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 Annual Percentage Rate 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:
Nominal and effective rates
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 |
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 friend's money goes into an account that pays 4.9 % compounded continuously. How long will it take 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 friend's 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)] = $2,196.15
Question 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)