### Week 5 - Part 1 - Compound Interest - Chapter 4: Sec. 4.8

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:

• 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

Interest comounded at discrete periods
• 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

YearAnnualMonthlyDaily
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
Let's look at a \$1,000 deposit made to an account that pays 5% interest per year. The table shows the balance in the account if we assume annual compounding (once a year), monthly compounding and daily compounding. As you can see from the table the accounts grow at different rates. The more frequent the compounding period the faster the account balance grows. This means that the effective yearly interest rates of the accounts must be different (the amount of interest actually paid during a year). They differ because your account earns interest on interest when compounding takes place. The effective interest rates can be calculated as follows.

• 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%

Interst compounded continuously
• The general equation for continuous compounding is B = P ert
• B is balance, P is the initial amount, r is the compounded continuously annual rate and t is time in years.

YearContinous
0\$1,000.00
1\$1,051.27
2\$1,105.17
3\$1,161.83
It we have \$1,000 and can invest is at 5% compounded continuously we can expect the following balances shown in the table on the right. The effective annual interest rate for this account will be e.05(1)=1.0513 or 5.13%. The nominal rate would be 5%. It is not by accident that the effective continuous compounding rate and the effective daily compounding rate are close, if you kept increasing the number of compounding events during the year you are approaching continuous compounding and the effective rates approach one another.

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?

• 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)

• 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.

1. What are the nominal and effective rates of this account?
2. What will be the balance in this account after 26 weeks?
3. What will her balance be in 2 years?

Question 2 - John has invested \$1,200 at 4.7% compounded continuously.

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

Question 1

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%
2. B = 2,000(1 + .0468/52)26 = \$2,047.33
3. B = 2000(1 + 0.468/52)[52(2)] = \$2,196.15

Question 2

1. The nominal rate if 4.7%, to get the effective rate first calculate e.047=1.0481, the effective rate is 4.81%
2. 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)