Factors Influencing Bond Price
A bond's present value (price) is determined by the following formula:
Price = {Coupon_1}/{(1+r)^1} + {Coupon_2}/{(1+r)^2} + ... + {Coupon_n}/{(1+r)^n} + {Face Value}/{(1+r)^n}
For example, find the present value of a 5% annual coupon bond with $1,000 face, 5 years to maturity, and a discount rate of 6%. You should work this problem on your own, but the solution is provided below so you can check your work.
Price = {50}/{(1.06)^1} + {50}/{(1.06)^2} +{50}/{(1.06)^3} +{50}/{(1.06)^4} + {50}/{(1.06)^5} + {1000}/{(1.06)^5} = 957.88
A change in any of these variables (coupon, discount rate, or time to maturity) will influence the price of the bond.
A higher coupon rate will increase the value of the bond.
Find the price of the above bond if the coupon rate changes to:
a. 4%
b. 6%
c. 7%
Price_a = {40}/{(1.06)^1} + {40}/{(1.06)^2} + {40}/{(1.06)^3} + {40}/{(1.06)^4} + {40}/{(1.06)^5} + {1000}/{(1.06)^5} = 915.75
Price_b = {60}/{(1.06)^1} + {60}/{(1.06)^2} + {60}/{(1.06)^3} + {60}/{(1.06)^4} + {60}/{(1.06)^5} + {1000}/{(1.06)^5} = 1,000
Price_c = {70}/{(1.06)^1} + {70}/{(1.06)^2} + {70}/{(1.06)^3} + {70}/{(1.06)^4} + {70}/{(1.06)^5} + {1000}/{(1.06)^5} = 1,042.12
The higher the coupon rate, the higher the value of the bond, all else equal. In the particular case where the coupon rate is equal to the discount rate, then the bond's price is the same as its par value (since the bond cannot command a premium or require a discount).
A higher discount rate will decrease the value of the bond
Find the price of the original bond (coupon rate = 5%, 5 years to maturity, $1,000 face value) if the discount rate changes to:
a. 4%
b. 5%
c. 7%
Price_a = {50}/{(1.04)^1} + {50}/{(1.04)^2} + {50}/{(1.04)^3} + {50}/{(1.04)^4} + {50}/{(1.04)^5} + {1000}/{(1.04)^5} = 1,044.52
Price_b = {50}/{(1.05)^1} + {50}/{(1.05)^2} + {50}/{(1.05)^3} + {50}/{(1.05)^4} + {50}/{(1.05)^5} + {1000}/{(1.05)^5} = 1,000.00
Price_c = {50}/{(1.07)^1} + {50}/{(1.07)^2} + {50}/{(1.07)^3} + {50}/{(1.07)^4} + {50}/{(1.07)^5} + {1000}/{(1.07)^5} = 918.00
The higher the discount rate, the lower the value of the bond, all else equal. Again, in the particular case where the coupon rate is equal to the discount rate, then the bond's price is the same as its par value (since the bond cannot command a premium or require a discount).
A longer term to maturity will decrease the value of the bond.
Find the price of the original bond (coupon rate = 5%, $1,000 face value, discount rate of 6%) if the term to maturity changes to:
a. 2 years
b. 10 years
c. 30 years
Price_a = {50}/{(1.06)^1} + {50}/{(1.06)^2} + {1000}/{(1.06)^2} = 981.67
Price_b = {50}/{(1.06)^1} + {50}/{(1.06)^2} + ... + {50}/{(1.06)^{10} + {1000}/{(1.06)^{10} = 926.40
Price_c = {50}/{(1.06)^1} + {50}/{(1.06)^2} + ... + {50}/{(1.06)^{30} + {1000}/{(1.06)^{30} = 862.35
The longer the term to maturity, the lower the value of the bond, all else equal. The bulk of a bond's value is derived from the face value paid at maturity -- the longer the time to maturity, the more the discount rate will reduce the present value of that face value.
