Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-1 Chapter Two Determinants of Interest Rates

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-2 Interest Rate Fundamentals Nominal interest rates - the interest rate actually observed in financial markets –directly affect the value (price) of most securities traded in the market –affect the relationship between spot and forward FX rates Nominal interest rates - the interest rate actually observed in financial markets –directly affect the value (price) of most securities traded in the market –affect the relationship between spot and forward FX rates

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-3 Time Value of Money and Interest Rates Assumes the basic notion that a dollar received today is worth more than a dollar received at some future date Compound interest –interest earned on an investment is reinvested Simple interest –interest earned on an investment is not reinvested Assumes the basic notion that a dollar received today is worth more than a dollar received at some future date Compound interest –interest earned on an investment is reinvested Simple interest –interest earned on an investment is not reinvested

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-4 Calculation of Simple Interest Value = Principal + Interest (year 1) + Interest (year 2) Example: $1,000 to invest for a period of two years at 12 percent Value = $1,000 + $1,000(.12) + $1,000(.12) = $1,000 + $1,000(.12)(2) = $1,240 Value = Principal + Interest (year 1) + Interest (year 2) Example: $1,000 to invest for a period of two years at 12 percent Value = $1,000 + $1,000(.12) + $1,000(.12) = $1,000 + $1,000(.12)(2) = $1,240

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-5 Value of Compound Interest Value = Principal + Interest + Compounded interest Value = $1,000 + $1,000(.12) + $1,000(.12) + $1,000(.12) = $1,000[1 + 2(.12) + (.12) 2 ] = $1,000(1.12) 2 = $1, Value = Principal + Interest + Compounded interest Value = $1,000 + $1,000(.12) + $1,000(.12) + $1,000(.12) = $1,000[1 + 2(.12) + (.12) 2 ] = $1,000(1.12) 2 = $1,254.40

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-6 Present Value of a Lump Sum PV function converts cash flows received over a future investment horizon into an equivalent (present) value by discounting future cash flows back to present using current market interest rate –lump sum payment a single cash payment received at the end of some investment horizon –annuity a series of equal cash payments received at fixed intervals over the investment horizon PVs decrease as interest rates increase PV function converts cash flows received over a future investment horizon into an equivalent (present) value by discounting future cash flows back to present using current market interest rate –lump sum payment a single cash payment received at the end of some investment horizon –annuity a series of equal cash payments received at fixed intervals over the investment horizon PVs decrease as interest rates increase

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-7 Calculating Present Value (PV) of a Lump Sum PV = FV n (1/(1 + i/m)) nm = FV n (PVIF i/m,nm ) where: PV = present value FV = future value (lump sum) received in n years i = simple annual interest rate earned n = number of years in investment horizon m = number of compounding periods in a year i/m = periodic rate earned on investments nm = total number of compounding periods PVIF = present value interest factor of a lump sum PV = FV n (1/(1 + i/m)) nm = FV n (PVIF i/m,nm ) where: PV = present value FV = future value (lump sum) received in n years i = simple annual interest rate earned n = number of years in investment horizon m = number of compounding periods in a year i/m = periodic rate earned on investments nm = total number of compounding periods PVIF = present value interest factor of a lump sum

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-8 Calculating Present Value of a Lump Sum You are offered a security investment that pays $10,000 at the end of 6 years in exchange for a fixed payment today. PV = FV(PVIF i/m,nm ) at 8% interest - = $10,000( ) = $6, at 12% interest - = $10,000( ) = $5, at 16% interest - = $10,000( ) = $4, You are offered a security investment that pays $10,000 at the end of 6 years in exchange for a fixed payment today. PV = FV(PVIF i/m,nm ) at 8% interest - = $10,000( ) = $6, at 12% interest - = $10,000( ) = $5, at 16% interest - = $10,000( ) = $4,104.42

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-9 Calculation of Present Value (PV) of an Annuity nm PV = PMT  (1/(1 + i/m)) t = PMT(PVIFA i/m,nm ) t = 1 where: PV = present value PMT = periodic annuity payment received during investment horizon i/m = periodic rate earned on investments nm = total number of compounding periods PVIFA = present value interest factor of an annuity nm PV = PMT  (1/(1 + i/m)) t = PMT(PVIFA i/m,nm ) t = 1 where: PV = present value PMT = periodic annuity payment received during investment horizon i/m = periodic rate earned on investments nm = total number of compounding periods PVIFA = present value interest factor of an annuity

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-10 Calculation of Present Value of an Annuity You are offered a security investment that pays $10,000 on the last day of every year for the next 6 years in exchange for a fixed payment today. PV = PMT(PVIFA i/m,nm ) at 8% interest - = $10,000( ) = $46, If the investment pays on the last day of every quarter for the next six years at 8% interest - = $10,000( ) = $189, You are offered a security investment that pays $10,000 on the last day of every year for the next 6 years in exchange for a fixed payment today. PV = PMT(PVIFA i/m,nm ) at 8% interest - = $10,000( ) = $46, If the investment pays on the last day of every quarter for the next six years at 8% interest - = $10,000( ) = $189,139.26

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-11 Future Values Translate cash flows received during an investment period to a terminal (future) value at the end of an investment horizon FV increases with both the time horizon and the interest rate Translate cash flows received during an investment period to a terminal (future) value at the end of an investment horizon FV increases with both the time horizon and the interest rate

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-12 Future Values Equations FV of lump sum equation FV n = PV(1 + i/m) nm = PV(FVIF i/m, nm ) FV of annuity payment equation (nm-1) FV n = PMT  (1 + i/m) t = PMT(FVIFA i/m, mn ) (t = 0) FV of lump sum equation FV n = PV(1 + i/m) nm = PV(FVIF i/m, nm ) FV of annuity payment equation (nm-1) FV n = PMT  (1 + i/m) t = PMT(FVIFA i/m, mn ) (t = 0)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-13 Calculation of Future Value of a Lump Sum You invest $10,000 today in exchange for a fixed payment at the end of six years –at 8% interest = $10,000( ) = $15, –at 12% interest = $10,000( ) = $19, –at 16% interest = $10,000( ) = $24, –at 16% interest compounded semiannually = $10,000( ) = $25, You invest $10,000 today in exchange for a fixed payment at the end of six years –at 8% interest = $10,000( ) = $15, –at 12% interest = $10,000( ) = $19, –at 16% interest = $10,000( ) = $24, –at 16% interest compounded semiannually = $10,000( ) = $25,181.70

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-14 Calculation of the Future Value of an Annuity You invest $10,000 on the last day of every year for the next six years, –at 8% interest = $10,000( ) = $73, If the investment pays you $10,000 on the last day of every quarter for the next six years, –FV = $10,000( ) = $304, If the annuity is paid on the first day of each quarter, –FV = $10,000( ) = $310, You invest $10,000 on the last day of every year for the next six years, –at 8% interest = $10,000( ) = $73, If the investment pays you $10,000 on the last day of every quarter for the next six years, –FV = $10,000( ) = $304, If the annuity is paid on the first day of each quarter, –FV = $10,000( ) = $310,303.00

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-15 Relation between Interest Rates and Present and Future Values Present Value (PV) Interest Rate Future Value (FV) Interest Rate

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-16 Equivalent Annual Return (EAR) If you invest in a security that matures in 75 days and offers a 7% annual interest rate: EAR = (1 + i/(365/h)) 365/h - 1 =(1 + (.07)/(365/75)) 365/ = 7.20% If you invest in a security that matures in 75 days and offers a 7% annual interest rate: EAR = (1 + i/(365/h)) 365/h - 1 =(1 + (.07)/(365/75)) 365/ = 7.20%

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-17 Discount Yields Money market instruments (e.g., Treasury bills and commercial paper) that are bought and sold on a discount basis i dy = [(P f - P o )/P f ](360/h) Where: P f = Face value P o = Discount price of security Money market instruments (e.g., Treasury bills and commercial paper) that are bought and sold on a discount basis i dy = [(P f - P o )/P f ](360/h) Where: P f = Face value P o = Discount price of security

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-18 Single Payment Yields Money market securities (e.g., jumbo CDs, fed funds) that pay interest only once during their lives: at maturity i bey = i spy (365/360) Money market securities (e.g., jumbo CDs, fed funds) that pay interest only once during their lives: at maturity i bey = i spy (365/360)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-19 Loanable Funds Theory A theory of interest rate determination that views equilibrium interest rates in financial markets as a result of the supply and demand for loanable funds

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-20 Supply of Loanable Funds Interest Rate Quantity of Loanable Funds Supplied and Demanded DemandSupply

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-21 Funds Supplied and Demanded by Various Groups (in billions of dollars) Funds Supplied Funds Demanded Net Households $30,857.3 $12,849.2 $18,002.1 Business - nonfinancial 9, , ,337.4 Business - financial 29, , ,975.8 Government units 10, , ,199.2 Foreign participants 8, , ,111.9 Funds Supplied Funds Demanded Net Households $30,857.3 $12,849.2 $18,002.1 Business - nonfinancial 9, , ,337.4 Business - financial 29, , ,975.8 Government units 10, , ,199.2 Foreign participants 8, , ,111.9

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-22 Determination of Equilibrium Interest Rates Interest Rate Quantity of Loanable Funds Supplied and Demanded D S I H i I L E Q

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-23 Effect on Interest rates from a Shift in the Demand Curve for or Supply curve of Loanable Funds Increased supply of loanable funds Quantity of Funds Supplied Interest Rate DD SS SS* E E* Q* i* Q** i** Increased demand for loanable funds Quantity of Funds Demanded DD DD* SS E E* i* i** Q*Q**

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-24 Factors Affecting Nominal Interest Rates Inflation –continual increase in price of goods/services Real Interest Rate –nominal interest rate in the absence of inflation Default Risk –risk that issuer will fail to make promised payment Inflation –continual increase in price of goods/services Real Interest Rate –nominal interest rate in the absence of inflation Default Risk –risk that issuer will fail to make promised payment (continued)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-25 Liquidity Risk –risk that a security can not be sold at a predictable price with low transaction cost on short notice Special Provisions –taxability –convertibility –callability Term to Maturity Liquidity Risk –risk that a security can not be sold at a predictable price with low transaction cost on short notice Special Provisions –taxability –convertibility –callability Term to Maturity

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-26 Inflation and Interest Rates: The Fisher Effect The interest rate should compensate an investor for both expected inflation and the opportunity cost of foregone consumption (the real rate component) i = RIR + Expected(IP) or RIR = i – Expected(IP) Example: 3.49% % = 1.89% The interest rate should compensate an investor for both expected inflation and the opportunity cost of foregone consumption (the real rate component) i = RIR + Expected(IP) or RIR = i – Expected(IP) Example: 3.49% % = 1.89%

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-27 Default Risk and Interest Rates The risk that a security’s issuer will default on that security by being late on or missing an interest or principal payment DRP j = i jt - i Tt Example: DRP Aaa = 6.61% % = 1.13% DRP Baa = 7.92% % = 2.44% The risk that a security’s issuer will default on that security by being late on or missing an interest or principal payment DRP j = i jt - i Tt Example: DRP Aaa = 6.61% % = 1.13% DRP Baa = 7.92% % = 2.44%

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-28 Tax Effects: The Tax Exemption of Interest on Municipal Bonds Interest payments on municipal securities are exempt from federal taxes and possibly state and local taxes. Therefore, yields on “munis” are generally lower than on equivalent taxable bonds such as corporate bonds. i m = i c (1 - t s - t F ) Where: i c = Interest rate on a corporate bond i m = Interest rate on a municipal bond t s = State plus local tax rate t F = Federal tax rate Interest payments on municipal securities are exempt from federal taxes and possibly state and local taxes. Therefore, yields on “munis” are generally lower than on equivalent taxable bonds such as corporate bonds. i m = i c (1 - t s - t F ) Where: i c = Interest rate on a corporate bond i m = Interest rate on a municipal bond t s = State plus local tax rate t F = Federal tax rate

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-29 Term to Maturity and Interest Rates: Yield Curve Yield to Maturity Time to Maturity (a) (b) (c) (a) Upward sloping (b) Inverted or downward sloping (c) Flat

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-30 Term Structure of Interest Rates Unbiased Expectations Theory –at a given point in time, the yield curve reflects the market’s current expectations of future short-term rates Liquidity Premium Theory –investors will only hold long-term maturities if they are offered a premium to compensate for future uncertainty in a security’s value Market Segmentation Theory –investors have specific maturity preferences and will generally demand a higher maturity premium Unbiased Expectations Theory –at a given point in time, the yield curve reflects the market’s current expectations of future short-term rates Liquidity Premium Theory –investors will only hold long-term maturities if they are offered a premium to compensate for future uncertainty in a security’s value Market Segmentation Theory –investors have specific maturity preferences and will generally demand a higher maturity premium

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 2-31 Forecasting Interest Rates Forward rate is an expected or “implied” rate on a security that is to be originated at some point in the future using the unbiased expectations theory _ _ 1 R 2 = [(1 + 1 R 1 )(1 + ( 2 f 1 ))] 1/2 - 1 where 2 f 1 = expected one-year rate for year 2, or the implied forward one-year rate for next year Forward rate is an expected or “implied” rate on a security that is to be originated at some point in the future using the unbiased expectations theory _ _ 1 R 2 = [(1 + 1 R 1 )(1 + ( 2 f 1 ))] 1/2 - 1 where 2 f 1 = expected one-year rate for year 2, or the implied forward one-year rate for next year