Almost every day you are bombarded with opportunities to borrow money. Credit card applications are everywhere. Many of these offers talk about the APR or **A**nnual **P**ercentage **R**ate that is associated with the money or credit offer. The APR is also called the "nominal rate" (meaning in name only). In this section we will address compound interest and look at interest that is compounded in discrete periods such as daily, weekly, monthly and yearly and we will look at interst that is compounded continuously. When you are done with this section you should be able to:

- Identify the nominal and effective annual rates of an account
- Calculate the balance of an account after any given period of time
- Calculate the time for the balance in an account to reach a given value.

Nominal and effective rates

- The general equation for the balance of an account is B = P(1 + r/n)
^{nt} - B is balance, P is the initial amount, r is the annual interest rate, n is the number of times per year that interest is compounded

Year | Annual | Monthly | Daily |
---|---|---|---|

0 | $1,000.00 | $1,000.00 | $1,000.00 |

1 | $1,050.00 | $1,051.16 | $1,051.27 |

2 | $1,102.50 | $1,104.94 | $1,105.16 |

3 | $1,157.63 | $1,161.47 | $1,161.82 |

- Annual compounding: Effective rate = (1 + .05/1) = 1.05 or 5%, the APR or nominal rate and the effective rate are the same.
- Monthly comounding: Effective rate = (1. +.05/12)
^{12}= 1.0512, the APR is 5% but the effective rate is 5.12% - Daily compounding: Effective rate = (1 +.05 .05/365)
^{365}= 1.0513, the APR is 5% but the effective rate is 5.13%

- The general equation for continuous compounding is B = P e
^{rt} - B is balance, P is the initial amount, r is the compounded continuously annual rate and t is time in years.

Year | Continous |
---|---|

0 | $1,000.00 |

1 | $1,051.27 |

2 | $1,105.17 |

3 | $1,161.83 |

**How long will it take our money to grow to a specified amount?**

You and a friend each have $1,000. You put yours in an account that pays 5% per year compounded semi-annually (twice a year). Your friends money goes into an account that pays 4.9 % compounded continuously. How long will it you each of the accounts to grow to $2,500?

Your account

- 2500 = 1000 (1 + .05/2)
^{(2t)}- 2,500/1000 = (1 + .05/2)
^{(2t)}- 2.5 = (1 + .05/2)
^{(2t)}- log(2.5) = 2t log(1+.05/2)
- log(2.5)/[2 log(1+.05/2)] = t
- t = 18.55 years (a little more than 18 and 1/2 years)

Your friends account

- 2500 = 1000 e
^{.05t}- 2500/1000 = e
^{.05t}- ln (2.5) = ln e
^{.05t}- ln(2.5) = .05 t ln e = ,05t (since ln e = 1)
- t = ln(2.5)/.05 = 18.33 years (about 18 and 1/3 years, a little faster than your account)
Practice Problems

Question 1 - Margaret invests $2,000 in a bank account that pays 4.68% compounded weekly.

- What are the nominal and effective rates of this account?
- What will be the balance in this account after 26 weeks?
- What will her balance be in 2 years?
Question 2 - John has invested $1,200 at 4.7% compounded continuously.

- What are the nominal and effective rates of this account?
- How long will it take for John's money to double?

Answers to questions

Question 1

- The nominal rate is 4.68%, to get the effective rates first calculate (1 + 0.468/52)
^{52}= 1.0479, effective rate = 4.79%- B = 2,000(1 + .0468/52)
^{26}= $2,047.33- B = 2000(1 + 0.468/52)
^{[52(2)] }= $2,196.15Question 2

- The nominal rate if 4.7%, to get the effective rate first calculate e
^{.047}=1.0481, the effective rate is 4.81%- 2(1200) = 1200 e
^{.047t}2400/1200 = e

^{.047}2 = e

^{.047t}ln 2 = .047t lne = .047 t (since ln e = 1)

t = ln 2/.047 = 14.75 years (about 14 years and 9 months)