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Today in Precalculus

This guide offers a comprehensive overview of key concepts in mathematics of finance, focusing on compound interest and annual percentage yield (APY). It includes detailed formulas and practical examples to illustrate how to calculate the final amount with compound interest and determine the required interest rate for desired outcomes. Additionally, it provides insight into APY for comparing investments with different compounding methods, enhancing your understanding of how interest rates affect personal finance decisions.

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Today in Precalculus

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  1. Today in Precalculus • Notes: Mathematics of Finance • Interest rates • Annual Percentage Yield • Homework

  2. Compound Interest • Compound interest: A: final amount P: principal r: interest rate k: number of payments per year t: number of years

  3. Example 1 P=$18,000 k = 2 t = 6 r= 0.075 =$27,998.18

  4. P=$18,000 k = 4 t = ? r= 0.075 A=$27,000 27,000=18,000(1.01875)4t 1.5=1.018754t log1.5 = 4tlog1.01875 4t=21.827 t = 5.457 years Example 1b

  5. Example 1c A=$31,500 P=$18,000 k = 12 t = 15 r= ? r= .0374 So an interest rate of 3.74% is required.

  6. ContinuouslyCompound Interest • Continuously Compounded interest: A=Pert A: final amount P: principal r: interest rate t: number of years

  7. Example 2 P=20,000 t = 8 r = 0.053 A=20,000e(8•.053) = $30,561.23

  8. Example 2b 100,000 = 20,000e.053t 5 = e0.053t ln5 = 0.053t A = $100,000 P = $20,000 r = 0.053 t = ? t = 30.367 years

  9. Example 2c A = $40,000 P = $20,000 r = ? t = 10 40,000 = 20,000e10r 2 = e10r ln2=10r r =.0693 An interest rate of 6.93% is needed

  10. Annual Percentage Yield (APY) • A common basis for comparing investments with different interest rates and methods of compounding. • The percentage rate that, if compounded annually, would yield the same return as the given interest rate with the given compounding.

  11. Example 1 Bob invests with the local bank at 2.12% interest compounded monthly. What is the equivalent APY? So the APY is 2.14%. This means his investment compounded monthly at 2.12% earns the same interest as if it earned 2.14%, compounded once a year. =.0214

  12. Which investment is more attractive, one that pays 8.5% compounded quarterly or another that pays 8.45% compounded monthly? x=.0877 Example 2 x=.0879 So the investment with the 8.45% rate compounded monthly is the better investment

  13. Example 3 The interest charge by your credit card company is 18.75% APY. What is the monthly interest charge on a balance of $5,000? r= .1731 $5000x.1731 = $865.43

  14. Homework • Pg 341: 3-9 odd, 21-39odd, 41-46all

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