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Économie & finance

Ch 6 financial markets. Teachers resource.

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Bba 2204 fin mgt week 6 bondsStephen Ong

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gitman_ch_6_pptx.pptxalizia94

- Chapter 6 Interest Rates And Bond Valuation
- Principles of Managerial Finance Sixteenth Edition Chapter 6 Interest Rates and Bond Valuation Copyright © 2022, 2019, 2015 Pearson Education, Inc. All Rights Reserved.
- Learning Goals (1 of 2) LG 1 Describe interest rate fundamentals, the term structure of interest rates, and risk premiums. LG 2 Review the legal aspects of bond financing and bond cost. LG 3 Discuss the general features, yields, prices, ratings, popular types, and international issues of corporate bonds. LG 4 Understand the key inputs and basic model used in the bond valuation process.
- Learning Goals (2 of 2) LG 5 Apply the basic valuation model to bonds, and describe the impact of required return and time to maturity on bond prices. LG 6 Explain yield to maturity (YTM), its calculation, and the procedure used to value bonds that pay interest semiannually.
- 6.1 Interest Rates and Required Returns (1 of 14) • Interest Rate Fundamentals – Interest Rate • Usually applied to debt instruments such as bank loans or bonds • the compensation paid by the borrower of funds to the lender • from the borrower’s point of view, the cost of borrowing funds • The interaction of supply and demand determines interest rates – Required Return • The return that the supplier of funds requires the demander to pay • Applies to almost any kind of investment
- Figure 6.1 Supply-Demand Relationship
- 6.1 Interest Rates and Required Returns (2 of 14) • Interest Rate Fundamentals – Inflation • A rising trend in the prices of most goods and services – Liquidity Preference • A general tendency for investors to prefer short- term (i.e., more liquid) securities
- 6.1 Interest Rates and Required Returns (3 of 14) • Interest Rate Fundamentals – Negative Interest Rates • When a loan carries an interest rate below zero, the lender essentially pays interest to the borrower rather than the other way around • A natural question is, why would anyone buy an investment if it paid an interest rate below zero? – The answer is that there is no good, safe alternative offering a better return
- Matter of Fact (1 of 2) Fear Turns T-Bill Rates Negative When the coronavirus outbreak reached the United States in early 2020, financial markets reacted in dramatic fashion. On March 25, interest rates on one-month and three-month Treasury bills briefly turned negative, meaning that investors paid more to the Treasury than the Treasury promised to pay back. Why would people put their money into an investment they know will lose money? Fear about the effects of COVID-19 on the economy sent stock prices tumbling, and investors moved their money into safer Treasury securities. The demand for those securities was so high, temporarily, that investors effectively paid the U.S. Treasury to hold their funds for a short time.
- 6.1 Interest Rates and Required Returns (4 of 14) • Interest Rate Fundamentals – Nominal and Real Interest Rates • Nominal Rate of Interest – The actual rate of interest charged by the supplier of funds and paid by the demander • Real Rate of Interest – The rate of return on an investment measured not in dollars but in the increase in purchasing power that the investment provides – The real rate of interest measures the rate of increase in purchasing power
- 6.1 Interest Rates and Required Returns (5 of 14) • Interest Rate Fundamentals – Nominal and Real Interest Rates * 1+ = 1+ 1+ (6.1) r r i * (6.1a) r r i • r = nominal interest rate • r* = real interest rate • i = expected inflation rate
- Personal Finance Example 6.1 (1 of 3) Burt Gummer is a survivalist who constantly worries that the apocalypse may happen any day now. To be prepared, Burt stores food with a long shelf life in his basement. Burt has $100 to add to his food stores, and he is considering the purchase of 100 cans of Spam for $1 each. Burt expects the inflation rate over the coming year to be 9%, so a can of Spam will cost $1.09 each in a year. Burt’s wife, Heather, has heard of an investment that will pay a 21% nominal return over the next year, so she thinks Burt should invest the money rather than use it to buy Spam.
- Personal Finance Example 6.1 (2 of 3) Equation 6.1a says that the approximate real return on Burt’s potential investment is 12%: 12% ≈ 21% − 9% On the basis of this calculation, Burt might expect that if he invests $100, he could buy 112 cans of Spam next year rather than 100 cans this year (a 12% increase in purchasing power). Suppose that Burt invests the money, earns a 21% rate of return, and one year later has $121. By that time, one can of Spam costs $1.09, so Burt is just barely able to purchase 111 cans (111 cans × $1.09 = $120.99).
- Personal Finance Example 6.1 (3 of 3) Burt’s purchasing power has increased by 11%, not by the 12% that he expected. Equation 6.1 reveals that the exact real return on Burt’s investment is 11%: (1 0.21) (1 *)(1 0.09) 1.21 1 * 1.09 1.110 1 0.11 11% * r r r
- 6.1 Interest Rates and Required Returns (6 of 14) • Interest Rate Fundamentals – Nominal Interest Rates, Inflation, and Risk • RF = Risk-free rate * (6.2) F R r i • rj = Nominal return on investment j • RPj = Risk premium above the risk-free rate for investment j (6.3) j F j r R RP
- Figure 6.2 Inflation and the Risk-free Interest Rates a Interest rate on a one-year Treasury bill issued in January of each calendar year. Data from selected Federal Reserve Bulletins. b Change in consumer price index in each calendar year. Data from U.S. Department of Labor Bureau of Labor Statistics.
- 6.1 Interest Rates and Required Returns (7 of 14) • Term Structure of Interest Rates – Yield Curves • Term Structure of Interest Rates – The relationship between the maturity and rate of return for bonds with similar levels of risk • Yield Curve – A graphic depiction of the term structure of interest rates • Yield to Maturity (YTM) – Compound annual rate of return earned on a debt security purchased on a given day and held to maturity – An estimate of the market’s required return on a particular bond
- 6.1 Interest Rates and Required Returns (8 of 14) • Term Structure of Interest Rates – Yield Curves • Normal Yield Curve – An upward-sloping yield curve indicates that long-term interest rates are generally higher than short-term interest rates • Inverted Yield Curve – A downward-sloping yield curve indicates that short-term interest rates are generally higher than long-term interest rates • Flat Yield Curve – A yield curve that indicates that interest rates do not vary much at different maturities
- Figure 6.3 Treasury Yield Curves Source: Data from Daily Treasury Yield Curve Rates. https://www.treasury.gov/resource-center/datachart- center/interest- rates/Pages/ TextView.aspx?data=yield
- 6.1 Interest Rates and Required Returns (9 of 14) • Term Structure of Interest Rates – Yield Curves • Deflation – A general trend of falling prices • The shape of the yield curve may affect the firm’s financing decisions – A financial manager who faces a downward- sloping yield curve may be tempted to rely more heavily on cheaper, long-term financing – When the yield curve is upward sloping, the manager may believe it wise to use cheaper, short-term financing
- Figure 6.4 The Slope of the Yield Curve and the Business Cycle
- Matter of Fact (2 of 2) Bond Yields Hit Record Lows On March 9, 2020, the 10-year Treasury note yield reached an all-time low of 0.318% as fears about the spreading coronavirus and its impact on the economy prompted investors to pour money into safe Treasury securities. Lower interest rates are usually good news for the housing market. Many mortgage rates are linked to rates on Treasury securities. For example, the traditional 30-year mortgage rate is typically linked to the yield on 10-year Treasury notes. Mortgage rates did reach new lows, but the slowing economy discouraged home buyers (and sellers) from taking advantage, though many homeowners did benefit from the falling rates when they refinanced their mortgages.
- 6.1 Interest Rates and Required Returns (10 of 14) • Term Structure of Interest Rates – Theories of the Term Structure • Expectations Theory – The theory that the yield curve reflects investor expectations about future interest rates; an expectation of rising interest rates results in an upward-sloping yield curve, and an expectation of declining rates results in a downward-sloping yield curve
- Example 6.2 (1 of 3) Suppose that a one-year T-bill currently offers a 3.5% return and a two-year Treasury note offers a 3.0% annual return. Thus, the short-term rate is higher than the long- term rate and the yield curve slopes down. According to the expectations theory, what belief must investors hold about the rate of return that a one-year T-bill will offer next year? First, recognize that by purchasing the two-year note, investors can earn a return of 6.09% over two years: 2 1 0.030 1.0609
- Example 6.2 (2 of 3) If the market is in equilibrium, then the expected return on a strategy of purchasing a sequence of two one-year T-bills must offer the same return, so we have 1 0.035 1 E r 1.0609 1 E r 1.0609 1.035 E r 0.025 2.5% ( ( )) ( ) ( )
- Example 6.2 (3 of 3) Investors believe the T-bill will offer a 2.5% return next year, which is lower than the 3.0% return currently offered by one-year T-bills. In other words, investors are indifferent between earning 3.0% for two consecutive years on the Treasury note or earning 3.5% this year and 2.5% next year on a sequence of two one-year T-bills. Thus, today’s downward-sloping yield curve implies that investors expect falling rates.
- 6.1 Interest Rates and Required Returns (11 of 14) • Term Structure of Interest Rates – Theories of the Term Structure • Liquidity Preference Theory – Theory suggesting that long-term rates are generally higher than short-term rates (hence, the yield curve is upward sloping) because investors perceive short-term investments as more liquid and less risky than long-term investments – Borrowers must offer higher rates on long- term bonds to entice investors away from their preferred short-term securities
- 6.1 Interest Rates and Required Returns (12 of 14) • Term Structure of Interest Rates – Theories of the Term Structure • Market Segmentation Theory – Theory suggesting that the market for loans is segmented on the basis of maturity and that the supply of and demand for loans within each segment determine its prevailing interest rate; the slope of the yield curve is determined by the general relationship between the prevailing rates in each market segment
- 6.1 Interest Rates and Required Returns (13 of 14) • Risk Premiums: Issuer and Issue Characteristics – The nominal rate of interest for security j is equal to the risk- free rate, consisting of the real rate of interest plus the expected inflation rate, plus the risk premium – The risk premium varies with characteristics of the company that issued the security as well as characteristics of the security itself
- Example 6.3 (1 of 3) The nominal interest rates on a number of classes of long-term securities in April 2020 were as follows: Security Nominal interest rate One-Year U.S. Treasury bills 0.2% Corporate bonds: Investment grade 3.0% High-yield 8.0%
- Example 6.3 (2 of 3) Using the one-year Treasury bill rate as our benchmark risk-free rate, we can calculate the risk premium of the other securities by subtracting the risk-free rate, 0.2%, from rates offered by each of the other corporate securities: Security Risk premium Corporate bonds: Investment grade 3.0% − 0.2% = 2.8% High-yield 8.0% − 0.2% = 7.8%
- Example 6.3 (3 of 3) Investment grade bonds pay a risk premium over Treasury bills in part because of their term (investment grade bonds have a longer term than Treasury bills), but also because they are issued by corporations and are not backed by the full faith and credit of the U.S. government as are Treasury securities. High-yield bonds pay an even larger risk premium because they are issued by less creditworthy corporations.
- 6.1 Interest Rates and Required Returns (14 of 14) • Risk Premiums: Issuer and Issue Characteristics – Several factors influence a bond’s risk premium • Default risk or credit risk – The likelihood that a corporation will fail to make interest payments or principle repayments on its bonds • Bond characteristics – e.g., Time to maturity • Interest rate risk – When interest rates in the economy change, bond prices move in the opposite direction – Some bonds are more sensitive to interest rate risk and therefore pay higher risk premiums to compensate investors
- 6.2 Government and Corporate Bonds (1 of 15) • Municipal Bond – A bond issued by a state or local government body • Corporate Bond – A long-term debt instrument indicating that a corporation has borrowed a certain amount of money and promises to repay it in the future under clearly defined terms
- 6.2 Government and Corporate Bonds (2 of 15) • Par Value, Face Value, Principal – The amount of money the borrower must repay at maturity, and the value on which periodic interest payments are based • Coupon Rate – The percentage of a bond’s par value that will be paid annually, typically in two equal semiannual payments, as interest
- 6.2 Government and Corporate Bonds (3 of 15) • Legal Aspects of Corporate Bonds – Bond Indenture • A legal document that specifies both the rights of the bondholders and the duties of the issuing corporation – Standard Debt Provisions • Provisions in a bond indenture specifying certain record- keeping and general business practices that the bond issuer must follow; normally, they do not place a burden on a financially sound business – Restrictive Covenants • Provisions in a bond indenture that place operating and financial constraints on the borrower
- 6.2 Government and Corporate Bonds (4 of 15) • Legal Aspects of Corporate Bonds – Subordination • In a bond indenture, the stipulation that subsequent creditors agree to wait until all claims of the senior debt are satisfied – Sinking-Fund Requirement • A restrictive provision often included in a bond indenture, providing for the systematic retirement of bonds prior to their maturity
- 6.2 Government and Corporate Bonds (5 of 15) • Legal Aspects of Corporate Bonds – Collateral • A specific asset against which bondholders have a claim in the event that a borrower defaults on a bond – Secured Bond • A bond backed by some form of collateral – Unsecured Bond • A bond backed only by the borrower’s ability to repay the debt
- 6.2 Government and Corporate Bonds (6 of 15) • Legal Aspects of Corporate Bonds – Trustee • A paid individual, corporation, or commercial bank trust department that acts as the third party to a bond indenture and can take specified actions on behalf of the bondholders if the terms of the indenture are violated
- 6.2 Government and Corporate Bonds (7 of 15) • Cost of Bonds to the Issuer – Impact of Bond Maturity • Usually, long-term debt pays higher interest rates than short-term debt – Impact of Offering Size • Bond flotation and administration costs per dollar borrowed are likely to decrease as offering size increases • However, the risk to the bondholders may increase, because larger offerings result in greater risk of default, all other factors held constant – Impact of the Issuer’s Risk • The greater the issuer’s default risk, the higher the interest rate
- 6.2 Government and Corporate Bonds (8 of 15) • Cost of Bonds to the Issuer – Impact of the Risk-free Rate • The risk-free rate in the capital market determines a floor on the cost of a bond offering • Generally, the rate on U.S. Treasury securities of equal maturity is the lowest cost of borrowing money – To that basic rate is added a risk premium that reflects the factors mentioned prior (maturity, offering size and issuer’s risk)
- 6.2 Government and Corporate Bonds (9 of 15) • General Features of a Bond Issue – Conversion Feature • A feature of convertible bonds that allows bondholders to change each bond into a stated number of shares of common stock – Call Feature • A feature included in nearly all corporate bond issues that gives the issuer the opportunity to repurchase bonds at a stated call price prior to maturity – Call Price • The stated price at which a bond may be repurchased, by use of a call feature, prior to maturity
- 6.2 Government and Corporate Bonds (10 of 15) • General Features of a Bond Issue – Call Premium • The amount by which a bond’s call price exceeds its par value – Stock Purchase Warrants • Instruments that give their holders the right to purchase a certain number of shares of the issuer’s common stock at a specified price over a certain period of time
- 6.2 Government and Corporate Bonds (11 of 15) • Bond Yields – Current Yield • A measure of a bond’s cash return for the year • Calculated by dividing the bond’s annual interest payment by its current price – Yield to Maturity (YTM) – Yield to Call (YTC)
- 6.2 Government and Corporate Bonds (12 of 15) • Bond Prices – Though corporate bonds are held mostly by institutional investors and are not as actively traded as stocks, it is still important to understand market conventions for quoting bond prices and yields. – Basis points • A way of quoting an interest rate such that 100 basis points equals one percentage point
- Table 6.1 Data on Selected Bonds Company Coupon Maturity Price Yield (YTM) Verizon 5.050% Mar. 15, 2034 123.88 2.915% Apple 3.850 Aug. 4, 2046 124.36 2.554 Tesla 5.300 Aug. 15, 2025 94.50 6.542 Safeway 7.450 Sep. 15, 2027 105.10 6.566 Amazon 4.250 Aug. 22, 2057 99.98 4.251 Bond data from http://finra-markets.morningstar.com/BondCenter/ActiveUSCorpBond.jsp, accessed on April 23, 2020.
- 6.2 Government and Corporate Bonds (13 of 15) • Bond Ratings – Independent agencies such as Moody’s, Fitch, and Standard & Poor’s assess the riskiness of publicly traded bond issues – These agencies derive their ratings by using financial ratio and cash flow analyses to assess the likely payment of bond interest and principal – Normally an inverse relationship exists between the quality of a bond and the rate of return that it must provide bondholders
- Table 6.2 Moody’s and Standard & Poor’s Bond Ratings Note: Some ratings may be modified to show relative standing within a major rating category; for example, Moody’s uses numerical modifiers (1, 2, 3), whereas Standard & Poor’s uses plus (+) and minus (−) signs. Sources: Based on Moody’s Investors Service, Inc., and based on Standard & Poor’s Corporation.
- 6.2 Government and Corporate Bonds (14 of 15) • Common Types of Bonds – Debentures – Subordinated Debentures – Income Bonds – Mortgage Bonds – Collateral Trust Bonds – Equipment Trust Certificates – Zero- (or Low-) Coupon Bonds – Junk (High-Yield) Bonds – Floating-Rate Bonds – Extendible Notes – Putable Bonds
- Table 6.3 Characteristics and Priority of Lender’s Claim of Traditional Types of Bonds (1 of 2) Bond type Characteristics Priority of lender’s claim Unsecured bonds Debentures Unsecured bonds that only creditworthy firms can issue. Convertible bonds are normally debentures. Claims are the same as those of any general creditor. May have other unsecured bonds subordinated to them. Subordinated debentures Claims are not satisfied until those of the creditors holding certain (senior) debts have been fully satisfied. Claim is that of a general creditor but not as good as a senior debt claim. Income bonds Payment of interest is required only when earnings are available. Commonly issued in reorganization of a failing firm. Claim is that of a general creditor. Are not in default when interest payments are missed because they are contingent only on earnings being available.
- Table 6.3 Characteristics and Priority of Lender’s Claim of Traditional Types of Bonds (2 of 2) Bond type Characteristics Priority of lender’s claim Secured Bonds Mortgage bonds Secured by real estate or buildings. Claim is on proceeds from sale of mortgaged assets; if not fully satisfied, the lender becomes a general creditor. The first-mortgage claim must be fully satisfied before distribution of proceeds to second- mortgage holders and so on. A number of mortgages can be issued against the same collateral. Collateral trust bonds Secured by stock and (or) bonds that are owned by the issuer. Collateral value is generally 25% to 35% greater than bond value. Claim is on proceeds from stock and/or bond collateral; if not fully satisfied, the lender becomes a general creditor. Equipment trust certificates Used to finance “rolling stock,” such as airplanes, trucks, boats, railroad cars. A trustee buys the asset with funds raised through the sale of trust certificates and then leases it to the firm; after making the final scheduled lease payment, the firm receives title to the asset. A type of leasing. Claim is on proceeds from the sale of the asset; if proceeds do not satisfy outstanding debt, trust certificate lenders become general creditors.
- Table 6.4 Characteristics of Contemporary Types of Bonds (1 of 2) Bond type Characteristicsa Zero- (or low-) coupon bonds Issued with no (zero) or a very low coupon (stated interest) rate and sold at a large discount from par. A significant portion (or all) of the investor’s return comes from gain in value (i.e., par value minus purchase price). Generally callable at par value. Junk (high-yield) bonds Debt rated Ba or lower by Moody’s or BB or lower by Standard & Poor’s. Commonly used by rapidly growing firms to obtain growth capital, most often as a way to finance mergers and takeovers. High-risk bonds with high yields, often yielding 2% to 3% more than the best-quality corporate debt. Floating-rate bonds Stated interest rate is adjusted periodically within stated limits in response to changes in specified money market or capital market rates. Popular when future inflation and interest rates are uncertain. Tend to sell at close to par because of the automatic adjustment to changing market conditions. Some issues provide for annual redemption at par at the option of the bondholder. Extendible notes Short maturities, typically one to five years, that can be renewed for a similar period at the option of holders. Similar to a floating-rate bond. An issue might be a series of three-year renewable notes over a period of 15 years; every three years, the notes could be extended for another three years, at a new rate competitive with market interest rates at the time of renewal.
- Table 6.4 Characteristics of Contemporary Types of Bonds (2 of 2) Bond type Characteristicsa Putable bonds Bonds that can be redeemed at par (typically, $1,000) at the option of their holder either at specific dates after the date of issue and every one to five years thereafter or when and if the firm takes specified actions, such as being acquired, acquiring another company, or issuing a large amount of additional debt. In return for its conferring the right to “put the bond” at specified times or when the firm takes certain actions, the bond’s yield is lower than that of a nonputable bond. aThe claims of lenders (i.e., bondholders) against issuers of each of these types of bonds vary, depending on the bonds’ other features. Each of these bonds can be unsecured or secured.
- 6.2 Government and Corporate Bonds (15 of 15) • International Bond Issues – Eurobond • A bond issued by an international borrower and sold to investors in countries with currencies other than the currency in which the bond is denominated – Foreign Bond • A bond that is issued by a foreign corporation or government and is denominated in the investor’s home currency and sold in the investor’s home market – Bulldog Bond » Foreign bond issued in Britain – Samurai Bond » Foreign bond issued in Japan
- 6.3 Valuation Fundamentals (1 of 2) • Valuation – The process that links risk and return to determine the worth of an asset • Key Inputs – Cash Flows – Timing – Risk and Required Return
- Personal Finance Example 6.4 (1 of 2) Celia Sargent wishes to estimate the value of three assets: common stock in Michaels Enterprises, an interest in an oil well, and an original painting by a well-known artist. Her cash flow estimates for each are as follows: Stock in Michaels Enterprises: Expects to receive cash dividends of $300 per year indefinitely starting in one year. Oil well: Expects to receive cash flows of $2,000 after one year, $4,000 after two years, and $10,000 after four years, when the well will run dry.
- Personal Finance Example 6.4 (2 of 2) Original painting: Expects to sell the painting in five years for $85,000. With these cash flow estimates, Celia has taken the first step in valuation.
- Personal Finance Example 6.5 (1 of 2) Let’s return to Celia Sargent’s task of placing a value on the original painting and consider two scenarios. Scenario 1: Certainty A major art gallery has contracted to buy the painting for $85,000 after five years. Because this contract is already signed and the art gallery is well established and reliable, Celia views this asset as “money in the bank,” and uses something close to the prevailing risk-free rate of 3% as the required return when calculating the painting’s value today.
- Personal Finance Example 6.5 (2 of 2) Scenario 2: High risk The values of original paintings by this artist have fluctuated widely over the past 10 years. Although Celia expects to sell the painting for $85,000, she realizes that its sale price could range between $30,000 and $140,000. Because of the high uncertainty surrounding the painting’s value, Celia believes that a 15% required return is appropriate. These two estimates illustrate how the valuation process accounts for risk through the discount rate. Although adjusting the discount rate for risk has a subjective element, analysts use historical data and forward-looking models to estimate the required return with as much precision as possible.
- 6.3 Valuation Fundamentals (2 of 2) • Basic Valuation Model – The value of an asset is the present value of all the future cash flows it is expected to provide. 1 2 0 1 2 (6.4) 1 1 1 n n CF CF CF V r r r • V0 = value of the asset at time zero • CFt = cash flow expected in year t • r = required return (discount rate) • n = time period (investment’s life or investor’s holding period)
- Personal Finance Example 6.6 (1 of 3) Celia Sargent values each asset by discounting its cash flows using Equation 6.4. Because Michael’s stock pays a perpetual stream of $300 dividends, Equation 6.4 reduces to Equation 5.7, which says that the present value of a perpetuity equals the dividend payment divided by the required return. Celia decides that a 12% discount rate is appropriate for this investment, so her estimate of the value of Michael’s Enterprises stock is $300 0.12 $2,500
- Personal Finance Example 6.6 (2 of 3) Next, Celia values the oil well investment, which she believes is the most risky of the three investments. Discounting the oil well’s cash flows using a 20% required return, Celia estimates the well’s value to be 1 2 4 $2,000 $4,000 $10,000 $9,266.98 (1 0.20) (1 0.20) (1 0.20)
- Personal Finance Example 6.6 (3 of 3) Finally, Celia estimates the value of the painting by discounting the expected $85,000 cash payment in five years at 15%: $85,000 ÷ (1 + 0.15)5 = $42,260.02 Note that, regardless of the pattern of the asset’s expected cash flows, Celia can use the basic valuation equation to determine the asset’s value.
- 6.4 Bond Valuation (1 of 11) • Bond Fundamentals – Bonds are long-term debt instruments used by business and government to raise large sums of money, typically from a diverse group of lenders – Most corporate bonds • pay interest semiannually (every six months) at a stated coupon rate • have an initial maturity of 10 to 30 years • and have a par value, principal, or face value, of $1,000 that the borrower must repay at maturity
- Example 6.7 The Mills Company just issued a 6% coupon rate, 10-year bond with a $1,000 par value that pays interest annually. Investors who buy this bond receive the contractual right to two types of cash flows: (1) $60 annual interest (6% coupon rate × $1,000 par value) distributed at the end of each year and (2) the $1,000 par value at the end of the tenth year.
- 6.4 Bond Valuation (2 of 11) • Bond Values for Annual Coupons 0 1 2 3 0 1 1 1 1 1 1 (6.5) 1 1 n n n t n t C C C C M B r r r r r C M B r r • B0 = value (or price) of the bond at time zero • C = annual coupon interest payment in dollars • n = number of years to maturity • M = par value in dollars • r = annual required return on the bond
- 6.4 Bond Valuation (3 of 11) • Bond Values for Annual Coupons 0 1 1 (6.5a) 1 1 n n C M B r r r • B0 = value (or price) of the bond at time zero • C = annual coupon interest payment in dollars • n = number of years to maturity • M = par value in dollars • r = annual required return on the bond
- Example 6.8 (1 of 4) Tim Sanchez wishes to determine the current value of the Mills Company bond. If the bond pays interest annually and the required annual return on the bond is 6% (equal to its coupon rate), then we can calculate the bond’s value using Equation 6.5a: 0 10 10 0 $60 1 $1,000 1 0.06 (1 0.06) (1 0.06 $1,000 0.44161 $558.39 $441.61 $558 ) .39 $1, [ 000 0 ] . 0 B B The timeline on the next slide depicts the computations involved in finding the bond value.
- Example 6.8 (2 of 4)
- Example 6.8 (3 of 4) Calculator use Using the Mills Company’s inputs shown at the left, you should find the bond value to be exactly $1,000. If you compare the calculator keystrokes to the ones we displayed when we used a calculator to find the present value of an ordinary annuity earlier in this text, you will see that there is an additional term here, namely, the $1,000 future value (FV). We must add that to our sequence of keystrokes because the bond pays out an annuity plus a $1,000 lump sum at the end. When we add the FV keystroke, we are capturing the value of that final lump sum payment when the bond matures. Note that the calculated bond value is equal to its par value, which will always be the case when the required return is equal to the coupon rate.
- Example 6.8 (4 of 4) Spreadsheet use We can also calculate the value of the Mills Company bond using Excel’s PV function as shown in the following Excel spreadsheet.
- 6.4 Bond Valuation (4 of 11) • Bond Values for Semiannual Coupons – As a practical matter, most bonds make semiannual rather than annual interest payments – Calculating the value for a bond paying semiannual interest requires three changes to the approach we’ve used so far: 1. Convert the annual coupon payment, C, to a semiannual payment by dividing C by 2 2. Recognize that if the bond has n years to maturity it will make 2n coupon payments (i.e., in n years there are 2n semiannual periods) 3. Discount each payment by using the semiannual required return calculated by dividing the annual required return, r, by 2
- 6.4 Bond Valuation (5 of 11) • Bond Values for Semiannual Coupons 0 1 2 3 2 2 2 0 2 1 2 2 2 2 1+ 1+ 1+ 1+ 1 2 2 2 2 2 2 (6.6) 1+ 1 2 2 n n n t n t C C C C M B r r r r r C M B r r
- 6.4 Bond Valuation (6 of 11) • Bond Values for Semiannual Coupons 0 2 2 / 2 1 1 (6.6a) / 2 1 1 2 2 n n C M B r r r • B0 = value (or price) of the bond at time zero • C = annual coupon interest payment in dollars • n = number of years to maturity • M = par value in dollars • r = annual required return on the bond
- Example 6.9 (1 of 3) Assuming that the Mills Company bond pays interest semiannually and that the required annual return, r, is 6%, Equation 6.6a indicates that the bond’s value is 0 2(10) 2(10 0 ) $60/2 1 $1,000 1 0.06/2 0.06 0.06 1 1 $1,000 0.44632 $553.68 $446.32 $553.68 $1,0 2 2 00 B B As before, because the required return equals the coupon rate, the bond sells at par value. We will soon see that when the required return does not equal the coupon rate, the bond may sell above or below par value.
- Example 6.9 (2 of 3) Calculator use When using a calculator to find the price of a bond that pays interest semiannually, N is the number of semiannual periods until maturity and PMT is the semiannual interest payment. For the Mills Company bond, N equals 20 semiannual periods (2 × 10 years), I/Y remains 6 for the annual required rate of return, PMT equals an interest payment of $30 ($60 ÷ 2), and FV is 1,000 for the par value. As shown on the left, this calculation verifies that the bond’s price is $1,000. Spreadsheet use The value of the Mills Company bond paying semiannual interest at an annual required return of 6% also can be calculated as shown in the following Excel spreadsheet.
- Example 6.9 (3 of 3)
- 6.4 Bond Valuation (7 of 11) • Changes in Bond Prices – When the required return rises, the bond price falls, and when the required return falls, the bond price rises – Required Returns and Bond Prices • Discount – The amount by which a bond sells below its par value » When the required return is greater than the coupon rate, the bond’s value will be less than its par value • Premium – The amount by which a bond sells above its par value » When the required return falls below the coupon rate, the bond’s value will be greater than par
- Example 6.10 (1 of 3) Let’s reconsider the Mills Company bond paying a 6% coupon rate and maturing in 10 years (assume annual interest payments for simplicity). Initially, we assumed that the required return on this bond was 6%, so the bond’s value was equal to par value, $1,000. Let’s see what happens to the bond’s value if the required return is higher or lower than the coupon rate. Table 6.5 shows that at an 8% required return, the bond sells at a discount with a value of $865.80, but if the required return is 4%, the bond sells at a premium with a value of $1,162.22.
- Example 6.10 (2 of 3) Calculator use Using the inputs shown at the left, you will find that when the required annual return is 8%, the Mill’s annual coupon bond sells for $865.80, which is a discount of $134.20 below par value. At a 4% required annual return, the bond sells for $1,162.22, a premium of $162.22. Figure 6.5 illustrates the inverse relationship between the required return and the price of the Mills Company bond.
- Example 6.10 (3 of 3) Spreadsheet use The values for the Mills Company bond at required returns of 8% and 4% also can be calculated as shown in the following Excel spreadsheet. Once this spreadsheet has been configured, you can calculate the bond price for any required return by simply changing the input values.
- Table 6.5 Bond Values for Various Required Returns (Mills Company’s 6% Annual Coupon Bond with a 10-Year Maturity and $1,000 Par Value) Required return, r Bond value, B0 Status 8% $ 865.80 Discount 6 1,000.00 Par value 4 1,162.22 Premium
- Figure 6.5 Bond Values and Required Returns
- 6.4 Bond Valuation (8 of 11) • Changes in Bond Prices – Interest Rate Risk • The chance that a bond’s required return will change and thereby cause a change in the bond’s price • Other things being equal, Long-term bond prices move more when rates change whereas short- term bond prices are less sensitive to rate changes – Long-term bonds are associated with more interest rate risk
- 6.4 Bond Valuation (9 of 11) • Changes in Bond Prices – Interest Rate Risk • Other things being equal, higher coupon bonds have less interest rate risk exposure compared to lower coupon bonds • Other things being equal, a bond with lower required return has greater interest rate risk compared with a bond that has a higher required return
- Figure 6.6 Interest Rate Risk and Bond Characteristics
- 6.4 Bond Valuation (10 of 11) • Changes in Bond Prices – The Passage of Time and Bond Prices • If a bond trades at a premium, and its required return remains constant, as the bond gets closer to maturity its price will fall, eventually reaching par value at maturity • If a bond sells at a discount and its required return remains fixed, as time passes the bond’s price will rise, eventually reaching par value at the moment before the bond matures • Bonds trading at par will remain at that value until maturity as long as the required return does not deviate from the coupon rate
- Example 6.11 Figure 6.7 depicts the behavior of the bond values calculated earlier and presented in Table 6.5 for Mills Company’s 6% coupon rate bond paying annual interest and having 10 years to maturity. Each of the three required returns—8%, 6%, and 4%—remains constant throughout the bond’s 10-year life. The bond’s value at both 8% and 4% approaches and ultimately equals the bond’s $1,000 par value at its maturity, as the discount (at 8%) or premium (at 4%) declines with the passage of time.
- Figure 6.7 Time to Maturity and Bond Prices
- 6.4 Bond Valuation (11 of 11) • Yield to Maturity (YTM) – The compound annual rate of return earned on a bond purchased assuming that all payments arrive on time and that the investor holds the bond to maturity – Mathematically, a bond’s YTM is the discount rate that equates the bond’s market price to the present value of its cash flows
- Personal Finance Example 6.12 (1 of 10) Jaylen Washington wishes to find the YTM on Mills Company’s bond. The bond currently sells for $929.76, has a 6% coupon rate and $1,000 par value, pays interest annually, and has 10 years to maturity. We can find the bond’s YTM using equation 6.5a and iteratively solving for the discount rate, r, at which the present value of cash flows equals the bond’s market price. Start by plugging the bond’s price and cash flows into Equation 6.5a and substituting YTM for r. 10 10 $60 1 $1,000 $929.76 1 YTM (1 YTM) (1 YTM)
- Personal Finance Example 6.12 (2 of 10) Now make an initial guess for the YTM and solve for the present value of the bond’s cash flows. Because the bond’s current price is less than its par value of $1,000 (i.e., it’s a discount bond) its YTM must be greater than its coupon rate. Make an initial guess for the YTM that is greater than the bond’s 6% coupon rate, say 10%. 10 10 $60 1 $1,000 $929.76? 1 0.10 (1 0.10) (1 0.10) $929.76 $600[0.61446] $385.54 $368.68 $385.54 $754.22
- Personal Finance Example 6.12 (3 of 10) If the discount rate is 10% the present value of cash flows equals $754.22, which is less than the bond’s current price, so the YTM must be less than 10%. Now try 8%. 10 10 $60 1 $1,000 $929.76 ? 1 0.08 (1 0.08) (1 0.08) $929.76 $750 [0.53681] $463.19 $402.61 $463.19 $865.80
- Personal Finance Example 6.12 (4 of 10) Here again the present value of $865.80 is less than the $929.76 current price, so the YTM must be lower still. Trying 7% gives the answer. 10 10 $60 1 $1,000 $929.76 ? 1 0.07 (1 0.07) (1 0.07) $929.76 $857.14 [0.49165] $508.35 $421.41 $508.35 $929.76 At a 7% discount rate the present value of the bond’s cash flows equals the current price, so 7% is the bond’s YTM. In other words, the market’s required annual rate of return for the Mills’ bond is 7%.
- Personal Finance Example 6.12 (5 of 10) Calculator use Financial calculators require either the present value (the bond price in this case) or the future cash flows (coupons and principal in this case) to be input as negative numbers to calculate yield to maturity. Using the inputs shown at the left, you find the YTM to be 7.0%.
- Personal Finance Example 6.12 (6 of 10) Spreadsheet use Excel makes it easier to calculate the yield to maturity. One way to solve for the YTM is to enter the bond’s cash flows and use Excel’s internal rate of return (IRR) function. Start by entering the bond’s price as a negative number, and then enter subsequent coupon and principal payments as positive values. The price is the cash outflow the investor must pay to receive the subsequent cash inflows that the bond provides.
- Personal Finance Example 6.12 (7 of 10) Spreadsheet use Excel’s IRR function simply finds the discount rate at which the present value of cash inflows (the coupon and principal payments) equals the present value of cash outflows (the bond’s price). But that is exactly the definition of the YTM—the discount rate at which the present value of a bond’s coupon and principal payments equals the bond’s price. Therefore, the IRR function solves for a bond’s YTM. Note also that the IRR function calculates the Y TM per period. In this example, payments arrive annually, so each period is one year, and the resulting IRR equals the YTM per year.
- Personal Finance Example 6.12 (8 of 10)
- Personal Finance Example 6.12 (9 of 10) Spreadsheet use Another way to calculate the YTM in Excel recognizes that the typical bond’s interest payments form an annuity, which allows you to solve for a bond’s yield to maturity using Excel’s RATE function.
- Personal Finance Example 6.12 (10 of 10)
- Personal Finance Example 6.13 (1 of 5) Now suppose Mills’ bond makes semiannual coupon payments but as in the previous example, the bond sells for $929.76. For that price investors receive 20 semiannual coupon payments of $30 (6% coupon rate × $1,000 par value ÷ 2 coupon payments per year)plus the $1,000 par value at maturity. Before using the RATE function, let’s use Equation 6.6a and iteratively solve for the YTM. Start by plugging the bond’s given information into Equation 6.6a and substituting YTM for r. 2(10) 2(10) $60 / 2 1 $1,000 $929.76 1 YTM/2 YTM YTM 1 1 2 2
- Personal Finance Example 6.13 (2 of 5) The YTM will be higher than the coupon rate because the bond sells below par. In the example with annual payments, the YTM was 7%, and intuitively it might seem that the same answer will prevail here. Plugging 7% into the equation for Y TM we find that the present value is a bit too low. 2(10) 2(10) $60 / 2 1 $1,000 $929.76 ? 1 0.07 / 2 0.070 0.070 1 1 2 2 $929.76 $857.14 [0.49743] $502.57 $928.94
- Personal Finance Example 6.13 (3 of 5) If the discount rate is 7%, the present value of cash flows is $928.94 which is barely below the market price, so the Y TM must be slightly lower than 7%. With more trial and error, the correct YTM of 6.988% emerges. 2(10) 2(10) $60 / 2 1 $1,000 $929.76 ? 1 0.06988 / 2 0.06988 0.06988 1 1 2 2 $929.76 $858.61 [0.49685] $503.15 $426.60 $503.15 $929.75 The discount rate of 6.988% results in a present value of the bond’s cash flows that is within $0.01 of the bond’s current price.
- Personal Finance Example 6.13 (4 of 5) Calculator use Using the inputs shown at the left, you will find a calculator solution of 6.988% for the annual YTM. Because your calculator was set to two periods per year (P/Y) it accounts for the semiannual payment frequency when determining the annual rate of return (I/Y). To find the semiannual required rate of return, simply divide the annual YTM by two.
- Personal Finance Example 6.13 (5 of 5) Spreadsheet use The RATE function in Excel calculates the YTM per period, or the YTM per half year in the case of the Mills bond. To convert the 3.494% semiannual YTM into an annual figure, simply multiply it by two.
- Review of Learning Goals (1 of 10) • LG 1 – Describe interest rate fundamentals, the term structure of interest rates, and risk premiums. • Equilibrium in the flow of funds between savers and borrowers produces the interest rate or required return • Most interest rates are expressed in nominal terms • The nominal interest rate represents the rate at which money grows over time, whereas the real interest rate represents the rate at which purchasing power grows over time • The difference between the nominal rate and the real rate is (approximately) the inflation rate (or the expected inflation rate)
- Review of Learning Goals (2 of 10) • LG 1 (Cont.) – Describe interest rate fundamentals, the term structure of interest rates, and risk premiums. • For risky assets, the nominal interest rate is the sum of the risk-free rate and a risk premium reflecting issuer and issue characteristics • The risk-free rate is the real rate of interest plus an inflation premium • For any class of similar-risk bonds, the term structure of interest rates is the relation between the rate of return and the time to maturity • Yield curves plot this relation on a graph and can be downward sloping (inverted), upward sloping (normal), or flat
- Review of Learning Goals (3 of 10) • LG 1 (Cont.) – Describe interest rate fundamentals, the term structure of interest rates, and risk premiums. • The expectations theory, liquidity preference theory, and market segmentation theory offer different explanations of the shape of the yield curve • Risk premiums for non-Treasury debt issues result from default or credit risk and other features such as maturity and exposure to interest rate risk
- Review of Learning Goals (4 of 10) • LG 2 – Review the legal aspects of bond financing and bond cost. • Corporate bonds are long-term debt instruments that the company must repay under clearly defined terms • Most bonds are issued with maturities of 10 to 30 years and a par value of $1,000 • The bond indenture, enforced by a trustee, states all conditions of the bond issue • It contains both standard debt provisions and restrictive covenants, which may include a sinking-fund requirement and/or a security interest • The cost of a bond to an issuer depends on its maturity, offering size, and issuer risk and on the risk-free rate
- Review of Learning Goals (5 of 10) • LG 3 – Discuss the general features, yields, prices, ratings, popular types, and international issues of corporate bonds. • A bond issue may include a conversion feature, a call feature, or stock purchase warrants • The return on a bond can be measured by its current yield, yield to maturity (YTM), or yield to call (YTC) • Bond prices are typically reported along with their coupon, maturity date, and yield to maturity (YT M). • Bond ratings by independent agencies indicate the risk of a bond issue
- Review of Learning Goals (6 of 10) • LG 3 (Cont.) – Discuss the general features, yields, prices, ratings, popular types, and international issues of corporate bonds. • Various types of traditional and contemporary bonds are available • Eurobonds and foreign bonds enable established creditworthy companies and governments to borrow large amounts internationally
- Review of Learning Goals (7 of 10) • LG 4 – Understand the key inputs and basic model used in the bond valuation process. • Key inputs to the valuation process include cash flows, timing, risk, and the required return • The value of any asset is equal to the present value of all future cash flows it is expected to provide over the relevant time period
- Review of Learning Goals (8 of 10) • LG 5 – Apply the basic valuation model to bonds, and describe the impact of required return and time to maturity on bond prices. • The value of a bond is the present value of its coupon payments plus the present value of its par value • The discount rate used to determine bond value is the required return, which may differ from the bond’s coupon rate • A bond can sell at a discount, at par, or at a premium, depending on whether the required return is greater than, equal to, or less than its coupon rate
- Review of Learning Goals (9 of 10) • LG 5 (Cont.) – Apply the basic valuation model to bonds, and describe the impact of required return and time to maturity on bond values. • The chance that interest rates will change and thereby alter the required return and bond value is called interest rate risk. • The shorter the amount of time until a bond’s maturity, the higher its coupon rate, or the higher its required return, the less responsive is its market value to a change in the required return. • The passage of time affects bond prices. The price of a bond will approach its par value as the bond moves closer to maturity.
- Review of Learning Goals (10 of 10) • LG 6 – Explain yield to maturity (YTM), its calculation, and the procedure used to value bonds that pay interest semiannually. • Yield to maturity is the rate of return investors earn if they buy a bond at a specific price and hold it until maturity • YTM can be calculated by using a financial calculator or by using an Excel spreadsheet • Bonds that pay interest semiannually are valued by using the same procedure used to value bonds paying annual interest except that the interest payments are one-half of the annual interest payments, the number of periods is twice the number of years to maturity, and the required return is one-half of the stated annual required return on similar-risk bonds
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- Long Description: The vertical axis represents the equilibrium interest rate and the horizontal axis represents funds supplied or demanded. The details of the graph are as follows: A supply line S naught slopes upward from the lower left corner to the upper right corner. Another supply line S1, also an upward sloping line, is parallel to S naught and is drawn below it. A demand line D slopes downward from the upper left corner to the lower right corner intersects both the supply lines. It intersects the line S naught at S naught equals D on the horizontal axis and r naught on the vertical axis. The demand line intersects line S1 at S1 equals D on the horizontal axis and r1 on the vertical axis. Dashed straight lines are drawn to the axes from the points of intersection. A downward arrow from r naught to r1 and a rightward arrow from S naught to S1 is visible between the dashed lines.
- Long Description: The vertical axis represents annual rate in percentage and ranges from 0 to 16 in increments of 2. The horizontal axis represents years and ranges from 1962 to 2019 in increments of 3. There are two lines in the graph, the details are as below: The line for the inflation rate starts at 1.25 percent for 1962 and shows a fluctuating trend over the years. The data shown is as follows: 1962: 1.25%; 1965: 2.0%; 1968: 4.0%; 1971: 3.0%; 1974: 12%; 1977: 6%; 1980: 13%; 1983: 3.5%; 1986: 1%; 1989: 4.5%; 1992: 3.0%; 1995: 2.5%; 1998: 1.5%; 2001: 1.75%; 2004: 3.0%; 2007: 4.0%; 2010: 1.5%; 2013: 1.5%; 2016: 2.0%; 2019: 2.0%. The line for the treasury bill rate starts at 3 percent for 1962 and shows a fluctuating trend over the years. The data shown is as follows: 1962: 3.0%; 1965: 4.0%; 1968: 5.0%; 1971: 4.0%; 1974: 7.0%; 1977: 5.0%; 1980: 12.0%; 1983: 9.0%; 1986: 8.0%; 1989: 9.0%; 1992: 4.0%; 1995: 7.0%; 1998: 5.0%; 2001: 6.0%; 2004: 1.0%; 2007: 5.0%; 2010: 0.5%; 2013: 0.1%; 2016: 0.5%; 2019: 2.5%. Wherever the treasury bill rate is higher than the inflation rate, the area between the lines is shaded and indicates positive real interest rates. Wherever the treasury bill rate is lesser than the inflation rate, the area between the lines is shaded and indicates negative real interest rates. Notes: All values are approximate. Interest rate on a 1-year Treasury bill issued in January of each calendar year. Change in consumer price index in each calendar year.
- Long Description: The vertical axis represents yield to maturity in percentage and ranges from 0 to 6 in increments of 1. The horizontal represents time to maturity in years and ranges from 0 to 35 in increments of 5. There are four lines in the graph and the details are as below: The line for February 2002 starts at 1.8 and with a high growth rate reaches to a value of 4 percent for 5 years and to a value of 5 percent for 10 years. The slopes of the line decreases thereafter and the line reaches a value of 5.5 percent by 15 years. The line remains almost in this range thereafter. The line for July 2006 starts at 5.2 percent and with a slight decline and falls down to 5.0 percent by 5 years. The line remains almost in this range thereafter. The line for December 2006 starts at 5.2 percent and with a sudden decline line falls down to 4.3 percent for 5 years. The line remains almost unchanged thereafter. The line for April 2020 starts at 0.1 percent and shows a growing trend. The line reaches a value of 1.5 percent for 5 years and 1.7 percent for 10 years. The slope of the line keeps going upward after 10 years. The line reaches a value of 1.2 percent for 20 years and 1.3 percent for 30 years. Note: All values are approximate.
- Long Description: The vertical axis represents the 5-year treasury bond yield to 1-year treasury bill yield in percentage and ranges from negative 3 to 3 in increments of 1. The horizontal axis represents the years and ranges from 1959 to 2019 in increments of 3. The details area as follows: The yield curve shows quick fluctuations over the years. The line starts at 0.4% in the year 1959 and with quick fluctuations and falls down to negative 0.2% in 1969 but recovers back to 1.6% between 1970 and 1971. The line falls down to an all-time low value of negative 2.4% in the year 1980. But, between 1980 and 1981 the line spikes back up to 1%, but plunges down to negative 1.8% in the year 1982. It recovers back to 1.4% in the year 1987. The line again falls to negative 0.2, but recovers to 2.5% in the year 1992. In the year 1995, the line falls to 2%, it drops to negative 1% in the year 1999. Between the year 2000 and 2001 the line is at negative 0.5 and maintains a similar trend till 2007, but peaks up in years 2001 and 2003 at 2.1% and 2% respectively. In the year 2007 the line dips to negative 0.4%, in 2009 it is at 2.1%, then between 2010 and 11 the line is at 1%. In the year 2011 the line recovers back up to 2%, but fall to 0.4% in 2012. In the year 2014 it rises up to 1.7%, but drops to 0.6 in 2015 and further dips to negative 0.3 in 2019. Note: All values are approximate.
- Long Description: The table lists Moody’s. Standard and Poor’s. and interpretation, respectively are as follows: investment grade bonds, Aaa, AAA, highest grade or lowest credit risk; Aa, AA, high grade; A, A, upper medium grade; Baa, BBB, medium grade; speculative or non-investment grade bonds, Ba, BB, lower medium grade or speculative; B, B, speculative; Caa, CCC, very speculative; Ca, CC, near default; C, C and D, likely in default.
- Long Description: The timeline is labeled "Year" and ranges from 0 through 10 in increments of 1. The bond value is depicted as the present value at Year 0, for each year. The cash flows is as follows: Year 1 to 10: 60 dollars each and totalling to 441.61 dollars. Future Value: 1,000 dollars. Year 0: 558.39 dollars. Arrows from all years are drawn and appear to converge at year 0. Text here reads: B naught is 1,000. 00 dollars, which is equal to PV.
- Long Description: The layout of keys on the calculator screen is as follows: Row 1: Top-left, 2 ND open square parentheses P Slash Y close square parentheses. Top-right, 1; Enter. Row 2: 2 ND open square parentheses Quit close square parentheses. Row 3: 10; N. Row 4: 6; I slash Y. Row 5: 60; PMT. Row 6: 1000; FV. Row7: CPT; PV; Negative 1,000.
- Long Description: The details are as follows: Par value: 1000 dollars Coupon rate: 6% Annual coupon: 60 dollars Required annual return: 6% Number of years to maturity: 10 Bond value: Negative 1,000.00 dollars. Text below reads: Entry in cell B4 is equal to B2 times B3. Entry in cell B7 is equal to PV open parentheses B5, B6, B4, B2, 0 close parentheses. The minus sign appears before the $1,000.00 in B7 because the bond’s price is a cost for the investor.
- Long Description: The layout of keys on the calculator screen is as follows: Row 1: Top-left, 2 ND open square parentheses P Slash Y close square parentheses. Top-right, 1; Enter. Row 2: 2 ND open square parentheses Quit close square parentheses. Row 3: 20; N. Row 4: 6; I slash Y. Row 5: 30; PMT. Row 6: 1000; FV. Row7: CPT; PV; Negative 1,000.
- Long Description: The details are as follows: Par value: 1000 dollars Coupon rate: 6% Annual coupon: 60 dollars Coupon payments per year: 2 Required annual return: 6% Number of years to maturity: 10 Bond value: Negative 1,000.00 dollars. Text below reads: Entry in cell B4 is equal to B2 times B3. Entry in cell B7 is equal to PV open parentheses B6 slash B5, B7 times B5, B4 slash B5, B2, 0 close parentheses. The minus sign appears before the $1,000.00 in B8 because the bond’s price is a cost for the investor.
- Long Description 1: The layout of keys on the calculator screen 1 are as follows: Row 1: Top-left, 2 ND open square parentheses P Slash Y close square parentheses. Top-right, 1; Enter. Row 2: 2 ND open square parentheses Quit close square parentheses. Row 3: 10; N. Row 4: 8; I slash Y. Row 5: 60; PMT. Row 6: 1000; FV. Row 7: CPT; PV; Negative 865.80. Long Description 2: The layout of keys on the calculator screen 2 are as follows: Row 1: Top-left, 2 ND open square parentheses P Slash Y close square parentheses. Top-right, 1; Enter. Row 2: 2 ND open square parentheses Quit close square parentheses. Row 3: 10; N. Row 4: 4; I slash Y. Row 5: 60; PMT. Row 6: 1000; FV. Row 7: CPT; PV; Negative 1,162.22.
- Long Description: The details are as follows: Par value: 1000 dollars; 1000 dollars Coupon rate: 6%; 6% Annual coupon: 60 dollars; 60 dollars Required annual return: 8%; 4% Number of years to maturity: 10; 10 Bond value: Negative 865.80 dollars; Negative 1,162. 22 dollars. Text below reads: Entry in cell B4 is equal to B2 times B3. Entry in cell B7 is equal to PV open parentheses B5, B6, B4, B2, 0 close parentheses. Note that the bond trades at a discount that is below par because the bond’s coupon rate is below investors’ required rate of return. Entry in cell C4 is equal to C2 times C3. Entry in cell C7 is equal to PV open parentheses C5, C6, C4, C2, 0 close parentheses. Note that the bond trades at a discount that is below par because the bond’s coupon rate is below investors’ required rate of return.
- Long Description: The vertical axis represents market value of bond, B naught in dollars and ranges from 0 to 1,600 in increments of 200. The horizontal axis represents required return, r in percentage and ranges from 0 to 18 in increments of 2. A curve slopes downward from the upper left corner to the lower right corner. The solid lines are drawn from a point on the curve to the required return of 6 percent on the horizontal axis and par value of 1000 dollars on the vertical axis. Dashed lines are drawn from a point on the curve above the par value to 4 percent required return and premium value of 1,162 dollars, and from a point below the par value to 8 percent required return and a discount value of 866 dollars.
- Long Description: The horizontal axis represents change in required rate in percentage points and ranges from negative 5 to 5, in unit increments. The vertical axis represents change in bond price in percentage and ranges from negative 60 to 140, in increments of 20. There are four lines in the graph and the details are as below: Line A starts at 20 % change in bond price and slopes down to negative 10%. Line B begins at 76% change in bond price and slopes down to negative 35%. Line C opens at 80% change in bond price and trails off to negative 38%. Line D begins at 124% change in bond price and steeps down negative 42%. All the lines intersect at value 0 on the horizontal axis. Note: All values are approximate. A table in the graph lists the following information: Bond A: Coupon,10%; Maturity, 5 years; Required return, 10%. Bond B: Coupon,10%; Maturity, 30 years; Required return, 10%. Bond C: Coupon, 6%; Maturity, 30 years; Required return, 10%. Bond D: Coupon,6%; Maturity, 30 years; Required return, 6%.
- Long Description: The vertical axis represents the market value of Bond, B naught in dollars and ranges from 866 to 1,162. The axis is truncated from 10 to 886. The horizontal axis represents time to maturity in years and ranges from 10 to 0 in decrements of 1. There are three lines in the graph. The details are as below: The line for par-value bond, required return, where r is equal to 6 % is a line parallel to the horizontal axis at the value of 1000 dollars. The line for discount bond, required return, where r is equal to 8 % is an upward sloping line and reaches to a value of 885 dollars in 8 years, 948 dollars in 3 years, and finally to 1000 dollars on maturity, which is 0 years. The line for premium bond, required return, where r is equal to 4% is a downward sloping line and falls down to a value of 1,135 dollars in 8 years, 1,055 dollars in 3 years, and finally to 1000 dollars on maturity, which is 0 years. For all the three lines, dashed lines from the axes are drawn at these intersection points to depict the same. All the three lines converge at a point M, which is at 0 years.
- Long Description: The equation reads 929.76 dollars equals 60 dollars over YTM times left parenthesis1 minus start fraction 1 over left parenthesis 1 plus YTM right parenthesis super 10 end fraction right parenthesis plus start 1000 dollars over left parenthesis 1 plus YTM right parenthesis super 10.
- Long Description: The equation reads 929.76 dollars questioned equal-to 60 dollars over 0.10 times left parenthesis 1 minus start fraction 1 over left parenthesis 1 plus 0.10 right parenthesis super 10 end fraction right parenthesis plus start 1000 dollars over left parenthesis 1 plus 0.10 right parenthesis super 10; 929.76 dollars not-equal-to 600 dollars left parenthesis 0.61446 right parenthesis plus 385.54 dollars equals 368.68 dollars plus 385.54 dollars equals 754.22 dollars.
- Long Description: The equation reads 929.76 dollars questioned equal-to 60 dollars over 0.08 times left parenthesis 1 minus start fraction 1 over left parenthesis 1 plus 0.08 right parenthesis super 10 end fraction right parenthesis plus start 1000 dollars over left parenthesis 1 plus 0.08 right parenthesis super 10.
- Long Description: The equation reads 929.76 dollars questioned equal-to 60 dollars over 0.07 times left parenthesis 1 minus start fraction 1 over left parenthesis 1 plus 0.07 right parenthesis super 10 end fraction right parenthesis plus start 1000 dollars over left parenthesis 1 plus 0.07 right parenthesis super 10. 929.76 dollars equals 857.14 dollars times 0.49165 plus 508.35 dollars equals 421.41 dollars plus 508.35 dollars equals 929.76 dollars.
- Long Description: The layout of keys on the calculator screen is as follows: Row 1: Top-left, 2 ND open square parentheses P Slash Y close square parentheses. Top-right, 1; Enter. Row 2: 2 ND open square parentheses Quit close square parentheses. Row 3: 10; N. Row 4: Negative 929.96; Plus, or minus PV. Row 5: 60; PMT. Row 6: 1,000; FV. Row 7: CPT; I slash Y; 7.0.
- Long Description: The details of the cash flow are as follows: Year 0: Negative 929.76 dollars Year 1 to Year 9: 60 dollars Year 10: 1,060 dollars YTM: 7.0%. Text at the bottom reads: Entry in cell B14 is equal to IRR open parentheses B3 is to B13.
- Long Description: The details are as below: Par value: 1000 dollars Coupon rate: 6% Annual coupon: 60 dollars Coupon payments per year: 1 Number of years to maturity: 10 Bond value: Negative 929.76 dollars Bond yield to maturity: 7%. Text below reads: Entry in cell B4 is equal to B2 times B3. Entry in cell B8 is equal to RATE open parentheses B timesB5, B4 slash B5, B7, B2, 0 close parentheses multiplied by B5. Because the Excel RATE function returns a periodic rate of return that corresponds to the payment frequency, the periodic rate must be multiplied times the annual payment frequency to find the annual YTM.
- Long Description: The equation reads 929.76 dollars equals 60 dollars over 2 over YTM over 2 times left parenthesis 1 minus start fraction 1 over left parenthesis 1 plus start fraction YTM over 2 end fraction right parenthesis super 2 times 10 end fraction right parenthesis plus start fraction 1000 dollars over left parenthesis 1 plus start fraction YTM over 2 end fraction right parenthesis super 2 times 10.
- Long Description: An equation reads 929.76 dollars questioned equal-to 60 dollars over 2 over 0.070 over 2 times left parenthesis 1 minus start fraction 1 over left parenthesis 1 plus start fraction 0.070 over 2 end fraction right parenthesis super 2 times 10 end fraction right parenthesis plus start fraction 1000 dollars over left parenthesis 1 plus start fraction 0.06988 over 2 end fraction right parenthesis super 2 times 10; 929.76 dollars not-equal-to 857.14 dollars left parenthesis 0.49743 right parenthesis plus 502.57 dollars equals 928.94 dollars.
- Long Description: The calculation reads 929.76 dollars questioned equal-to 60 dollars over 2 over 0.06988 over 2 times left parenthesis 1 minus start fraction 1 over left parenthesis 1 plus start fraction 0.06988 over 2 end fraction right parenthesis super 2 times 10 end fraction right parenthesis plus start fraction 1000 dollars over left parenthesis 1 plus start fraction 0.06988 over 2 end fraction right parenthesis super 2 times 10; 929.76 dollars approximately equals 858.61 dollars left parenthesis 0.49685 right parenthesis plus 503.15 dollars equals 426.60 dollars plus 503.15 dollars equals 929.75 dollars.
- Long Description: The layout of keys on the calculator screen is as follows: Row 1: Top-left, 2 ND open square parentheses P Slash Y close square parentheses. Top-right, 2; Enter. Row 2: 2 ND open square parentheses Quit close square parentheses. Row 3: 20; N. Row 4: Negative 929.76; Plus, or minus PV. Row 5: 30; PMT. Row 6: 1,000; FV. Row 7: CPT; I slash Y; 6.988.
- Long Description: The details are as below: Par value: 1000 dollars Coupon rate: 6% Annual coupon: 60 dollars Coupon payments per year: 2 Number of years to maturity: 10 Bond value: Negative 929.76 dollars Bond yield to maturity: 6.988%. Text below reads: Entry in cell B4 is equal to B2 times B3. Entry in cell B8 is equal to RATE open parentheses B times B5, B4 slash B5, B7, B2, 0 close parentheses multiplied by B5. Because the Excel RATE function returns a periodic rate of return that corresponds to the payment frequency, the periodic rate must be multiplied times the annual payment frequency to find the annual YTM.

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