Chapter 4 Understanding Interest Rates Measuring Interest Rates Present Value (present discounted value): A dollar paid to you one year from now is less valuable than a dollar paid to you today Why? A dollar deposited today can earn interest and become $1 x (1+i) one year from today. 4-2 Dr. Ayman Haddad 1
Discounting the Future Let i =.10 In one year $100 X (1+ 0.10) = $110 In two years $110 X (1 + 0.10) = $121 or 100 X (1 + 0.10) In three years $121 X (1 + 0.10) = $133 or 100 X (1 + 0.10) In n years $100 X (1 + i) n 2 3 4-3 Simple Present Value PV = today's (present) value CF = future cash flow (payment) i = the interest rate PV = CF (1 + i ) n 4-4 Dr. Ayman Haddad 2
The timeline tells us that we can also work backward from future amounts to the present: for example, $133 = $100 (1 + 0.10)^3 three years from now is worth $100 today, so that: (Discounting the future) 100 $133 (1 0.10) 3 4-5 Time Line $100 $100 $100 $100 Year 0 1 2 n PV 100 100/(1+i) 100/(1+i) 2 100/(1+i) n 4-6 Dr. Ayman Haddad 3
let s assume that you just hit the $20 million jackpot in the New York State Lottery, which promises you a payment of $1 million for the next twenty years. You didn t really win $20 million, but instead won less than half as much $9.4 million. 4-7 Four Types of Credit Market Instruments Simple Loan which we have already discussed, in which the lender provides the borrower with an amount of funds, which must be repaid to the lender at the maturity date along with an additional payment for the interest. Many money market instruments are of this type: for example, commercial loans to businesses. 4-8 Dr. Ayman Haddad 4
Fixed Payment Loan (which is also called a fully amortized loan) in which the lender provides the borrower with an amount of funds, which must be repaid by making the same payment every period (such as a month), consisting of part of the principal and interest for a set number of years. For example, if you borrowed $1,000, a fixed-payment loan might require you to pay $126 every year for 25 years. Installment loans (such as auto loans) and mortgages are frequently of the fixed-payment type. 4-9 Coupon Bond pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount (face value or par value) is repaid. A coupon bond is identified by three pieces of information. First is the corporation or government agency that issues the bond. Second is the maturity date of the bond. 4-10 Dr. Ayman Haddad 5
Third is the bond s coupon rate, the dollar amount of the yearly coupon payment expressed as a percentage of the face value of the bond. In our example, the coupon bond has a yearly coupon payment of $100 and a face value of $1,000. The coupon rate is then $100/$1,000 0.10, or 10%. Capital market instruments such as U.S. Treasury bonds and notes and corporate bonds are examples of coupon bonds. 4-11 Discount Bond (also called a zero-coupon bond) is bought at a price below its face value (at a discount), and the face value is repaid at the maturity date. Unlike a coupon bond, a discount bond does not make any interest payments; it just pays off the face value. For example, a discount bond with a face value of $1,000 might be bought for $900; in a year s time the owner would be repaid the face value of $1,000. U.S. Treasury bills, U.S. savings bonds, and long-term zerocoupon bonds are examples of discount bonds. 4-12 Dr. Ayman Haddad 6
Yield to Maturity The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today 4-13 Simple Loan PV = amount borrowed = $100 CF = cash flow in one year = $110 n = number of years = 1 $110 $100 = (1 + i ) (1 + i) $100 = $110 $110 (1 + i) = $100 i = 0.10 = 10% For simple loans, the simple interest rate equals the yield to maturity 1 4-14 Dr. Ayman Haddad 7
Fixed Payment Loan The same cash flow payment every period throughout the life of the loan LV = loan value FP = fixed yearly payment n = number of years until maturity FP FP FP FP LV =... + i i i i 2 3 1 + (1 + ) (1 + ) (1 + ) n 4-15 Fixed Payment Loan In the case of our earlier example, the loan is $1,000 and the yearly payment is $126 for the next 25 years. The present value is calculated as follows: At the end of one year, there is a $126 payment with a PV of $126/(1 + i); at the end of two years, there is another $126 payment with a PV of $126/(1 + i)^2; and so on until at the end of the twenty-fifth year, the last payment of $126 with a PV of $126/(1 + i)^25 is made. Making today s value of the loan ($1,000) equal to the sum of the present values of all the yearly payments gives us: (Use financial Calculator) 4-16 Dr. Ayman Haddad 8
1000 $126 (1 i) 1 $126 (1 i) 2... $126 (1 i) 25 4-17 Coupon Bond Using the same strategy used for the fixed-payment loan: P = price of coupon bond C = yearly coupon payment F = face value of the bond n = years to maturity date C C C C F P =... + i i i i +) n i 2 3 n 1+ (1+ ) (1+ ) (1+ ) (1 4-18 Dr. Ayman Haddad 9
The present value of a $1,000-face-value bond with ten years to maturity and yearly coupon payments of $100 (a 10% coupon rate) can be calculated as follows: At the end of one year, there is a $100 coupon payment with a PV of $100/(1+i ); at the end of the second year, there is another $100 coupon payment with a PV of $100/(1+i )^2; and so on until at maturity, there is a $100 coupon payment with a PV of $100/(1+ i )^10 plus the repayment of the $1,000 face value with a PV of $1,000/(1 +i )^10. Setting today s value of the bond (its current price, denoted by P) equal to the sum of the present values of all the payments for this bond gives: 4-19 $100 $100 $100 Price of Bond... 1 2 10 (1 i) (1 i) (1 i) 1000 10 (1 i) the coupon payment, the face value, the years to maturity, and the price of the bond are known quantities, and only the yield to maturity is not. 4-20 Dr. Ayman Haddad 10
Table 1 Yields to Maturity on a 10%- Coupon-Rate Bond Maturing in Ten Years (Face Value = $1,000) Three facts from the Table above: When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate The price of a coupon bond and the yield to maturity are negatively related The yield to maturity is greater than the coupon rate when the bond price is below its face value 4-21 Consol or Perpetuity Special case of a coupon bond A bond with no maturity date that does not repay principal but pays fixed coupon payments forever P C / i C i c c c P price of the consol yearly interest payment yield to maturity of the consol can rewrite above equation as this : i C / c P c For coupon bonds, this equation gives the current yield, an easy to calculate approximation to the yield to maturity 4-22 Dr. Ayman Haddad 11
One nice feature of consols is that you can immediately see that as i goes up, the price of the bond falls. For example, if a consol pays $100 per year forever and the interest rate is 10%, its price will be $1,000 $100/0.10. If the interest rate rises to 20%, its price will fall to $500 $100/0.20. For example, with a consol that pays $100 yearly and has a price of $2,000, the yield to maturity is easily calculated to be 5% ($100/$2,000). 4-23 Discount Bond For any one year discount bond i = F - P P F = Face value of the discount bond P = current price of the discount bond The yield to maturity equals the increase in price over the year divided by the initial price. As with a coupon bond, the yield to maturity is negatively related to the current bond price. 4-24 Dr. Ayman Haddad 12
If the current purchase price of this bill is $900, then equating this price to the present value of the $1,000 received in one year, then 900 $1000 (1 i) $ 1000 900 i 900 4-25 Our calculations of the yield to maturity for a variety of bonds reveal the important fact that current bond prices and interest rates are negatively related: When the interest rate rises, the price of the bond falls, and vice versa. 4-26 Dr. Ayman Haddad 13
Other measure of Interest (less accurate measures) The yield to maturity is the most accurate measure of interest rates; this is what economists mean when they use the term interest rate. Mostly the terms interest rate and yield to maturity are used synonymously. The less accurate measure Yield on a discount basis (discount Yield). 4-27 Discount Yield i db F P F X 360 days to maturity 4-28 Dr. Ayman Haddad 14
Discount Yield On our one-year bill, which is selling for $900 and has a face value of $1,000, the yield on a discount basis would be as follows: i db : yield on a discount basis; F: face value, P: Purchase price i db 1000 900 X 1000 360 365 9.9% 4-29 Discount Yield When the bill has one year to maturity, the second term on the right-hand side of the formula is 360/365=0.986 rather than 1.0. The more serious source of the understatement, however, is the use of the percentage gain on the face value rather than on the purchase price. 4-30 Dr. Ayman Haddad 15
The Distinction Between Interest Rates and Returns (rate of return) Rate of Return: The payments to the owner plus the change in value expressed as a fraction of the purchase price RET = C + P - P t 1 t P t P t RET = return from holding the bond from time t to time t + 1 P t = price of bond at time t P t 1 = price of the bond at time t + 1 C = coupon payment C P t = current yield = i c P t 1 - P t P t = rate of capital gain = g 4-31 let us see what the return would look like for a $1,000-face-value coupon bond with a coupon rate of 10% that is bought for $1,000, held for one year, and then sold for $1,200. The payments to the owner are the yearly coupon payments of $100, and the change in its value is $1,200 $1,000 $200. Adding these together and expressing them as a fraction of the purchase price of $1,000 gives us the one-year holding-period return for this bond: 100 200 1000 30 % 4-32 Dr. Ayman Haddad 16
The Yield to maturity was 10% only. This demonstrates that the return on a bond will not necessarily equal the interest rate on that bond. The paym ents to the owner plus the change in value expressed as a fraction of the purchase price C P RET = + t 1 - P t P t P t RET = return from holding the bond from tim e t to tim e t + 1 P t = price of bond at tim e t P t 1 = price of the bond at tim e t + 1 C = coupon paym ent C P t = current yield = i c P t 1 - P t P t = rate of capital gain = g 4-33 Table 2 One-Year Returns on Different- Maturity 10%-Coupon-Rate Bonds When Interest Rates Rise from 10% to 20% 4-34 Dr. Ayman Haddad 17
The Distinction Between Interest Rates and Returns (cont d) The return equals the yield to maturity only if the holding period equals the time to maturity A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if time to maturity is longer than the holding period The more distant a bond s maturity, the greater the size of the percentage price change associated with an interest-rate change 4-35 The Distinction Between Interest Rates and Returns (cont d) The more distant a bond s maturity, the lower the rate of return the occurs as a result of an increase in the interest rate Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise 4-36 Dr. Ayman Haddad 18
Interest-Rate Risk Prices and returns for long-term bonds are more volatile than those for shorter-term bonds There is no interest-rate risk for any bond whose time to maturity matches the holding period (See table above). 4-37 The Distinction Between Real and Nominal Interest Rates Nominal interest rate makes no allowance for inflation Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing Ex ante real interest rate is adjusted for expected changes in the price level Ex post real interest rate is adjusted for actual changes in the price level 4-38 Dr. Ayman Haddad 19
Fisher Equation r i = nominal interest rate i r i i e = real interest rate e = expected inflation rate When the real interest rate is low, there are greater incentives to borrow and fewer incentives to lend. The real interest rate is a better indicator of the incentives to borrow and lend. 4-39 let us first consider a situation in which you have made a one-year simple loan with a 5% interest rate (i 5%) and you expect the price level to rise e by 3% over the course of the year ( =3%). As a result of making the loan, at the end of the year you will have 2% more in real terms, that is, in terms of real goods and services you can buy. In this case, the interest rate you have earned in terms of real goods and services is 2%; that is, i r 5 % 3 % 2 % 4-40 Dr. Ayman Haddad 20
Now what if the interest rate rises to 8%, but you expect the inflation rate to be 10% over the course of the year? Although you will have 8% more dollars at the end of the year, you will be paying 10% more for goods; the result is that you will be able to buy 2% fewer goods at the end of the year and you are 2% worse off in real terms. i r 8 % 10 % 2 % 4-41 As a lender, you are clearly less eager to make a loan in this case, because in terms of real goods and services you have actually earned a negative interest rate of 2%. By contrast, as the borrower, you fare quite well because at the end of the year, the amounts you will have to pay back will be worth 2% less in terms of goods and services you as the borrower will be ahead by 2% in real terms. When the real interest rate is low, there are greater incentives to borrow and fewer incentives to lend. 4-42 Dr. Ayman Haddad 21
Figure 1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953 2011 Sources: Nominal rates from www.federalreserve.gov/releases/h15 and inflation from ftp://ftp.bis.gov/special.requests/cpi/cpia.txt. The real rate is constructed using the procedure outlined in Frederic S. Mishkin, The Real Interest Rate: An Empirical Investigation, Carnegie-Rochester Conference Series on Public Policy 15 (1981): 151 200. This procedure involves estimating expected inflation as a function of past interest rates, inflation, and time trends and then subtracting the expected inflation measure from the nominal interest rate. 4-43 Dr. Ayman Haddad 22