An six-year annual-pay coupon bond was issued with a face value of $1000 and a c

Question

An six-year annual-pay coupon bond was issued with a face value of $1000 and a coupon rate of 12%. It is now 1.25 years later and the yield-to-maturity is 9%. (Keep in mind that the cash flows happen 0.75 years, 1.75 years, 2.75 years, etc. from now.)

You may use a computer for computations, but please show the basic algebra.

(a) What is the dirty price" of the bond? (Remember, the dirty price is the current value of the bond. It includes all interest accrued since the last coupon date.)

(b) How much accrued interest does the bond contain? (Convention is to accrue interest at a constant dollar amount per day - simple interest, NOT compound interest.)

(c) What is the clean price of the bond?

(d) What is the Macaulay Duration of the bond?

(e) What is the interest rate sensitivity of the bond?

Explanation / Answer

A) The dirty price of the bond can be calculated as follows

1141.009

The above table shows the calculations for the dirty price of the bond. The first column represents the time from now when the cash flows will occur. Second column is the amount of cash flows ie 5 coupon payments and fv at the end. Third column is the discount factor calculated using the YTM. it is simply 1/(1+YTM)^n where YTM = 9% and n is the term or column 1 in this case. Fourth column is the product of 2 and 3 and summing it up will give you the dirty price of the bond.

B) Accrued interest is simply 0.25*120 = $30

0.25 is the years for which the interest has been accrued as it is mentioned it is now 1.25 years later which means that the last coupon payment occured 0.25 years earlier. and 120 is the coupon payment.

C) Clean price = Dirty price - Accrued interest = 1141-30 = $1111

D) Macaulay Duration can be calculated as

3.84

Here the table is same as in the first case the last column is just the product of column 1 and 4 and duration is the sum of column 5 / sum of column 4

E) interest rate sensitiviy of the bond is the change in price of the bond when the interest rates change by 1 bps . So simply replace the YTM to 9.01% in the above calculations and the price becomes 1140.61 which is a decrease of -0.035%

Year CF df PV(CF) 0.75 120 0.937411 112.4893 1.75 120 0.86001 103.2012 2.75 120 0.789 94.68003 3.75 120 0.723853 86.86241 4.75 1120 0.664086 743.776

1141.009

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