Chapter 6Debt Valuation and  Interest Rates

Chapter 6 Outline

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6.1 The Basics of

Bonds• Bond

terminology• Security and

protective provisions in bonds

• Credit ratings

6.2 Bond Valuation

• Bond valuation

• Bond prices• Zero-coupon

bonds

6.3 Bond Yields

• Yield to maturity

• Yield to call• Current

yield

6.4 Government

Debt Obligations

• Sovereign debt ratings

• U.S. Treasury obligations

• Treasury bills

6.5 Interest Rates

• Base interest rates

• Fisher effect• Yield

Premiums

Definitions: Debt - a promise by the borrower to repay the

amount borrowed, plus interest. Bill or paper - a short-term debt obligation with

an initial maturity less than 1 year. Note - an obligation with maturity between 1 and

7 years. Bond - a debt obligation with a term longer than

7 years.

6.1 The Basics of Bonds

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The key feature of a bond is that the borrower typically agrees to pay the lender a regular series of cash payments, and to repay the full principal amount—that is, the amount borrowed—by the maturity date. We often refer to the borrower as the issuer of the bond and the

lender or creditor as the bondholder. These promises are stipulated in the bond contract and are a fixed

contractual commitment. The traditional bond provides for identical payments at regular intervals (usually semi-annually or annually), with the full principal to be repaid at the stated maturity date.

We refer to the interest payment as a coupon because at one time bonds literally had coupons attached to them, and the investor had to cut the coupon from the bond certificate and send it for payment.

The Basics of Bonds

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A typical bond has interest payments throughout its life and a principal repayment at maturity.

If the loan is amortized, you see the pattern in Panel A, and if the loan is a bond, you see the pattern in Panel B.

In the case of the amortized loan, principal is paid off over time, whereas in the case of the bond the principal is paid off at the bond’s maturity.

The Basics of Bonds

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Bond indenture or indenture agreement - a legal document held and administered by a trust company that specifies the payment requirements, all other salient matters relating to the issue (such as any assets that might serve as security or collateral for the bond), any protective provisions, and other additional features.

Par value (or face value or maturity value) - represents the amount that is paid at maturity for traditional bonds. The par value of most bonds is $1,000, though bond prices are typically quoted based on a par value of 100. In other words, if the price of a bond is quoted at 99.583, a $1,000 par value

bond would be selling for $995.83. Term to maturity of the bond - the time remaining to the maturity date Regular interest payment (or coupon) - determined by multiplying the

coupon rate (which is stated on an annual basis) by the par value of the bond. For example, a bond with a coupon rate of 6 percent and a par value of $1,000

would pay coupons of $60 if they are paid annually, or $30 every six months if they are paid semi-annually.

Most U.S. corporate bonds pay interest semi-annually; that is, every six months.

Bond Terminology

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A bond secured by real property is a mortgage bond, and a bond secured by a pledge of other financial assets, such as common shares, bonds, or Treasury bills, is a collateral trust bond.

An equipment trust certificate is secured by equipment, such as the rolling stock of a railway.

A debenture is a debt obligation that is not secured by a specific asset, but rather is secured by a general claim on the company’s unencumbered assets, that is, those assets that have not been pledged as security for other debt obligations.

A protective covenant is any clause in the bond indenture that restricts the actions of the issuer. A negative covenant prohibits certain actions; for example, a company may be

restricted from making a dividend payment larger than a certain amount or prevented from pledging its assets to another lender.

A positive covenant specifies actions that the company agrees to undertake, for example, to provide quarterly financial statements or maintain certain working capital levels.

Security and Protective Provisions in Bonds

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Credit-rating firms perform detailed analyses of bond issuers and issues to determine the likelihood that the issuer will make the payments of interest and principal repayments on a debt obligation when promised.

In the United States, these firms are known as Nationally Recognized Statistical Rating Organizations (NRSROs) and are registered with the Securities and Exchange Commission. The largest NRSROs are Standard & Poor’s (S&P), Moody’s, and Fitch, but

thanks to changes in the regulation of NRSROs, the field has widened and includes DBRS, Ltd.; A. M. Best Company, Inc.; and Kroll Bond Rating Agency, among others.

The issuer of a debt obligation contracts with one or more of these NRSROs for a rating based on the creditworthiness of the issuer, tempered by the features of the debt obligation.

Credit Ratings

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NRSROs use a classification system starting with the highest-rated debt, AAA or Aaa, and then lesser ratings are AA or Aa, A, BBB or Baa, and so on.

Credit Ratings

9Debt-Rating Categories for Standard & Poor’s and Moody’s

Sinking fund and purchase fund provision - requirement that the issuer set aside funds each year to be used to pay off the debt at maturity.

Embedded options - may provide the issuer more flexibility, the investor more flexibility, or simply make a bond more attractive to investors and, hence, lower the return required by investors. A callable bond, for example, gives the issuer the option to “call” or

repurchase outstanding bonds at a predetermined call price (generally at a premium over par) at specified times.

Floating-rate bonds - debt obligations that have adjustable coupons that are usually tied to some variable short-term rate.

Additional Bond Features

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The value of a straight bond is the present value of the future cash flows on the bond, which consist of interest payments and the par value repaid at maturity. We can use the following equation to value the bond:

In this equation, PV0 = bond value, PMTt = periodic interest (or coupon) payments, rb = bond discount rate (or market rate), N = number of periods remaining to maturity, and FVN = face (par) value of the bond.

6.2 Bond Valuation

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Problem:Consider a semi-annual bond with a face value of $1,000, matures in 5 years, has a Coupon rate of 6.5%, and is priced to yield 5%. The inputs are:

PMTt = ($1,000 × 0.065) ÷ 2 = $32.50 rb = 0.05 ÷ 2 = 0.025 or 2.5% N = 5 × 2 = 10 six-month periods FV = $1,000

Bond Valuation

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Solution:We can solve this as the sum of the parts (that is, the present value of the FV, plus the present value of the annuity of PMTt),

Bond Valuation

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Bond prices are typically quoted in terms of value per $100 of face value. For example, if a $1,000 face value bond sells at $990, its bond quote

is 99. As another example, if a $500 face value bond sells for $495, its bond

quote is $495 ÷ $500 = 99. Using bond quotes, we eliminate the need to know a bond’s face

value. Problem:Consider the bond that matures in 5 years, has a coupon rate of 6.5%, and is priced to yield 8%. What is the appropriate quote for this bond?

Bond Prices

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Solution:

You will notice that the only inputs changing are the coupon and the future value. In equation form, the value of the bond is:

The quote of 93.9, therefore, is stating the price as a percentage of face value. If the face value is $1,000, the value of the bond is $939.

Bond Prices

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Bond Value Illustration

16Value of a $1,000 Face Value Bond, with a 5% CouponMaturing in 15 Years, for Different Yields to Maturity

Zero-coupon bonds are easy to evaluate by using a variation of the equation used to value a bond in which we drop the first term because there are no interest payments to discount. This leaves us with the following:

Valuing Zero-Coupon Bonds

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Example:Consider a zero-coupon bond that has 5 years remaining to maturity. If the bond is priced to yield 6%, the value of the bond, using bond quote terms, is:

Using a calculator, the inputs are: FV= 100 i = 3% N = 10

Zero-Coupon Bonds

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Up until now, we have provided the market interest rate, the yield that is needed to value a bond today. However, different types of yields are actually associated with bonds: Yield to maturity - the yield that an investor would realize if he or

she bought the bond at the current price, held it to maturity, received all the promised payments on their scheduled dates, and reinvested all the cash flows received at this yield.

Yield to call - the return on a callable bond, assuming that the bond is called away before maturity.

Current yield - the ratio of the annual coupon interest divided by the current market price.

6.3 Bond Yields

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Consider a semi-annual bond that has a face value of $1,000, matures in 5 years, and has a coupon rate of 6.5%. If the bond has a current market value of $980, what is the bond’s yield to maturity? Assuming that the bond has semi-annual interest, which we assume unless specified otherwise, PMT = ($1,000 × 0.065) ÷ 2 = $32.50 N = 5 × 2 = 10 six-month periods FV = $1,000 PV = $980

Solving this would require a trial-and-error approach—solving for the YTM that causes the right-hand side of this equation to be equal to the left-hand side, $980:

The six-month rate is 3.4%, so the yield to maturity is 6.9%.

Yield to Maturity (YTM)

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Bonds are callable at a specified price (the call price), but there is often a schedule of call prices associated with a given callable bond. The call price is usually stated as a percentage of par value. Therefore, a call price of 105 indicates that a

$1,000 face value bond is callable at $1,050.Equation for estimating yield to call:

Yield to Call (YTC)

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Problem:Consider a semi-annual pay bond with 10 years remaining to maturity, a 6%coupon rate, and a face value of $1,000, that is priced to yield 5%. If the bond is callable in 2 years at 110, what is the bond’s yield to call? Solution:First, we need to solve for the current price (PV) based on the yield to maturity of 5%:

PMTt = ($1,000 × 0.06) ÷ 2 = $30 rb = 0.05 ÷ 2 = 0.025 or 2.5% N = 10 × 2 = 20 six-month periods FV = $1,000

Solving for PV, the current value of the bond is $1,077.95.Solving for the yield to call requires using trial and error to solve for YTC in the following:

Or, PMTt = ($1,000 x 0.06) / 2 = $30.00 N = 2 x 2 = 4 six-month periods FV = $1,100 PV = $1,077.95

Yield to Call

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Current yield (CY) - the ratio of the annual coupon interest divided by the current market price. Not a true measure of the return to a bondholder

because it disregards the bond’s purchase price relative to all the future cash flows and uses just the next year’s interest payment.

Also sometimes referred to as the flat or cash yield Equation for calculating current yield:

Current Yield

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We refer to a debt obligation of a national government as a Treasury security and refer to these obligations as sovereign debt.

National governments issue debt obligations that, in theory, are considered to be risk free because the government can print money or raise revenues through taxes to meet the cash flow obligations of this sovereign debt. We say “in theory” because there have been situations in which this

debt has been viewed as not risk free; Venezuela, Russia, Argentina, the Ukraine, and Belize, among other nations, have defaulted on their debt.

In some cases the debt is restructured, allowing the nation to repay the debt under more favorable terms.

6.4 Government Debt Obligations

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Sovereign debt ratings consider: Fiscal performance Debt burden External liquidity and international investment position Institutional effectiveness Political risks Monetary flexibility Economic structure Growth prospects

Sovereign Debt Ratings

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Greece’s credit rating has deteriorated over time, as the economic conditions in the country have worsened:

Example: Greece (Hellenic Republic)

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Date Debt Rating

March 26, 1997 A-

March 13, 2001 A

June 10, 2003 A+

November 17, 2004 A

January 14, 2009 A-

December 16, 2009 BBB+

April 27, 2010 BB+

March 29, 2011 BB-

May 9, 2011 B

June 13, 2011 CCC

July 27, 2011 CC

February 27, 2012 D

May 2, 2012 CCC

Country Rating as of June 30, 2011

Rating as of January 13, 2012

Germany AAA AAA

Iceland BBB- BBB-

India BBB- BBB-

Israel AA- AA-

Italy A+ BBB+

Japan AA- AA-

Mexico A A-

People’s Republic of China AA- AA-

Russian Federation BBB+ BBB+

Sweden AAA AAA

United Kingdom AAA AAA

United States AAA AA+

Credit Ratings of Other Countries

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Obligations of the U.S. government include: Short-term securities, which we refer to as Treasury bills (or T-bills) Medium-term debt that we refer to as notes Long-term debt (bonds)

Treasury securities also include Treasury Inflation Protected Securities (TIPS), which protect the investor against the effects of inflation.

U.S. Treasury Obligations

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U.S. Treasury Securities OutstandingIn billions

1996

1997

1998

1999

2000

2001

2002

2003

2004

2005

2006

2007

2008

2009

2010

2011

2012

4153

0$0

$2,000

$4,000

$6,000

$8,000

$10,000

$12,000

$14,000Bills Notes Bonds TIPS

Jun-

50

Jun-

54

Jun-

58

Jun-

62

Jun-

66

Jun-

70

Jun-

74

Sep-

78

Sep-

82

Sep-

86

Sep-

90

Sep-

94

Sep-

98

Sep-

02

Sep-

06

Sep-

10$0

$2,000,000

$4,000,000

$6,000,000

$8,000,000

$10,000,000

$12,000,000

$14,000,000

$16,000,000

$18,000,000

U.S. National DebtIn millions

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Treasury bill (T-bill) - a short term government debt obligation that matures in 1 year or less. The available maturities are 4 weeks, 13 weeks, 26 weeks, and 52 weeks. Partly because of this short term to maturity, they do not make regular

interest payments but rather are sold at a discount from their par (or

face) value, which is paid on the maturity date. Treasury securities, including bills, are sold through auctions.

Bidding in a government securities auction consists of competitive and noncompetitive bids. Competitive bids comprise 35% of an offering, in which the bidder

specifies the yield that is acceptable. Noncompetitive bids accept the rate that results from the competitive

bids. The rate set by the auction process is the result of a Dutch auction.

Treasury Bills

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In a Dutch auction, bidders submit the rate and amount that they are willing to buy. The Treasury then determines the rate at which it can sell the amount of securities, accepting all bids at or below the clearing rate.

Example: suppose the Treasury wants to sell $50 million of securities and receives the following bids:

The rate in this case would be set to 5% because at that rate the Treasury would sell the $25 + 10 + 15 = $50 million in securities. Bidders 1, 2, and 3 would receive a rate of 5%, and bidders 4 and 5 would be left out.

Treasury Bills

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T-bills can be priced by estimating the present value of the expected future payment (i.e., the par value that is to be repaid at maturity).

The value of a T-bill today, P, is the discounted face value, using the annualized rate adjusted to the proportion of the year.

In the equation for the valuation of a bill, rBEY is the bond equivalent yield, FV is the face value, and n is the term to maturity expressed as number of days:

If the bond equivalent yield is 5% and the maturity is 90 days, the present value of this bond for each $1 face value is:

Treasury Bills

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Interest rates are usually quoted on an annual percentage basis. The interest rate is the price of money, which is determined by

the laws of supply and demand, as for any other commodity. All else constant, as the demand for loanable funds decreases,

so does the price, and as a result interest rates increase; conversely, interest rates decrease as the supply of loanable funds increases.

The interest rate that we have been discussing so far is the nominal interest rate because this is the rate charged for lending today’s dollars in return for getting dollars back in the future, without taking into account the purchasing power of those future dollars.

6.5 Interest Rates

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We refer to the base rate as the risk-free rate (rf), which is the compensation for the time value of money:

This relationship is an approximation of the direct relationship between inflation and interest rates that is often referred to as the “Fisher relationship,” after Irving Fisher.

Using this relationship, if the risk-free rate of interest is 3% and the expected inflation rate is 1.8% , the real rate is 3% – 1.8% = 1.2%.

Base Interest Rates

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Investors attempt to protect themselves from the loss in purchasing power caused by inflation by increasing their required nominal yield. As a result, interest rates will be low when expected inflation is low and high when expected inflation is high.

More precisely, the Fisher relationship is the following:

Applying a bit of algebra, we see that the risk-free rate is not only the sum of the real rate and the expected inflation rate, but also the cross-product term, real rate × expected inflation:

Fisher Effect

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Interest Rates and Inflation, January 1960-July 2011

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In addition to differences in terms to maturity, the yield on bonds will differ from the risk-free rate because of additional risks or features associated with these instruments.

The difference between any two interest rates is the spread. The spread is often quoted in terms of a basis point (bp),

where one basis point is 1/100th of a percent. Therefore, if the yields on two investments are 4% and 5%,

respectively, the spread is 100 basis points, or 100 bps.

Yield Premiums

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Yield Premiums

38U.S. Government Yield Curve

The traditional coupon-paying bond has a series of cash payments of interest every six months, with the principal repayment at maturity.

However, bonds may also have features such as sinking funds, embedded options, and floating rates.

In addition, some bonds do not pay interest and, hence, are priced at a discount from their face value.

Chapter Summary

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We value debt obligations as we do any stream of cash flows. The key for bond valuation is that we understand the timing of the bond’s cash flows and the appropriate discount rate.

One of the most important factors affecting bond prices is the level of interest rates. There is an inverse relation between the yield on a bond and its

value. Treasury securities include short-term securities, notes, and

bonds. The short-term securities, Treasury bills, are priced at a discount

because these securities do not bear interest.

Chapter Summary

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