Question

A bond has four years to maturity, an 8% annual coupon and a par value of $100. The bond pays a continuously compounded interest of 5%.

a. What would the actual percentage change in the price of the bond be if the interest rate goes up from 5% to 6%?

b. What would be the percentage change in the price of the bond implied by the duration approximation?

c. What would be the percentage change in the price of the bond implied by the duration plus convexity approximation?

d. Why does adding the convexity term to the approximation improve it?

Please do not use excell tables or excell formulas. Our answers must handwritten. Thank you.

Answer 1

a). For continuous compounding with interest rate y,

Present Value (PV) = CFt*e^(-y*t) where CFt = cash flow at time t.

Using this, bond price is calculated as follows:

Bond price = 110.16

If continuous interest rate is 6% then:

So, percentage change in price = (new price - current price)/current price = (106.27-110.16)/110.16 = -3.53%

b). Bond duration and convexity calculation:

Bond duration = sum of weighted time = 3.599 years

For continuous compounding, modified duration (MD) equals the Macaulay duration so MD = 3.599 years

Change in price = - MD*current price*change in yield = -3.599*110.16*0.01 = -3.96

Percentage change in bond price = change in bond price/current price = -3.96/110.16 = -3.60%

c). Convexity (for continuous compounding) = sum of (PV*n^2)/current price = 1,513.30/110.16 = 13.74

Change in price = ((- MD*change in yield) + ((convexity/2)*change in yield^2))*current price

= ((-3.599*0.01) + ((13.74/2)*0.01^2))*110.16 = -3.89

Percentage change in bond price = change in bond price/current price = -3.89/110.16 = -3.53%

d). Bond price change with respect to change in interest rate is not linear but follows a convex path. Duration is the first approximation of the price to change in interest rate so for larger changes in interest, it does not provide a close approximation as it is linear. Convexity is the second derivative of price with respect to interest rate change so it is a curvature and provides the adjustment to the duration so that the change in price estimate is more accurate.