Investment Science: The price of a bond and it's relationship to the Present Value formula

Monday, September 6, 2010

In this article, we will see the relationship between the price of a bond and the present value (of an investment/annuity) formula, we will take both types of bonds into account, namely the zero-coupon bond and coupon bond.

We know that the price of a bond is given by


where

P = Price of the bond
F = Face value of the bond
C = Coupon payment per year
m = Number of periods in an year the coupon is paid (Typically m = 2 or coupons are paid every 6 months)
n = Number of coupon periods left (with m coupon periods per year)
λ = Yield to maturity

For a zero coupon bond (C = 0), therefore the above formula becomes


We know the relationship between the Present value and the Future value of an investment (compounded at 'm' periods per year at an interest rate 'r' for 'n' years) is given by



Note the similarities, the price of a zero coupon bond is the face value discounted by it's yield to maturity (YTM),  something similar to the above present value/future value relationship.

The zero coupon bonds typically have longer durations and they are more sensitive to changes in yield than the coupon bonds.

For a coupon bond, the price is given by



We know that the Present value (P) of an annuity which pays an amount A every individual period (an year) for 'n' years at an yearly interest rate 'r' is given by


The first term in the price of the coupon bond can be considered as a part which discounts the face value of the bond (similar to the present value/future value relationship as shown above) and the second term can be considered as the part which discounts the coupon payments which occurs every period (till the bond maturity date), similar to the Present value of an annuity formula.

The above similarities will help one easily understand how the price of a bond is derived.

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