Question

Consider two bonds with $1,000 face values that carry coupon rates of 5%, make annual coupon payments, and exhibit similar risk characteristics. The first bond has two years to maturity whereas the second has three years to maturity. The yield to maturity of these investments is 5%. If the yield to maturity rises by half a percentage point, what are the respective percentage price changes of the two bonds? Find the exact answer and approximate answers using the duration rule and the duration-convexity rule. Discuss the quality of the approximation.

Answer 1

First we have to find out the duration of both the bonds

I will try to make it simple in tabular form and will also give the formula also:

where D is the duration

t is the periods or maturity

y is the yield

Bond A	Period or years of cash flow	Cash flow	PV of CF @ 5%	weight = PV of cash flow/bond price	weight x period
	1	50	47.6	0.0476	0.0476
	2	1050	952.35	0.9524	1.9048
total		Bond price	999.95	1.0000	1.9524
Bond B	1	50	47.6	0.04759	0.04759
	2	50	45.35	0.04534	0.09068
	3	1050	907.2	0.90707	2.72121
total		Bond price	1000.15	1.0000	2.85948

So as we have now the duration of both the bonds that is 1.9524 for A and 2.85948 for B we can use the duration formula to find out the respective price change in the bond due to interest or yield change.

where D* = D/(1+y)

So for Bond A = - (1.9524/1.05) x 0.5% = 0.93% change in price

for Bond B = - ( 2.85948/1.05) x 0.5% = 1.362% change in price

These are the formulaes of convexity

you can find out the convexity and then the respective change in the price

The only difference in both the methods is that convexity one is more accurate than the only Duration one formulae

Also when the change in interest rate or yield is small it would matter less. And the relation between interest rate and price is negative so there is negative sign there.

Actually the change in the price can be predicted upto a certain level where the convexity curve and the bond curve lie one above the other.

A large changement may be differ due to other factors also.