Understanding Interest Rates

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1 Money & Banking Notes Chapter 4 Understanding Interest Rates Measuring Interest Rates Present Value (PV): 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)(1+ i) one year from today, where i is the annual interest rate. For example, Year 0 Year 1 Year 2 Year 3 Year n $100 ($100)(1+0.10) = $110 ($110)(1+0.10) = $121 ($121)(1+0.10) = $133 Simple Present Value (example) PV = today s Present Value =? CF = future (yearly) cash flow payment = $250 i = the (usually annual) interest rate = 10 % n = number of years (or periods) = 2 years ($100)(1 + i) n PV = $ double check for yourself from PV table: i = 10%, n = 2 periods

2 Four Types of Credit Market Instruments - Simple Loan: the lender provides the borrower with an amount of funds (the principal) which must be repaid at the maturity date (the ending period of the loan) together with the interest owed (e.g., a commercial loan to a business). - Fixed Payment Loan: the lender provides the borrower with an amount of funds (the principal) which must be repaid by making equal payments (say, monthly) consisting of the principal and interest owed for a set number of years (e.g., a mortgage or auto loan). - Coupon Bond: it pays the bondholder (the lender) a fixed interest payment (called coupon payment) every year (or six months) until the maturity date at which point the face value (or par value) is repaid (e.g., a 5-year $1,000 coupon bond with a coupon rate of 7 % would pay its holder $70 every year for 5 years); e.g., Treasury bonds and corporate bonds. - Discount Bond: also called a zero-coupon bond, it is bought at a price below its face value (i.e., at a discount) and the face value is repaid at maturity, thus, it makes no interest payments (e.g., U.S. Treasury-bills). where did this come from? Yield to Maturity (ytm) = is the interest rate at which the present value of cash flow payments (received from a debt instrument) is equal to its value today (PV).

3 When a bond sells at par = it means at face value Simple Loan PV = amount borrowed = $100 CF = cash flow in one year = $110 n = number of years = 1 $100 = $110 (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 Fixed Payment Loan The same cash flow payment every period throughout also PV n the life of the loan LV = loan value FP = fixed yearly payment = number of years until maturity FP FP FP FP LV = i (1 + i) (1 + i) (1 + i) n 1

.. + 2 3 n 1+ i (1+ i) (1+ i) (1+ i) (1 +) n i Discount Bond (Bond prices and interest rates are negatively related.

4 Coupon Bond Using the same strategy used for the fixed-payment loan: P = price of coupon bond C = yearly coupon payment also PV F = face value of the bond n = years to maturity date also FV C C C C F P = n 1+ i (1+ i) (1+ i) (1+ i) (1 +) n i Discount Bond (Bond prices and interest rates are negatively related.) The Distinction Between Interest Rates and Returns Rate of return = payments to the owner of the security plus the change in its value, expressed as a fraction of its purchase price (current yield + capital gains yield).

5 The Distinction Between Interest Rates and Returns (cont.) current selling price future selling price The return on a bond equals the yield to maturity only if the holding period equals the ytm (put differently, the return on a bond will not necessarily equal the ytm on that bond). A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if the ytm is longer than the holding period.

6 The more distant a bond s maturity, the greater the size of the percentage change in price associated with an interest-rate change. The more distant a bond s maturity, the lower the rate of return that 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. 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 (i.e., when you hold the bond until its maturity, there is no interest-rate risk). The Distinction Between Real and Nominal Interest Rates - Nominal interest rate makes no allowance (adjustment) for inflation

in the price level Fisher Equation When the real interest rate is low, there are greater incentives to borrow and fewer incentives to

7 - Real interest rate is adjusted for changes in price level (inflation rate) 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 Fisher Equation When the real interest rate is low, there are greater incentives to borrow and fewer incentives to lend. Why? Suppose: e = 10 % = expected inflation rate i = 8 % = nominal interest rate What is the real interest rate? Answer = 2 % figure this out

8 Exercises 1. What is the present value of $500 to be paid each time in two years (Year 1 and Year 2) if the interest rate is 5 percent? (Hint: you can t add up the two 500s.) Fig. 1. Timeline of the present value problem i = 5 % today Year 0 Year 1 Year 2 PV =? $500 $500 Setting up the equation, PV 500 (1 0.05) cash flow 1 cash flow (1 0.05) 2 So that, PV FV interest factor, period = 1, i = 5 % for denominator, use FV interest factor from table FV interest factor, period = 2, i = 5 %

9 Alternatively, use PV interest factor as shown below PV = 500( ) + 500( ) PV interest factor, period = 1, i = 5 % PV interest factor, period = 2, i = 5 % Answer: = $ What is the present value of $500 to be paid in 2 years (Year 2) if the interest rate is 5 percent? i = 5 % today 0 PV =? Year 1 Year 2 $500 cash flow year 2 PV 500 (1 0.05) 2 (500)( ) Answer: $ Don t believe this. Prove it to yourself.

10 3. If a security pays $55 in year one and $133 in year three, its present value is $150 if the interest rate is how much? i =? % today 0 PV = $150 Year 1 Year 2 $55 Year 3 $133 Setting up the equation, cash flow year 1 cash flow year (1 133 ) 1 i) (1 i 3 What is the interest rate (i) that would make the cash flows of $55 (in yr. 1) and $133 (in yr. 3) equal the present value of $150?

11 Using a trial and error solution, first try i = 12%: 150? 55 (1 0.12)? ? Should a higher or lower interest rate be assumed next? Now try i = 10%: (1 0.12) ? 55 (1 0.10)? ? Answer: i = 10 % (1 0.10) 3 FV interest factor, period = 3, i = 12 % 3 FV interest factor, period = 3, i = 10 % close enough

12 4. If a $5,000 coupon bond has a coupon rate of 13 percent, then the coupon payment every year is how much? Coupon payment = (coupon rate)(bond s face value) = (0.13)($5,000) = $ An $8,000 coupon bond with a $400 coupon payment every year has a coupon rate of how much? coupon payment Coupon rate = face value $400 = $8,000 face or par value Answer: coupon rate = 0.05 or 5 % 6. For a 3-year simple loan of $10,000 at 10 percent, the amount to be repaid is how much? That is, after 3 years, how much is the future value of $10,000 today at 10 %?

13 today 0 PV = $10,000 i = 10 % Year 1 Year 2 Year 3 FV =? FV = $10,000( ) 3 = $10,000(1.331) Answer: repayment = $13,310 amt. to be repaid year 3 FV interest factor, period = 3, i = 10 % This is like depositing money in a bank and withdrawing the (accumulated) money after 3 years.

14 7. If a security pays $110 next year and $121 the year after that, what is its yield to maturity if it sells today for $200? PV i =? % today 0 Year 1 Year 2 PV = $200 $110 $ (1 121 ) 1 i) (1 i 2 Solve by trial and error like in Problem #3. Answer: i = 10 % Prove this answer to yourself.

15 8. What is the price of a perpetuity that has a coupon (payment) of $100 per year and a yield to maturity of 5 %? The ytm formula for a perpetuity is: where C = yearly (coupon) payment P = price of perpetuity (consol) i = yield to maturity (ytm) which can be rewritten as: Answer: P = $2,000 P i Prove this answer to yourself. 9. What is the yield to maturity (ytm) of a $1,000 discount bond that matures in one year and currently sells for $900? C P (Solve this problem and bring your solution to class.) C i

4. Understanding.. Interest Rates. Copyright 2007 Pearson Addison-Wesley. All rights reserved. 4-1

4. Understanding.. Interest Rates. Copyright 2007 Pearson Addison-Wesley. All rights reserved. 4-1 4. Understanding. Interest Rates Copyright 2007 Pearson Addison-Wesley. All rights reserved. 4-1 Present Value A dollar paid to you one year from now is less valuable than a dollar paid to you today Copyright