valuing zero coupon bonds

Many investors assume that the value of a zero coupon bond, or 'zero', is derived in the same manner as the yield to maturity on a par coupon bond, such as a Government of Canada bond. This is not the case. Yields on zero coupon bonds involve greater mathematical sophistication in order to account for the individual cash flows and unique maturity dates. This is where the 'bootstrapping' method of deriving a yield curve comes to bear rewards for investors and dealers alike.

As we are aware by this point in our travels through The Financial Pipeline, a traditional interest bearing bond is comprised of the principal portion, which will be repaid to the holder of the bond, in full, at maturity and the interest portion of the bond, consisting of coupon payments which the holder of the bond receives at regular intervals, usually every 6 months. Since the traditional par bond is made up of unique cash flows, it is possible to separate the two components, or strip, the bond into its constituent components. This process is done by taking the interest bearing coupons as one distinct and unique security, known simply as 'coupons', and the principal, the residual, as another distinct security.

The traditional measure for comparing the value of bond investments has been the yield to maturity. For example, when calculating the price on a traditional interest bearing bond the yield to maturity (for this example assume 7%) is utilized on all cash flows to discount the value of those cash flows from the maturity date to the present date in order to determine the present value of all the cash flows. Since each of the coupon related cash flows are received at different distinct points of time in the future, each of these cash flows is subject to reinvestment risk. In order to compensate for the reinvestment risk, a 'spot' curve must be constructed. This will allow the investor to utilize the discount rate appropriate to the specific date associated with each cash flow.

Since the residual and coupons are the constituent parts of the original bond, the 'spot' curve should theoretically value the individual cash flows to equal the price of the regular par bond at a given yeild to maturity. The sum of the parts should be equal to the whole. Since we know that the yield to maturity calculation for a par bond utilizes the same discount rate for all of the cash flows associated with the bond, namely the yield to maturity, to derive the price of the par bond, a series of yields must be derived which will make all the cash flows associated with the distinct stripped securities equal to the yield to maturity of the par bond. The derivation of the unique discount rates, or yields, at various intervals along the yield curve is referred to as the term structure of interest rates. It is this 'spot' term structure which we are interested in deriving in order to value zero coupon bonds. The spot curve may be determined through a method known as bootstrapping.

The term structure of interest rates refers to the relationship between bonds of different terms. When interest rates of bonds are plotted against their terms, this is called the yield curve. Economists and investors believe that the shape of the yield curve reflects the market's future expectation for interest rates and the conditions for monetary policy.

Bootstrapping is an iterative process which determines an appropriate discount rate associated with a unique maturity solving for the unknown 'zero' rate. By starting at the front end of the yield curve with a known 6 month t-bill rate and a known one year Government bond yield, a forward rate may be determined to equate a single 1 year security with two six month securities: one starting today and maturing in six months, and one starting in six months and maturing one year from now. This same methodology is utilized along the yield curve in six month intervals to derive the 'spot' curve.

If the six month yield to maturity is 6%, and the 1 year yield to maturity is 7%, then the 1 year spot rate will be approximately 7.02%. This difference in observed yields versus derived spot yields becomes more pronounced the further out the yield curve the spot rate is calculated. Let the following table act as an example of a fictitious yield curve and the associated spot curve.

If the yield curve is positively slopped, then the theoretical spot curve will lie above the yield curve, as in the example above. This is due to the fact that the greater the maturity, the greater the yield. As the maturity increases along the yield curve, the appropriate discount rate associated with each distinct cash flow must also increase in order to maintain the yield to maturity/zero coupon value equilibrium. When the equilibrium price becomes out of line arbitrage opportunities exist to either strip more bonds or reconstitute current stripped bonds. The opposite is true when the yield curve is inverted: the spot curve will lie below the yield curve.

Due to the nature of the distinct cash flows associated with various coupon bearing bonds there are certain bonds which are more likely to be stripped than others. The effect of the coupon size on the present value of future cash flows has a direct bearing on stripping. A high coupon bond (example: 15%) has a larger portion of its present value derived from its coupon cash flows than a low coupon bond (example: 4%). This is due to the fact that every six months the high coupon bond holder will receive $7.50 for each $100 of par value they hold, while the low coupon bond holder will receive $2.00. Over the life of a ten year bond the total cash flow received from the high coupon bond is $150, where as the low coupon pays only $40. Due to the significant difference in cash flows, the economics of the spot curve favours stripping high coupon bonds over low coupon bonds.

When considering adding zero coupon bonds to your portfolio remember that 'bootstrapping' is the way to determine the value of the security you are considering. It is not a simple task to construct the 'spot' curve of iterated forward spot yields, but the effort can pay-off in a better understanding of the value of your securites.