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Understanding Compound Interest and Continuous Compounding

Learn about compound interest, discrete and continuous compounding, effective annual yield, calculating values, yields, and rates with examples. Exploring logarithms for compound interest calculations.

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Understanding Compound Interest and Continuous Compounding

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  1. Project 2- Stock Option Pricing • Mathematical Tools -Today we will learn Compound Interest

  2. Compounding • Suppose that money left on deposit earns interest. • Interest is normally paid at regular intervals, while the money is on deposit. • This is called compounding.

  3. Compound Interest • Discrete Compounding -Interest compounded n times per year • Continuous Compounding -Interest compounded continuously

  4. Compound Interest Discrete Compounding P- dollars invested r -an annual rate n- number of times the interest compounded per year t- number of years F- dollars after t years.

  5. Yield for Discrete Compounding • The annual rate that would produce the same amount as in discrete compounding for one year. • Such a rate is called an effective annual yield,annual percentage yield, or just the yield. Compunded n times for one year Compounded once a year for one year

  6. Yield for Discrete Compounding Interest at an annual rate r, compounded n times per year has yield y.

  7. Discrete CompoundingExample 1 • What is the value of $74,000 after • 3-1/2 years at 5.25%,compounded monthly? • (ii) What is the effective annual yield?

  8. Example1 (i) Using Discrete Compounding formula Given P=$74,000 r=0.0525 n=12 t=3.5 Goal- To find F

  9. Example 1 (ii) Using yield formula Given r=0.0525 n=12 Goal- To find y

  10. Discrete CompoundingExample 2 (i)What is the value of $150,000 after 5 years at 6.2%, compounded quarterly? (ii) What is the effective annual yield?

  11. Example 2 (i) Using Discrete Compounding formula Given P=$150,000 r=0.062 n=4 t=5 Goal- To find F

  12. Example 2 (ii) Using yield formula Given r=0.062 n=4 Goal- To find y

  13. Annual rate for Discrete Compounding

  14. Annual rate for Discrete Compounding Interest compounded n times per year at a yield y, has an annual rate r.

  15. Discrete CompoundingExample 3 • What rate, r, compounded monthly, • will yield 5.25%?

  16. Example 3 (i) Using Annual rate formula Given y=0.0525 n=12 Goal- To find r

  17. The value of P dollars after t years, when compounded continuously at an annual rate r,is F = Pert Compound Interest Continuous Compounding

  18. Yield for Continuous Compounding Interest at an annual rate r, compounded continuously has yield y.

  19. ContinuousCompoundingExample 1 (i)Find the value, rounded to whole dollars, of $750,000 after 3 years and 4 months, if it is invested at a rate of 6.1% compounded continuously. (ii) What is the yield, rounded to 3 places, on this investment?

  20. Example1 • Using Continuous • Compounding formula • Given • P=$750,000 • r=0.061 • t=(40/12) • Goal- To find F F = Pert F = 750,000e0.061(40/12) =$ 919,111

  21. Example 1 (ii) Using yield formula Given r=0.061 Goal- To find y

  22. Logarithms • Why do we need logarithms for compound interest ? • To find r (since r is an exponent) Recall: yield formula for continuous compounding

  23. Review of Logarithms • For any base b, the logarithm function logb (x) • The equations u = bv and v = logbu are equivalent • Eg: 100=102 and 2=log10100 are equivalent • Two types -Common Logarithms (base is 10) -Natural Logartihms (base is e)- Notation: ln

  24. Review of Logarithms 1.The logarithm logb(x) function is the INVERSE of expb(x) 2. logb(x) is defined for any positive real number x

  25. Review of Logarithms The basic properties of exponents, yield properties for the logarithm functions. bubv = bu+v and (bu)v = buv, logb(uv) = logbu + logbv logb(u/v) = logbu logbv logbuv = vlogbu.

  26. Review of Logarithms • ln u = ln v if and only if u=v • Most commonly used to obtain solution of equations • We can transform an equation into an equivalent form by taking ln of both sides

  27. Review of LogarithmsExample1 Find the annual rate, r, that produces an effective annual yield of 6.00%, when compounded continuously.

  28. Example 1 (ii) Using yield formula Given y=6.00% Goal- To find r Taking ln on both sides

  29. Review of LogarithmsExample 2 Find the annual rate, r, that produces an effective annual yield of 5.15%, when compounded continuously. Round your answer to 3 places.

  30. Example 2 (ii) Using continuous compounding formula Given y=5.15% Goal- To find r Taking ln on both sides

  31. Review of LogarithmsExample 3 How long will it take $10,000 to grow to $15,162.65 if interest is paid at an annual rate of 2.5% compounded continuously?

  32. Example 3 (ii) Using yield formula Given F=$15,162.65 P=$10,000 r=0.025 Goal- To find t

  33. Example 3

  34. Value of Money Discrete compounding Recall • Present value (P) and Future value(F) of money • We need to rearrange the formula to find P The present value of money for discrete compounding

  35. Value of Money Continuous compounding Recall • Present value (P) and Future value(F) of money • We need to rearrange the formula to find P The present value of money for continuous compounding

  36. Ratio (R) • Under continuous compounding-The ratio of the future value to the present value • This allows us to convert the interest rate for a given period to a ratio of future to present value for the same period

  37. Recall- Class Project We suppose that it is Friday, January 11, 2002. Our goal is to find the present value, per share, of a European call on Walt Disney Company stock. • The call is to expire 20 weeks later • strike price of $23. • stock’s price record of weekly closes for the past 8 years(work basis). • risk free rate 4% (this means that on Jan 11,2002 the annual interest rate for a 20 week Treasury Bill was 4% compounded continuously)

  38. Project Focus I • Walt Disney- r =4%, compounded continuously The weekly risk-free rate for the Walt Disney The risk-free weekly ratio for the Walt Disney

  39. Project Focus II • Suppose we know the future value (fv) for our 20 week option at the end of 20 weeks • risk-free rate annual interest 4% • Can find the Present value (pv)

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