Zero Coupon Bonds

A zero-coupon bond makes no coupon payments, only making a single cash flow at the end. It always sells at a discount (price lower than face value), so they are called pure discount bonds.

• Treasury bills are U.S. government zero-coupon bonds with a maturity of up to one year.

Suppose that a one-year, risk-free, zero-coupon bond with a $100,000facevaluehasaninitialpriceof$96,618.36. The cash flows would be:

The rate of return on the bond would be:

$$96,618.36=\frac{100,000}{(1+r_1)}⟶r_1=3.5\%$$

The Yield to Maturity (YTM) on the bond is the discount rate that sets the present value of the promised payment from the bond equal to the current market price of the bond.

• Intuitively, the yield to maturity for a zero-coupon bond is the return you will earn as an investor by buying the bond at its current market price, holding the bond to maturity, and receiving the promised face value payment.

$$\begin{array}{rl}96,618.36& =\frac{100,000}{(1+{\text{YTM}}_1)}\\ 1+{\text{YTM}}_1& =\frac{100,000}{96,618.36}=1.05⟶{\text{YTM}}_1=3.5\%\end{array}$$

In general, the yield to maturity of an $n$-year zero-coupon bond can be found by:

$$1+{\text{YTM}}_n={\left(\frac{\text{Face Value}}{\text{Price}}\right)}^{1/n}$$

• The per-period rate of return for holding the bond from today till maturity on date $n$
• Solving for the YTM of a zero-coupon bond is the same process we used to solve for the internal rate of return – YTM is the IRR of buying the bond.

The price of a zero-coupon bond:

$$P=\frac{\text{FV}}{(1+{\text{YTM}}_n{)}^n}$$

Risk-Free