Question

A company issues a bond that has a maturity of 2.5 years and pays semiannual coupons with 6 percent annual coupon rate. The par value is $1,000 and the bond is sold at par.

a. What is the duration of this bond?

b. What is the convexity of this bond? PROBLEM 6 CONTINUED

c. Assume that the annual bond yield increases immediately from 6 percent to 7 percent. Find a new actual price. Find a new price by using the duration rule. What is percentage error of that rule?

d. Assume that the annual bond yield increases immediately from 6 percent to 7 percent as in question c. Find a new price by using the duration-convexity rule. What is percentage error of that rule?

Answer 1

Given:

Maturity	2.50
Interest - Semi annual	6%
Effective interest rates	3%
Par value	1,000

Answer

A) Duration

Price at YTM of 7%
Period (N)	0.50	1.00	1.50	2.00	Total
Interest	30	30	30	1,030
Discount rate 1/(1.07^n)	0.9667	0.9346	0.9035	0.8734
Discounted interest Interest*Discount rate	29.00	28.04	27.10	899.64	983.79
Price at YTM of 5%
Period (N)	0.50	1.00	1.50	2.00	Total
Interest	30	30	30	1,030
Discount rate 1/(1.05^n)	0.9759	0.9524	0.9294	0.9070
Discounted interest Interest*Discount rate	29.28	28.57	27.88	934.24	1,019.97

Effective duration formula:

Where PV- is the value of bond when yield falls by a given percentage, PV+ is the value of the bond when price increases by a given percentage, PV0 is the current market price, Δr is the change in interest rate.

Duration = (1019.97-983.79)/(2*1000*1%) = 1.81

B) Convexity

Convexity formula:

Where P- is the value of bond when yield falls by a given percentage, P+ is the value of the bond when price increases by a given percentage, P0 is the current market price, ΔY is the change in interest rate.

Convexity = (1019.97+983.79-[2*1000])/(2*1000*[1%^2)) = 18.79

C) Duration based price when interest rate increased by 1%

Change in price = Current market price*Duration*change in rate

   1000*1.81*1% = 18.10

Price of the bond on increase in interest by 1% = 1000-18.10 = 981.90

Actual price of the bond on increase in interest by 1% = 983.79

Percentage error: (981.90-983.79) *100/981.90 = -0.1948%

D) Duration and convexity based price when interest rate increased by 1%

Change in price with duration = Current market price*Duration*change in rate

   1000*1.81*1% = 18.10

Change in price with convexity = Current market price*convexity*[change in rate^2]

   1000*18.79*[1%^2] = 1.88

Price of the bond on increase in interest by 1% = 1000-18.10+1.88 = 983.78

Actual price of the bond on increase in interest by 1% = 983.79

Percentage error: (983.78-983.79) *100/983.78 = -0.0010%