Zero-coupon payments

Exhibit 16.5 shows very clearly the price responses for three 20-year bonds offering 10%, 6% annual coupon, and zero-coupon payments. One feature worth noting is that the curvilinear, price/yield relationship is not constant. It varies at different points on the curve for each type of... 

As above, assuming a constant average inflation rate, which is then used to calculate the value of the bond s coupon and redemption payments. The duration of the cash flow is then calculated by observing the effect of a parallel shift in the zero-coupon yield curve. By assuming a constant inflation rate and constant increase in the cash flow stream, a further assumption is made that the parallel shift in the yield curve is as a result of changes in real yields, not because of changes in inflation expectations. Therefore, this duration measure becomes in effect a real yield duration ... 

To obtain the price of an inflation-linked bond, it is necessary to determine the value of coupon payments and principal repayment. Inflation-linked bonds can be structured with a different cash flow indexation. As noted above, duration, tax treatment and reinvestment risk, are the main factors that affect the instrument design. For instance, index-aimuity bmids that give to the investor a fixed annuity payment and a variable element to compensate the inflation have the shortest duration and the highest reinvestment risk of aU inflation-linked bonds. Conversely, inflation-linked zero-coupon bonds have the highest duration of all inflation-linked bonds and do not have reinvestment risk. In addition, also the tax treatment affects the cash flow structure. In some bond markets, the inflation adjustment on the principal is treated as current income for tax purpose, while in other markets it is not. 

Thus far our coverage of valuation has been on fixed-rate coupon bonds. In this section we look at how to value credit-risky floaters. We begin our valuation discussion with the simplest possible case—a default risk-free floater with no embedded options. Suppose the floater pays cash flows quarterly and the coupon formula is 3-month LIBOR flat (i.e., the quoted margin is zero). The coupon reset and payment dates are assumed to coincide. Under these idealized circumstances, the floater s price will always equal par on the coupon reset dates. This result holds because the floater s new coupon rate is always reset to reflect the current market rate (e.g., 3-month LIBOR). Accordingly, on each coupon reset date, any change in interest rates (via the reference rate) is also reflected in the size of the floater s coupon payment. 

The source of dollar return called reinvestment income represents the interest earned from reinvesting the bond s interim cash flows (interest and/or principal payments) until the bond is removed from the investor s portfolio. With the exception of zero-coupon bonds, fixed income securities deliver coupon payments that can be reinvested. Moreover, amortizing securities (e.g., mortgage-backed and asset-backed securities) make periodic principal repayments which can also be invested. 

Let us partition the total dollar return for this bond into its three components coupon payments, capital gain/loss, and reinvestment income. The coupon payments contribute 70 of the total euro return. The capital gain/loss component is zero because the bond is purchased at par and held to maturity. Lastly, the remainder of the total euro return ( 26.72) must be due to reinvestment income. 

A principal strip is a zero-coupon bond created from a bond s principal payment. 

Many Euro government bonds can be stripped, breaking them down into each of the single payments that they involve, that is, one flow for each remaining coupon payment and another one for the principal. With this procedure an n-year maturity coupon-bearing bond is transformed into n + 1 strips (zero coupon bonds), which can be traded separately in the market. Yet this market is much less liquid in the Eurozone than in the United States. 

In a securitisation structure, a bankruptcy remote SPV uses the proceeds of the debt issue to acquire the properties. It then leases them back to the seller for a term that will equal or exceed the tenor of the debt issue. Lease payments will service the debt in one of various ways The debt may be fully amortised over the term (although this gives rise to a significant tax mismatch) the debt may be partially amortised, which requires refinancing or a sale of the property to ensure repayment of the debt at maturity or the issue size may be increased to fund the purchase of a zero coupon bond to repay the principal at maturity. A typical structure is illustrated in Exhibit 12.4. 

We now revisit the earlier Vasicek example for short interest rates to consider the case where the underlying bond pays an annual coupon at a 5% rate (p = 0.05), all the other characteristics remain as before. In order to calculate the call price of the coupon-bond European option first we need to calculate the interest rate such that the present value at the maturity of the option of all later cash flows on the bond equals the strike price. This is done by trial and error using equation (18.48) and the value we get here is = 22.30%. Next, we map the strike price into a series of strike prices via equation (18.50) that are then associated with coupon payments considered as zero-coupon bonds and calculate the value of the European call options contingent on those zero-coupon bonds as in the above example. The calculations are described in Exhibit 18.7. 

In the United States, all bonds make periodic coupon payments except for one type the sero-coupon. Zero-coupon bonds do not pay any coupon. Instead investors buy them at a discount to face value and redeem them at... 

As noted above, the bond market includes securities, known as zero-coupon bonds, or strips, that do not pay coupons. These are priced by setting C to 0 in the pricing equation. The only cash flow is the maturity payment, resulting in formula (1.18)... 

The market convention is sometimes simply to double the semiannual yield to obtain the annualized yields, despite the fact that this produces an inaccurate result. It is only acceptable to do this for rough calculations. An annualized yield obtained in this manner is known as a hand equivalent yield. It was noted earlier that the one disadvantage of the YTM measure is that its calculation incorporates the unrealistic assumption that each coupon payment, as it becomes due, is reinvested at the rate rm. Another disadvantage is that it does not deal with the situation in which investors do not hold their bonds to maturity. In these cases, the redemption yield will not be as great. Investors might therefore be interested in other measures of return, such as the equivalent zero-coupon yield, considered a true yield. 

All bonds except zero-coupon bonds accrue interest on a daily basis that is then paid out on the coupon date. As mentioned earlier, the formulas discussed so far calculate bonds prices as of a coupon payment date, so that no accrued interest is incorporated in the price. In all major bond markets, the convention is to quote this so-called clean price. 

Accrued interest compensates sellers for giving up all the next coupon payment even though they will have held their bonds for part of the period since the last coupon payment. A bond s clean price moves with market interest rates. If the market rates are constant during a coupon period, the clean price will be constant as well. In contrast, the dirty price for the same bond will increase steadily as the coupon interest accrues from one coupon payment date until the next ex-dividend date, when it falls by the present value of the amount of the coupon payment. The dirty price at this point is below the clean price, reflecting the fact that accrued interest is now negative. This is because if the bond is traded during the ex-dividend period, the seller, not the buyer, receives the next coupon, and the lower price is the buyer s compensation for this loss. On the coupon date, the accrued interest is zero, so the clean and dirty prices are the same. 

As discussed in chapter 1, there are two types of fixed-income securities zero-coupon bonds, also known as discount bonds or strips, and coupon bonds. A zero-coupon bond makes a single payment on its maturity date, while a coupon bond makes interest payments at regular dates up to and including its maturity date. A coupon bond may be regarded as a set of strips, with the payment on each coupon date and at maturity being equivalent to a zeto-coupon bond maturing on that date. This equivalence is not purely academic. Before the advent of the formal market in U.S. Treasury strips, a number of investment banks traded the cash flows of Treasury securities as separate zero-coupon securities. 

A zero-coupon bond is the simplest fixed-income security. It makes no coupon payments during its lifetime. Instead, it is a discount instrument, issued at a price that is below the face, or principal, amount. The rate earned on a zero-coupon bond is also referred to as the spot interest rate. The notation P t, T) denotes the price at time r of a discount bond that matures at time T, where T >t - The bond s term to maturity, T - t, is... 

In the academic literature, the risk-neutral price of a zero-coupon bond is expressed in terms of the evolution of the short-term interest rate, r t)—the rate earned on a money market account or on a short-dated risk-free security such as the T-bill—which is assumed to be continuously compounded. These assumptions make the mathematical treatment simpler. Consider a zero-coupon bond that makes one payment, of 1, on its maturity date T. Its value at time ris given by equation (3.14), which is the redemption value of 1 divided by the value of the money market account, given by (3.12). 

In (3.33) the regressor parameters are the coupons paid on each coupon-payment date, and the coefficients are the prices of the zero-coupon bonds Pj wherey = 1,2,..., 7V. The values are obtained using OLS as long as the term structure is complete and I > N. 

As explained in chapter 3, zero-coupon, or spot, rates are true interest rates for their particular terms to maturity. In zero-coupon swap pricing, a bank views every swap, even the most complex, as a series of cash flows. The zero-coupon rate for the term from the present to a cash flows payment date can be used to derive the present value of the cash flow. The sum of these present values is the value of the swap. 

For an underlying coupon-paying bond, the equation must be modified by reducing P by the present value of all coupons paid during the life of the option. This reflects the fact that prices of call options on couponpaying bonds are often lower than those of similar options on zero-coupon bonds because the coupon payments make holding the bonds themselves more attractive than holding options on them. 

Figure 11.2 is a one-period binomial interest rate tree, or lattice, for the six-month interest rate. From this lattice, the prices of six-month and 1-year zero-coupon bonds can be calculated. As discussed in chapter 3, the current price of a bond is equal to the sum of the present values of its future cash flows. The six-month bond has only one future cash flow its redemption payment at face value, or 100. The discount rate to derive the present value of this cash flow is the six-month rate in effect at point 0. This is known to be 5 percent, so the current six-month zero-coupon bond price is 100/(1 + [0.05/2]), or 97.56098. The price tree for the six-month zero-coupon bond is shown in FIGURE 11.3. 

Deriving the one-year bonds price at period 0 is straightforward. Once again, there is only one future cash flow— the period 2 redemption payment at face value, or 100—and one possible discount rate the one-year interest rate at period 0, or 5.15 percent. Accordingly, the price of the one-year zero-coupon bond at point 0 is 100/(1 + [0.0515/2] ), or 95 0423-At period 1, when the same bond is a six-month piece of paper, it has two possible prices, as shown in figure 11.4, which correspond to the two possible sbc-month rates at the time 5.50 and 5.01 percent. Since each interest rate, and so each price, has a 50 percent probability of occurring, the avert e, or expected value, of the one-year bond at period 1 is [(0.5 x 97.3236) + (0.5 x 97.5562)], or 97.4399. 

In the United States, Canada, and New Zealand, indexed bonds can be stripped, allowing coupon and principal cash flows to be traded separately. This obviates the need for specific issues of zero-coupon indexed securities, since the market can create products such as deferred-payment indexed bonds in response to specific investor demand. In markets allowing stripping of indexed government bonds, a strip is simply a single cash flow with an inflation adjustment. An exception to this is in New Zealand, where the cash flows are separated into three components the principal, the principal inflation adjustment, and the inflation-linked coupons—the latter being an indexed annuity. 

Reinvestment risk. Like holders of a conventional bond, investors in a coupon indexed bond are exposed to reinvestment risk because they cannot know in advance what rates will be in effect when the bond s coupon payments are made, investors cannot be sure when they purchase their bond what yield they will earn by holding it to maturity. Bonds, such as indexed annuities, that pay more of their return in the form of coupons carry more reinvestment risk. Indexed zero-coupon bonds, like their conventional counterparts, carry none. 

The Z-chtss, or Z, bond ranks below all other classes in the CMO s structure and pays no cash flows for part of its life, functioning essentially like a zero-coupon bond (hence its name). When the CMO is issued, the Z bond, also known as an accrual, or accretion, bond, has a relatively small nominal value. At the start of its life, it pays out cash flows on a monthly basis, as determined by its coupon. However, when the Z bond itself is not receiving principal payments, its cash flows are used to retire some of the principal of the other classes in the structure. In their place, the bond receives credits, which increase its face value by the amount of the forgone coupon. As a result, its principal amount is higher at the end of its life than at the start. When all classes of bond ahead of the Z bond have been retired, the Z bond itself starts to pay out principal and interest cash flows. 

The PO bond is similar to a zero-coupon in that it is issued at a discount to par value. The PO bondholders return is a function of the rapidity at which prepayments are made the quicker the prepayment, the higher the return. This is like the buyer of a zero-coupon bond receiving the maturity payment ahead of the redemption date. The highest possible return for the bondholder would occur if all the mortgages were prepaid the instant after the PO bond was bought. A low return occurs if all the mortgages are held until maturity, so that there are no prepayments. 

As already discussed, lOs, which receive the interest payments of the underlying collateral, and POs, which receive principal payments, exhibit different price behavior from pass-throughs and from each other. Figure 14.5 (page 263) showed that when interest rates are very high and prepayments, accordingly, unlikely, POs act as if repayable at par on maturity, like zero-coupon bonds. When interest rates decline and prepayments... 

As noted in chapter 2, a Treasury bond can be seen as a bundle of individual zero-coupon securities, each maturing on one of the bond s cash flow payment dates. In this view, the Treasury s price is the sum of the present values of all the constituent zero-coupon bond yields. Assume that the spot rates for the relevant maturities—ri,r2,rg,.rj f—can be observed. If a bond pays a semiannual coupon computed at an annual rate of C from period 1 to period N, its present value can be derived using equation (16.7). 

When the bond yield curve is flat, the spot curve is too. When the yield curve is inverted, the theoretical zero-coupon curve must lie below it. This is because the rates discounting coupon bonds earlier cash flows are higher than the rate discounting their final payments at redemption. In addition, the spread between zero-coupon and bond yields should decrease with maturity. 

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