Question

1. ABC Inc., and XYZ Inc., both have 7% coupon bonds outstanding, with semiannual interest payments, and both are currently priced at the par value of $1000. The ABC Inc., bond has 4 years to maturity, whereas the XYZ Inc., bond 25 years maturity. If interest rates suddenly rise by 2 percent, what is the percentage change in the price of these bonds? If interest rates were to suddenly fall by 2% instead, what would the percentage change in the price of these bonds be then? Illustrate your answers by graphing bond prices versus YTM. What does this problem tell you about the interest rate risk of longer-term bonds

Answer 1

Face Vlaue	1000	1000
Coupon	7%	7%
Period	4 years	25 years
Price	1000	1000
YTM	7%	7%

As the price of bond and face face value are same at $ 1000 therefore the YTM equal Coupn rate = 7%

Now

If interest rate rises by 2% then the YTM will increase by 2% to 9% (7 + 2)

We calculate the price or (PV the of bond) at new yield of 9%

Using Excel PV function

PV(Rate,nper,Pmt,Fv)

here

For Bond with 4 years of maturity

Rate = YTM = 9%/2 as semi annual interest payments are made

Nper = period = 4 *2 = 8 (as semi annual interest payments are made)

Pmt = $ 35 ( 1000 * 7%/2)

Fv = Redemption value of the bond => $ 1000

=PV(9%/2,8,35,1000)

= $ -934.04

Price of 4 year bond = 934.04

% Change in the price = (Price at 9% YTM - Price at 7% YTM) / Price at 7% YTM *100

= (934.04 - 1000) * 100 => - 6.59%

For Bond with 25 years of maturity

Rate = YTM = 9%/2 as semi annual interest payments are made

Nper = period = 25 *2 = 50 (as semi annual interest payments are made)

Pmt = $ 35 ( 1000 * 7%/2)

Fv = Redemption value of the bond => $ 1000

=PV(9%/2,50,35,1000)

= $ -802.38

Price of 25 year bond =$ 802.38

% Change in the price = (Price at 9% YTM - Price at 7% YTM) / Price at 7% YTM *100

= (802.38 - 1000) * 100

=> - 19.76%

Similar calculation is done in case of interest rates decreasing by 2%

New YTM = 5% (7 - 2)

Prices of both the bonds: using same excel function: PV(Rate,nper,Pmt,Fv)

4 years Bond

=PV(5%/2,8,35,1000)

=> $ 1071.70

% change in the price = (1071.7 - 1000) / 1000 => 7.17%

25 years Bond

=PV(5%/2,50,-35,-1000)

=> $ 1283.62

% change in the price = (1283.7 - 1000) / 1000 =>28.37%

Graphical Summary of Bond Prices versus YTM

		4 year Bond	25 Year Bond	% change
	YTM	Price	Price	4 year Bond	25 year Bond
	7%	1000	1000
Scenario 1	9%	934.04	802.38	-6.60%	-19.76%
Scenario 2	5%	1071.7	1283.62	7.17%	28.37%

The bond with long maturity has more interest rate risk than the bonds with short maturity as depicted in the above problem.

The 25 year bond price rises and declines more in % terms than that of 4 year bond that shows the interest rate risk is more in long maturity bonds.