Question

4. a) Compute the modified duration of a 10% coupon, 4-year corporate bond with a yield to maturity of 8%. b) Using the modified duration, If the market yield drops by 25 basis points, there will be a __________% (increase/decrease) in the bond's price.

Answer 1

To calculate Modified duration we first need to calculate Macaulay duration:

• Lets assume that the par value of the bond is 100 so 10% coupon will be 10
• CF in year 4 = coupon +par value = 10+100=110
• YTM =8%, we discount all the coupons and principal payments to time 0 using YTM as the discount rate
• Next we multiply the PV with t or the corresponding year value
• Then we sum all PV x t and PV
• We divide the sum of PV x t by sum of PV to get the Macaulay duration
• Then we divide it by (1+YTM) to calculate Modified duration. As it is an annual coupon bond, n = 1 in (1+YMT/n) in the formula of modified duration

Year	CF	PV	PV x t
1	10	10/1.08 = 9.259259259	9.259259x1=9.259259
2	10	(10/1.08)2 = 8.573388203	8.573388203x2=17.14678
3	10	(10/1.08)3 = 7.93832241	7.93832241x3=23.81497
4	10+100= 110	(110/1.08)4 = 80.85328381	80.85328381x4=323.4131
	Sum	9.259259259+8.573388203+7.93832241+80.85328381=106.6242537	9.259259+17.14678+23.81497+323.4131=373.6341
		Macaulay Duration	373.6341/106.6242537 = 3.504213
		Modified Duration	3.504213/(1.08) = 3.244642

• Duration states the percentage impact in the price of the bond to a percentage change in the interest rate in the market.
• Price of the bond is inversely related to the interest rates. When interest rates rise, price falls and vice versa
• if YTM falls by 25bps the price of the bond should increase
• We use modified duration to calculate the percentage increase
  • 3.244642 of modified duration states that for every 1% change in the interest rate of the bond, the price changes by 3.244642% in the opposite direction.
  • 25bps =0.25% or 1/4 %
  • So if interest rate falls by 1/4% the price of the bond increases by 3.244642/4 % = 0.812%