Question

Consider two bonds, both with 8% coupon rates (assume annual coupon payments) one with 10 years to maturity and the other with 20 years to maturity. Assume that current market rates of interest are 8%. Calculate the difference in the change of the price of the two bonds if interest rates decrease to 6% one year after purchasing the bond. Repeat the procedure assuming that interest rates increase to 10% one year after purchase. Explain the major bond pricing principle that is being illustrated here

Answer 1

	Bond A	Bond B
Lets Par Value (p)	$1000	$1000
Coupon Rate (annual)	8%	8%
Coupon (c)	8%*1000 =$80	8%*1000 =$80
years to maturity (n)	10	20

1. If Interest Rate(r) = 8% per annum

Price of Bond A = (c/r)[1-(1/(1+r)ⁿ)] + p/(1+r)ⁿ

Price of Bond A = (80/0.08)[1-(1/(1.08)¹⁰)] + 1000/(1.08)¹⁰

Price of Bond A = 100*0.5368 + 463.2 = $1000

Price of Bond B = (c/r)[1-(1/(1+r)ⁿ)] + p/(1+r)ⁿ

Price of Bond B = (80/0.08)[1-(1/(1.08)²⁰)] + 1000/(1.08)²⁰

Price of Bond A = 100*0.7854 + 214.6 = $1000

Difference between price of two bonds = 1000-1000 = $0

2. If Interest Rate(r) = 6% per annum

Price of Bond A = (c/r)[1-(1/(1+r)ⁿ)] + p/(1+r)ⁿ

Price of Bond A = (80/0.06)[1-(1/(1.06)¹⁰)] + 1000/(1.06)¹⁰

Price of Bond A = 1333.33*0.4416 + 558.4 = $1147.2

Price of Bond B = (c/r)[1-(1/(1+r)ⁿ)] + p/(1+r)ⁿ

Price of Bond B = (80/0.06)[1-(1/(1.06)²⁰)] + 1000/(1.06)²⁰

Price of Bond A = 1333.33*0.6881 + 311.80 = $1229.26

Difference in Price of two bonds = 1229.26-1147.2 = $82.06

3. If Interest Rate(r) = 10% per annum

Price of Bond A = (c/r)[1-(1/(1+r)ⁿ)] + p/(1+r)ⁿ

Price of Bond A = (80/0.10)[1-(1/(1.10)¹⁰)] + 1000/(1.10)¹⁰

Price of Bond A = 800*0.6144 + 385.54 = $877.06

Price of Bond B = (c/r)[1-(1/(1+r)ⁿ)] + p/(1+r)ⁿ

Price of Bond B = (80/0.10)[1-(1/(1.1)²⁰)] + 1000/(1.1)²⁰

Price of Bond A = 800*0.8513 + 148.64 = $829.68

Difference in price of two bonds = 829.68-877.06 = -$47.38

This shows that

If Coupon Rate > Interest Rate , bond with higher maturity sells at higher price as compared to same bond with less maturity

If Coupon Rate < Interest Rate , bond with higher maturity sells at lower price as compared to same bond with less maturity