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?

Answer 1

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?

Ans.

Given,

Maturity period (M) = 4 years

Annual coupons=8 %

Par value=$100

Compounded interest rate =5% annum

Formula,

Macauley Duration=( \sumt=1. t*C/(1+y)^t - n*M/(1+y)^n

C= Periodic coupon Payment

Y= Periodic yield

M= the bond Maturity value

n= Duration of bond in periods

Modified Duration = Macauley Duration/ (1+ y+M/n)

y= yield of Maturity

n = no. Of coupon period / year

The Macauley Duration is calculated to be 7.37 years( (1*80)/(1+0.08)+(2*80)/(1+0.08)^2+(3*80)/(1+0.08)^3+(4*80)/(1+0.08)^4+(4*100)/(1+0.08)^4/80*(1-(1+0.08)^-4)/0.08+100/1+0.08^4)

A. Modified Duration for this bond with a maturity of 5% for one coupon period is 7.01 years (7.37/(1+0.05)/1) Therefore,

If the yield of the maturity of 5% to 6% the Duration of the bond will decrease by 0.36 (7.37-7.01)

The formula to calculate the percentage change in the price of the bond is the change in yield multiplied by 100% . This resulting % change in the bond for an interest rate increase from 5 % to 6%. Is to be calculated 7.01% (0.01* -7.01*100%)

Therefore if the interest rate increases 1 % the price of the bond is expected to drop by 7.01%

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