A zero-coupon bond is a bond that does not pay any periodic interest except on maturity, where the principal or face value is repaid. Zero-coupon bonds are also called pure discount bonds.
A zero-coupon bond can be valued as follows:
Here, 
is the annually compounded yield or rate of the bond, and 
is the time remaining to the maturity of the bond.
Let's take a look at an example of a 5-year zero-coupon bond with a face value of $100. The yield is 5 percent, compounded annually. The price can be calculated as follows:
A simple Python zero-coupon bond calculator can be used to illustrate this example:
def zero_coupon_bond(par, y, t):
"""
Price a zero coupon bond.
Par - face value of the bond.
y - annual yield or rate of the bond.
t - time to maturity in years.
"""
return par/(1+y)**tUsing the preceding example, we get the following result:
>>> print zero_coupon_bond(100, 0.05, 5) 78.3526166468
In the preceding example, we assumed that the investor is able to invest $78.35 at the prevailing annual interest rate of 5 percent for 5 years, compounded annually.
As the compounding frequency increases (say, from compounded yearly to compounded daily), the future value of money reaches an exponential limit. That is to say, the present value of $100 today will reach a future value of 
when it is invested at a continuously compounded rate R for a period of time, T. Discounting these values for a security that pays $100 at a future time T with a continuously compounded discount rate R, its value at time zero is 
. This rate is known as the spot rate.
Spot rates represent the current interest rates for several maturities, should we want to borrow or lend money now. Zero rates represent the internal rate of return of zero-coupon bonds.
We can use spot rates and zero rates of bonds of different maturities to construct the present yield curve.