Question

Consider three bonds with maturities of 3, 6, and 9 years. All three bonds have a coupon rate of 7% and have face values of $1,000. Assume semiannual coupon payments. Use this information to answer the following questions:

a) What would be the market price of each bond if their YTM was 5%?

b) What would be the market price of each bond if their YTM was 9%?

c) Graph the relationship between bond prices and the yields-to-maturity for the three bonds. What conclusions can you draw regarding the relationship between time to maturity and the sensitivity of bond prices to changes in interest rates?

Answer 1

a) 3 year bond when YTM or r = 5%

.

Price = Sum of Discounted coupons +7 (1 + Frequency)" x frequency 1-(1+ Frequency) Price = Coupon x - frequency - + 1000 (1 + 0,05)3x2 0.07 Price = ( 1-(1 + 0.05 ) -3x2 - x 1000) X 1000 (1 + 0.05 )3x2 Price = 1,055.08

6 year bond when YTM or r = 5%

Price = Sum of Discounted coupons +7 (1 + Frequency)" x frequency )-nx frequency 1- (1+ Price = Coupon x - - + 1000 (1 + 0,05 )6x2 frequency 0.07 Price = ( 1-(1 + 0,05) -6x2 - x 1000) X + 1000 (1 + 0,05 )6x2 1 Price = 1, 102.58

9 year bond when YTM or r = 5%

FV Price = Sum of Discounted coupons +7 (1 + Frequency)" x frequency 1- (1+ Price = Coupon x - )-nx frequency 1000 *(1 + 0,05)9x2 frequency 0.07 Price = ( 1-(1 + 0.05 ) -9x2 - x 1000) X 1000 (1 + 0.05)9x2 Price = 1, 143.53

b) 3 year bond when YTM or r = 9%

.

FV Price = Sum of Discounted coupons +7 (1 + Frequency)" x frequency )-nx frequency 1- (1+ Price = Coupon x - 1000 - + frequency (1 + 0,093x2 0.07 Price = G x 1000) 1-(1 + 0,09) -3x2 = 1000 *(1 + 0.09,3x2 Price = 948.42

6 year bond when YTM or r = 9%

Price = Sum of Discounted coupons +7 (1 + Frequency)" x frequency )-nx frequency 1- (1+ Price = Coupon x - 1000 (1 + 0.09 )6x2 frequency 0.07 Price = G x 1000) 1-(1 + 0,09) -6x2 = 1000 (1 + 0.09) 6x2 Price = 908.81

9 year bond when YTM or r = 9%

FV Price = Sum of Discounted coupons +7 (1 + Frequency)" x frequency 1-(1+ Frequency) Price = Coupon x - frequency - + 1000 (1 + 0,09)9x2 0.07 1-(1 + 0.09 )-9x2 1000 Price = x 1000) X - *(1 + 0.09,9x2 Price = 878.40

c) Data table is

Year	3	6	9
5%	$   1,055.08	$   1,102.58	$   1,143.53
9%	$      948.42	$      908.81	$      878.40

$1,500.00 $1,055.08 $1,102.58 $1,143.53 $1,000.00 Price $948.42 $908.81 $878.40 $500.00 Maturity - 5% - 9%

We can see from this graph that the distance between the blue line and red line goes on increasing as the maturity increases, so price of higher maturity bonds is more sensitive to a change in yield than the low maturity bonds