Question

Q21. Consider a 2-year bond. The coupon rate of the bond is 10%, and the bond pays coupons semiannually. The bond is selling at a yield to maturity of 8.0% annually, or 4.0% semi-annually.

(a) What is the duration of the bond measured in half-years and in years? (10 points)

(b) If the semi-annually yield changes from 4.0% to 5.0%, what is the predicted change in the price of the bond (a dollar amount) using duration? (7 points)

(c) Suppose you are the company that is issuing this coupon bond. To immunize your liability, you would like to invest in a portfolio consists of one-year zero-coupon bonds and three-year zero-coupon bonds. What are the durations of the one-year zero and the three-year zero, respectively? What weight of the one-year zero will you need to hold for immunization? (5 points)

Answer 1

a)

Here i am sharing the excel working and formulaes of getting bond duration.

Manual Formulaes:

where D is the Duration

The whole manual working is done in excel please look into it and the formulaes below.

Excel Formulaes:

B) Now the relation bewteen price and bond duration is given below.

Where P is the price and y is the YTM and D is the Duration.

So.

Change in Y is 1%

Also the Price of bond today is 103.63 as calculated in excel table.

y is the 4%

D is the 1.86

so

Change inP/103.63 = - 1.86 [ 1.05-1.04]/1.04

Change in P /103.63 = -0.018

Change in p = -1.85

so expected Price will be = 103.63-1.85 = 101.78 approx.

c) For immunisation we have to understand that it means to invest in the portfolio in such a way that the Duration of asset class and their weights mulplity will be equal to the duration of our liability that is the the issue of bond of 2 years which has duration of 1.86

our liability will arise in 2 years with duration 1.86 so we have to invest in 1 year zero coupon and 3 year zero coupon in such a way that

w x 1 + (1-w) x 3 = 1.86 where w is the weight in asset and the duration of zero coupon bond is always their maturity period as there is no cash flow in between.

So

w + 3 - 3w = 1.86

or 3-1.86=2w

or 1.14 = 2w

or w= 0.57

so we should invest 57% in 1 year and 43% in 3 year zero coupon bond.

thaknks

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