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ERIC WANG
Chapter101.pptx
10.Bond prices and yields
Instructor: Seongcheol Paeng
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Table
10.1 Bond Characteristics
10.2 Bond Pricing
10.3 Bond Yields
10.4 Bond Prices Over Time
10.5 Default Risk and Bond Pricing
10.6 the Yield Curve
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10.1 Bond Characteristics
A bond is a security that is issued in connection with a borrowing arrangement. The borrower issues (i.e., sells) a bond to the lender for some amount of cash; the bond is in essence the “IOU” (I owe you) of the borrower.
The arrangement obligates the issuer to make specified payments to the bondholder on specified dates. A typical coupon bond obligates the issuer to make semiannual payments of interest, called coupon payments, to the bondholder for the life of the bond.
To illustrate, a bond with a par value of $1,000 and a coupon rate of 8% might be sold to a buyer for $1,000. The issuer then pays the bondholder 8% of $1,000, or $80 per year, for the stated life of the bond, say, 30 years.
The $80 payment typically comes in two semiannual installments of $40 each. At the end of the 30-year life of the bond, the issuer also pays the $1,000 par value to the bondholder.
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10.1 Bond Characteristics
Sometimes, however, zero-coupon bonds are issued that make no coupon payments. In this case, investors receive par value at the maturity date but receive no interest payments until then: The bond has a coupon rate of zero.
These bonds are issued at prices considerably below par value, and the investor’s return comes solely from the difference between issue price and the payment of par value at maturity.
Treasury Bonds and Notes
Figure 10.1 is an excerpt from the listing of Treasury issues from the The Wall Street Journal Online. Treasury notes are issued with original maturities between 1 and 10 years, while Treasury bonds are issued with maturities ranging from 10 to 30 years.
Both bonds and notes may be purchased directly from the Treasury in denominations of only $100, but denominations of $1,000 are far more common. Both make semiannual coupon payments.
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10.1 Bond Characteristics
Treasury Bonds and Notes
The highlighted issue in Figure 10.1 matures in August 2023.
Its coupon rate is 6.25%. Par value is $1,000; thus, the bond pays interest of $62.50 per year in two semiannual payments of $31.25.
Payments are made in February and August of each year.
Although bonds are typically sold in denominations of $1,000 par value, the bid and ask prices are quoted as a percentage of par value.
Therefore, the ask price is 132.992% of par value, or $1,329.92. The minimum price increment, or tick size, in The Wall Street Journal listing is 1/128, so this bond may also be viewed as selling for % of par value.
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Figure 10.1 Prices and yields of U.S. Treasury bonds on August 15, 2014
10.1 Bond Characteristics
Treasury Bonds and Notes
Accrued Interest and Quoted Bond Prices:
The bond prices that you see quoted in the financial pages are not actually the prices that investors pay for the bond. This is because the quoted price does not include the interest that accrues between coupon payment dates.
Example 10.1 Suppose that the coupon rate is 8%.
Then the semiannual coupon payment is $40.
Because 30 days have passed since the last coupon payment, the accrued interest on the bond is $40 × = $6.59.
If the quoted price of the bond is $990, then the invoice price will be $990 + $6.59 = $996.59.
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10.1 Bond Characteristics
Corporate Bonds
Like the government, corporations borrow money by issuing bonds. Figure 10.2 is a sample of corporate bond listings. Although some bonds trade electronically on the NYSE Bonds platform, most bonds are traded over the counter in a network of bond dealers linked by a computer quotation system.
The “Rating” column is the estimation of bond safety given by the three major bond rating agencies, Moody’s, Standard & Poor’s, and Fitch. Bonds with A ratings are safer than those rated B or below. As a general rule, safer bonds with higher ratings promise lower yields to maturity.
Call Provisions on Corporate Bonds: Some corporate bonds are issued with call provisions, allowing the issuer to repurchase the bond at a specified call price before the maturity date.
For example, if a company issues a bond with a high coupon rate when market interest rates are high, and interest rates later fall, the firm might like to retire the high-coupon debt and issue new bonds at a lower coupon rate to reduce interest payments.
The proceeds from the new bond issue are used to pay for the repurchase of the existing higher-coupon bonds at the call price. This is called refunding. Callable bonds typically come with a period of call protection, an initial time during which the bonds are not callable.
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Figure 10.2 Listing of corporate bonds
10.1 Bond Characteristics
Corporate Bonds
To compensate investors for this risk, callable bonds are issued with higher coupons and promised yields to maturity than noncallable bonds.
Convertible Bonds: A bond with an option allowing the bondholder to exchange the bond for a specified number of shares of common stock in the firm.
Puttable Bonds: While the callable bond gives the issuer the option to extend or retire the bond at the call date, the extendable or put bond gives this option to the bondholder.
Floating-Rate Bonds: Floating-rate bonds make interest payments that are tied to some measure of current market rates. For example, the rate might be adjusted annually to the current T-bill rate plus 2%. If the one-year T-bill rate at the adjustment date is 4%, the bond’s coupon rate over the next year would then be 6%.
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10.1 Bond Characteristics
Preferred Stock
Although preferred stock strictly speaking is considered to be equity, it often is included in the fixed-income universe. This is because, like bonds, preferred stock promises to pay a specified stream of dividends. However, unlike bonds, the failure to pay the promised dividend does not result in corporate bankruptcy.
Other Domestic Issuers
For example, state and local governments issue municipal bonds. Government agencies, such as the Federal Home Loan Bank Board, the Farm Credit agencies, and the mortgage pass-through agencies Ginnie Mae, Fannie Mae, and Freddie Mac also issue considerable amounts of bonds.
International Bonds
International bonds are commonly divided into two categories: foreign bonds and Eurobonds.
Foreign bonds are issued by a borrower from a country other than the one in which the bond is sold. The bond is denominated in the currency of the country in which it is marketed.
A Eurobond is a debt instrument that's denominated in a currency other than the home currency of the country or market in which it is issued.
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10.1 Bond Characteristics
Innovation in the Bond Market
Issuers constantly develop innovative bonds with unusual features; these issues illustrate that bond design can be extremely flexible.
INVERSE FLOATERS These are similar to the floating-rate bonds we described earlier, except that the coupon rate on these bonds falls when the general level of interest rates rises.
ASSET-BACKED BONDS Miramax has issued bonds with coupon rates tied to the financial performance of Pulp Fiction and other films.
PAY-IN-KIND BONDS Issuers of pay-in-kind bonds may choose to pay interest either in cash or in additional bonds. If the issuer is short on cash, it will likely choose to pay with new bonds rather than scarce cash.
CATASTROPHE BONDS Oriental Land Co., which manages Tokyo Disneyland, has issued bonds with a final payment that depends on whether there has been an earthquake near the park.
More recently, FIFA (the Fédération Internationale de Football Association) issued catastrophe bonds with payments that would have been halted if terrorism forced the cancellation of the 2006 World Cup.
These bonds are a way to transfer “catastrophe risk” from insurance companies to the capital markets.
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10.1 Bond Characteristics
Innovation in the Bond Market
INDEXED BONDS Indexed bonds make payments that are tied to a general price index or the price of a particular commodity.
The United States Treasury started issuing such inflation-indexed bonds in January 1997. They are called Treasury Inflation Protected Securities (TIPS).
By tying the par value of the bond to the general level of prices, coupon payments, as well as the final repayment of par value, on these bonds increase in direct proportion to the consumer price index.
Therefore, the interest rate on these bonds is a risk-free real rate.
To illustrate how TIPS work, consider a newly issued bond with a three-year maturity, par value of $1,000, and a coupon rate of 4%.
Assume that inflation turns out to be 2%, 3%, and 1% in the next three years. Table 10.1 shows how the bond cash flows will be calculated.
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10.1 Bond Characteristics
Innovation in the Bond Market
The nominal rate of return on the bond in the first year is
Nominal return = (Interest + Price appreciation) / Initial price
= (40.80 + 20) / 1,000 = 6.08%
Real return = (1 + Nominal return) / (1 + Inflation) − 1
= 1.0608 / 1.02 − 1 = .04, or 4%
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10.2 Bond Pricing
Bond value = Present value of coupons + Present value of par value
If we call the maturity date T and call the discount rate r, the bond value can be written as
(10.1)
(10.2)
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10.2 Bond Pricing
Example 10.2 We discussed earlier an 8% coupon, 30-year maturity bond with par value of $1,000 paying 60 semiannual coupon payments of $40 each. Suppose that the interest rate is 8% annually, or r = 4% per six-month period. Then the value of the bond can be written as
n=60: The bond has a maturity of 30 years, so it makes 60 semiannual payments.
i=4: The semiannual interest rate is 4%.
FV=1,000: The bond will provide a one-time cash flow of $1,000 when it matures.
PMT=40: Each semiannual coupon payment is $40.
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10.2 Bond Pricing
Example 10.2 We discussed earlier an 10% coupon, 30-year maturity bond with par value of $1,000 paying 60 semiannual coupon payments of $50 each. Suppose that the interest rate is 10% annually, or r = 5% per six-month period. Then the value of the bond can be written as
n=60: The bond has a maturity of 30 years, so it makes 60 semiannual payments.
i=5: The semiannual interest rate is 5%.
FV=1,000: The bond will provide a one-time cash flow of $1,000 when it matures.
PMT=50: Each semiannual coupon payment is $40.
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10.2 Bond Pricing
Figure 10.3 shows the price of the 30-year, 8% coupon bond for a range of interest rates including 8%, at which the bond sells at par, and 10%, at which it sells for $810.71.
The negative slope illustrates the inverse relationship between prices and yields.
The shape of the curve in Figure 10.3 implies that an increase in the interest rate results in a price decline that is smaller than the price gain resulting from a decrease of equal magnitude in the interest rate.
This property of bond prices is called convexity because of the convex shape of the bond price curve. This curvature reflects the fact that progressive increases in the interest rate result in progressively smaller reductions in the bond price.
Therefore, the price curve becomes flatter at higher interest rates.
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Figure 10.3 The inverse relationship between bond prices and yields: Price of an 8% coupon bond with 30-year maturity making semiannual coupon payments
10.2 Bond Pricing
As a general rule, keeping all other factors the same, the longer the maturity of the bond, the greater the sensitivity of its price to fluctuations in the interest rate.
For example, consider Table 10.2, which presents the price of an 8% coupon bond at different market yields and times to maturity.
For any departure of the interest rate from 8% (the rate at which the bond sells at par value), the change in the bond price is greater for longer times to maturity.
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10.2 Bond Pricing
Bond Pricing between Coupon Dates
As we pointed out earlier, bond prices are typically quoted net of accrued interest.
These prices, which appear in the financial press, are called flat prices.
The actual invoice price that a buyer pays for the bond includes accrued interest.
Thus, Invoice price = Flat price + Accrued interest
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10.3 Bond Yields
EXAMPLE 10.4 Yield to Maturity Suppose an 8% coupon, 30-year bond is selling at $1,276.76.
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10.3 Bond Yields
EXAMPLE 10.5
If the coupons were paid only annually, we would change the entry for payments per year to 1 (see cell D8), and the yield would fall slightly to 5.99%.
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10.3 Bond Yields
Yield to maturity differs from the current yield of a bond, which is the bond’s annual coupon payment divided by the bond price.
For example, for the 8%, 30-year bond currently selling at $1,276.76, the current yield would be $80/$1,276.76 = .0627, or 6.27% per year.
In contrast, recall that the effective annual yield to maturity is 6.09%.
For this bond, which is selling at a premium over par value ($1,276 rather than $1,000), the coupon rate (8%) exceeds the current yield (6.27%), which exceeds the yield to maturity (6.09%).
This example illustrates a general rule: For premium bonds (bonds selling above par value), coupon rate is greater than current yield, which in turn is greater than yield to maturity. For discount bonds (bonds selling below par value), these relationships are reversed.
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10.3 Bond Yields
Figure 10.4 illustrates the risk of call to the bondholder.
The colored line is the value of a “straight” (that is, noncallable) bond with par value of $1,000, an 8% coupon rate, and a 30-year time to maturity as a function of the market interest rate.
If interest rates fall, the bond price, which equals the present value of the promised payments, can rise substantially.
Now consider a bond that has the same coupon rate and maturity date but is callable at 110% of par value, or $1,100.
When interest rates fall, the present value of the bond’s scheduled payments rises, but the call provision allows the issuer to repurchase the bond at the call price.
If the call price is less than the present value of the scheduled payments, the issuer can call the bond at the expense of the bondholder.
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Figure 10.4 Bond prices: Callable and straight debt. Coupon = 8%; maturity = 30 years; semiannual payments
10.3 Bond Yields
The dark line in Figure 10.4 is the value of the callable bond.
At high market interest rates, the risk of call is negligible because the present value of scheduled payments is substantially less than the call price; therefore, the values of the straight and callable bonds converge.
At lower rates, however, the values of the bonds begin to diverge, with the difference reflecting the value of the firm’s option to reclaim the callable bond at the call price. At very low market rates the present value of schedule payments significantly exceeds the call price, so the bond is called. Its value at this point is simply the call price, $1,100.
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Figure 10.4 Bond prices: Callable and straight debt. Coupon = 8%; maturity = 30 years; semiannual payments
10.3 Bond Yields
Example 10.7 Yield to Call
Suppose the 8% coupon, 30-year maturity bond sells for $1,150 and is callable in 10 years at a call price of $1,100. Its yield to maturity and yield to call would be calculated using the following inputs: Yield to call is then 6.64%. Yield to maturity is 6.82%
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10.3 Bond Yields
Realized Compound Return versus Yield to Maturity
Consider for example, a two-year bond selling at par value paying a 10% coupon once a year. The yield to maturity is 10%. If the $100 coupon payment is reinvested at an interest rate of 10%, the $1,000 investment in the bond will grow after two years to $1,210, as illustrated in Figure 10.5, Panel A.
The coupon paid in the first year is reinvested and grows with interest to a second-year value of $110, which, together with the second coupon payment and payment of par value in the second year, results in a total value of $1,210.
To summarize, the initial value of the investment is $1,000. The final value in two years is = $1,210. The compound rate of return, therefore, is calculated as follows.
$1,000 =$1,210
r = .10 = 10%
With a reinvestment rate equal to the 10% yield to maturity, the realized compound return equals yield to maturity.
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Figure 10.5 Bond prices: Growth of invested funds. In Panel A, interest payments are reinvested at 10%, the bond’s yield to maturity. In Panel B, the reinvestment rate is only 8%.
10.3 Bond Yields
Realized Compound Return versus Yield to Maturity
Example 10.8 If the interest rate earned on the first coupon is less than 10%, the final value of the investment will be less than $1,210, and the realized compound yield will be less than 10%.
Suppose the interest rate at which the coupon can be invested is only 8%. The following calculations are illustrated in Panel B of Figure 10.5.
The realized compound return is the compound rate of growth of invested funds, assuming that all coupon payments are reinvested.
The investor purchased the bond for par at $1,000, and this investment grew to $1,208.
$1,000= $1,208
r = .0991 = 9.91%
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Figure 10.5 Bond prices: Growth of invested funds. In Panel A, interest payments are reinvested at 10%, the bond’s yield to maturity. In Panel B, the reinvestment rate is only 8%.
10.3 Bond Yields
Realized Compound Return versus Yield to Maturity
Horizon Analysis: Analysis of bond returns over a multiyear horizon, based on forecasts of the bond’s yield to maturity and the reinvestment rate of coupons.
Example 10.9 Suppose you buy a 30-year, 7.5% (annual payment) coupon bond for $980 (when its yield to maturity is 7.67%) and you plan to hold it for 20 years. Your forecast is that the bond’s yield to maturity will be 8% when it is sold and that the reinvestment rate on the coupons will be 6%.
+…+
$966.45 + $2,758.92 = $3,725.37. This corresponds to an annualized compound return of 6.90%:
$980 = $3,725.37
r = .0690 = 6.90%
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10.4 Bond Prices Over Time
Example 10.10
That present value is
$70 × Annuity factor(8%, 3) + $1,000 × PV factor(8%, 3) = $974.23
which is less than par value. In another year, after the next coupon is paid and remaining maturity falls to two years, the bond would sell at
$70 × Annuity factor(8%, 2) + $1,000 × PV factor(8%, 2) = $982.17
thereby yielding a capital gain over the year of $7.94.
If an investor had purchased the bond at $974.23, the total return over the year would equal the coupon payment plus capital gain, or $70 + $7.94 = $77.94.
This represents a rate of return of $77.94/$974.23, or 8%, exactly the current rate of return available elsewhere in the market.
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10.4 Bond Prices Over Time
Figure 10.6 traces out the price paths (net of accrued interest) of two bonds, one with a coupon above the yield to maturity and another with a coupon below yield to maturity, as time to maturity approaches, at least for the case in which the market interest rate is constant.
The low-coupon bond enjoys capital gains as price steadily approaches par value, while the high-coupon bond suffers capital losses.
Clearly, longer-term bonds at this time offered higher promised yields, a common pattern, and one that reflects the relative risks of the bonds.
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Figure 10.6 Price path of two 30-year maturity bonds each selling at a yield to maturity of 8%. Bond price approaches par value as maturity date approaches.
10.4 Bond Prices Over Time
Yield to Maturity versus Holding-Period Return
Example 10.11 Consider a 30-year bond paying an annual coupon of $80 and selling at par value of $1,000. The bond’s initial yield to maturity is 8%.
If the yield remains at 8% over the year, the bond price will remain at par, so the holding-period return also will be 8%. But if the yield falls below 8%, the bond price will increase.
Suppose the yield falls and the price increases to $1,050. Then the holding-period return is greater than 8%:
Holding-period return = {$80 + ($1,050 − $1,000)} / $1,000 = .13 or 13%
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10.4 Bond Prices Over Time
Zero-Coupon Bonds and Treasury STRIPS
The Treasury program under which coupon stripping is performed is called STRIPS (Separate Trading of Registered Interest and Principal of Securities), and these zero-coupon securities are called Treasury strips.
To illustrate, consider a zero with 30 years until maturity, and suppose the market interest rate is 10% per year. The price of the bond today is $1,000/ = $57.31. Next year, with only 29 years until maturity, if the yield to maturity is still 10%, the price will be $1,000/ = $63.04, a 10% increase over its previous-year value.
Because the par value of the bond is now discounted for one less year, its price has increased by the one-year discount factor.
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Figure 10.6 Price path of two 30-year maturity bonds each selling at a yield to maturity of 8%. Bond price approaches par value as maturity date approaches.
10.4 Bond Prices Over Time
Zero-Coupon Bonds and Treasury STRIPS
Figure 10.7 presents the price path of a 30-year zero-coupon bond for an annual market interest rate of 10%. The bond’s price rises exponentially, not linearly, until its maturity.
Exponential Function
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Figure 10.6 Price path of two 30-year maturity bonds each selling at a yield to maturity of 8%. Bond price approaches par value as maturity date approaches.
10.4 Bond Prices Over Time
After-Tax Returns
Example 10.12 If the interest rate originally is 10%, the 30-year zero would be issued at a price of $1,000/ = $57.31.
The following year, the IRS calculates what the bond price would be if its yield were still 10%. This is $1,000/ = $63.04.
Therefore, the IRS imputes interest income of $63.04 − $57.31 = $5.73. This amount is subject to tax.
Notice that the imputed interest income is based on a “constant yield method” that ignores any changes in market interest rates.
If interest rates actually fall, let’s say to 9.9%, the bond price will be $1,000/ = $64.72.
If the bond is sold, then the difference between $64.72 and $63.04 will be treated as capital gains income and taxed at the capital gains tax rate.
If the bond is not sold, then the price difference is an unrealized capital gain and does not result in taxes in that year.
In either case, the investor must pay taxes on the $5.73 of imputed interest at the ordinary income tax rate.
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10.5 Default Risk and Bond Pricing
If the company goes bankrupt, the bondholders will not receive all the payments they have been promised. Therefore, the actual payments on these bonds are uncertain, for they depend to some degree on the ultimate financial status of the firm.
Those rated BBB or above (S&P, Fitch) or Baa and above (Moody’s) are considered investment grade bonds, while lower-rated bonds are classified as speculative grade or junk bonds.
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10.5 Default Risk and Bond Pricing
Junk Bonds
At the height of Drexel’s difficulties, the high-yield bond market nearly dried up but eventually rebounded dramatically.
However, the average credit quality of newly issued high-yield debt today is higher than the average quality in the boom years of the 1980s.
Of course, junk bonds are more vulnerable to financial distress than investment grade bonds.
During the financial crisis of 2008–2009, prices on these bonds fell dramatically, and their yields to maturity rose equally dramatically.
The spread between yields on Treasury bonds and B-rated bonds widened from around 3% in early 2007 to an astonishing 19% by the start of 2009.
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10.5 Default Risk and Bond Pricing
Determinants of Bond Safety
1. Coverage ratios. Ratios of company earnings to fixed costs. For example, the times-interest earned ratio is the ratio of earnings before interest payments and taxes to interest obligations. The fixed-charge coverage ratio includes lease payments and sinking fund payments with interest obligations to arrive at the ratio of earnings to all fixed cash obligations. Low or falling coverage ratios signal possible cash flow difficulties.
2. Leverage ratio (Debt-to-equity ratio). A too-high leverage ratio indicates excessive indebtedness, signaling the possibility the firm will be unable to earn enough to satisfy the obligations on its bonds.
3. Liquidity ratios. The two common liquidity ratios are the current ratio (current assets/ current liabilities) and the quick ratio (current assets excluding inventories/current liabilities). These ratios measure the firm’s ability to pay bills coming due with its most liquid assets.
4. Profitability ratios. Measures of rates of return on assets or equity. Profitability ratios are indicators of a firm’s overall performance. The return on assets (earnings before interest and taxes divided by total assets) or return on equity (net income/equity) are the most popular of these measures. Firms with higher return on assets or equity should be better able to raise money in security markets because they offer prospects for better returns on the firm’s investments.
5. Cash flow-to-debt ratio. This is the ratio of total cash flow to outstanding debt.
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10.5 Default Risk and Bond Pricing
Determinants of Bond Safety
Standard & Poor’s periodically computes median values of selected ratios for firms in several rating classes, which we present in Table 10.3.
Of course, ratios must be evaluated in the context of industry standards, and analysts differ in the weights they place on particular ratios.
Nevertheless, Table 10.3 demonstrates the tendency of ratios to improve along with the firm’s rating class.
EBIT: Earnings Before Interest and Taxes
EBITDA: Earnings Before Interest and Taxes, Depreciation, Amortization
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10.5 Default Risk and Bond Pricing
Bond Indentures
In addition to specifying a payment schedule, the bond indenture, which is the contract between the issuer and the bondholder, also specifies a set of restrictions that protect the rights of the bondholders.
SINKING FUNDS: Bonds call for the payment of par value at the end of the bond’s life. This payment constitutes a large cash commitment for the issuer. To help ensure that the commitment does not create a cash flow crisis, the firm may agree to establish a sinking fund to spread the payment burden over several years.
SUBORDINATION OF FURTHER DEBT: One of the factors determining bond safety is the total outstanding debt of the issuer. If you bought a bond today, you would be understandably distressed to see the firm tripling its outstanding debt tomorrow. Your bond would be riskier than it appeared when you bought it. To prevent firms from harming bondholders in this manner, subordination clauses restrict the amount of their additional borrowing.
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10.5 Default Risk and Bond Pricing
Bond Indentures
DIVIDEND RESTRICTIONS: Covenants also limit the dividends firms may pay. These limitations protect the bondholders because they force the firm to retain assets rather than pay them out to stockholders. A typical restriction disallows payments of dividends if cumulative dividends paid since the firm’s inception exceed cumulative retained earnings plus proceeds from sales of stock.
COLLATERAL Some bonds are issued with specific collateral behind them. Collateral is a particular asset that the bondholders receive if the firm defaults. If the collateral is property, the bond is called a mortgage bond. Collateralized bonds generally are considered safer than general debenture bonds, which are unsecured, meaning they do not provide for specific collateral; credit risk of unsecured bonds depends on the general earning power of the firm.
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10.5 Default Risk and Bond Pricing
Bond Indentures
Figure 10.9 shows the terms of a bond issued by Mobil as described in Moody’s Industrial Manual.
The terms of the bond are typical and illustrate many of the indenture provisions we have mentioned.
The bond is registered and listed on the NYSE. It was issued in 1991, but it was not callable until 2002.
Although the call price started at 105.007% of par value, it falls gradually until it reaches par after 2020.
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10.5 Default Risk and Bond Pricing
Yield to Maturity and Default Risk
Example 10.13 Suppose a firm issued a 9% coupon bond 20 years ago. The bond now has 10 years left until its maturity date, but the firm is having financial difficulties.
Investors believe that the firm will be able to make good on the remaining interest payments but that at the maturity date, the firm will be forced into bankruptcy and bondholders will receive only 70% of par value. The bond is selling at $750. Yield to maturity (YTM) would then be calculated using the inputs shown in the following table.
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10.5 Default Risk and Bond Pricing
Yield to Maturity and Default Risk
The yield to maturity based on promised payments is 13.7%. Based on the expected payment of $700 at maturity, however, the yield would be only 11.6%. The stated yield to maturity is greater than the yield to maturity investors actually expect to receive.
Example 10.13 suggests that when a bond becomes more subject to default risk, its price will fall, and therefore its promised yield to maturity will rise. Therefore, the default premium, the spread between the stated yield to maturity and that on otherwise comparable Treasury bonds, will rise.
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10.5 Default Risk and Bond Pricing
Yield to Maturity and Default Risk
Example 10.14 Suppose that the condition of the firm in Example 10.13 deteriorates further, and investors now believe that the bond will pay off only 55% of face value at maturity.
Investors now demand an expected yield to maturity of 12% (i.e., 6% semiannually), which is .4% higher than in Example 10.13.
But the price of the bond will fall from $750 to $688 [n = 20; i = 6; FV = 550; PMT = $45].
At this price, the stated yield to maturity based on promised cash flows is 15.2%. While the expected yield to maturity has increased by .4%, the drop in price has caused the promised yield to maturity (and the default premium) to rise by 1.5%.
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10.5 Default Risk and Bond Pricing
Yield to Maturity and Default Risk
To compensate for the possibility of default, corporate bonds must offer a default premium. The default premium is the difference between the promised yield on a corporate bond and the yield of an otherwise identical government bond that is riskless in terms of default.
The pattern of default premiums offered on risky bonds is sometimes called the risk structure of interest rates. The greater the default risk, the higher the default premium.
Figure 10.10 shows spreads between yields to maturity of bonds of different risk classes since 1997. You can see here clear evidence of default-risk premiums on promised yields. Notice, for example, the incredible run-up of credit spreads during the crisis of 2008–2009.
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Figure 10.10 Yield spreads between corporate and 10-year Treasury bonds
10.5 Default Risk and Bond Pricing
Credit Default Swaps
A credit default swap (CDS) is in effect an insurance policy on the default risk of a bond or loan.
This insight suggests how CDS contracts should be priced. If a BB-rated bond bundled with insurance via a CDS is effectively equivalent to a AAA-rated bond, then the fair price of the swap ought to approximate the yield spread between AAA-rated and BB-rated bonds. The risk structure of interest rates and CDS prices ought to be tightly aligned.
Figure 10.11 shows the prices of five-year CDS contracts on Greek government debt between 2009 and 2010 as well as the spread between yields on Greek and German government bonds. As expected, the credit spread and the CDS prices move almost in lockstep.
You can see in Figure 10.11 that both the credit spread and CDS price started to increase dramatically toward the end of 2009.
As perceptions of Greece’s credit risk increased, so did the price of insuring its debt. Ultimately, in what amounted to the largest-ever sovereign default, lenders agreed in 2012 to reduce Greece’s debt by around $130 billion.
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Figure 10.11 Prices of five-year credit default swaps: On May 3, 2010, Greek Government formally asks IMF for a Stand-By Arrangement agreement.
10.6 The Yield Curve
The graphical relationship between the yield to maturity and the term to maturity is called the yield curve.
The relationship also is called the term structure of interest rates because it relates yields to maturity to the term (maturity) of each bond.
Four such sets of curves are reproduced in Figure 10.12. Figure 10.12 illustrates that a wide range of yield curves may be observed in practice.
Panel A is an essentially flat yield curve.
Panel B is an upward-sloping curve, and Panel C is a downward-sloping, or “inverted,” yield curve.
Finally, the yield curve in Panel D is hump-shaped, first rising and then falling.
Rising yield curves are most commonly observed.
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10.6 The Yield Curve
The Expectations Theory
Suppose everyone in the market believes firmly that while the current one-year interest rate is 8%, the interest rate on one-year bonds next year will rise to 10%.
This notion is the essence of the expectations hypothesis of the yield curve, which asserts that the slope of the yield curve is attributable to expectations of changes in short-term rates.
Expectations Hypothesis: The theory that yields to maturity are determined solely by expectations of future short-term interest rates.
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Figure 10.13 Returns to two two-year investment strategies
10.6 The Yield Curve
The Liquidity Preference Theory
Liquidity Preference Theory: The theory that investors demand a risk premium on long-term bonds.
Liquidity Premium: The extra expected return demanded by investors as compensation for the greater risk of longer-term bonds.
A Synthesis
Of course, we do not need to make an either/or choice between expectations and risk premiums. Both influence the yield curve, and both should be considered in interpreting it.
Figure 10.14 shows two possible yield curves. In Panel A, rates are expected to rise over time. This fact, together with a liquidity premium, makes the yield curve steeply upward-sloping.
In Panel B, rates are expected to fall, which by itself would make the yield curve slope downward. However, the liquidity premium lends something of an upward slope. The net effect of these two opposing factors is a “hump-shaped” curve.
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Figure 10.14 Illustrative yield curves Panel A: Increasing expected short rates combined with increasing liquidity premium. The result is a sharply rising yield curve. Panel B: Declining expected short rates combined with constant liquidity premium. The result is a hump-shaped yield curve.
10.6 The Yield Curve
A Synthesis
These two examples make it clear that the combination of varying expectations and liquidity premiums can result in a wide array of yield-curve profiles.
For example, an upward-sloping curve does not in and of itself imply expectations of higher future interest rates, because the slope can result either from expectations or from risk premiums.
A curve that is more steeply sloped than usual might signal expectations of higher rates, but even this inference is perilous.
Figure 10.15 presents yield spreads between 90-day T-bills and 10-year T-bonds since 1970. The figure shows that the yield curve is generally upward-sloping in that the longer-term bonds usually offer higher yields to maturity, despite the fact that rates could not have been expected to increase throughout the entire period.
This tendency is the empirical basis for the liquidity premium doctrine that at least part of the upward slope in the yield curve must be due to a risk premium.
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Figure 10.15 Term spread: Yields on 10-year versus 90-day Treasury securities
Assignments
Problem Sets (Paraphrase with your own words.)
Explain Bond Characteristics (general perspective).
We discussed earlier an 7% coupon, 30-year maturity bond with par value of $1,000 paying 60 semiannual coupon payments of $40 each. Suppose that the interest rate is 6% annually, or r = 3% per six-month period. Then the value of the bond can be written as
Deadline: 7/24
Submit it via email to [email protected]
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