Understanding Interest Rates in Money, Banking and Financial Markets, Study notes of Health sciences

Download Understanding Interest Rates in Money, Banking and Financial Markets and more Study notes Health sciences in PDF only on Docsity! Lecture Notes on MONEY, BANKING, AND FINANCIAL MARKETS Peter N. Ireland Department of Economics Boston College irelandp@bc.edu http://www2.bc.edu/~irelandp/ec261.html Chapter 4: Understanding Interest Rates 1. Types of Credit Market Instruments Simple Loan Fixed-Payment Loan Coupon Bond Discount Bond 2. Present Value 3. Yield to Maturity Simple Loan Fixed-Payment Loan Coupon Bond Discount Bond 4. Other Measures of Interest Rates Current Yield Yield on a Discount Basis 5. Interest Rates versus Returns This chapter begins the second part of the course, which focuses on the behavior of interest rates. The analysis starts on familiar ground, by reviewing some of the types of credit market instruments that we discussed earlier, in our overview of the financial system. In particular, it usefully classifies those credit market instruments into four types: simple loans, fixed-payment loans, coupon bonds, and discount bonds. 1 1.4 Discount Bond Also called a “zero-coupon bond.” Bought at a price below face value (bought at a discount); makes no interest payments, but returns face value at maturity. Example: Discount bond with $1,000 face value and one-year maturity sells for $900 today and returns $1000 in one year. The $100 difference between the purchase price and face value represents the interest payment. US Treasury bills are of this type. 2 Present Value The various types of credit market instruments all require payments at different times: simple loans and discount bonds make payments only at maturity, while fixed-payment loans and coupon bonds make regular payments until maturity. How can we find a unified approach to measuring interest rates on these various types of instruments? The key to answering this question lies with the concept of present value. Present value captures the idea that a dollar received in the future is less valuable than a dollar received today. To see how the concept of present value works, let’s consider a series of examples. Example 1: A simple loan of $100 requires the borrower to repay $100 principal plus $10 interest one year from now. For this simple loan, the interest payment expressed as a percentage of the principal is a sensible way of measuring the interest rate. In fact, when we measure the interest rate in this way, we are computing the simple interest rate: Simple Interest Rate = i = Interest Principal = $10 $100 = 0.10 = 10% Example 2: If you make a simple loan of $100 at the simple interest rate i = 0.10 for one year, you get $100 + $100× i = $100× (1 + i) = $100× (1.10) = $110 4 at the end of the year. If you lend this $110 out again for another year at the same simple interest rate i = 0.10, you get $110 + $110× i = $110× (1 + i) = $110× (1.10) = $121 at the end of the second year. Equivalently, we can write $100× (1 + i)× (1 + i) = $100× (1 + i)2 = $100× (1.10)2 = $121. If you lend the $121 out for a third year at the simple interest rate i = 0.10, you get $121× (1 + i) = $100× (1 + i)3 = $100× (1.10)3 = $133.10 at the end of the third year. Example 3: Starting with $100, if the simple interest rate on one-year loans is i, and if you make these loans for n consecutive years, you get $100× (1 + i)n at the end of those n years. Example 4: Working backwards, if the simple interest rate is i = 0.10, then: $110 = $100× (1 + i) received next year is worth $100 = $110/(1 + i) today. $121 = $100 × (1 + i)2 received two years from now is worth $100 = $121/(1 + i)2 today. $133.10 = $100×(1+i)3 received three years from now is with $100 = $133.10/(1+i)3 today. This process of working backwards is called “discounting the future.” Example 5: If the simple interest rate is i, then the present value of $1 received n years from now is defined as Present value of $1 received n years from now = $1 (1 + i)n < $1. Thus, the idea of present value captures the face that $1 received in the future is worth less than $1 received today. More generally, Present value of $X received n years from now = $X (1 + i)n . 5 3 Yield to Maturity Yield to Maturity = the simple interest rate that equates the present value of payments received from a debt instrument to the price or value of that debt instrument today. Also sometimes called the “internal rate of return.” This is the most accurate and widely-applicable measure of interest rates. In fact, we can use this definition to compute the yield to maturity on each of our four types of credit market instruments. 3.1 Simple Loan Consider a simple loan of $100 that repays $100 principal plus $10 interest, for a total of $110, in one year. The yield to maturity i must satisfy Value today = $100 = Present value of future payments = $110 1 + i . Let’s solve for i: $100 = $110 1 + i $100× (1 + i) = $110 $100 + $100× i = $110 $100× i = $110− $100 i = $110− $100 $100 = $10 $100 = 0.10 = 10%. Thus, for simple loans, the yield to maturity equals the simple interest rate. 3.2 Fixed-Payment Loan Consider a fixed payment loan of $1,000 that requires payments of $126 per year for 25 years. The yield to maturity i must satisfy Value today = $1000 = Present value of future payments = $126 1 + i + $126 (1 + i)2 + $126 (1 + i)3 + ...+ $126 (1 + i)25 . 6 $900× (1 + i) = $1000 $900 + $900× i = $1000 $900× i = $1000− $900 i = $1000− $900 $900 = $100 $900 = 0.111 = 11.1% More generally, for any discount bond, if P = today’s bond price F = face value n = years to maturity i = yield to maturity then P = F (1 + i)n From this general formula, we can see that for a discount bond, too, the bond price and the yield to maturity are negatively related: when the yield to maturity falls, the bond price rises; and when the yield to maturity rises, the bond price falls. In the special case where n = 1: P = F (1 + i) P × (1 + i) = F P + P × i = F P × i = F − P i = F − P P . Suppose we rewrite this last formula as i = F − P P = F P − P P = F P − 1. Once again, the bond price and the yield to maturity are negatively related: when P falls, i rises; and when P rises, i falls. 9 4 Other Measures of Interest Rates Although the yield to maturity is the most accurate and widely-applicable measure of interest rates, it is often difficult to calculate. For this reason, a couple of other measures of interest rates are frequently used: the current yield and the yield on a discount basis. 4.1 Current Yield Applies only to coupon bonds. Recall that the yield to maturity on a coupon bond usually cannot be found without the help of a computer. As a result, the yield to maturity on a coupon bond is often approximated by the current yield. To calculate the current yield, let P = today’s bond price C = annual coupon payment ic = current yield then the current yield ic is defined as ic = C P . Four facts about the current yield: 1. ic better approximates i when today’s bond price P is closer to the face value F and when the maturity of the bond is longer. 2. In fact: If P = F , then ic = i. If P > F , then ic > i. If P <: F , then ic < i. 3. ic and i always move in the same direction: ic rises when i rises, and ic falls when i falls. 4. The bond price P and the current yield ic are negatively related: when P rises, ic falls; and when P falls, ic rises. 10 4.2 Yield on a Discount Basis Applies only to discount bonds. Although it is fairly easy to calculate the yield to maturity on a one-year discount bond, it is more difficult to calculate the yield to maturity on a discount bond with maturity less than or greater than one year. For this reason, interest rates on discount bonds, including US Treasury bills, are often quoted in terms on the yield on a discount basis. To calculate the yield on the discount basis for a discount bond, let P = today’s bond price F = face value idb = yield on a discount basis then the yield on a discount basis idb is defined as idb = F − P F × 360 days to maturity . Recall that for a discount bond with one-year maturity yield to maturity = i = F − P P . Thus, the yield on a discount basis calculation has two peculiarities: 1. It uses the percentage gain on face value, (P − F )/F , rather than the percentage gain on the purchase price (F − P )/P . 2. It considers the year to be 360 days long instead of 365. As a result of both of these peculiarities, the yield on a discount basis always understates the yield to maturity. Example: Discount bond with $1,000 face value and one-year maturity sells for $900 today: P = $900 F = $1000 n = 1 11