Coupon bonds interest rate payments

The yield to maturity is the discount rate that is used to determine the present value of all future cash flows to be received. The yield is reported on an annual basis but is an add-on interest rate. That is, one half of the reported yield is the correct rate to use per six-month period for coupon bonds with semiannual payments of interest. 

The assumptions of this study are premised on the commitment to a multi trillion dollar, centralized H2 production and delivery system in the U.S. over a thirty-year time period. Therefore, it is believed that the capital structure assumptions of 30% equity capital and 70% debt are more realistic for the assumed scale of capital investments. In addition, there are cash flow benefits to financing capital budgeting projects with debt capital rather than equity capital because interest on debt is tax deductible whereas dividends payments are not. The 7% interest rate for 30-year coupon bonds is a reasonable assumption for the assumed scale of investments, particularly so if a national H2 plan is adopted with government regulation and guaranteed bond issues. 

We obtain a closed-form solution for the special case of a coupon bond option containing only one payment date (see section 5.3.1). Furthermore, there exists a closed-form solution assuming one-factor interest rate models (see Jamishidian [42]). 

A fixed-rate bond pays fixed coupons during the bond s life known with certainty. Conversely, a floating-rate note ox floater pays variable coupons linked to a reference rate. This makes the coupon payments uncertain. The main pim-pose of this debt instrument is to hedge the risk of rising interest rates. Although the financial crisis and liquidity provided by central banks have decreased the level of interest rates, they will at some point of course rise in future years. 

In contrast, for putable bonds, the right to exercise the option is held by the bondholder. In fact, putable bonds allow the bondholder to sell the bond back before maturity. Conversely to callable bonds, this happens when interest rates go up (risk-free rate increases, or the issuer s credit quality decreases). In fact, the bondholders may have the advantage to sell the bond and buy another one with higher coupon payments. 

Yield to call This method calculates the yield for the next available call date. The yield to call is determined assuming the coupon payment until the call date and the principal repayment at the call date. For instance, the yield to first call is the rate of return calculated assuming cash flow payments until first call date. When interest rates are less than the ones at issue, the yield to call is useful because most probably the bond will be called at next call date ... 

The option-adjusted spread (OAS) is the most important measure of risk for bonds with embedded options. It is the average spread required over the yield curve in order to take into account the embedded option element. This is, therefore, the difference between the yield of a bond with embedded option and a government benchmark bond. The spread incorporates the future views of interest rates and it can be determined with an iterative procedure in which the market price obtained by the pricing model is equal to expected cash flow payments (coupons and principal). Also a Monte Carlo simulation may be implemented in order to generate an interest rate path. Note that the option-adjusted spread is influenced by the parameters implemented into the valuation model as the yield curve, but above all by the volatility level assumed. This is referred to volatility dependent. The higher the volatility, the lower the option-adjusted spread for a callable bond and the higher for a putable bond. 

To calculate the value of these bonds, it is preferable to use the binomial tree model. The value of a straight bond is determined as the present values of expected cash flows in terms of coupon payments and principal repayment. For bonds with embedded options, since the main variable that drives their values is the interest rate, the binomial tree is the most suitable pricing model. 

Several factors affect the decision if exercising the option or not. The first one is the asymmetric profit-loss profile. The potential gain of the option holder is unlimited when the price of the underlying asset rises, and losing only the initial investment if the price decreases. The second one is the time of value. In fact, in callable bonds, usually the price decreases as the bmid approaches maturity. This incentives the option holder to delay the exercise for a lower strike price. However, coupon payments with lower interest rates can favour the early exercise. 

Step-up callable notes are a particular type of structured fixed income products. These bonds offer a coupon payment that increase during the bond s life. Moreover, they include a call option, that as we discussed earlier, the issuer has the right to redeem the bond early. The question, whether a callable step-up note will be called or not always depends on the evolution of interests rates. Therefore, the inclusion of these two characteristics makes the bond attractive to investors with higher performance than a conventional bond. The added variable coupon element acts for an investor as cushion compared to a conventional callable bond. In fact, the increasing coupon payment increases the value of a callable bond. However, if interest rates go down and coupon payments increase, the incentive of the issuer to redeem the bond early is greater than a simple callable. 

As noted, the coupon rate is the interest rate the issuer agrees to pay each year. The coupon rate is used to determine the annual coupon payment which can be delivered to the bondholder once per year or in two or more equal installments. As noted, for bonds issued in European bond markets and the Eurobond markets, coupon payments are made annually. Conversely, in the United Kingdom, United States, and Japan, the usual practice is for the issuer to pay the coupon in two semiannual installments. An important exception is structured products (e.g., asset-backed securities) which often deliver cash flows more frequently (e.g., quarterly, monthly). 

Determining a bond s value involves computing the present value of the expected future cash flows using a discount rate that reflects market interest rates and the bond s risks. A bond s cash flows come in two forms— coupon interest payments and the repayment of principal at maturity. 

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. 

To illustrate, consider once again the 5.25% coupon BMW Finance described in Exhibit 3.10. From the yield analysis screen in Exhibit 3.11, we can locate the bond s full price under the heading Payment Invoice on the right-hand side of the screen. The full price is 1,117,726.69 (labeled Total ) for a 1 million par value position. The cash flows of the bond are (1) annual payments of 5,250 for the next four years and (2) a payment of 1,000,000 at maturity. The interest rate that makes these cash flows equal to the full price is 2.793%. 

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. 

Swaptions are options that allow the buyer to obtain at a future time one position in a swap contract. It is quite elementary that an interest rate swap, fixed for floating, can be understood as a portfolio of bonds.To consider this assume that the notional principal is 1. Then the claim on the fixed payments is the same as a bond paying coupons with the rate p and no principal. Let X be the time when the swap is conceived. The claim on the fixed income stream is worth, at time X,... 

A bond s term to maturity is crucial because it indicates the period during which the bondholder can expect to receive coupon payments and the number of years before the principal is paid back. The principal of a bond—also referred to as its redemption value, maturity value, par value, or face value—is the amount that the issuer threes to repay the bondholder on the maturity, or redemption, date, when the debt ceases to exist and the issuer redeems the bond. The coupon rate, or nominal rate, is the interest rate that the issuer agrees to pay during the bond s term. The annual interest payment made to bondholders is the bond s coupon. The cash amount of the coupon is the coupon rate multiplied by the principal of the bond. For example, a bond with a coupon rate of 8 percent and a principal of 1,000 will pay an annual cash amount of 80. 

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. 

Duration is a measure of price sensitivity to interest rates—that is, how much a bond s price changes in response to a change in interest rates. In mathematics, change like this is often expressed in terms of differential equations. The price-yield formula for a plain vanilla bond, introduced in chapter 1, is repeated as (2.1) below. It assumes complete years to maturity, annual coupon payments, and no accrued interest at the calculation date. 

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). 

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. 

The price of a PO bond fluctuates with mortgage interest rates. As noted earlier, the majority of mortgages are fixed-rate loans. If mortgage rates fall below the PO bond s coupon rate, the volume of prepayments should increase as the individuals holding the underlying loans refinance them, speeding the stream of payments to the bondholder. The PO s price will rise both because of the faster cash flows and because the flows are now discounted at a lower rate. The opposite happens when mortgage rates rise. 

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... 

A bond s yield to maturity will understate (or overstate) the realized compounded yield when the true reinvestment rate is greater than (or less than) the calculated yield to maturity. Figure A4-6 illustrates this relationship for a 10 percent coupon bond that pays 30 in interest every 6 months, has 10 years until it matures, and is originally priced to sell at par (that is, its yield to maturity is equal to the coupon rate). If the annual reinvestment rate is also 10 percent (5 percent per 6-month period), the terminal value of the cash flows received plus the interest earned from the reinvestment of those cash flows will be equal to 2,653.30 1,000 from the maturity value of the bond, 1,000 to be received in the form of coupon payments, and 653.30 from reinvesting the coupons every 6 months to earn a 5 percent, 6-month rate. Given the starting value of 1,000 and the terminal value of 2,653.30, the terminal value ratio is equal to... 

To differentiate redemption yield from other yield and interest rate measures described in this book, it will be referred to as rm. Note that this section is concerned with the gross redemption yield, the yield that results from payment of coupons without deduction of any withholding tax. The net redemption yield is what will be received if the bond is traded in a market where bonds pay coupon net, without withholding tax. It is obtained by multiplying the coupon rate Cby (1 — marginal tax rate). The net redemption yield is always lower than the gross redemption yield. 

The principal is the amount of money that you lend to the issuer of the bond, most of the time, this principal is set at a relatively simple 1,000 so that institutions can more easily market an issue (it makes it easier for us individual investors, tool). Also included in the indenture is the coupon, which is the stated interest rate that the borrower promises to pay you. To make things easier, many bond investors think of coupons as the annual rate of interest expressed as a percentage of the face value of the bond. This rate is fixed when the bond is first issued (although there are some bonds with floating coupons), most issuers make semiannual payments based on this fixed rate. 

Treasury notes have maturities of between 2 and 10 years. Because of their longer maturities, these notes have more interest rate risk associated with them and so their prices fluctuate more than T-bill prices. The U.S. government used to issue Treasury bonds, which carried maturities of 15, 20, and 30 years. The 30-year Treasury bond was just retired in November 2001. Another Treasury security is a "strip" or zero-coupon Treasury security created by separating the income streams of coupon payments and principal, wherein the holder receives no coupon payments, buys the bond at a discount, and is returned the principal at par. There is a high degree of volatility associated with strips. 

U.S. Treasury notes and bonds are coupon bonds that pay interest semiannually. For example, if the bond s coupon rate is 10 percent, a 1,000 investment will give the investor 100, paid out in two semiannual payments of 50. This represents a 10 percent return on investment. 

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