Question

1.a.  Calculate the price and duration for the following bond when the going rate of interest is 8%.  The bond offers  7.5% coupon rate, matures in 3 years and has a par value of $1,000.  Show full calculations and fill the table below.

YR	PV of  $ 1	Bond Cash Flows	PV (Cash Flows)	Year * Present Value of Cash Flow
1
2
3
3
Total

Price= __________

Duration=____________

b. What would be the new price if the market rate of interest rises to 9%? Show by using the duration only and show all calculations.

c.  What would be the new price if the market rate is 7.5 %?

Answer 1

Solution:

1a.

Given,

Par Value = $1,000

Coupon rate = 7.5%

Coupon Payment = 7.5% of 1000 = 75

Maturity = 3 years

Interest rate,r = 8%

So, the bond will make 3 annual coupon payments of 75 and a final principal payment of 1000.

For year 1, Pv of $1 = 1/1.08 = 0.925926

For year 2, Pv of $1 = 1/(1.08)^2 = 0.857339

For year 3, Pv of $1 = 1/(1.08)^3 = 0.793832

Table:

YR	PV of $ 1	Bond Cash Flows	PV (cashflows)	Year * Present Value of Cash flow
1	0.925926	75	0.925926*75=	69.44444	1*69.44444=	69.44444
2	0.857339	75	0.857339*75=	64.30041	2*64.30041=	128.60082
3	0.793832	75	0.793832*75=	59.53742	3*59.53742=	178.61225
3	0.793832	1000	0.793832*1000=	793.8322	3*793.8322=	2381.49672
Total				987.1145		2758.15424

To calculate Maculay duration we need to take divide sum of all Year * Present Value of Cash flow by sum of all PV(cashflows) = 2758.15424/987.1145 = 2.794158

Price= Total of all PV(Cashflows) = 987.1145

Duration= Modified duration is Maculay duration divided by 1+r = 2.794158/(1+0.08) =2.587184

1b.

If interest rate rises to 9% from 8% i.e. change of 1%.

Change in price of bond = -Duration*change in interest rate = -2.587184 * 1% = -2.587184%

So, new price of bond = (1-2.587184%) * old price = (1-2.587184%) * 987.1145 = (1-0.02587184) * 987.1145 = 961.57605

1c.

If interest rate fall to 7.5% from 8% i.e. change of -0.5%.

Change in price of bond = -Duration*change in interest rate = -2.587184 * -0.5% = 1.293592%

So, new price of bond = (1+1.293592​​​​​​​%) * old price = (1+1.293592​​​​​​​%) * 987.1145 = (1+0.01293592) * 987.1145 = 999.88375