Lecture 11 Fixed-Income Securities: An Overview

Transcription

1 1 Lecture 11 Fixed-Income Securities: An Overview Alexander K. Koch Department of Economics, Royal Holloway, University of London January 11, 2008 In addition to learning the material covered in the reading and the lecture, students should In addition to learning the material covered in the reading and the lecture, students should know the main sources of debt financing; be familiar with the various bond features, as well as the risks they may entail; understand what a yield to maturity is and how it relates to a coupon yield. Required reading: Bodie, Kane, and Marcus (2008) (Chapter 5 [some background material] and 14) Supplementary reading: Grinblatt and Titman (2002) (Chapter 2) Alexander.Koch@rhul.ac.uk.

2 2 1 Discounting Simple interest interest calculation based only on the original principal amount. Example: 100 principal is invested at 8 percent simple interest for ten years. This means that it yields 8 per year. Compound interest gives the interest earned on original principal plus interest on interest earned earlier during the holding period. To highlight the difference, consider the following example. Example 1 Suppose you have 100 to invest for a year. Your bank offers you a deal where you receive 5 percent of the amount invested every six months and these payments are reinvested under the same conditions. What is your return on the investment in the first year? Based on simple interest the interest rate p.a. ( per annum - per year) would be 10 percent (= ). However, simple interest ignores the interest that you earn on your interest. Let s look at the cash flow from this investment: date today 6m 12M reinvestment Thus, the compound or effective annual rate of return is percent (= ). Often interest rates are quoted based on simple interest. Let R denote the simple interest rate p.a.. Then, we can define the periodic interest rate as R/m, where 1/m is the fraction of a year after which interest accrues, i.e., m is the frequency of compounding. Among the so-called benchmark interest rates are the LIBOR (London Interbank Offered Rate) interest rates. They are stated as simple annual interest rates, from which a periodic interest rate for an investment period of less than one year is computed according to the following convention periodic interest rate= LIBOR period in days 360. Example 2 The one-month (1M) LIBOR is quoted at 4 3/8 percent. An investment of 100 for 1 month at LIBOR will yield a payment of principal and interest at maturity of [ ] =

3 Compounding frequency Value of P = 1, 000, 000 after (R=10 percent p.a.) m 1 year 10 years formula annually 1 1,100,000 2,593,742 P (1 + R) n quarterly 4 1,103,813 2,685,064 P (1 + R/4) 4 n weekly 52 1,105,065 2,715,673 P (1 + R/52) 52 n daily 365 1,105,156 2,717,910 P (1 + R/365) 365 n continuous 1,105,171 2,718,282 P e R n Table 1: Investment returns for different compounding frequencies Additional benchmark rates are among others: the bank prime rate 1 and rates derived from government debt (e.g., Federal Funds rate (1 day), T-bills (1-6 months), constant maturity T-Notes (1-10 years), T-bonds (10-30 years)...). Suppose that principal P is invested for n years at rate R and interest payments occur m times per year. Then, if interest payments are reinvested at the periodic investment rate, the value of the investment at the end of the holding period, S n, is given by S n = P ( 1 + R m) m n. If one increases the frequency of interest payments from daily, to every minute, to every second, etc., we can use the following mathematical result that captures the snowball effect of compound interest as we approach the limit of continuous compounding: 2 ( lim 1 + R m n = e m m) R n, e = This yields the formula for continuous compounding: S n = P e R n. Continuous compounding is used extensively in pricing derivatives such as options and futures. The higher the compounding frequency for a given simple interest rate, the more compounded interest the investor earns. We can see the snowball effect of different compounding frequencies for a holding period of 1 year and 30 years by looking at table 1 1 The rate offered by banks to its best and most creditworthy commercial customers. 2 For more on this, e.g., see the section on the exponential function in Chiang (1984). 3

4 4 2 Debt instruments Debt or fixed income instruments are IOU s, i.e., contracts where the borrower promises to pay a fixed cash flow stream to the lender. The contract specifies the size and the timing of interest payments and a schedule for the repayment of the amount owed on the loan (principal). Moreover, it may contain other financial requirements and restrictions that the borrower must adhere to, the rights of the lender in case the borrower defaults (i.e., violates any of the key terms of the contract, particularly the promise to pay). These contracts can permit the sale of the right to the cash flow to another investor (be negotiable) or exclude this (be non-negotiable). Types of debt instruments (see readings) bank loans (including lines of credit and loan commitments) leases commercial paper bonds (preferred stock - [located between equity and fixed income instruments]) 2.1 Bonds A bond is a security issued by the borrower in connection with a debt contract. This contract, called bond indenture, includes details such as coupon rate, maturity date, par (or face) value, and bond covenants, which we will discuss in the following. Example 3 On , IOWE PLC issues debt of 100 million. Debt is in bonds issued at par value 1,000 with a coupon rate 8 percent and maturity date The coupon rate is a simple interest rate that specifies the annual interest payments in proportion to the par (face) value of the bond: annual interest payments=coupon rate face value. Typically, bonds involve semiannual coupon payments, i.e., in the example above IOWE promises to pay every 6 months coupon rate 2 face value = , 000 = 40.

5 5 At maturity the issuer repays the face value of the bond. Prices for bonds are quoted as percentage of face value and net of accrued interest. Accrued interest is the prorated share of the upcoming coupon payments that the seller is implicitly entitled to: accrued interest= annual coupon payment 2 days elapsed since last coupon date. days in coupon period When buying a bond, the purchaser has to pay the quoted price (flat price) of the bond plus accrued interest: purchase price= flat price + accrued interest. Example 4 Suppose the BUYME bond which has a face value of 1,000 and a coupon of 6 percent is quoted at / days have passed since the last coupon payment and 150 days are left in the semiannual coupon period. Therefore, accrued interest on a bond is The purchase price is = , = 1, Types of bonds (see readings for more details) While some bonds are traded on formal exchanges, most bond trading occurs in over-thecounter (OTC) markets formed by networks of bond dealers. Types of bonds: government bonds Examples: Treasury (T-) Notes (maturities 1-10 years), T-Bonds (10-30 years). municipal bonds, bonds from government agencies corporate bonds Rating agencies assess default risk of an issuer (be it a company or a country). The two best known rating agencies are Moody s Investors Service and Standard & Poor s Corporation. They rate bonds on a scale from best quality Aaa/AAA (Moody s/s&p) to default D/D. So-called investment grade bonds have a rating of Baa/BBB or higher. Lower rated bonds

6 6 are considered speculative (Junk Bonds). 3 U.S. government debt is regarded as having (virtually) no default risk. Note, however that even governments might default on their loans (examples of defaults: Russia s default on debt incurred under Czar Nicholas II in 1917 and, more recently, Argentina). In this context default risk is often termed sovereign risk. Moreover, an investment grade rating for a company is no guarantee either. For example, WorldCom issued bonds with such a rating worth $11.78 bn in May 2001 and filed for bankruptcy only a year later, resulting in losses to bond holders of more than 80 percent of their investment. The credit-worthiness of a bond relative to government debt is reflected in the spread, i.e., the difference in yield (to maturity) (see below for more on yields) between the bond and a comparable-maturity government bond. The spread and interest rates are commonly measured in basis points (bp) by practitioners, where 1bp= 0.01 percentage points. For example, a bond with a spread of 378 bp means that the yield on the bond is the yield on the comparable government bond The bond indenture might include other provisions, which we will briefly review: Option provisions call provisions: the issuer is granted the right to repurchase (during specified periods) the bond at an agreed call price before the maturity date. This option (as the ones discussed below) has a value that should be priced into the bond, since it allows the firm to buy back bonds and refinance the debt at lower interest rates should market rates fall sufficiently (refunding). Valuing this right requires special techniques which we will discuss in Lectures convertibility (convertible bonds): the bondholder has the option to exchange the bond for an agreed number of shares or hold onto the bond until maturity. Example 5 SWAPME issues bonds at par value 1,000 which are convertible into 20 shares of the firm s stock. The current share price is 35. Market conversion value: value if option to exchange bond for shares were exercised immediately, i.e., = 700. Conversion premium: Market value of bond - market conversion value, i.e., 1, = Junk Bonds were widely used financing vehicles in leveraged buy-outs and hostile takeovers in the 1980s. The boom of Junk Bonds collapsed amid an insider trading scandal involving the most prominent player in this market Michael Milken of Drexel Burnham Lambert. These events are the subject of James B. Steward s book Den of Thieves.

7 7 extendability (extendable (or put) bonds): the bondholder has the option to demand repayment of the face value of the bond before maturity, i.e. can choose at any time whether or not to extend the contract beyond the present date. The option to retire is exercised if the current comparable market yield is greater than the coupon rate on the bond. Then the bondholder can invest the principal at a higher interest rate. Bond covenants (see readings) subordination clauses The debt may be junior to other debt in the event of a default. That is, claims to the assets of the company of holders of senior debt are satisfied first before the ones of holders of subordinated debt, whose claims precede those of equity holders. collateral A firm might pledge to bondholders collateral in form of a mortgage (mortgage bond), other securities (collateral trust bond), or equipment (e.g., equipment trust certificates used to finance transportation companies), stocks of merchandise, etc. dividend restrictions The firm may be restricted in the amount of profits it can pay out to equity holders in form of dividends. This is aimed at retaining an equity cushion to absorb losses and make default less likely. sinking funds clauses Requires a certain portion of the debt to be retired before maturity of all the bonds in the issue. For example, by repurchasing of randomly selected bonds (typically by exercising a call provision). 3 Bond pricing (basics) To price bonds, we will again use the idea that well functioning financial markets should lead to prices that exclude arbitrage opportunities (called fair value). To start off and understand the basics, we will abstract from default risk and assume that investors can borrow or lend at an interest rate R p.a., regardless of the investment horizon. 4 Thus, the fair price of a 4 In the next lecture we will drop this assumption and move to pricing based on different rates for different investment horizons obtained from the yield curve.

8 8 bond is its present value computed by discounting all cash flows, using the interest rate R. It is easy to see why an arbitrage opportunity would arise if a bond were not priced at its present value. Take for example, a one-year bond, which has present value PV=[coupon + par value]/[1+r]. If the price>pv, then sell bond and invest the money at the rate R for one year. If the price<pv, buy bond and finance this with a one-year loan at rate R. Perpetuity bond (consol) A perpetuity bond lasts for ever and only pays interest at regular intervals. 5 Given a discount rate of R per period, the fair value of a perpetuity paying x every period is 6 Annuity factor fair value = t=1 x (1 + R) t = x R. We can use the previous present value formula to derive the annuity factor, useful in computing the present value of a constant stream of coupon payments x lasting T periods. The cash-flow pattern of such a bond can be replicated using two perpetuities, for which we know the pricing formula: period PV T-1 T T+1... long perpetuity starting in t=1 short perpetuity starting in t=t ( ) x 1 R 1 1 (1+R) T - x R x R +x +x +x +x +x +x +x 1 (1+R) T -x -x +x +x +x +x +x This yields, annuity factor T = 1 R Straight bond (bullet bond) ( ) [ 1 1 (1 + R) T = T t=1 ] 1 (1 + R) t. A straight bond produces a cash-flow pattern of regular (typically semiannual) coupon payments and a payment equal to the par value at the end of the maturity. If R is the appropriate discount rate for the length of the coupon payment period at any maturity, the fair value of a 5 One of the rare examples of perpetuities actually issued are the consols used by the British Treasury to finance the Napoleonic Wars in For a derivation see, e.g., Appendix A of Copeland, Weston, and Shastri (2004).

9 9 bond with maturity T periods from now is fair value = P V (bond cashflows) = P V (coupon payments) + P V (par value) T coupon = t par value + (1 + R) t (1 + R) T. t=1 If we have constant coupon payments, we can use the annuity factor: fair value = coupon 1 ( ) 1 par value 1 R (1 + R) T + (1 + R) T. Example 6 Compute the fair value of a bond with par value 5, 000, coupon 7 percent (semiannual payments) and 10-year maturity for the following cases for annual simple interest: (i) 0.06, (ii) 0.07, (iii) We have twenty semiannual periods: T=20, 6M rate: R=annual rate/2. The present values are given below: Present values annual rate coupon payments 2, , , payment at T 2, , , fair value 5, , , From the preceding example, we see that a bond that pays a coupon rate equal to the discount rate is priced fairly at par (value). If the discount rate is above the coupon rate, the fair price is below par because a buyer must be compensated for not receiving the interest that an alternative investment would yield. Similarly, if the discount rate is below the coupon rate the bond is fairly priced above par to compensate the seller for giving up the right to payments at the higher coupon rate relative to an alternative investment. Thus, the major risk component (besides credit risk) of holding fixed income securities stems from uncertainty about future interest rates. Bond prices increase when interest rates decline and decrease when interest rates rise (relative to the rates used in computing the old bond price). Zero coupon bonds These bonds carry no coupons and pay only their face value at maturity. Thus, zero-coupon bonds provide all returns in form of appreciation in bond value (which at maturity is equal to the face value). Bond dealers can break down coupon bonds into zero-coupon bonds by issuing

10 10 a separate zero-coupon bond for each cash flow. This process is called stripping (see next example). We will see that stripping is a useful framework to think in when pricing bonds. Example 7 A 10-year coupon bond with coupon C (semiannual payments) and face value FV can be stripped into 6M zero-coupon bond with face value C 1Y zero-coupon bond with face value C 18M zero-coupon bond with face value C 2Y zero-coupon bond with face value C... 10Y zero-coupon bond with face value C 10Y zero-coupon bond with face value FV 4 Bond yields The current yield on a bond current yield = annual coupon bond price is a measure of the actual annual income that a bond provides relative to its price (as opposed to the coupon yield, which is simply the coupon rate, and which measures the annual income in relation to par value). Example 8 An 8 percent (coupon rate with semiannual payments) bond with face value 1,000 and maturity date 01/01/2014 (i.e., 10 years maturity) is traded at on 01/01/2004, i.e., has market value 725. Then the current yield is percent (=80/725). As a measure of the average return that a bond provides if held on to until maturity, yield to maturity (YTM) (also known as internal rate of return) is commonly used. The YTM is the interest rate that makes the present value of a bond s cash flow equal to its market price. These computations are usually done by computer. Try out the YIELD function for Excel described on p.462 in the book for the above example to reproduce the result stated here (see screenshot in figure 1): 7 market price = 20 t=1 Y T M = percent. 40 (1 + Y T M/2) t + 1, 000 (1 + Y T M/2) 20

11 Figure 1: Excel YIELD function applied to example settlement date 01/01/2004 =DATE(2004,01,01) maturity date 01/01/2014 =DATE(2014,01,01) annual coupon rate 0.08 bond price (% of face value) 72.5 redemption value (% of face va 100 coupon payments per year 2 Yield to maturity Yield to maturity M period payment present value formula =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A =$B11/(1+$D$9/2)^$A30 bond price 725 =SUM(C11:C30)

12 12 As we see, the current yield and the yield to maturity are not the same (even though they are close in this example). Realized compound yield versus yield to maturity The YTM will equal the realized compound return over the investment period only if the investor can actually reinvest coupon payments at an interest rate equal to the bond s YTM. Example 9 Compute the yield to maturity and the realized compound return of a 1-year bond selling at par value, for 1, 000, and with 10 percent coupon (semiannual payments). Consider the following cases for interest rates available for reinvesting coupon payments, quoted as annual simple interest: (i) 0.08, (ii) 0.10, (iii) Since the bond is trading at par, its YTM equals the coupon rate of 10 percent. The annualized realized compound yield (RCY) is computed by solving (1 + RCY/2) 2 1, 000 = end of period value (1) To solve this quadratic equation it is convenient to reformulate the problem in terms of a 6M return x = RCY/2 and the total return over the holding period T R = end of period value/1, Equation (1) is equivalent to 8 (1 + x) 2 = 1 + T R x x T R = 0 solution is the positive root: x = T R. The different yields are given in table 2. All three investments have a current yield equal to the coupon yield of 10 percent. The yield to maturity is also 10 percent. However, the realized compound yield equals the YTM only if the reinvestment rate is equal to the YTM. If it is lower (higher) than the YTM then the RCY < (>) YTM. In general, the (anticipated) reinvestment rates that will be available in the future are going to differ from the current interest rate for a given maturity. The next example explores how this affects the price of a bond: Example 10 Consider the bond in example 9 and suppose now that today s 6M interest rate is 10 percent p.a. it is anticipated that the 6M interest rate in six months is going to be 8 percent p.a.. What is the fair value of the bond? 7 You may have to use the add-in manager for Excel first: Tools/Add-Ins/Analysis ToolPak. ( 8 Recall that the roots of the quadratic equation x 2 + p x + q = 0 are x 1,2 = p ± p ) 2 q. 2 2

13 13 t=0 t=6m t=12m total realized yield cash flow -1, return compound to reinvestments yield maturity at 0.08 p.a sum -1,000 1, at 0.10 p.a sum -1,000 1, at 0.12 p.a sum -1,000 1, Table 2: Yields for different reinvestment rates To answer this question, it is useful to think of the cash flows provided by the bond and reinvestment, as separate cash-flow streams. That is, mentally we strip the bond into zero-coupon bonds that replicate the cash-flow pattern of the investment. t=6m t=12m bond cash flows ,050 reinvestment of coupon zero-coupon bonds (ZB) long 1Y ZB with FV 1,050 +1,050 long 6M ZB with FV short 6M ZB with FV long 1Y ZB with FV discount rate 0.10/2 0.08/2 The present values for the different zero-coupon bonds then are: 1Y ZB with FV 1,050 We can think of computing the present value as discounting the 12M payment to the date in 6M at 4 percent (= 0.08/2) and then discounting this value at 5 percent: 1, long 6M ZB with FV 50 cancels out against short position (both have an absolute present value of 50/ ).

14 14 1Y ZB with FV Thus, the bond has a fair value of 1, and should be traded at a price of (percent of face value). To check your understanding of the logic applied to valuing this bond, ponder why the present value of the 1Y zero-coupon bond paying 52 is the same as that for the 6M zero-coupon bond paying 50, namely The logic underlying the last example is crucial for pricing fixed income securities (and other securities). We will explore this in more detail in the next lecture where we look at the implications for bond pricing of different interest rates at different maturities. References Bodie, Zvi, Alex Kane, and Alan J. Marcus, 2008, Investments (Irwin McGraw-Hill: Chicago). Chiang, Alpha C., 1984, Fundamental Methods of Mathematical Economics (McGraw- Hill/Irwin: New York) 3rd edn. Copeland, Thomas, J. Weston, and Kuldeep Shastri, 2004, Financial Theory and Corporate Policy (Addison-Wesley: Reading, Mass). Grinblatt, Mark, and Sheridan Titman, 2002, Financial Markets and Corporate Strategy (Boston, Mass.: Irwin McGraw-Hill).