This document discusses compound interest and continuous compound interest. It provides examples of calculating future values given present values, rates, and time periods. Key concepts covered include compound interest formulas using principal, interest rate, number of periods, and the continuous compound interest formula relating future value, present value, rate, and time. Worked examples demonstrate applying the formulas to calculate unknown values.

1. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Math 1300 Finite Mathematics
Section 3.2 Compound Interest
Jason Aubrey
Department of Mathematics
University of Missouri
university-logo
Jason Aubrey Math 1300 Finite Mathematics

2. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
If at the end of a payment period the interest due is reinvested
at the same rate, then the interest as well as the original
principal will earn interest at the end of the next payment
period. Interest payed on interest reinvested is called
compound interest.
university-logo
Jason Aubrey Math 1300 Finite Mathematics

3. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
university-logo
Jason Aubrey Math 1300 Finite Mathematics

4. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
A = P(1 + rt)
university-logo
Jason Aubrey Math 1300 Finite Mathematics

5. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
A = P(1 + rt)
1
= 1, 000 1 + 0.08
4
university-logo
Jason Aubrey Math 1300 Finite Mathematics

6. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
A = P(1 + rt)
1
= 1, 000 1 + 0.08
4
= 1, 000(1.02) = $1, 020
university-logo
Jason Aubrey Math 1300 Finite Mathematics

7. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
A = P(1 + rt)
1
= 1, 000 1 + 0.08
4
= 1, 000(1.02) = $1, 020
This is the amount at the end
of the first quarter. Then:
university-logo
Jason Aubrey Math 1300 Finite Mathematics

8. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
Second quarter:
= $1,020(1.02) =
A = P(1 + rt) $1,040.40
1
= 1, 000 1 + 0.08
4
= 1, 000(1.02) = $1, 020
This is the amount at the end
of the first quarter. Then:
university-logo
Jason Aubrey Math 1300 Finite Mathematics

9. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
Second quarter:
= $1,020(1.02) =
A = P(1 + rt) $1,040.40
1 Third quarter:
= 1, 000 1 + 0.08
4 = $1,040.40(1.02) =
= 1, 000(1.02) = $1, 020 $1,061.21
This is the amount at the end
of the first quarter. Then:
university-logo
Jason Aubrey Math 1300 Finite Mathematics

10. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Suppose you invest $1,000 in a bank that pays 8%
compounded quarterly. How much will the bank owe you at the
end of the year?
Second quarter:
= $1,020(1.02) =
A = P(1 + rt) $1,040.40
1 Third quarter:
= 1, 000 1 + 0.08
4 = $1,040.40(1.02) =
= 1, 000(1.02) = $1, 020 $1,061.21
Fourth quarter:
This is the amount at the end = $1,061.21(1.02) =
of the first quarter. Then: $1,082.43
university-logo
Jason Aubrey Math 1300 Finite Mathematics

11. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Look at the pattern:
university-logo
Jason Aubrey Math 1300 Finite Mathematics

12. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Look at the pattern:
First quarter: = $1, 000(1.02)
university-logo
Jason Aubrey Math 1300 Finite Mathematics

13. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Look at the pattern:
First quarter: = $1, 000(1.02)
Second quarter: = $1, 000(1.02)2
university-logo
Jason Aubrey Math 1300 Finite Mathematics

14. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Look at the pattern:
First quarter: = $1, 000(1.02)
Second quarter: = $1, 000(1.02)2
Third quarter: = $1, 000(1.02)3
university-logo
Jason Aubrey Math 1300 Finite Mathematics

15. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Look at the pattern:
First quarter: = $1, 000(1.02)
Second quarter: = $1, 000(1.02)2
Third quarter: = $1, 000(1.02)3
Fourth quarter: = $1, 000(1.02)4
university-logo
Jason Aubrey Math 1300 Finite Mathematics

16. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount: Compound Interest)
A = P(1 + i)n
university-logo
Jason Aubrey Math 1300 Finite Mathematics

17. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount: Compound Interest)
A = P(1 + i)n
where i = r /m and
A = amount (future value) at the end of n periods
university-logo
Jason Aubrey Math 1300 Finite Mathematics

18. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount: Compound Interest)
A = P(1 + i)n
where i = r /m and
A = amount (future value) at the end of n periods
P = principal (present value)
university-logo
Jason Aubrey Math 1300 Finite Mathematics

19. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount: Compound Interest)
A = P(1 + i)n
where i = r /m and
A = amount (future value) at the end of n periods
P = principal (present value)
r = annual nominal rate
university-logo
Jason Aubrey Math 1300 Finite Mathematics

20. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount: Compound Interest)
A = P(1 + i)n
where i = r /m and
A = amount (future value) at the end of n periods
P = principal (present value)
r = annual nominal rate
m = number of compounding periods per year
university-logo
Jason Aubrey Math 1300 Finite Mathematics

21. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount: Compound Interest)
A = P(1 + i)n
where i = r /m and
A = amount (future value) at the end of n periods
P = principal (present value)
r = annual nominal rate
m = number of compounding periods per year
i = rate per compounding period
university-logo
Jason Aubrey Math 1300 Finite Mathematics

22. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount: Compound Interest)
A = P(1 + i)n
where i = r /m and
A = amount (future value) at the end of n periods
P = principal (present value)
r = annual nominal rate
m = number of compounding periods per year
i = rate per compounding period
n = total number of compounding periods
university-logo
Jason Aubrey Math 1300 Finite Mathematics

23. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: P = 800; i = 0.06; n = 25; A =?
university-logo
Jason Aubrey Math 1300 Finite Mathematics

24. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: P = 800; i = 0.06; n = 25; A =?
A = P(1 + i)n
university-logo
Jason Aubrey Math 1300 Finite Mathematics

25. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: P = 800; i = 0.06; n = 25; A =?
A = P(1 + i)n
A = $800(1.06)25
university-logo
Jason Aubrey Math 1300 Finite Mathematics

26. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: P = 800; i = 0.06; n = 25; A =?
A = P(1 + i)n
A = $800(1.06)25
A = $3, 433.50
university-logo
Jason Aubrey Math 1300 Finite Mathematics

27. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $18, 000; r = 8.12% compounded monthly;
n = 90; P =?
university-logo
Jason Aubrey Math 1300 Finite Mathematics

28. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $18, 000; r = 8.12% compounded monthly;
n = 90; P =?
r 0.0812
First we compute i = m = 12 = 0.0068. Then,
university-logo
Jason Aubrey Math 1300 Finite Mathematics

29. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $18, 000; r = 8.12% compounded monthly;
n = 90; P =?
r 0.0812
First we compute i = m = 12 = 0.0068. Then,
A = P(1 + i)n
university-logo
Jason Aubrey Math 1300 Finite Mathematics

30. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $18, 000; r = 8.12% compounded monthly;
n = 90; P =?
r 0.0812
First we compute i = m = 12 = 0.0068. Then,
A = P(1 + i)n
$18, 000 = P(1.0068)90
university-logo
Jason Aubrey Math 1300 Finite Mathematics

31. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $18, 000; r = 8.12% compounded monthly;
n = 90; P =?
r 0.0812
First we compute i = m = 12 = 0.0068. Then,
A = P(1 + i)n
$18, 000 = P(1.0068)90
$18, 000 ≈ P(1.84)
university-logo
Jason Aubrey Math 1300 Finite Mathematics

32. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $18, 000; r = 8.12% compounded monthly;
n = 90; P =?
r 0.0812
First we compute i = m = 12 = 0.0068. Then,
A = P(1 + i)n
$18, 000 = P(1.0068)90
$18, 000 ≈ P(1.84)
P ≈ $9, 782.61
university-logo
Jason Aubrey Math 1300 Finite Mathematics

33. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition (Amount - Continuous Compound Interest)
If a principal P is invested at an annual rate r (expressed as a
decimal) compounded continuously, then the amount A in the
account at the end of t years is given by
A = Pert
university-logo
Jason Aubrey Math 1300 Finite Mathematics

34. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: P = $2, 450; r = 8.12%; t = 3 years; A =?
university-logo
Jason Aubrey Math 1300 Finite Mathematics

35. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: P = $2, 450; r = 8.12%; t = 3 years; A =?
A = Pert
university-logo
Jason Aubrey Math 1300 Finite Mathematics

36. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: P = $2, 450; r = 8.12%; t = 3 years; A =?
A = Pert
A = $2, 450e(0.0812)(3) = $3, 125.79
university-logo
Jason Aubrey Math 1300 Finite Mathematics

37. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $15, 875; P = $12, 100; t = 48 months; r =?
university-logo
Jason Aubrey Math 1300 Finite Mathematics

38. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $15, 875; P = $12, 100; t = 48 months; r =?
A = Pert
university-logo
Jason Aubrey Math 1300 Finite Mathematics

39. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $15, 875; P = $12, 100; t = 48 months; r =?
A = Pert
$15, 875 = $12, 100e4r
university-logo
Jason Aubrey Math 1300 Finite Mathematics

40. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $15, 875; P = $12, 100; t = 48 months; r =?
A = Pert
$15, 875 = $12, 100e4r
1.311 = e4r
university-logo
Jason Aubrey Math 1300 Finite Mathematics

41. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $15, 875; P = $12, 100; t = 48 months; r =?
A = Pert
$15, 875 = $12, 100e4r
1.311 = e4r
ln(1.311) = 4r
university-logo
Jason Aubrey Math 1300 Finite Mathematics

42. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $15, 875; P = $12, 100; t = 48 months; r =?
A = Pert
$15, 875 = $12, 100e4r
1.311 = e4r
ln(1.311) = 4r
ln(1.311)
=r
4
university-logo
Jason Aubrey Math 1300 Finite Mathematics

43. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A = $15, 875; P = $12, 100; t = 48 months; r =?
A = Pert
$15, 875 = $12, 100e4r
1.311 = e4r
ln(1.311) = 4r
ln(1.311)
=r
4
r = 0.068
university-logo
Jason Aubrey Math 1300 Finite Mathematics

44. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
university-logo
Jason Aubrey Math 1300 Finite Mathematics

45. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03
university-logo
Jason Aubrey Math 1300 Finite Mathematics

46. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03
n = 5x2 = 10
university-logo
Jason Aubrey Math 1300 Finite Mathematics

47. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03
n = 5x2 = 10
A = P(1 + i)n
university-logo
Jason Aubrey Math 1300 Finite Mathematics

48. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03
n = 5x2 = 10
A = P(1 + i)n
$10, 000 = P(1.03)10
university-logo
Jason Aubrey Math 1300 Finite Mathematics

49. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03
n = 5x2 = 10
A = P(1 + i)n
$10, 000 = P(1.03)10
$10, 000 ≈ P(1.3439)
university-logo
Jason Aubrey Math 1300 Finite Mathematics

50. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03
n = 5x2 = 10
A = P(1 + i)n
$10, 000 = P(1.03)10
$10, 000 ≈ P(1.3439)
P ≈ $7, 441.03
university-logo
Jason Aubrey Math 1300 Finite Mathematics

51. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03 i = r /m = 0.06/2 = 0.03
n = 5x2 = 10
A = P(1 + i)n
$10, 000 = P(1.03)10
$10, 000 ≈ P(1.3439)
P ≈ $7, 441.03
university-logo
Jason Aubrey Math 1300 Finite Mathematics

52. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03 i = r /m = 0.06/2 = 0.03
n = 5x2 = 10 n = 10x2 = 20
n
A = P(1 + i)
$10, 000 = P(1.03)10
$10, 000 ≈ P(1.3439)
P ≈ $7, 441.03
university-logo
Jason Aubrey Math 1300 Finite Mathematics

53. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03 i = r /m = 0.06/2 = 0.03
n = 5x2 = 10 n = 10x2 = 20
n
A = P(1 + i) A = P(1 + i)n
$10, 000 = P(1.03)10
$10, 000 ≈ P(1.3439)
P ≈ $7, 441.03
university-logo
Jason Aubrey Math 1300 Finite Mathematics

54. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03 i = r /m = 0.06/2 = 0.03
n = 5x2 = 10 n = 10x2 = 20
n
A = P(1 + i) A = P(1 + i)n
$10, 000 = P(1.03)10 $10, 000 = P(1.03)20
$10, 000 ≈ P(1.3439)
P ≈ $7, 441.03
university-logo
Jason Aubrey Math 1300 Finite Mathematics

55. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03 i = r /m = 0.06/2 = 0.03
n = 5x2 = 10 n = 10x2 = 20
n
A = P(1 + i) A = P(1 + i)n
$10, 000 = P(1.03)10 $10, 000 = P(1.03)20
$10, 000 ≈ P(1.3439) $10, 000 ≈ P(1.806)
P ≈ $7, 441.03
university-logo
Jason Aubrey Math 1300 Finite Mathematics

56. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: If an investment company pays 6% compounded
semiannually, how much should you deposit now to have
$10,000?
5 years from now? 10 years from now?
i = r /m = 0.06/2 = 0.03 i = r /m = 0.06/2 = 0.03
n = 5x2 = 10 n = 10x2 = 20
n
A = P(1 + i) A = P(1 + i)n
$10, 000 = P(1.03)10 $10, 000 = P(1.03)20
$10, 000 ≈ P(1.3439) $10, 000 ≈ P(1.806)
P ≈ $7, 441.03 P ≈ $5, 536.76
university-logo
Jason Aubrey Math 1300 Finite Mathematics

57. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A zero coupon bond is a bond that is sold now at a
discount and will pay it’s face value at some time in the future
when it matures - no interest payments are made. Suppose
that a zero coupon bond with a face value of $40,000 matures
in 20 years. What should the bond be sold for now if its rate of
return is to be 5.124% compounded annually.
university-logo
Jason Aubrey Math 1300 Finite Mathematics

58. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A zero coupon bond is a bond that is sold now at a
discount and will pay it’s face value at some time in the future
when it matures - no interest payments are made. Suppose
that a zero coupon bond with a face value of $40,000 matures
in 20 years. What should the bond be sold for now if its rate of
return is to be 5.124% compounded annually.
Compounded annually means i = m = 0.05124 = 0.05124. Here
r
1
A = $40, 000 and we want to find P. We also know that n = 20.
university-logo
Jason Aubrey Math 1300 Finite Mathematics

59. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A zero coupon bond is a bond that is sold now at a
discount and will pay it’s face value at some time in the future
when it matures - no interest payments are made. Suppose
that a zero coupon bond with a face value of $40,000 matures
in 20 years. What should the bond be sold for now if its rate of
return is to be 5.124% compounded annually.
Compounded annually means i = m = 0.05124 = 0.05124. Here
r
1
A = $40, 000 and we want to find P. We also know that n = 20.
A = P(1 + i)n
university-logo
Jason Aubrey Math 1300 Finite Mathematics

60. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A zero coupon bond is a bond that is sold now at a
discount and will pay it’s face value at some time in the future
when it matures - no interest payments are made. Suppose
that a zero coupon bond with a face value of $40,000 matures
in 20 years. What should the bond be sold for now if its rate of
return is to be 5.124% compounded annually.
Compounded annually means i = m = 0.05124 = 0.05124. Here
r
1
A = $40, 000 and we want to find P. We also know that n = 20.
A = P(1 + i)n
$40, 000 = P(1.05124)20
university-logo
Jason Aubrey Math 1300 Finite Mathematics

61. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: A zero coupon bond is a bond that is sold now at a
discount and will pay it’s face value at some time in the future
when it matures - no interest payments are made. Suppose
that a zero coupon bond with a face value of $40,000 matures
in 20 years. What should the bond be sold for now if its rate of
return is to be 5.124% compounded annually.
Compounded annually means i = m = 0.05124 = 0.05124. Here
r
1
A = $40, 000 and we want to find P. We also know that n = 20.
A = P(1 + i)n
$40, 000 = P(1.05124)20
$40, 000
P= = $14, 723.89
(1.05124)20
university-logo
Jason Aubrey Math 1300 Finite Mathematics

62. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Definition
If a principal is invested at the annual (nominal) rate r
compounded m times a year, then the annual percentage yield
is
r m
APY = 1 + −1
m
If a principal is invested at the annual (nominal) rate r
compounded continuously, then the annual percentage yield is
APY = er − 1
The annual percentage yield is also referred to as the effective
rate or the true interest rate.
university-logo
Jason Aubrey Math 1300 Finite Mathematics

63. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded daily
for a bond that has an APY of 6.8%.
r m
APY = 1 + −1
m
university-logo
Jason Aubrey Math 1300 Finite Mathematics

64. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded daily
for a bond that has an APY of 6.8%.
r m
APY = 1 + −1
m
r 365
0.068 = 1 + −1
365
university-logo
Jason Aubrey Math 1300 Finite Mathematics

65. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded daily
for a bond that has an APY of 6.8%.
r m
APY = 1 + −1
m
r 365
0.068 = 1 + −1
365
r 365
1.068 = 1 +
365
university-logo
Jason Aubrey Math 1300 Finite Mathematics

66. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded daily
for a bond that has an APY of 6.8%.
r m
APY = 1 + −1
m
r 365
0.068 = 1 + −1
365
r 365
1.068 = 1 +
365
√
365 r
1.068 = 1 +
365
university-logo
Jason Aubrey Math 1300 Finite Mathematics

67. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded daily
for a bond that has an APY of 6.8%.
r m
APY = 1 + −1
m
r 365
0.068 = 1 + −1
365
r 365
1.068 = 1 +
365
√
365 r
1.068 = 1 +
365
r
0.00018 =
365
university-logo
Jason Aubrey Math 1300 Finite Mathematics

68. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded daily
for a bond that has an APY of 6.8%.
r m
APY = 1 + −1
m
r 365
0.068 = 1 + −1
365
r 365
1.068 = 1 +
365
√
365 r
1.068 = 1 +
365
r
0.00018 =
365
r = 0.0658
university-logo
Jason Aubrey Math 1300 Finite Mathematics

69. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded
continuously has the same APY as 6% compounded monthly?
r m
APY = 1 + −1
m
university-logo
Jason Aubrey Math 1300 Finite Mathematics

70. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded
continuously has the same APY as 6% compounded monthly?
r m
APY = 1 + −1
m
0.06 12
APY = 1+ − 1 = 0.0617
12
university-logo
Jason Aubrey Math 1300 Finite Mathematics

71. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded
continuously has the same APY as 6% compounded monthly?
r m
APY = 1 + −1
m
0.06 12
APY = 1 + − 1 = 0.0617
12
APY = er − 1
university-logo
Jason Aubrey Math 1300 Finite Mathematics

72. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded
continuously has the same APY as 6% compounded monthly?
r m
APY = 1 + −1
m
0.06 12
APY = 1 + − 1 = 0.0617
12
APY = er − 1
0.0617 = er − 1
university-logo
Jason Aubrey Math 1300 Finite Mathematics

73. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded
continuously has the same APY as 6% compounded monthly?
r m
APY = 1 + −1
m
0.06 12
APY = 1 + − 1 = 0.0617
12
APY = er − 1
0.0617 = er − 1
ln(1.0617) = r
university-logo
Jason Aubrey Math 1300 Finite Mathematics

74. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: What is the annual nominal rate compounded
continuously has the same APY as 6% compounded monthly?
r m
APY = 1 + −1
m
0.06 12
APY = 1 + − 1 = 0.0617
12
APY = er − 1
0.0617 = er − 1
ln(1.0617) = r
r = 0.05987
university-logo
Jason Aubrey Math 1300 Finite Mathematics

75. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: Which is the better investment and why: 9%
compounded quarterly or 9.3% compounded annually?
university-logo
Jason Aubrey Math 1300 Finite Mathematics

76. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: Which is the better investment and why: 9%
compounded quarterly or 9.3% compounded annually?
m 4
r 0.09
APY1 = 1 + −1= 1+ − 1 = 0.09308
m 4
university-logo
Jason Aubrey Math 1300 Finite Mathematics

77. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: Which is the better investment and why: 9%
compounded quarterly or 9.3% compounded annually?
m 4
r 0.09
APY1 = 1 + −1= 1+ − 1 = 0.09308
m 4
APY2 = 0.093
university-logo
Jason Aubrey Math 1300 Finite Mathematics

78. Compound Interest
Continuous Compound Interest
Growth and Time
Annual Percentage Yield
Example: Which is the better investment and why: 9%
compounded quarterly or 9.3% compounded annually?
m 4
r 0.09
APY1 = 1 + −1= 1+ − 1 = 0.09308
m 4
APY2 = 0.093
The first offer is better because its APY is larger.
university-logo
Jason Aubrey Math 1300 Finite Mathematics