Question

13. Given a 3 year, 5.0%, $1000 bond, with a 6% YTM,

1) What will be this bond’s price in the market?

2) What will be this bond’s duration?

3)What will be this bond’s price in the market, if the interests are paid semiannually?

4)What. what will happen to this bond’s price in the market, if the interest rate(YTM) decreases to 5%?

Answer 1

Answer : Calculation of Price of Bond :

Price of Bond = [Coupon * PVAF @YTM for n years] + [Face Value * PVF @YTM for nth year]

here YTM % = 6%

n is the number of years to maturity i.e 3 years

Coupon = 1000 * 5% = 50

Price of Bond = [50 * PVAF @6% for 3 years] + [1000 * PVF @6% for 3th year]

= [50 * 2.67301194942] + [1000 * 0.83961928301]

= 133.650597471 + 839.61928301

= 973.27

(2.) Calculation of Duration of Bond :

Duration of Bond will always be less then the maturity years of Bond as coupon paymnet are made in between.

Therefore using Duartion Formula , where

Duration = Sum of (Weights * Discounted Cash Flows) / Sum of Discounted Cash Flows

Below is the table showing calculations :

Year (Weights)	Cah Flows	PVF @6%	Discounted Cash Flows	Weights * Discounted Cash Flows
1	50	0.943396226	47.16981132	47.16981132
2	50	0.88999644	44.499822	88.999644
3	1050	0.839619283	881.6002472	2644.800742
		Total	973.2698805	2780.970197

Duration = 2780.970197 / 973.2698805

= 2.86 years.

Answer : 3) Calculation of Bond Price if if the interests are paid semiannually :

Price of Bond = [Coupon * PVAF @YTM for n years] + [Face Value * PVF @YTM for nth year]

here YTM % = 6% / 2 = 3% [Divided by 2 , as interest paid semiannually]

n is the number of years to maturity i.e 3 years * 2 = 6 [Multiplied by 2 , as interest paid semiannually]

Coupon = 1000 * 5% = 50 / 2 = 25 [Divided by 2 , as interest paid semiannually]

Price of Bond = [25 * PVAF @3% for 6 half years] + [1000 * PVF @3% for 6th year]

= [25 * 5.41719144374] + [1000 * 0.83748425665]

= 135.429786093 + 837.48425665

= 972.91

4.) Calculation of Price of Bond if YTM decreases to 5%

There is inverse relationship betwwen price of Bond and interest rate if Interest rate decreases price of Bond increases.

Price of Bond = [Coupon * PVAF @YTM for n years] + [Face Value * PVF @YTM for nth year]

here YTM % = 5%

n is the number of years to maturity i.e 3 years

Coupon = 1000 * 5% = 50

Price of Bond = [50 * PVAF @5% for 3 years] + [1000 * PVF @5% for 3th year]

= [50 * 2.72324802935] + [1000 * 0.86383759852]

= 136.162401467 + 863.83759852

= 1000

In this case as YTM is equal to coupon rate price of Bond is equal to par value.