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The Role of Fixed-Income Securities in the Financial Markets and Portfolio Management

Par vs. Spot Yield Curves (Cont.)

- A par value yield curve summarizes the yields for coupon-bearing instruments where the coupon rate is equal to the yield-to-maturity. Assuming that the above example is based a collection of Eurobonds (i.e., bonds that pay an annual coupon), the 10% yield for the three-year instrument can be interpreted as the average annual return that the investor can expect if he or she:

(i) Holds the bond until maturity,

(ii) Reinvests all intermediate cash flows (i.e., the first two coupons) at the same 10% rate for the remaining time until maturity.

- A spot, or zero coupon, yield curve summarizes the yields for non-coupon-bearing instruments (i.e., pure discount bonds). These yields can therefore be interpreted as more of a "pure" return since there is no concern about having to reinvest intermediate coupon cash flows. For example, if the above yield curve corresponded to zero coupon securities, the 10%, three-year yield would represent the average annual price appreciation in the bond if it were held to maturity.

Uses for Implied Forward Rates

- Predictions of Future Spot Rates: This assumes that investors set yield curves with unbiased expectations, which is seldom true. Generally, implied forward rates are upward-biased predictions of future spot rates because of liquidity premiums attached to yields of longer-term maturity bonds relative to shorter-term instruments
- Maturity Choice Decisions: Helps fixed-income investors decide on appropriate maturity structure for a bond portfolio by quantifying the reinvestment rate embedded in longer-term securities compared to shorter-term ones
- Pricing Interest Rate Derivatives: Sets the arbitrage boundaries for the rates attached to actual forward agreements (e.g., bond futures, interest rate swaps)

Basics of Bond Valuation

- Bonds are simply loans from bondholder to issuer (e.g., firm or government). Just like loans, bonds require interest payments and repayment of principal (also called face value or par) at a pre-specified future date. Interest payments are called coupon payments and bond principal repayments are usually non-amortizing (i.e., paid all at once at maturity)
- The current market value of a fixed-income bond is the present value of its future coupon and principal cash flows. In theory, the interest rates used to discount those future cash flows are the zero-coupon (or pure discount)rates corresponding to the dates of each cash flow.

Basics of Bond Valuation (cont.)

- Consider a five-year, 9% (annual coupon payment) Eurobond. The market value of the bond, 103.99 (% of par value), can be obtained by calculating the present value of each scheduled cash flow using a sequence of zero-coupon rates commensurate with the riskiness of the bond.

Basics of Bond Valuation (cont.)

- The yield to maturity (y) of the bond is the constant interest rate per period that solves the following equation:
- The yield-to-maturity is the internal rate of return of all cash flows. It is the rate such that the present values of the cash flows, each discounted by that same rate, exactly equal the market value of the bond. The yield to maturity of this bond turns out to be 8.00%.

Basics of Bond Valuation (cont.)

- The yield to maturity is a statistic about the rate of return on the bond that includes both the coupon cash flows as well as any inevitable capital gain or loss if the bond is held to maturity (a gain if the bond is purchased at a discount below par value, a loss if the bond is purchased at a premium above par value).
- Therefore, it contains more information than the current yield, which is simply the coupon rate divided by the current price, e.g., 9 103.99 = .0865 . The current yield of 8.65% overstates the investor’s rate of return since it neglects the capital loss.

Basics of Bond Valuation (cont.)

- Notice that the yield to maturity can be interpreted as a "weighted average" of the sequence of zero-coupon rates, with most of the weight placed on the last cash flow since that is when the principal is redeemed, in that both deliver the same present value:
- Clearly, the yield to maturity must lie within the range of the zero-coupon rates.

Current Coupon, Premium, and Discount Bonds

- A Current Coupon (or Par-Value) Bond is one for which the current market price equals the face value. In that case, the coupon rate (C/F) will equal the current yield (C/P), which will equal the yield-to-maturity (y).

P = F <===> C/F = C/P = y

The bond is priced at par value since its coupon rate is "fair" in that it equals the current market interest rate as represented by the yield-to-maturity.

- A Premium Bond has a current market price that exceeds the face value. In this case, the coupon rate will be higher than the current yield, which in turn will be higher than the yield-to-maturity.

P > F <===> C/F > C/P > y

The bond is priced at a premium above par value since its coupon rate is "high" given current market rates. A par-value, current coupon bond would have a lower coupon rate, so the premium represents the value of the "excessive" coupon cash flows. In fact, the amount of the premium is the present value of the annuity represented by the difference between the coupon rate and the bond's yield, discounted at that yield.

- A Discount Bond has a current market price that is less than the face value. The coupon rate will be less than the current yield, which will be less than the yield-to-maturity.

P < F <===> C/F < C/P < y

The bond is priced at a discount below par value since its coupon rate is "low" given current market rates. The amount of the discount is the present value of the annuity represented by the difference between the yield and the coupon rate. For example, a zero-coupon bond will usually be at a deep discount to par value.

Current Coupon, Premium, and Discount Bonds (cont.)

- Example: Calculate the yield-to-maturity statistic on a seven-year, 6-3/4% Treasury note priced at 98.125. Assume that a semi-annual coupon payment has just been made so that exactly 14 periods remain until the principal is refunded at maturity.
- Algebraically, the yield is the solution “y” to the following equation:
- Solving for the periodic yield (i.e., y/2) on a financial calculator (such as the HP 12C) obtains 3.5472 [100 FV, 14 n, 3.375 PMT, -98.125 PV, i …. 3.5472].
- The annualized yield-to-maturity would then be reported as y = 7.0944% (i.e., 3.5472 x 2).

Sources of Bond Risk

- Primary:
- Default: Will the borrower honor its promise to repay?
- Interest Rate: How will changing market conditions affect the value of the bond?
- Price risk component
- Reinvestment risk component
- Secondary:
- Call: Will the borrower refinance the loan under conditions that are disadvantageous to investor?
- Liquidity: How easily can bond be bought or sold?
- Tax: Will changes in the tax code affect bond values?

Bond Yields, Pricing, and Volatility

- Theorem #1: Bond prices are inversely related to bond yields.

Implication: When market rates fall, bond prices rise, and vice versa.

- Theorem #2: Generally, for a given coupon rate, the longer is the term to maturity, the greater is the percentage price change for a given shift in yields. (The maturity effect)

Implication: Long-term bonds are riskier than short-term bonds for a given shift in yields, but also have more potential for gain if rates fall.

- Theorem #3: For a given maturity, the lower is the coupon rate, the greater is the percentage price change for a given shift in yields. (The coupon effect)

Implication: Low-coupon bonds are riskier than high-coupon bonds given the same maturity, but also have more potential for gain if rates fall.

- Theorem #4: For a given coupon rate and maturity, the price increase from a given reduction in yield will always exceed the price decrease from an equivalent increase in yield. (The convexity effect)

Bond Yields, Pricing, and Volatility (cont.)

Implication: There are potential gains from structuring a portfolio to be more convex (for a given yield and market value) since it will outperform a less convex portfolio in both a falling yield market as well as a rising yield

Price

Convex Price-Yield Curve

Yield

Bond Yields, Pricing, and Volatility: Example

- Consider the following bonds:
- Initial Prices:

Bond Yields, Pricing, and Volatility: Example (cont.)

- Prices after yields increase by 50 bp:
- Percentage price changes:

Bond A: (906.43 - 924.18) / (924.18) = -1.92% (least)

Bond B: (668.78 - 705.46) / (705.46) = -5.20% (most)

Bond C: (952.68 - 1000.00) / (1000.00) = -4.73% (middle)

Bond Yields, Pricing, and Volatility: Example (cont.)

- Question: Where would Bond D, which has a coupon rate of 6% and a maturity of 19 years, fit into this price sensitivity spectrum? (Assume its initial yield is also 8%.)

Initial:

After:

So, percentage change:

Bond D: (768.31 - 807.93) / (807.93) = -4.90%

Calculating the Duration Statistic

- The duration of a bond is a weighted average of the payment dates, using the present value of the relative cash payments as the weights:
- This statistic is the Macaulay duration, named after Frederick Macaulay who first developed it, and can be interpreted as the point in the life of the bond when the average cash flow is paid.

Calculating the Duration Statistic: Example

- Consider a five-year, 12% annual payment bond having a face value of $1,000. Suppose that the bond is priced at a premium to yield 10% (p.a.). The price of the bond is $1,075.82 and the Macaulay duration is 4.074:

or:

Duration as a Measure of Price Volatility

- Basic Price-Yield Elasticity Relationship:
- Convert to “Volatility Prediction” Equation:
- Prediction Equation in Modified Form (% price change):

Duration as a Measure of Price Volatility (cont.)

- Convert to dollar (or cash) sensitivity:

DMV ~ -(Mod D)( Dy)(MV)

- Sensitivity to a one bp yield change (i.e., Dy = 0.0001):

DMV ~ -(Mod D)(0.0001)(MV) = Basis Point Value = BPV

Duration and Price Volatility: Example

- Consider again the five-year, 12% coupon bond with a yield to maturity of 10%:
- Macaulay D: 4.074
- Modified D: 3.704 (= 4.074 / 1.1)
- This means that an increase in yields of 100 bp will change the bond’s price by approximately 3.7% in opposite direction
- Basis Point Value: $0.0398 [= (3.704)(.0001)(107.582)]
- This means that a one bp change in yields will cause the bond’s price to move by about 4 cents per $100 of par value (which would correspond to a 40 cent movement for a bond with a par value of $1000)

Convexity Trades: An Example(Source: R. Dattareya and F. Fabbozi)

- Consider the following hypothetical U.S. Treasury bonds:
- Consider two different bond portfolios:
- Bullet Portfolio: 100% of Bond C
- Barbell Portfolio: 50.2% of Bond A, 49.8% of Bond B
- Notice the following:
- Duration of Barbell: (.502)(4.005)+(.498)(8.882) = 6.434
- Same as Bullet Portfolio
- Convexity of Barbell: (.502)(19.82)+(.498)(124.17) = 71.7846
- Greater than Bullet Portfolio

Relative Performance (Bullet Rtn – Barbell Rtn) Over Six-Month Period

Bond Swaps

- Another type of active trade is a bond swap. This involves liquidating a current position and simultaneously buying a different issue in its place with similar attributes, but a chance of improved returns.
- Notable examples of bond swaps include:
- Pure Yield Pickup Swaps: Swapping out of a low-coupon bond into a comparable higher-coupon bond to realize an automatic and instantaneous increase in current yield and yield to maturity.
- Substitution Swaps: Swapping comparable bonds that are trading at different yields; based on the premise that the credit market is temporarily out of balance.
- Tax Swaps: Trades motivated by prevailing tax codes and accumulated capital gains in a portfolio (e.g., selling a bond with a capital loss to offset one with a capital gain).

Bond Swap Example

- Evaluate the following pure yield pickup swap: You are currently holding a 20-year, Aa-rated, 9.0 percent coupon bond priced to yield 11.0 percent.
- As a swap candidate, you are considering a 20-year, Aa-rated, 11.0 percent coupon bond priced to yield 11.5 percent
- You can assume that all cash flows are reinvested at 11.5 percent.

An Overview of Equity Alternatives

- As we have seen, debt and equity securities are the fundamental cornerstones of the capital markets. They represent the most prevalent securities that companies use to raise external funds and that investors purchase to hold in their portfolios.
- Often, however, there will be cases when either investors or issuers will want to do a transaction involving securities with an equity-like payoff structure, but they may choose not (or otherwise be unable) to use “plain vanilla” equity directly. Some reasons why conventional stock shares may not be appropriate even when an equity payoff is desired include:
- A corporation seeking to raise additional capital may find the market for its common stock to be unreceptive, perhaps due to other recent issuances.
- An institutional investor may be restricted from holding equity directly but can purchase a debt instrument with a equity-like principal payoff at maturity.
- A company may be able to lower the present cost of a debt financing by structuring a bond contract that allows investors the right to convert the debt into common equity at a future date.
- We will look at two alternative forms of equity along these lines: (i) convertible securities, and (ii) structured notes

Notion of Convertible Bonds

- A convertible bond can be viewed as a pre-packaged portfolio containing two distinct securities: (i) a regular bond and (i) an option to exchange the bond for a pre-specified number of shares of the issuing firm’s common stock. Thus, a convertible bond represents a hybrid investment involving elements of both the debt and equity markets.
- The option involved can be viewed as either a put (i.e., the investor has the right to sell the bond back to the issuer and receive a fixed number of shares) or a call (i.e., the investor can buy a fixed number of shares from the issuing company, paid for with the bond).
- From the investor’s standpoint, there are both advantages and disadvantages to this packaging. Specifically, while buyer receives equity-like returns with a “guaranteed” terminal payoff equal to the bond’s face value, he or she must also pay the option premium, which is embedded in the price of the security.
- Conversely, the issuer of a convertible bond increases the company’s leverage while providing a potential source of equity financing in the future. This arrangement may be particularly useful as a means for low-rated issuers to borrow money more cheaply in the present than with a “straight” debt issue while creating a potential demand for their shares if future conditions are favorable.

Convertible Bond Example: Cypress Semiconductor

- As an example of how one such issue is structured and priced, consider the 4.00 percent coupon convertible subordinated notes (“sub cv nt”) maturing in February of 2005 issued by a NYSE-traded company, Cypress Semiconductor Corporation (CY). Cypress Semiconductor designs, develops, manufactures and markets a broad line of high-performance digital and mixed-signal integrated circuits for a range of markets, including data communications, telecommunications, computers and instrumentation systems.
- The Bloomberg screen on the next slide shows the issue’s CUSIP identifier, contract terms and default rating, (i.e., B1), and indicates that this bond pays interest semi-annually on February 1 and August 1. The bond issue has $283 million outstanding and is callable at 101 percent of par.
- At the time of this report (i.e. February 2001), the listed price of the convertible was 92 percent of par and the price of Cypress Semiconductor common stock was 27.375 per share.

CY Convertible Bond Example (cont.)

- As spelled out at the top of this display, each $1,000 face value of this bond can be converted into 21.6216 shares of Cypress Semiconductor common stock. This statistic is called the instrument’s conversion ratio. At the current share price of $27.375, an investor exercising her conversion option would have received only $591.89 (= $27.375 21.6216) worth of stock, an amount considerably below the current market value of the bond.
- In fact, the conversion parity price (i.e. the common stock price at which immediate conversion would make sense) is equal to $42.55, which is the bond price of $920 divided by the conversion ratio of 21.6216. The prevailing market price of 27.375 is far below this parity level, meaning that the conversion option is currently out of the money. Of course, if the conversion parity price ever fell below the market price for the common stock, an astute investor could buy the bond and immediately exchange it into stock with a greater market value.

CY Convertible Bond Example (cont.)

- Most convertible bonds are also callable by the issuer. Of course, a firm will never call a bond selling for less than its call price (which is the case with the Cypress Semiconductor note). In fact, firms often wait until the bond is selling for significantly more than its call price before calling it. If the company calls the bond under these conditions, investors will have an incentive to convert the bond into the stock that is worth more than they would receive from the call price; this situation is referred to as forcing conversion.
- Two other factors also increase the investors’ incentive to convert their bonds. First, some instruments have conversion prices that step up over time according to a predetermined schedule. Since a stepped up conversion price leads to a lower number of shares received, it becomes more likely that investors will exercise their option just before the conversion price increases. Second, a firm can help to encourage conversion by increasing the dividends on the stock, thereby making the income generated by the shares more attractive relative to the income from the bond.

CY Convertible Bond Example (cont.)

- Another important characteristic when evaluating convertible bonds is the payback or break-even time, which measures how long the higher interest income from the convertible bond (compared to the dividend income from the common stock) must persist to make up for the difference between the price of the bond and its conversion value (i.e., the conversion premium). The calculation is as follows:
- For instance, the annual coupon yield payment on the Cypress Semiconductor convertible bond is $40, while the firm’s dividend yield is zero. Thus, assuming you sold the bond for 920 and used the proceeds to purchase 33.607 shares (= $920/$27.375) of Cypress Semiconductor stock, the payback period would be:

CY Convertible Bond Example (cont.)

- It is also possible to calculate the combined value of the investor’s conversion option and issuer’s call feature that are embedded in the note. In the Cypress Semiconductor example, with a market price of $920, the convertible’s yield-to-maturity can be calculated as the solution to:
- or y = 6.29 percent. This computation assumes 8 semi-annual coupon payments of $20 (= 40 2). Since the yield on a Cypress Semiconductor debt issue with no embedded options and the same (B1) credit rating and maturity was 8.5 percent, the present value of a “straight” fixed-income security with the same cash flows would be:
- This means that the net value of the combined options is $69.94, or $920 minus $850.06. Using the Black-Scholes valuation model, it is easily confirmed that a four-year call option to buy one share of Cypress Semiconductor stock – which does not pay a dividend – at an exercise price of $42.55 (i.e. the conversion parity value) is equal to $6.35. Thus the value of the investor’s conversion option – which allows for the acquisition of 21.6216 shares – must be $137.26 (= 21.6216 $6.35). This means that the value of the issuer’s call feature under these conditions must be $67.32 (= $137.26 – $69.94).

Notion of Structured Notes

- Generally speaking, structured notes are debt issues that have their principal or coupon payments linked to some other underlying variable. Examples would include bonds whose coupons are tied to the appreciation of an equity index such as the S&P 500 or a zero-coupon bond with a principal amount tied to the appreciation of an oil price index.
- There are several common features that distinguish structured notes from regular fixed-income securities, two of which are important for our discussion. First, structured notes are designed for (are targeted to) a specific investor with a very particular need. That is, these are not "generic" instruments, but products tailored to address an investor's special constraints, which are often themselves created by tax, regulatory, or institutional policy restrictions.
- Second, after structuring the financing to meet the investor's needs, the issuer will typically hedge that unique exposure with swaps or exchange-traded derivatives. Inasmuch as the structured note itself most likely required an embedded derivative to create the desired payoff structure for the investor, this unwinding of the derivative position by the issuer generates an additional source profit opportunity for the bond underwriter.

Equity Index-Linked Note Example: MITTS

- In July of 1992, Merrill Lynch & Co. raised USD 77,500,000 by issuing 7,750,000 units of an S&P 500 Market Index Target-Term Security, or "MITTS" for short, at a price of USD 10 per unit. These MITTS units had a maturity date of August 29, 1997, making them comparable in form to a five-year bond even though they traded on the New York Stock Exchange. Indeed, Merrill Lynch issued them as a series of Senior Debt Securities making no coupon payments prior to maturity.
- At maturity, a unit holder received the original issue price plus a "supplemental redemption amount," the value of which depended on where the Standard & Poor's 500 index settled relative to a predetermined initial level. Given that this supplemental amount could not be less than zero, the total payout to the investor at maturity can be written:
- where the initial S&P value was specified as 412.08.

MITTS Example (cont.)

- From the preceding description, recognize that the MITTS structure combines a five-year, zero-coupon bond with an S&P index call option, both of which were issued by Merrill Lynch. Thus, the MITTS investor essentially owns a "portfolio" that is: (i) long in a bond and (ii) long in an index call option position.
- This particular security was designed primarily for those investors who wanted to participate in the equity market but, for regulatory or taxation reasons, were not permitted to do so directly. For example, the manager of a fixed-income mutual fund might be able to enhance her return performance by purchasing this "bond" and then hoping for an appreciating stock market.
- Notice that the use of the call option in this design makes it fairly easy for Merrill Lynch to market to its institutional customers in that it is a "no lose" proposition; the worst-case scenario for the investor in that she simply gets her money back without interest in five years. (Of course, the customer does carry the company's credit risk for this period.) Thus, at origination the MITTS issue had no downside exposure to stock price declines.

MITTS Example (cont.)

- The call option embedded in this structure is actually a partial position. To see this, we can rewrite the option portion of the note's redemption value as:
- Thus, given that a regular index call option would have a terminal payoff of Max[0, (Final S&P – X)], where X is the strike price, the derivative in the MITTS represents 2.79% of this amount.

MITTS Example (cont.)

- On February 28, 1996, the closing price for the MITTS issue was USD 15.625, while the S&P 500 closed at 644.75. Further, the semi-annually compounded yield of a zero-coupon (i.e., "stripped") Treasury bond on this date was 5.35%.
- Assuming a credit spread of 30 basis points to be appropriate for Merrill Lynch's credit rating (i.e., A+ and A1 by Standard & Poor's and Moody's, respectively) and the remaining time to maturity (i.e., one-and-a-half years, or three half-years), the bond portion of the MITTS issue should be worth:
- This means that the investor is paying $6.43 (= 15.63 - 9.20) for the embedded index call.

MITTS Example (cont.)

- Without reproducing the full calculations, it is interesting to note that the theoretical value on February 28, 1996 of an S&P index call option expiring on August 29, 1997 with an exercise price of 412.08 is $243.19.
- Thus, since the MITTS option feature represents 0.0279 of this amount, the call option embedded in the MITTS issue is worth $6.78 (= 243.19 x 0.0279). Thus, on this particular date the MITTS issue was priced in the market below its theoretical value, presenting investors with a potential buying opportunity depending on their transaction costs. In fact, the embedded call is actually priced below the index option’s intrinsic value of $6.49 (= [644.75 – 412.08] x 0.0279), making the issue that much more attractive to investors.

MITTS Example (cont.)

- This MITTS transaction can be illustrated as follows:

Max(0,% SPX Rtn)

$10

August 1992

February 1996

August 1997

Zero-Coupon Bond

$9.20

$10

SPX Index Call Option

$6.43

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