Maturity present value

MATURITY DATE YEARS TO MATURITY CASH FLOW PRESENT VALUE AT 8% YIELD TO MATURITY (%) PRESENT VALUE AT YIELD TO MATURITY... 

Fatty acid composition may differ from sample to sample, depending on the place of production, the latitude, the climate, the variety, and the stage of maturity of the fruit. Greek, Italian, and Spanish olive oils are low in linoleic and palmitic acids and have a high percentage of oleic acid. Tunisian olive oils are higher in linoleic and palmitic acids and lower in oleic acid. Table 9.2 presents values of fatty acid composition for Greek olive oils from a study of the State Chemical Labaratory in Athens. 

For the calculation, we cancel out the principal payments of par at maturity. We assume that cash flows are annual and take place on the same coupon dates. The breakeven asset-swap spread A is calculated by setting the present value of all cash flows equal to 0. From the perspective of the asset swap seller, the present value is ... 

Under these four assumptions, the price of an asset can be described in present value terms relative to the value of the risk-free cash deposit M, and, in fact, the price is described as a Q-martingale. A European-style contingent liability with maturity date t is therefore valued at time 0 under the risk-neutral probability as... 

The bond price equation is usually given in terms of discount factors, with the present value of each coupon payment and the maturity payment being the product of multiplying them by their relevant discount factors. This allows us to set the price equation as shown by Equation (5.2),... 

The price of an inflation-linked bmid is determined as the present value of future coupon payments and principal at maturity. Like a conventional bond, the valuation depends on the cash flow structure. We can have three main cash flow structures of index-linked bonds. 

This means that p f) is the expected value of the present value of the bond s cash flows, that is, the expected yield gained by buying the bond at the price p f) and holding it to maturity is r. If our required yield is r, for example this is the yield on the equivalent-maturity government bond, then we are able to determine the coupon rate C for which p r) is equal to 100. The default-risk spread that is required for a corporate bond means that C will be greater than r. Therefore, the theoretical default spread is C — r basis points. If there is a zero probability of default, then the default spread is 0 and C = r. 

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. 

To illustrate the process, let s value a 4-year, 6% coupon bond with a maturity value of 100. The coupon payments are 6 for the next four years. In addition, on the maturity date, the investor receives the repayment of principal ( 100). The value of a nonamortizing bond can be divided in two components (1) the present value of the coupon payments (i.e., an annuity) and (2) the present value of the maturity value (i.e., a lump sum). Therefore, when a single discount rate is employed, a bond s value can be thought of as the sum of two presents values—an annuity and a lump sum. 

The present value of the maturity value is just the present valne of a lump sum and is equal to... 

Simply put, this number tells us how much the coupon payments contribute to the bond s valne. In addition, the bondholder receives the maturity value when the bond matures so the present value of the maturity value must be added to the present value of the coupon payments. The present value of the maturity value is... 

This number ( 79.21) tells us how much the bond s maturity value contributes to the bond s value. The bond s value is the sum of these two present values which in this case is 100 ( 20.79 + 79.21). 

As the bond moves toward maturity with no change in the discount rate, the price has declined from 103.546 to 102.7278. What are the mechanics of this result The value of a coupon bond can thought of as the sum of two present values—the present value of the coupon payments and the present value of the maturity value. What happens to each of these present values as the bond moves toward maturity with no change in the discount rate The present value of the coupon payments falls for the simple reason that there are fewer coupon payments remaining. Correspondingly, the present value of the maturity value rises because it is one year closer to the present. What is the net effect The present value of the coupon payments fall by more than the present value of the maturity value rises so the bond s value declines or is pulled down to par. 

The intuition for the result reveals a great deal about bond valuation. Why does the present value of the coupon payments fall by more than the present value of the maturity value rises Recall why a coupon... 

Why does the present value of the maturity value rise by more than the present value of the coupon payments falls A coupon bond sells at a discount because it offers a lower coupon rate (6%) than new comparable bonds issued at par (7%). So, relative to a bond selling at par, the repayment of the principal at maturity is a relatively more important cash flow. To be sure, it is the capital gain we obtain from this payment... 

It is important to stress that this result holds regardless of the path 3-month LIBOR takes in the future. To see this, we replicate the process described in Exhibit 3.5 once again with one important exception. Rather than remaining constant, we assume that 3-month LIBOR forward rates increase by 1 basis point per quarter until the floater s maturity. These calculations are displayed in Exhibit 3.6. As before, the present value of the floater s projected cash flows is 100. When the market s required margin equals the quoted margin, any increase/decrease in the floater s projected cash flows will result in an offsetting increase/... 

The most common measure of yield in the bond market is the yield to maturity. The yield to maturity is simply a bond s internal rate of return. Specifically, the yield to maturity is the interest rate that will make the present value of the bond s cash flows equal to its market price plus accrued interest (i.e., the full price). To find the yield to maturity, we must first determine the bond s expected future cash flows. Then we search by trial and error for the interest rate that will make the present value of the bond s cash flows equal to the market price plus accrued interest. 

The zero-volatility spread, also referred to as the Z-spread or static spread, is a measure of the spread that the investor would realize over the entire benchmark spot rate curve if the bond were held to maturity. Unlike the nominal spread, it is not a spread at one point on the yield curve. The Z-spread is the spread that will make the present value of the cash flows from the nongovernment bond, when discounted at the benchmark rate plus the spread, equal to the nongovernment bond s market price plus accrued interest. A trial-and-error procedure is used to compute the Z-spread. 

Exhibit 3.16 presents the calculation of the discount margin for this security. Each period in the security s life is enumerated in Column (1), while the Column (2) shows the current value of the reference rate. Column (3) sets forth the security s cash flows. For the first 11 periods, the cash flow is equal to the reference rate (10%) plus the quoted margin of 80 basis points multiplied by 100 and then divided by 2. In last 6-month period, the cash flow is 105.40—the final coupon payment of 5.40 plus the maturity value of 100. Different assumed margins appear at the top of the last five columns. The rows below the assumed margin indicate the present value of each period s cash flow for that particular... 

PVCF = present value of the cash flow in period t discounted at the yield to maturity where t = 1,2,..., n... 

The third and final step is to calculate the percentage change in the bond s portfolio value when each key rate and neighboring spot rates are changed. There will be as many key rate durations as there are preselected key rates. Let s illustrate this process by calculating the key rate duration for a coupon bond. Our hypothetical 6% coupon bond has a maturity value of 100 and matures in five years. The bond delivers coupon payments semiannually. Valuation is accomplished by discounting each cash flow using the appropriate spot rate. The bond s current value is 107.32 and the process is illustrated in Exhibit 4.27. The initial hypothetical (and short) spot curve is contained in column (3). The present values of each of the bond s cash flows is presented in the last column. 

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. 

The fair value for the TRS will be the value of the spread for which the present value of the LIBOR +/- spread leg equals the present value of the returns on the underlying reference asset. The present value of the returns on the underlying reference asset may be determined by evolving the underlying reference asset. The expected value of the TRS payoff at maturity should be discounted to the valuation date. 

For each moment in the life of the default swap we can sum up all of the instantaneous chances of actually receiving a default payout of 1 - R by integrating the above equation from time 0 to the maturity date, T. However, we need to make one adjustment, and that is to weight each chance of default by its present value. It is only appropriate that receiving a payout of 1 - R is worth more if that payout occurs tomorrow rather than next year, so after weighting each potential payout by its appropriate discount factor the value of the default protection becomes... 

Bond prices are expressed per 100 nominal —that is, as a percentage of the bond s face value. (The convention in certain markets is to quote a price per 1,000 nominal, but this is rare.) For example, if the price of a U.S. dollar-denominated bond is quoted as 98.00, this means that for every 100 of the bond s face value, a buyer would pay 98. The principles of pricing in the bond market are the same as those in other financial markets the price of a financial instrument is equal to the sum of the present values of all the future cash flows from the instrument. The interest rate used to derive the present value of the cash flows, known as the discount rate, is key, since it reflects where the bond is trading and how its return is perceived by the market. All the factors that identify the bond—including the nature of the issuer, the maturity date, the coupon, and the currency in which it was issued—influence the bond s discount rate. Comparable bonds have similar discount rates. The following sections explain the traditional approach to bond pricing for plain vanilla instruments, making certain assumptions to keep the analysis simple. After that, a more formal analysis is presented. 

The first bond in figure 1.2 matures in precisely six months. Its final cash flow will be 103.50, comprising the final coupon payment of 3-50 and the redemption payment of 100. The price, or present value, of this bond is 101.65- Using this, the six-month discount factor may be calculated as follows ... 

This technique of calculating discount factors, known as bootstrapping, is conceptually neat, but may not work so well in practice. Problems arise when you do not have a set of bonds that mature at precise six-month intervals. Liquidity issues connected with individual bonds can also cause complications. This is true because the price of the bond, which is still the sum of the present values of the cash flows, may reflect liquidity considerations (e.g., hard to buy or sell the bond, difficult to find) that do not reflect the market as a whole but peculiarities of that... 

Note that the discount factors in figure 1.3 decrease as the bond s maturity increases. This makes intuitive sense, since the present value of something to be received in the future diminishes the farther in the future the date of receipt lies. 

The average time until receipt of a bond s cash flows, weighted according to the present values of these cash flows, measured in years, is known as duration or Macaulays duration, referring to the man who introduced the concept in 1938—see Macaulay (1999) in References. Macaulay introduced duration as an alternative for the length of time remaining before a bond reached maturity. 

Duration varies with maturity, coupon, and yield. Broadly, it increases with maturity. A bonds duration is generally shorter than its maturity. This is because the cash flows received in the early years of the bond s life have the greatest present values and therefore are given the greatest weight. That shortens the avetj e time in which cash flows are received. A zero-coupon bond s cash flows are all received at redemption, so there is no present-value weighting. Therefore, a zero-coupon bond s duration is equal to its term to maturity. 

---