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## Chapter 4 – Interest Rates

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**Chapter 4 – Interest Rates**• Learning Objectives • Quoting Interest Rates (APR) • Effective Annual Rate (EAR) • TVM formulas with periodic interest rates • Monthly Amortization vs. Annual • Nominal and Real Interest Rates • Risk-free Rate and Premiums • Default and Maturity Premiums • Brief History of Interest Rates**Annual and Periodic Rates**• Annual Percentage Rate (APR) • The standard way to quote interest rate • Actual rate may be different • Period or Periodic Interest Rate • The rate for the period • Quarterly, Monthly, Daily, etc. • Annual Percentage Rate divided by Compounding periods per year • Effective Annual Rate (EAR) is the periodic rate compounded over a year**Annual and Periodic Rates**• Periodic Rates with 5.0% APR • Semi-Annual Rate (compounds twice a year) • 5.0% / 2 = 2.5% • Monthly Rate (compounds twelve times a year) • 5.0% / 12 = 0.04167% • Effective Annual Rates • EAR = (1 + [APR / (C/Y)])C/Y – 1, equation 4.2 • Semi-Annual Rate of 2.5%, EAR = 5.0625% • Monthly Rate of 0.04167%, EAR = 5.1162%**Annual and Periodic Rates**• Effective Annual Rate is the rate that you earn with your investment and it increases with the number of compounding periods per year (C/Y) • Maximum compounding is continuous compounding and • EAR = eAPR -1 • EAR = e0.05 -1= 5.12711% • e is the exponential function**Impact on TVM**• The r in the TVM equations is the periodic rate • It is the APR when compounding is annually • The n in the TVM equation matches r and is the number of periods (compounding periods per year times number of years) • Changing calculator mode for P/Y and C/Y**Impact on TVM**• Increasing the number of compounding periods changes results (changes effective interest rate) • Basic Mortgage Payment (Interest and Principal as you go on ordinary annuity) • Example 4.1 page 80 • Loan Amount $190,000 (PV) • Interest quoted at 8% APR • Payments are monthly (ordinary annuity) for 30 years or 360 monthly payments • Monthly Payment for P & I is $1,394.15**Impact on TVM**• Example 4.2 – Annual compounding versus Monthly compounding • $1,000,000 retirement goal at 9% APR • 30 years to retirement • Annual Payment with annual compounding, $7,336.35 • Monthly Payment with monthly compounding, $546.23 ($546.23 x 12 = $6,554.76) • EAR difference, Annual is 9%, Monthly is 9.381%**Consumer Loans and Monthly Amortization Schedule**• Most Consumer Loans are annuity stream fixed payments with interest accruing at end of month • Payment is for monthly interest expense and remainder is for principal reduction • Use TVM equation with monthly interest rate as r, number of payments or periods as n, and the amount of the loan as PV to find the monthly payment required.**Consumer Loans and Monthly Amortization Schedule**• Monthly amortization schedule for car loan of $25,000 for 72 months at 8% APR (Table 4.3)**Nominal and Real Interest Rates**• APR and Periodic Rates are nominal rates • Nominal Rates have two components • Real Rate • Expected Inflation Rate • Nominal ≈ Real Rate + Expected Inflation • Real Rate is reward for saving • Example 4.5 – 21 books next year versus 20 books now • Real Rate = 21/20 – 1 = 5% • Expected Inflation is the rising price of a good**Nominal and Real Interest Rates**• Fisher Effect • Relationship between real rate, expected inflation, and nominal rate • (1+r) = (1+r*) x (1+h) • r is the nominal rate • r* is the real rate • h is expected inflation • r = r* + h + (r* x h) • r* is the same the world round**Risk-free Rate and Premiums**• Risk-free rate (a nominal or real rate) with a guaranteed return • Nominal risk-free rate such as Treasury Bill • Real risk-free rate (excludes inflation) • Premiums impact the interest rates on different types of investments • Default Risk • Maturity**Risk-free Rate and Premiums**• Default Risk • Different Investments have different default risk based on the issuers ability to meet future promised payments • Low risk – U.S. Government • High risk – New Start-up Company • Maturity Premium – Investors demand more compensation for waiting longer**Risk-free Rate and Premiums**• Summary of Interest Rates • TVM equation uses a period nominal interest rate • The nominal interest rate is made up of two components, the real rate and inflation • Different investments have different nominal rates due to potential default (dp), and maturity (mp) • r ≈ rf* + inflation + dp + mp**Brief History of Interest Rates**• Four Different Investments over 50 Years • 3-Month Treasury Bill (risk-free rate) • Range 1% to 15% • Inflation in United States • Range -0.5% to 13% • Long-Term Treasury Bonds • Range -9% to 32% • Large U.S. Company Stocks • Range -27% to 52%**Homework**• Problem 3 – Effective Annual Rate • Problem 6 – PV with Periodic Rates • Problem 12– Amortization Schedule • Problem 18 – Nominal Rate