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How to determine Bond Price?


The bond price does not remain stable. The price of bond is important for investor who wants to invest in bond market or who wants to sell the bond before maturity.
To determine the fair price of the bond is calculate by discounting the cash flow to ascertain the present value of all the cash flows.

Formula:
P = Ʃ n n=1 C / (1+r) t +... M / (1+r) n
There is another method which makes easy to calculate bond price is:
P = C [1-(1/ (1+r) n) / r] + M / (1+r) n
Or
P = C /r [1-(1/ (1+r) n)] + M / (1+r) n
Where,
P = Bond Price
C = Coupon amount
r = required rate of return
M = Maturity value
n = Number of years
If the coupon bond paid in semi-annually then the formula will be:
P = C /r [1- (1/ (1+0.5r) 2n)] + M/ (1+0.5r) 2n
Or
P = C /2 [1- (1/ (1+0.5r) 2n)/ r/2] + M/ (1+0.5r) 2n

Zero coupon bonds are a bond which does not give any interest during the bond period. The benefit of the bond is that the investor can purchase it at discount on fair price. It is a long term bond which helps to fulfil the investors’ long term requirements.

Formula:
P = FV / (1+r) n
P = Zero coupon bond
FV = Face value of bond
r = required rate of return
n = number of years

Example: A 10year bond whose face value is Rs. 1, 000 and the bond bear 8% annual interest rate. Find out the bond price if the required rate of return is 7.8%.

Solution: P = C /r [1-(1/1+r) n)] + M / (1+r) n
= 80 /0.078 [1-(1/(1+0.078) 10)] + 1000 / (1+0.078) 10
= 1, 025.64 [1-(1/2.119)] + 1000 / 2.119
= 1, 025.64 [1-0.4719] + 471.92
= 1, 025.64*0.5281 + 471.92
= Rs.541.64 + 471.92
= 1, 013.56
Example: Mr. Y owns 20 year bond and the face value of the bond is Rs. 1, 000. The coupon rate is 8.9% semi-annually. The required rate of return is 7%. Find out the bond price.
Solution: P = C /r [1- (1/ (1+0.5r) 2n)] + M/ (1+0.5r) 2n
= 89 /0.07 [1- (1/ (1+0.5*0.07) 2*20)] + 1, 000/ (1+0.5*0.07) 2*20
= 1271.42 [1- (1/(1+0.035) 40)] + 1, 000/ (1+0.035) 40
= 1271.42 [1- (1/3.959)] + 1, 000/ 3.959
= 1271.42 [1- 0.252] + 252.589
= 1271.428 * 0.748 + 252.589
= 951.02 + 252.589
= 1203
Example: An investor owns zero coupon bonds whose face value is Rs. 1, 000 and the required rate of return is 5%. The bond matures after 12 years. Find out the price of a zero coupon bond.

Solution: P = FV / (1+r) n
= 1, 000/ (1+0.05) 12
= 1, 000 / 1.795
= 557

Example: Find out the bond price with the help of given information:
·         A 15 year bond whose face value is Rs. 10, 000 and the interest rate is 10% annually. The required return is 9%.
·         Find out the bond price if interest pays semi-annually in above information.

Solution: Interest paid annually:
P = C [1-(1/ (1+r) n) / r] + M / (1+r) n
= 1000 [(1-(1/ (1.09) 15) / 0.09] + 10, 000 / (1.09) 15
= 1000 [1-(1/3.642) / 0.09] + 10, 000 / 3.642
= 1000 [1-0.2745/ 0.09] + 2745.74
= 1000 [0.7255 / 0.09] + 2745.74
= 1000 * 8.061 + 2745.74
= 10, 806.74

Interest paid semi-annually:
P = C /2 [1- (1/ (1+0.5r) 2n) / r/2] + M/ (1+0.5r) 2n
= 1000/2 [1- (1/ (1+0.5*0.09) 2*15)/ 0.09/2] + 10, 000/ (1+0.5*0.09) 2*15
= 500 [1- (1/ (1.045) 30)/ 0.045] + 10, 000/ (1.045) 30
= 500 [1- (1/3.745) / 0.045] + 10, 000/ 3.745
= 500 [1- 0.2670 / 0.045] + 2670.22
= 500 [0.733 / 0.045] + 2670.22
= 500 * 16.28 + 2670.22
= 10, 810.22


Example: Suppose you invested in a bond whose value is Rs.1000 and mature in 15 years. The interest rate is 6% semi annually and required rate of return is 13%. Find out the current price of a bond.
Solution: C/2*[1-(1/ (1+i/2) 2n)/I/2] + FV/ (1+i/2) 2n   (semi- annually)
C = 1000*6/100 = Rs.60 annually
= 60 / 2 * [1-(1/ (1+0.13/2) 2*15)/I/2] + [1000/ (1+0.13/2) 2*15]
= 30 *[1-(1/6.614)/0.065]+[1000/6.614]
=30*13.06 + 151.19
=Rs.542.99

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