FM24 Lecture Note 4: Interest Rates and Bonds PDF

Lecture Note 4. Interest Rates and Bonds Financial Management Main Contents Interest Rate Quotes and Adjustments Application: Discount Rates and Loans The Determinants of Interest Rates The Opportunity Cost of Capital Bond Valuation Bond Terminology Zero-Coupo...

Lecture Note 4. Interest Rates and Bonds Financial Management Main Contents Interest Rate Quotes and Adjustments Application: Discount Rates and Loans The Determinants of Interest Rates The Opportunity Cost of Capital Bond Valuation Bond Terminology Zero-Coupon Bonds Coupon Bonds Why Do Bond Prices Change? Corporate Bonds Financial Management 2 Lecture Note 4. Interest Rates and Bonds Learning Objectives Understand the different ways interest rates are quoted Use quoted rates to calculate loan payments and balances Know how inflation, expectations, and risk combine to determine interest rates See the link between interest rates in the market and a firm’s opportunity cost of capital Understand bond terminology and compute the price and yield to maturity of a zero-coupon bond and a coupon bond Analyze why bond prices change over time Know how credit risk affects the expected return from holding a corporate bond Financial Management 3 Lecture Note 4. Interest Rates and Bonds 1. Interest Rate Quotes and Adjustments Interest rate is the price of using money Interest rates may be quoted for different time intervals such as daily, monthly, semiannual, or annual It is often necessary to adjust the interest rate to a time period that matches that of our cash flows Effective annual rate (EAR) or annual percentage yield (APY) The total amount of interest that will be earned at the end of one year With an EAR of 5%, a $100 investment grows to $100  (1 + r) = $100  (1.05) = $105 After two years it will grow to $100  (1 + r)2 = $100  (1.05)2 = $110.25  Earning an EAR of 5% for two years is equivalent to earning 10.25% in total interest over the entire period Financial Management 4 Lecture Note 4. Interest Rates and Bonds 1. Quotes and Adjustments (Cont.) Adjusting the discount rate to different time periods In general, by raising the interest rate factor (1 + r) to the appropriate power, we can compute an equivalent interest rate for a longer (or shorter) time period We can use the same method to find the equivalent interest rate for periods shorter than one year In this case, we raise the interest rate factor (1+r) to the appropriate fractional power Ex: (1.05)0.5 = $1.0247, so an annual rate of 5%, is equivalent to a rate of 2.47% every half of a year A discount rate of r for one period can be converted to an equivalent discount rate for n periods Equivalent n-period discount rate = (1+r)n – 1 (4.1) When computing present or future values, you should adjust the discount rate to match the time period of the cash flows Financial Management 5 Lecture Note 4. Interest Rates and Bonds 1. Quotes and Adjustments (Cont.) Problem for EAR Suppose your bank account pays interest monthly with an effective annual rate of 6%. What amount of interest will you earn each month? From Eq. 4.1, a 6% EAR is equivalent to earning (1.06)1/12 – 1 = 0.4868% per month  The exponent in this equation is 1/12 because the period is 1/12 th of a year (a month) If you have no money in the bank today, how much will you need to save at the end of each month to accumulate $100,000 in 10 years? Timelines for the above saving plan  We can view the savings plan as a monthly annuity with 10  12 = 120 monthly payments We have the future value of the annuity ($100,000), the length of time (120 months), and the monthly interest rate (0.4868% per month) from the first question, therefore, FV (annuity) $100, 000 C= = = $615.47 per month 1 [(1 + r ) − 1] n 1 [(1.004868)120 − 1] r 0.004868 Financial Management 6 Lecture Note 4. Interest Rates and Bonds 1. Quotes and Adjustments (Cont.) Annual Percentage Rates (APR) It indicates the amount of simple interest earned in one year, that is, the amount of interest earned without the effect of compounding It is the most common way to quote interest rates Because it does not include the effect of compounding, the APR quote is typically less than the actual amount of interest that you will earn To compute the actual amount you will earn in one year, you must convert the APR to an EAR Because the APR does not reflect the true amount you will earn over one year, the APR itself cannot be used as a discount rate Instead, the APR is a way of quoting the actual interest earned each compounding period Interest rate per compounding period = APR / m  where, m is the number of compounding period per year Converting an APR to an EAR 𝑚 𝐴𝑃𝑅 1 + EAR = 1 + 𝑚 Financial Management 7 Lecture Note 4. Interest Rates and Bonds 1. Quotes and Adjustments (Cont.) Problem: converting the APR to a discount rate Your firm is purchasing a new telephone system that will last for four years You can purchase the system for an upfront cost of $150,000, or you can lease the system from the manufacturer for $4,000 paid at the end of each month The lease price is offered for a 48-month lease with no early termination—you cannot end the lease early and your firm can borrow at an interest rate of 6% APR with monthly compounding. Should you purchase the system outright or pay $4,000 per month? Solution  The cost of leasing the system is a 48-month annuity of $4,000 per month  The monthly compounding rate is 6%/12 = 0.5%, then it’s present value is 1  1  PV = 4000  1 −  = $170,321.27 0.005  1.00548   The PV of the cost of the lease is greater than purchasing value, $150,000, so, you should purchase the system outright Financial Management 8 Lecture Note 4. Interest Rates and Bonds 2. Application: Discount Rates and Loans Computing loan payments Consider a $30,000 car loan which is an amortizing loan with these terms: 6.75% APR with monthly compounding for 5 years → 6.75%/12 = 0.5625% per month  Amortizing loan: a loan on which the borrower makes monthly payments that include interest on the loan plus some part of the loan balance The timeline for a $30,000 car loan The monthly payment for a $30,000 car loan Financial Management 9 Lecture Note 4. Interest Rates and Bonds 2. Application (Cont.) Computing the outstanding loan balance The outstanding balance (or principal) on an amortizing loan is different each time It is equal to the PV of the remaining future loan payments  We calculate the outstanding loan balance by determining the PV of the remaining loan payments using the loan rate as the discount rate Example Let’s say that you are now 3 years into a $30,000 car loan (at 6.75% APR, originally for 60 months) and you decide to sell the car. When you sell the car, you will need to pay whatever the remaining balance is on your car loan. After 36 months of payments, how much do you still owe on your car loan?  We have already determined that the monthly payments on the loan are $590.50  The remaining balance on the loan is the present value of the remaining 2 years, or 24 months, of payments and the monthly discount rate is 0.5625%. The outstanding loan balance, therefore, is 1  1  Balance with 24 months remaining = $590.50  1 −  = $13, 222.32 0.005625  1.00562524  Financial Management 10 Lecture Note 4. Interest Rates and Bonds 3. The Determinants of Interest Rates How are interest rates determined? Fundamentally, interest rates are determined by market forces based on the relative supply and demand of funds There are various factors that may influence interest rates Inflation, current economic activity, expectations of future growth, etc. Inflation: real vs. nominal rates Nominal interest rates The rate at which your money will grow if invested for a certain period Real interest rate The rate of growth of your purchasing power, after adjusting for inflation 1+Nominal rate Growth of Money Growth in purchasing power=1+Real rate= = 1+Inflation rate Growth of Prices Real rate = (Nominal rate – Inflation rate)/(1+Inflation rate) ≈ Nominal rate – Inflation rate Financial Management 11 Lecture Note 4. Interest Rates and Bonds 3. The Determinants (Cont.) Investment and interest rate policy Interest rates affect not only individual’s propensity to save, but also firms’ incentive to raise capital and invest When the costs of an investment precede the benefits, an increase in the interest rate will make the investment less attractive The central bank attempts to use the relationship between interest rates and investment incentives when trying to guide the economy The central bank will often lower or raise interest rates in an attempt to stimulate investment if the economy is slowing or to reduce investment if the economy is overheating Financial Management 12 Lecture Note 4. Interest Rates and Bonds 3. The Determinants (Cont.) The yield curve and discount rates The interest rates that banks offer on investments or charge on loans depend on the horizon, or term, of the investment or loan The relationship between the investment term and the interest rate is called the term structure of interest rates We can use the term structure to compute the PVs and FVs of a risk-free cash flow over different investment horizons  Yield curve: a plot of bond yields as a function of the bonds’ maturity date  Risk-free interest rate: the interest rate at which money can be borrowed or lent without risk over a given period Present value of a cash flow stream using a term structure of discount rates C1 C2 CN PV = + +  + 1+ 0 r1 (1+ 0 r2 )2 (1+0 rN )N Financial Management 13 Lecture Note 4. Interest Rates and Bonds 3. The Determinants (Cont.) Term structure of risk-free U.S. interest rates, November 2006, 2007, and 2008 Problem: compute the present value of a risk-free five-year annuity of $1,000 per year, given the yield curve for November 2008 in the above term structure 1,000 1,000 1,000 1,000 1,000 PV = + + + + = $4,775 1.0091 1.00982 1.01263 1.01694 1.02015 Financial Management 14 Lecture Note 4. Interest Rates and Bonds 4. Opportunity Cost of Capital Opportunity cost of capital The best available expected return offered in the market on an investment of comparable risk and term to the cash flow being discounted The opportunity cost of capital is the return the investor forgoes when the investor takes on a new investment and it is the minimum return required by investor for an investment with the same risk and term For a risk-free project, it will typically correspond to interest rate on Treasury securities with a similar term (or risk-free interest) But, for a risky project, it will determined by risk-free interest rate plus risk premium required by investor for taking a risk Financial Management 15 Lecture Note 4. Interest Rates and Bonds 4. Opportunity Cost of Capital (Cont.) Opportunity cost of capital (Cont.) Problem Suppose a friend offers to borrow $100 from you today and in return pay you $110 one year from today Looking in the market for other options for investing your money, you find your best alternative option for investing the $100 that view as equally risky as lending it to your friend That option has an expected return of 8%. What should you do? Your decision depends on what the opportunity cost is of lending your money to your friend Your opportunity cost is at best an 8% expected return and it is less than the expected return of your friend’s offer. So, you will better off to make the loan for your friend Financial Management 16 Lecture Note 4. Interest Rates and Bonds 5.1 Bond Terminology Bond A security sold by government and corporations to raise funds from investors today in exchange for promised future payments (interests and principal) It is simply a loan. When an investor buys a bond from an issuer, the investor is lending money to the bond issuer and will receive promised interests and principal in the future Bond terminology Bond certificate  It states the terms of a bond as well as the amounts and dates of all payments to be made Face value, par value or principal amount: the notional amount  The face value is used to compute the interest payments and typically, it is repaid at maturity Maturity date: the final repayment date of a bond  Term: the time to maturity or the time remaining until the final repayment date of a bond Coupons: the promised interest payments of a bond  It is determined by multiplying the face value and the coupon rate of a bond and dividing it with the number of coupon payments in a year  It is paid periodically until the maturity date of the bond Financial Management 17 Lecture Note 4. Interest Rates and Bonds 5.1 Bond Terminology (Cont.) Types of bonds According to coupon payment ways Perpetuity bond, coupon bond (fixed or floating) and zero-coupon bond (pure discount bond) According to issuing prices Premium bond, par bond and discount bond According to issuers Government bond and public bond, corporate bond, financial bond, specific laws bond, private loan Whether or not there is a collateral or a guarantee Secured bond / unsecured bond, guaranteed bond / non-guaranteed bond According to option type embedded in a bond Callable bond and puttable bond Convertible bond (CB), Exchange bond (EB), Bond with warrant (BW) Financial Management 18 Lecture Note 4. Interest Rates and Bonds 5.2 Zero-Coupon Bonds Zero-coupon bonds or zeros Bonds not paying a coupon during the life of bonds That means the coupon rate of them is zero In trading a zeros, there are only two cash flows The bond’s market price at the time of purchase The bond’s face value at maturity Timeline for a zeros and its valuation 0 (Today) 1 2  T (Maturity) FV FV  B0 = (1+ 0 rT )T Problem What is the fair price of a one-year(from now), risk-free, zero-coupon bond with a $100,000 face value if the corresponding risk-free interest rate (or market interest rate) is 3.5% APR? FV 100,000 B0 = (1+0 r1 ) (1.035) = $96,618.36 = Financial Management 19 Lecture Note 4. Interest Rates and Bonds 5.2 Zero-Coupon Bonds (Cont.) Yield to maturity (YTM) of a zero-coupon bond The rate of return of an investment in a bond that is held to its maturity date The discount rate that sets the present value of the promised bond payments equal to the current market price of the bond YTM of an n-year zeros 1/n FV 1+YTMn = Market price Problem Suppose the following zero-coupon bonds are trading at the prices shown below per $100 face value. Determine the corresponding yield to maturity for each bond Maturity 1 year 2 years 3 years 4 years Price $96.62 $92.45 $87.63 $83.06  YTM1 = (100/96.62) – 1 = 3.50%, YTM2 = (100/92.45)1/2 - 1 = 4.00%  YTM3 = (100/87.63)1/3 - 1 = 4.50%, YTM4 = (100/83.06)1/4 - 1 = 4.75% Financial Management 20 Lecture Note 4. Interest Rates and Bonds 5.2 Zero-Coupon Bonds (Cont.) Risk-free interest rates A default-free zero-coupon bond that matures on date n provides a risk-free return over that period We often refer to the YTM of a default-free zero-coupon bond that matures on date n as the risk-free interest rate for that period Spot interest rates Default-free, zero-coupon yields In the yield curve, which plots the risk-free interest rate for different maturities, rates are the yields of risk-free zero-coupon bonds Financial Management 21 Lecture Note 4. Interest Rates and Bonds 5.3 Coupon Bonds Coupon bonds The value of coupon bond is the sum of PV of all future coupon payments and PV of the face value at maturity date Coupon bonds make regular coupon interest payments and pay the face value at maturity date Timeline for a coupon bond and its valuation 0 (Today) 1 2  T (Maturity) I1 I2  IT + FV I1 I2 IT FV  B0 = + +  + + (1+ 0 r1 )1 (1+ 0 r2 )2 (1+ 0 rT )T (1+ 0 rT )T Return on a coupon bond comes from the difference between the purchase price and the principal value and periodic coupon payments To compute the YTM of a coupon bond, we need to know the current market price of the bond, the coupon interest payments and when they are paid Financial Management 22 Lecture Note 4. Interest Rates and Bonds 5.3 Coupon Bonds (Cont.) Coupon bonds (Cont.) Problem Assume that it is May 15, 2010 and the U.S. Treasury has just issued securities with May 2015 maturity, $1,000 par value and a 2.2% coupon rate with semiannual coupons. The first coupon payment will be paid on November 15, 2010. What cash flows will you receive if you hold this note until maturity?  The annual coupon amount = $1,000×0.022 = $22  The semiannual coupon means the bond pays the coupon every 6 months (two times in a year). So, each coupon amount for every 6 months is a half of the annual coupon amount, $11 The timeline of future cash flows from the bond Financial Management 23 Lecture Note 4. Interest Rates and Bonds 5.3 Coupon Bonds (Cont.) Valuation of a coupon bond Using the following term structure information, computing the fair value of a coupon bond Maturities Rates (APR, %) Maturities Rates (APR, %) 6 months (0r0.5) 5.8% 24 months (0r2.0) 6.4% 12 months (0r1.0) 6.0% 30 months (0r2.5) 6.6% 18 months (0r1.5) 6.2% 36 months (0r3.0) 6.8% A 3-year(from now), 5% coupon bond with a $1,000 face value If the coupon rate is annual rate and interest rates in the term structure are also annual rates, then the fair price of the coupon bond is 50 50 1,050 50 50 1,050 B0 = + + = + + = $953.27 (1+ 0 r1 ) (1+ 0 r2 ) 2 (1+ 0 r3 ) 3 (1.06)1 (1.064)2 (1.068)3 Financial Management 24 Lecture Note 4. Interest Rates and Bonds 5.3 Coupon Bonds (Cont.) Valuation of a coupon bond (Cont.) Using the following term structure information, computing the fair value of a coupon bond Maturities Rates (APR, %) Maturities Rates (APR, %) 6 months (0r0.5) 5.8% 24 months (0r2.0) 6.4% 12 months (0r1.0) 6.0% 30 months (0r2.5) 6.6% 18 months (0r1.5) 6.2% 36 months (0r3.0) 6.8% A 3-year(from now), 5% coupon bond with a $1,000 face value If the coupon rate and interest rates in the term structure are all semiannual rates, how to change the fair price of the coupon bond?  In this case, coupon payment is $25(=$1,000×5%÷2) per 6 months (0.5 years) 25 25 25 25 25 1,025 𝐵0 = 𝑟 + 2+ 3+ 4+ 5+ 6 1 + 0 0.5 𝑟 𝑟 𝑟 𝑟 𝑟 2 1 + 0 1.0 1 + 0 1.5 1 + 0 2.0 1 + 0 2.5 1 + 0 3.0 2 2 2 2 2 = $952.66 Financial Management 25 Lecture Note 4. Interest Rates and Bonds 5.3 Coupon Bonds (Cont.) Coupon bond price quotes Prices and yields are often used interchangeably Bond traders usually quote yields rather than prices One advantage is that the yield is independent of the face value of the bond When prices are quoted in the bond market, they are conventionally quoted per $100 face value in U.S and per ₩10,000 face value in Korea Financial Management 26 Lecture Note 4. Interest Rates and Bonds 6. Why Do Bond Prices Change? After the issue date, the market price of a bond generally changes over time At any point in time, changes in market interest rates affect the bond’s YTM and its price As time passes, the bond gets close to its maturity date and the price converge to the face value Interest rate changes and bond prices If a bond sells at par, the only return investors will earn is from the coupons that the bond pays. Therefore, the bond’s coupon rate will exactly equal its yield to maturity As interest rates fluctuate, the yields that investors demand will also change There is a inverse relation between changes of interest rates and changes of bond prices Financial Management 27 Lecture Note 4. Interest Rates and Bonds 6. Why Do Bond Prices Change? (Cont.) The relation of interest rate and bond price 1,000,000 900,000 B o 800,000 n 700,000 d 600,000 P r 500,000 i 400,000 c e 300,000 200,000 ( ₩ ) 100,000 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Interest rates or YTM (%) Financial Management 28 Lecture Note 4. Interest Rates and Bonds 6. Why Do Bond Prices Change? (Cont.) Time and bond prices The bond price slowly rising as a coupon payment nears and then dropping abruptly after the payment is made This pattern continues for the life of the bond and finally the bond price converges to the face value as the time goes to maturity date Figure: the effect of time on bond prices Financial Management 29 Lecture Note 4. Interest Rates and Bonds 6. Why Do Bond Prices Change? (Cont.) Interest rate risk and bond prices Effect of time on bond prices is predictable, but unpredictable changes in interest rates also affect prices Bonds with different characteristics will respond differently to changes in interest rates Investors view long-term bonds to be riskier than short-term bonds because the price changes of long-term bonds due to interest rate changes are more sensitive than those of short-term bonds Bonds with higher coupon rates are less sensitive to interest rate changes then otherwise identical bonds with lower coupon bond because bonds with higher coupon rates pay higher cash flows upfront Bond prices in practice Bond prices are subject to the effects of both passage of time and changes in interest rates Prices converge to face value due to the time effect, but simultaneously move up and down because of changes in bond yields Financial Management 30 Lecture Note 4. Interest Rates and Bonds 7. Corporate Bonds Credit Risk Credit risk is the risk of default by the issuer of any bond that is not default free It is an indication that the bond’s cash flows are not known with certainty Treasury bonds issued by government are widely regarded to be risk-free but corporate bonds are not Corporate bonds may fail to pay the promised cash flows due to the financial distress of issuing firms. We call this possibility credit risk, default risk or counterparty risk Corporations with higher default risk will need to pay higher coupons to attract buyers to their bonds Corporate bond yields How does the credit risk of default affect bond prices and yield? YTM of a defaultable bond is not equal to the expected return of investing in the bond  YTM is calculated under an assumption that all promised cash flows are normally paid but corporate bonds with credit risk cannot guarantee this assumption A higher YTM does not necessarily imply that a bond’s expected return is higher Financial Management 31 Lecture Note 4. Interest Rates and Bonds 7. Corporate Bonds (Cont.) Bond ratings or credit ratings Several companies rate the creditworthiness of bonds Three best-known bond-rating companies are Standard & Poor’s, Moody’s and Fitch Ratings These ratings help investors assess creditworthiness The bond rating depends on the risk of bankruptcy and bondholders’ claim to assets in the event of bankruptcy Investment-grade bonds: bonds in the top 4 categories of credit worthiness with a low risk of default (Ex: AAA, AA, A, and BBB) Speculative bonds, junk bonds or high-yield bonds: bonds in below investment grade, having a high risk of default (Ex: BB, B, CCC, CC, C) Financial Management 32 Lecture Note 4. Interest Rates and Bonds 7. Corporate Bonds (Cont.) The definition of Long-term issue credit ratings - S&P Financial Management 33 Lecture Note 4. Interest Rates and Bonds 7. Corporate Bonds (Cont.) The definition of Short-term issue credit ratings - S&P Financial Management 34 Lecture Note 4. Interest Rates and Bonds 7. Corporate Bonds (Cont.) The definition of short- and long-term credit ratings suggested by S&P’s Financial Management 35 Lecture Note 4. Interest Rates and Bonds 7. Corporate Bonds (Cont.) Corporate yield curves We can plot a yield curve for corporate bonds just as we can for Treasury bonds Default spread or credit spread The difference between the yields of corporate bonds and Treasuries The magnitude of the credit spread will depend on investors’ assessment of the likelihood that a particular firm will default Figure: corporate yield curves for various ratings, March 2010 Financial Management 36 Lecture Note 4. Interest Rates and Bonds