In the example developed in Topic 3.4, Pricing Relative to the Yield Curve, a synthetic coupon bond also can be created. This is a portfolio that does not include the true coupon bond but that exactly replicates the timing and magnitude (and default risk) of cash flows that an investor would receive from the coupon bond. The coupon bond can then be priced directly from arbitrage arguments applied to this synthetic security.

You can create a synthetic coupon bond from a portfolio of zero-coupon bonds. The portfolio is constructed so that the cash flows from the face amounts of the portfolio of zero-coupon debt contracts exactly replicate the cash flows from ten coupon bonds.

The replicating portfolio is straightforward to construct. Consider a portfolio that consists of +1 one year zeroes, +1 two year zeroes and 11 three year zeroes. Observe that this portfolio provides the following cash flows:

This portfolio exactly replicates the cash flows provided from a portfolio of ten coupon bonds. The market value of this portfolio of zero-coupon bonds determines the arbitrage-free value of the coupon bond.

The value of the synthetic portfolio we have created is $1,012.49. This implies an arbitrage-free value of $101.25 for one coupon bond, which is exactly what we concluded in Topic 3.4.

The yield to maturity from the coupon bond is computed numerically by solving for the interest rate r that equates the present value of the cash inflows to the bond’s price.

You can verify that if the price of the bond is $101.25, the yield to maturity is 9.502% per year. Before you can actually earn this yield to maturity, however, you must reinvest all cash proceeds (the coupons) prior to maturity at the yield to maturity. If you invest them for less, your actual return will be lower.

Online, the yield to maturity is automatically computed and displayed as follows:

Typically, you will not be able to reinvest the intermediate coupons at the same interest rate. In fact, the yield curve tells you the interest rate that is available for the reinvestment of coupon payments.

In Figure 3.6, you can see that we are really interested in the interest rate that is applicable for the time periods Year 1 to Year 2 and Year 2 to Year 3. These are not the spot interest rates but instead the forward interest rates. These rates are the focus of our discussion in Topic 3.7, Forward Interest Rates.

(C) Copyright 1999, OS Financial Trading System