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FNCE 3020 Financial Markets and Institutions Fall Semester 2005 Lecture 2: Understanding Interest Rates and Calculation of Returns Interest Rate What do you think of when you hear this term? Interest Rate Defined “Two-sided” Definition: Borrowing: the cost of borrowing or the price paid for the rental of funds. From the standpoint of “deficit” entities. Saving/Investing: the return from lending funds. From the standpoint of “surplus” entities. Both concepts usually expressed as a percentage per year (per annum). Interest Rates: Borrowing and Savings Rates Yields in percent per annum Sep 2, 2004 Aug 29, 2005 Change Direct Market Aaa Industrial Bonds: 5.55% 5.01% - 54 Indirect (Intermediary) Market CDs (6 months) 1.76% 4.03% +227 Prime Loans 4.50% 6.50% +200 Spread 2.74% 2.47% - 27 Source: http://www.federalreserve.gov/releases/h15/update/ Savings (Investing) versus Lending Rates Important Terms in Lending Loan Principal: the amount of funds the lender provides to the borrower. Maturity Date: the date the loan must be repaid Loan Term: the time period from initiation to maturity date. Interest Payment: the cash amount that the borrower must pay the lender for the use of the loan principal. Simple Interest Rate: the interest payment divided by the loan principal. the percentage of principal that must be paid as interest to the lender. Convention is to express on an annual basis, irrespective of the loan term. Types of Debt Instruments Simple Loan: Principal and all interest both paid at maturity (when loan comes due). Borrow $1,000 today at 5% and in 6 months pay $1,025. Commercial bank loans to businesses. Fixed-payment Loan: Equal monthly payments representing a portion of the principal borrowed plus interest. Paid for a set number of years, at which time the principal amount is fully repaid. Home mortgages (conventional), automobile loans. Amortization Example Mortgage Loan Amount $500,000 Years: 30 (Monthly payments) Interest rate: 7% (fixed) Monthly Payment $3,326.51 (for 360 months) First Month Payment (n = 1): Principal: $409.84; Interest: $2,916.67 (or, $3,326.51) Last Month Payment (n = 360): Principal: $3,307.22; Interest: $19.29 (or, $3,326.51) Any Ideas as to why this Cartoon Would get a Bigger Laugh in England. Types of Debt Instruments Discount Bond (Zero-coupon Bond): Purchased at a price below face (or par) value with face value paid at maturity. There are no interest payments. Treasury bills Coupon Bond: periodic interest payments (stated as the coupon rate) for a specified period of time after which the total principal (face or par value) is repaid. Treasury and corporate bonds. These may be callable! Treasury Bill Market Treasury bills (along with notes and bonds) are marketable securities the U.S. government sells in order to pay off maturing debt and raise the cash needed to run the federal government. Treasury securities are sold in primary markets through: Competitive bid. In this form of bidding, investors specify the rate or yield they will accept. Treasury will either accept or reject the bid. Noncompetitive bid. By bidding noncompetitively investors agree to accept whatever rate or yield is determined at the auction . Returns on Discount Instruments 91-day Treasury Bills Issue date: September 1, 2005 Price 99.117 (per $100; or $991.17 on a $1,000 T-bill) At maturity (12/1/05) the bill pays $1,000.00 (face value) Gain = $8.83 Two Calculated Yields on T-Bills Discount rate (yield): 3.49% The discount rate (yield) takes into account the return as a percent of the face value ($1,000) of the T-bill. Investment rate (yield): 3.57% The investment rate (yield) relates the investor's return to the purchase price ($996.12) of the T-bill. Source: http://wwws.publicdebt.treas.gov/AI/OFBills Note: 14 day, 28 day and 91 day T-bills are auctioned every week. Calculating Treasury T-Bill Yields Computations of yields on Treasury bills depend on the face value, purchase price and maturity of the issue. There are two methods for determining yields: The discount method relates the investor's return to the bill's face value; The investment method relates the investor's return to the bill's purchase price. Thus, the investment discount method tends to overstate yields (results in higher calculated yields) relative to those computed by the discount method. Purchase price will always be lower than the face value. Discount Yield Formula The following formula is used to determine the discount yield for T-bills: Discount yield = [(FV - PP)/FV] * [360/M] FV = face value PP = purchase price M = maturity of bill. M = number of days to maturity 360 = the number of days used by banks to determine short-term interest rates. Investment Yield Formula The following formula is used to calculate the investment yield: Investment yield = [(FV - PP)/PP] * [365 or 366/M] FV = face value PP = purchase price M = maturity of bill Note: use 366 for leap year calculations. When comparing the return on investment in T-bills to other short-term investment options, the investment yield method is generally used. This yield is alternatively called the bond equivalent yield, the coupon equivalent rate, the effective yield and the interest yield. Example: Calculating T-Bill Yields Using data for September 1, 2005 (see previous slide) Discount yield = [(FV - PP)/FV] * [360/M] = [$1,000 -991.17/1,000]*[360/91] = 3.49% Investment yield = [(FV - PP)/PP] * [365/M] = [$1,000 – 991.17/991.17]*[365/91] = 3.57% For a discussion of T-bills rate calculations see: http://www.fednewyork.org/aboutthefed/fedpoint/fed28.ht ml Returns on Coupon Bonds Assume a 10 year, $1,000 face value, coupon bond with a coupon rate of 10% (paid annually). You would receive $100 per year for ten years as the annual interest payment. At the end of ten years (on maturity date) you would receive $1,000 (face value) Bonds and Present Value Concept Defined: Present value is today’s value of a payment (or series of payments) to be received in the future. $100 to be received in 1 year is worth $95.24 today if we assume the interest to be earned is 5% If you invested $95.24 for 1 year at a rate of 5%, at the end of the year you would have $100.00 Importance: provides a mechanism for determining: Today’s price of a credit market instrument. Thus, allows us to compare the prices of different instruments with different payment schedules. Equivalent (comparable) interest rates on different instruments. Determination of Market Price Assume a financial instrument offers the following 2 year payment stream: Annual (end of the year) interest payments of $70 (two payments each of $70), or a 7% coupon rate. A principal (face value) repayment of $1,000 at the end of the second year. Assume interest rates on financial instruments of similar risk to the one above are offering returns of 10% Question: How much should you pay (i.e., what is the market price) for the financial instrument in question? Present Value (“Market Price”) of a Security The price the market should pay for the security noted on the previous slide is equal to the present value of the expected future income stream discounted at a rate of interest of 10% (this is the current opportunity cost), or: PV = $70/(1+.10) + $1,070/(1+.10)2 PV = $63.64 + $1,070/1.21 PV = $63.64 + $884.30 PV = $947.94 (this is the market price!) Note: The bond is selling at a price below its par value. What If? What if the opportunity cost (market interest rate) is less than the coupon rate? What will the market price of the bond be if the discount rate is 5%? PV = $70/(1+.05) + $1,070/(1+.05)2 PV = $66.67 + $1,070/1.1025 PV = $66.67 + $970.52 PV = $1037.19 (this is the market price!) Note: Now the bond is selling at a price above par Rules #1 and #2 #1: When the market interest rate rises above the coupon rate on a bond, the price of the bond falls (sells at a discount of par). #2: When the market interest rate falls below the coupon rate on a bond, the price of the bond rises (sells at a premium of par) Thus: there is an inverse relationship between market interest rates and bond prices. What If? What if the opportunity cost (market interest rate) is equal to the coupon rate? What will the market price of the bond be? PV = $70/(1+.07) + $1,070/(1+.07)2 PV = $65.42 + $1,070/1.1449 PV = $65.42 + $934.58 PV = $1000.00 (this is the market price!) Note: Rule #3 -- The price will always equal par if the market rate equals the coupon rate. What if We Vary the Time to Maturity and the Market Rate Rises Assume a one year bond. PV = $1,070/(1.10) PV = $972.72 (this is the market price) Compare to the two year bond. PV = $70/(1+.10) + $1,070/(1+.10)2 PV = $63.64 + $1,070/1.21 PV = $63.64 + $884.30 PV = $947.94 (this is the market price!) Note: The longer the term to maturity, the greater the price change (i.e., decline). What if We Vary the Time to Maturity and the Market Rate Falls Assume a one year bond. PV = $1,070/(1.05) PV = $1,019.05 (this is the market price) Compare to the two year bond. PV = $70/(1+.05) + $1,070/(1+.05)2 PV = $66.67 + $1,070/1.1025 PV = $66.67 + $970.52 PV = $1037.19 (this is the market price!) Note: The longer the term to maturity, the greater the price change (i.e., increase). Rule #4 Rule #4: The greater the term to maturity, the greater the change in price for a given change in market interest rates. This becomes very important when developing a bond maturity strategy to incorporate your expected changes in interest rates. What if you think interest rates will fall? Where should you concentrate the maturity of your bonds? What if you think interest rates will rise? Where should you concentrate the maturity of your bonds? Interest Rate Measures There are three important ways of calculating the interest rate on a financial instrument. These include: Discount Yield and Investment Yield: Used to measure the yield on T-bills (and other discounted securities) and T-notes and Bonds. See Specific formulas noted earlier. Yield to Maturity: The interest rate that equates the future payments to be received from a financial instrument with its market price today (present value). Current Yield: Coupon payment divided by the current market price of a financial instrument. Can be a rough approximation of yield to maturity. Yield to Maturity Uses the concept of present value in the determination of the yield to maturity. Yield to maturity (i) is calculated in the formula below where: P = market price (present value) C = coupon payments F = face value (at maturity) n = years to maturity C C C C F P 2 3 ... n 1 i 1 i 1 i 1 i 1 i n Current Yield: Coupon payment divided by the current market price of a financial instrument. C ic P Two Characteristics of Current Yield The nearer the market price is to par and longer is the maturity of bond, the better approximation to yield to maturity. Change in the current yield always signals change in same direction as the yield to maturity. Web Site for Calculating Yields Visit the following web site. It allows you to calculate the current yield and yield to maturity for data you input. http://www.moneychimp.com/calculator/bond _yield_calculator.htm Issues of Risk Price Risk: Interest rates may move against you and produce losses on holdings of fixed income securities. Specifically when interest rates rise (fixed income security prices will fall). Greatest risk (potential price change) the longer the maturity of the fixed income security. Reinvestment Risk: Potential if holding short term fixed income securities. Need to roll them over at maturity. Interest rate at which you will reinvest is uncertain. Duration Issue: The fact that two bonds have the same term to maturity, does not mean that they carry the same interest rate risk. Assume A 10 year, 10% coupon bond and A 10 year zero coupon bond. Which has the greatest interest rate (price) risk for a given change in interest rates? The zero coupon bond because none of its payment occur until maturity. Why: The present value impact will be greater than with the coupon issue. Using Duration Duration is an estimate of the average lifetime of a security’s stream of payments. Everything else equal: The longer the term to maturity, the longer duration. The lower the coupon rate, the longer the duration. And, the greater the duration of a security, the greater the interest rate (price) risk. Distinction Between Real and Nominal Interest Rates Real interest rate: The interest rate that is adjusted for expected changes in the price level, and is calculated: Ir = I - Pe Where: Ir = real rate of interest I = market (nominal) rate of interest Pe = expected rate of inflation, i.e., price level changes (over the maturity of the financial asset) Real Interest Rate Impacts Real interest rate more accurately reflect the true cost of borrowing and returns (e.g., on lending, investing, savings). When the real rate is low (or negative), there is a greater incentive to borrow and less incentive to lend. Why? When real rate is high, there is less incentive to borrow and more incentive to lend. Why? Calculation of Real Interest Rates If I = 10% and Pe = 1% then Ir = 10% - 1% = 9% If I = 10% and Pe = 8% then Ir = 10% - 8% = 2% Note: When real rate is high, less incentive to borrow and more to lend. Note: When real rate is low (or negative), greater incentive to borrow and less to lend. Are Real Rates Subject to Change? Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2004 Are There Differences Globally? United States 2 year government bond rate 4.26% Expected rate of inflation (2006) 2.60% Real interest rate 2.06% Japan 2 year government bond rate 1.62% Expected rate of inflation (2006)* 0.00% Real interest rate 1.62% Australia 2 year government bond rate 5.04% Expected rate of inflation (2006)* 2.60% Real interest rate 2.44% Source of date: The Economist, August 30, 2005 Internet Source of Interest Rate Date Historical and Current Data for U.S. http://www.federalreserve.gov/releases/h15/update/ Real Time Data (U.S. and other major countries) http://www.bloomberg.com Go to Market Data and then to Rates and Bonds Other Countries: Economist.com (both web source or hard copy)

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