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Understanding Interest Rates: Fundamentals, Present and Future Value Calculations

This chapter explores the determinants of interest rates, focusing on nominal rates and their impact on security values in financial markets. It examines the time value of money, emphasizing the principle that money today is worth more than in the future due to interest. Key concepts include compound and simple interest, present value calculations for lump sums and annuities, and future value determinations. The relationship between interest rates and present and future values is highlighted, demonstrating their significance in financial decision-making.

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Understanding Interest Rates: Fundamentals, Present and Future Value Calculations

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  1. Chapter Two Determinants of Interest Rates McGraw-Hill/Irwin

  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 McGraw-Hill/Irwin

  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 McGraw-Hill/Irwin

  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 McGraw-Hill/Irwin

  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 = $1,254.40 McGraw-Hill/Irwin

  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 • annuity • PVs decrease as interest rates increase McGraw-Hill/Irwin

  7. Calculating Present Value (PV) of a Lump Sum PV = FVn(1/(1 + i/m))nm = FVn(PVIFi/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 McGraw-Hill/Irwin

  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(PVIFi/m,nm) • at 8% interest - = $10,000(0.630170) = $6,301.70 • at 12% interest - = $10,000(0.506631) = $5,066.31 • at 16% interest - = $10,000(0.410442) = $4,104.42 McGraw-Hill/Irwin

  9. Calculation of Present Value (PV) of an Annuity nm PV = PMT  (1/(1 + i/m))t = PMT(PVIFAi/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 McGraw-Hill/Irwin

  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(PVIFAi/m,nm) at 8% interest - = $10,000(4.622880) = $46,228.80 If the investment pays on the last day of every quarter for the next six years at 8% interest - = $10,000(18.913926) = $189,139.26 McGraw-Hill/Irwin

  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 McGraw-Hill/Irwin

  12. Future Values Equations • FV of lump sum equation • FVn = PV(1 + i/m)nm = PV(FVIF i/m, nm) • FV of annuity payment equation • (nm-1) • FVn = PMT (1 + i/m)t = PMT(FVIFAi/m, mn) • (t = 0) McGraw-Hill/Irwin

  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(1.586874) = $15,868.74 • at 12% interest = $10,000(1.973823) = $19,738.23 • at 16% interest = $10,000(2.436396) = $24,363.96 • at 16% interest compounded semiannually • = $10,000(2.518170) = $25,181.70 McGraw-Hill/Irwin

  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(7.335929) = $73,359.29 • If the investment pays you $10,000 on the last day of every quarter for the next six years, • FV = $10,000(30.421862) = $304,218.62 • If the annuity is paid on the first day of each quarter, • FV = $10,000(31.030300) = $310,303.00 McGraw-Hill/Irwin

  15. Relation between Interest Rates and Present and Future Values Present Value (PV) Future Value (FV) Interest Rate Interest Rate McGraw-Hill/Irwin

  16. Effective or Equivalent Annual Return (EAR) Rate earned over a 12 – month period taking the compounding of interest into account. EAR = (1 + r)c – 1 Where c = number of compounding periods per year McGraw-Hill/Irwin

  17. 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 McGraw-Hill/Irwin

  18. Supply of Loanable Funds Demand Supply Interest Rate Quantity of Loanable Funds Supplied and Demanded McGraw-Hill/Irwin

  19. Funds Supplied and Demanded by Various Groups (in billions of dollars) Funds SuppliedFunds DemandedNet Households $34,860.7 $15,197.4 $19,663.3 Business - nonfinancial 12,679.2 30,779.2 -12,100.0 Business - financial 31,547.9 45061.3 -13,513.4 Government units 12,574.5 6,695.2 5,879.3 Foreign participants 8,426.7 2,355.9 6,070.8 McGraw-Hill/Irwin

  20. Determination of Equilibrium Interest Rates D S Interest Rate I H i E I L Q Quantity of Loanable Funds Supplied and Demanded McGraw-Hill/Irwin

  21. Effect on Interest rates from a Shift in the Demand Curve for or Supply curve of Loanable Funds Increased supply of loanable funds Increased demand for loanable funds DD* Interest Rate SS SS DD DD SS* i** E* E i* E i* E* i** Q* Q** Q* Q** Quantity of Funds Supplied Quantity of Funds Demanded McGraw-Hill/Irwin

  22. Factors Affecting Nominal Interest Rates • Inflation • Real Interest Rate • Default Risk • Liquidity Risk • Special Provisions • Term to Maturity McGraw-Hill/Irwin

  23. 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% - 1.60% = 1.89% McGraw-Hill/Irwin

  24. 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 DRPj = ijt - iTt Example for December 2003: DRPAaa = 5.66% - 4.01% = 1.65% DRPBaa = 6.76% - 4.01% = 2.75% McGraw-Hill/Irwin

  25. Term to Maturity and Interest Rates: Yield Curve (a) Upward sloping (b) Inverted or downward sloping (c) Flat Yield to Maturity (a) (c) (b) Time to Maturity McGraw-Hill/Irwin

  26. Term Structure of Interest Rates • Unbiased Expectations Theory • Liquidity Premium Theory • Market Segmentation Theory McGraw-Hill/Irwin

  27. 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 __ 1R2 = [(1 + 1R1)(1 + (2f1))]1/2 - 1 where 2 f1 = expected one-year rate for year 2, or the implied forward one-year rate for next year McGraw-Hill/Irwin

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