Question

Interest Rate Risk. 

Bond J has a coupon rate of 4 percent. Bond S has a coupon rate of 14 percent. Both bonds have 13 years to maturity, make semiannual payments, a par value of $1,000, and have a YTM of 8 percent. If interest rates suddenly rise by 2 percent, what is the percentage price change of these bonds? What if rates suddenly fall by 2 percent instead? What does this problem tell you about the interest rate risk of lower-coupon bonds?

Answer 1

BOND J
Par/Face value	1000
Annual Coupon rate	0.04
Annual coupon	40
semi-annual coupon	20
Present Value = Future value/[(1+(r/m))^mt]
r is the yield to maturity that is 8%.
m is the compounding period that is 2
mt is the time period.
price of the bond = sum of present values of future cash flows
r/2	0.04
mt	1	2	3	4	5	6	13
future cash flow	20	20	20	20	20	20	20
present value	19.23077	18.49112	17.77993	17.09608	16.43854	15.80629	12.01148
mt	14	15	16	17	18	19	26
future cash flow	20	20	20	20	20	20	1020
present value	11.5495	11.10529	10.67816	10.26746	9.872562	9.492848	367.903
sum of present values	680.3446
b) The price of BOND J is $680.34.
Do the same analysis for Bond J when YTM increases by 2% and decreases by 2%.
Then calculate the % change in price.
r/2	0.05
mt	1	2	3	4	5	6	13
future cash flow	20	20	20	20	20	20	20
present value	19.04762	18.14059	17.27675	16.45405	15.67052	14.92431	10.60643
mt	14	15	16	17	18	19	26
future cash flow	20	20	20	20	20	20	1020
present value	10.10136	9.620342	9.16223	8.725934	8.310413	7.914679	286.8655
sum of present values	568.7444
When the yield to maturity rises by 2%, the price of BOND J is $568.74.
% change in price when interest rates rise by 2% is (568.74-680.34)/680.34
% change in price when interest rates rise by 2% is -16.40%.
r/2	0.03
mt	1	2	3	4	5	6	13
future cash flow	20	20	20	20	20	20	20
present value	19.41748	18.85192	18.30283	17.76974	17.25218	16.74969	13.61903
mt	14	15	16	17	18	19	26
future cash flow	20	20	20	20	20	20	1020
present value	13.22236	12.83724	12.46334	12.10033	11.74789	11.40572	472.9686
sum of present values	821.2316
When the yield to maturity falls by 2%, the price of BOND J is $821.23.
% change in price when interest rates fall by 2% is (821.23-680.34)/680.34
% change in price when interest rates fall by 2% is 20.71%.
SAME ANALYSIS FOR BOND S AS WAS FOR BOND J
Par/Face value	1000
Annual Coupon rate	0.14
Annual coupon	140
semi-annual coupon	70
b) The price of BOND S is $1479.48.
Do the same analysis for Bond S when YTM increases by 2% and decreases by 2%.
Then calculate the % change in price.
When the yield to maturity rises by 2%, the price of BOND S is $1287.50
% change in price when interest rates rise by 2% is (1287.50-1479.48)/1479.48
% change in price when interest rates rise by 2% is -12.98%.
When the yield to maturity falls by 2%, the price of BOND S is $1715.074.
% change in price when interest rates fall by 2% is (1715.07-1479.48)/1479.48
% change in price when interest rates fall by 2% is 15.92%.
This problem tells us that lower coupon bonds have higher interest rate risk.