Calculating yield and price static cash flow model

Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Duration, Modified Duration, and Convexity . . . . . . . . . . . . . . . . . . . . . 31 Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Properties of Macaulay Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Modified Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3 Bond Pricing and Spot and Forward Rates 47

Yield Curve Smoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 Smoothing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 Cubic Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Non-Parametric Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Spline-Based Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Nelson and Siegel Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Comparing Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

PARTTWOSELECTED CASHAND DERIVATIVE INSTRUMENTS

Non–Plain Vanilla Interest Rate Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . 119 Swaptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 Interest Rate Swap Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Corporate and Investor Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 Hedging Bond Instruments Using Interest Rate Swaps . . . . . . . . . . . . . . 127

8 Options 133

Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Credit Risk and Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Applications of Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Credit Derivative Instruments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Credit Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Credit-Linked Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 Total Return Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 Investment Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Capital Structure Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Exposure to Market Sectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Credit Spreads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 Funding Positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Credit-Derivative Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 Pricing Total Return Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Asset-Swap Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 Credit-Spread Pricing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

13 Hybrid Securities 227

Floating-Rate Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 Inverse Floating-Rate Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 Hedging Inverse Floaters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 Indexed Amortizing Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 Advantages for Investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 Synthetic Convertible Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 Investor Benefits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Interest Differential Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 Benefits for Investors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Motivation Behind CDO Issuance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284 Balance Sheet–Driven Transactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Investor-Driven Arbitrage Transactions . . . . . . . . . . . . . . . . . . . . . . . . . . 285 Analysis and Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Portfolio Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Cash Flow Analysis and Stress Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 Originator’s Credit Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Operational Aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Review of Credit-Enhancement Mechanisms . . . . . . . . . . . . . . . . . . . . . 288 Legal Structure of the Transaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 Expected Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289

PARTTHREESELECTED MARKET TRADING CONSIDERATIONS

Appendix: The Black-Scholes Model in Microsoft Excel . . . . . . . . . . . 331 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

The Bond Instrument

B informs viewers where the main stock market indexes closed that day and which is the cornerstone of the U.S. economy. All evening televi-sion news programs contain a slot during which the newscaster onds are the basic ingredient of the U.S. debt-capital market,

no longer necessarily the case. Asset-backed bonds, for instance, are issued in a number of tranches—related securities from the same issuer—each of which pays a different fi xed or fl oating coupon. Nevertheless, this is still commonly referred to as the fi xed-income market.

Basic Features and Definitions

One of the key identifying features of a bond is its issuer, the entity that is borrowing funds by issuing the bond in the market. Issuers generally fall into one of four categories: governments and their agencies; local govern-ments, or municipal authorities; supranational bodies, such as the World Bank; and corporations. Within the municipal and corporate markets there are a wide range of issuers that differ in their ability to make the interest payments on their debt and repay the full loan. An issuer’s ability to make these payments is identifi ed by its credit rating.

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

Since fi xed-income instruments are essentially collections of cash fl ows, it is useful to begin by reviewing two key concepts of cash fl ow analysis: discounting and present value. Understanding these concepts is essential. In the following discussion, the interest rates cited are assumed to be the market-determined rates.

Financial arithmetic demonstrates that the value of $1 received today is not the same as that of $1 received in the future. Assuming an interest rate of 10 percent a year, a choice between receiving $1 in a year and re-ceiving the same amount today is really a choice between having $1 a year from now and having $1 plus $0.10—the interest on $1 for one year at 10 percent per annum.

The investor will receive $1.479 in interest at the end of the term. The total value of the deposit after the three months is therefore $100 plus $1.479. To calculate the terminal value of a short-term investment—that is, one with a term of less than a year—accruing simple interest, equation (1.1) is modifi ed as follows:

Note that, in the sterling markets, the number of days in the year is taken to be 365, but most other markets—including the dollar and euro markets—use a 360-day year. (These conventions are discussed more fully below.)
Now consider an investment of $100, again at a fi xed rate of 6 per-cent a year, but this time for three years. At the end of the fi rst year, the investor will be credited with interest of $6. Therefore for the second year the interest rate of 6 percent will be accruing on a principal sum of $106. Accordingly, at the end of year two, the interest credited will be $6.36. This illustrates the principle of compounding: earning interest on interest. Equation (1.3) computes the future value for a sum deposited at a compounding rate of interest:

As the example above illustrates, more frequent compounding results in higher total returns. FIGURE 1.1 shows the interest rate factors cor-responding to different frequencies of compounding on a base rate of 6 percent a year.

This shows that the more frequent the compounding, the higher the annualized interest rate. The entire progression indicates that a limit can be defi ned for continuous compounding, i.e., where m = infi nity. To defi ne the limit, it is useful to rewrite equation (1.4) as (1.5).

In (1.6) e rn is the exponential function of rn. It represents the continuously compounded interest rate factor. To compute this factor for an interest rate of 6 percent over a term of one year, set r to 6 percent and n to 1, giving

Just as future and present value can be derived from one another, given an investment period and interest rate, so can the interest rate for a period be calculated given a present and a future value. The basic equation is merely rearranged again to solve for r. This, as will be discussed below, is known as the investment’s yield.

Discount factors can also be used to calculate the future value of a present investment by inverting the formula. In the example above, the six-month discount factor of 0.98756 signifi es that $1 receivable in six months has a present value of $0.98756. By the same reasoning, $1 today would in six months be worth

Using this six-month discount factor, the one-year factor can be derived from the second bond in fi gure 1.2, the 8 percent due 2001. This bond pays a coupon of $4 in six months and, in one year, makes a payment of $104, consisting of another $4 coupon payment plus $100 return of principal.

The price of the one-year bond is $101.89. As with the 6-month bond, the price is also its present value, equal to the sum of the present values of its total cash fl ows. This relationship can be expressed in the following equation:

specifi c bond. The approach, however, is still worth knowing.

Note that the discount factors in fi gure 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.