1,000 x 0.031 = $ 31

r = rate of interest = 8.3 % p.a. or 0.0415 semiannually.

n = no. of periods = 8 x 2 = 16

Bond price = $ 31 x [1-{1/ (1+0.0415)¹⁶}]/0.0415 + $100/ (1+0.0415)¹⁶

                     = $ 31 x [1-{1/ (1.0415)¹⁶}]/0.0415 + $100/ (1.0415)¹⁶

                    = $ 31 x [1-{1/ 1.916675}]/0.0415 + $100/ 1.916675

                    = $ 31 x (1-0.521737)/0.0415 + $100/ 1.916675

                   = $ 31 x 0.478263 /0.0415 + $100/ 1.916675

                   = $ 31 x 11.52441 + $ 521.737

                = $ 357.2567 + $ 521.737

                = $ 878.9937 or $ 878.99

If YTM increases to 8.7 %, bond price will change as:

r = 8.7 % p.a. or 0.087/2 = 0.0435 semiannually

Bond price = $ 31 x [1-{1/ (1+0.0435)¹⁶}]/0.0415 + $100/ (1+0.0435)¹⁶

                     = $ 31 x [1-{1/ (1.0435)¹⁶}]/0.0435 + $100/ (1.0435)¹⁶

                    = $ 31 x [1-{1/ 1.97642}]/0.0415 + $100/ 1.97642

                    = $ 31 x (1 - 0.505965)/0.0415 + $100/ 1.916675

                   = $ 31 x 0.494035/0.0415 + $100/ 1.916675

                   = $ 31 x 0.494035 + $ 505.9653

                = $ 352.0707 + $ 505.9653

                = $ 858.0360 or $ 858.04

Change in bond price = $ 878.9937 - $ 858.0360 = 20.9577 or $ 20.96

Hence option “C. The price of the bond will fall by $20.96” is correct answer.