Topic 9: Personal Finances

Interest is money that a borrower pays for using the lender's money.

Interest that is computed on the principal AND accumulated interest is called compounded interest. The interest paid on savings accounts is normally compound interest.
p is the principal or amount of money borrowed,
r is the interest rate, and
t is the duration of the loan,
and r and t must be expressed in the same time increment. (Note that t is only given in years in the compound interest formula below).

Example 9.2
A student saves $2,000 from his summer job and invests it in an account at an annual interest rate of 3% compounded quarterly. How much will be in the account at the end of one year?

p=2000, r=0.03, t=4

Quarter

 

Interest earned this quarter

 

 

Balance at the end of this quarter

 

 

Formula to calculate the Balance

 

1 0.0075 * 2000 = 15 2000 + 15 = 2015 (1 + 0.03/4) * 2000
2 0.0075 * 2015 = 15.11 2015 + 15.11 = 2030.11 (1 + 0.03/4)2 * 2000
3 0.0075 * 2030.11 = 15.23 2030.11 + 15.23 = 2045.34 (1 + 0.03/4)3 * 2000
4 0.0075 * 2045.34 = 15.34 2045.34 + 15.34 = 2060.68 (1 + 0.03/4)4 * 2000

Note that since the interest is paid quarterly, we use r/4, which here is 0.03 / 4 = 0.0075, to calculate the interest earned each quarter.

This leads us to the compound interest formula, with
A = amount of money,
APR = annual percentage rate,
t = number of years, and
n = number compounding periods per year

A = (1 + APR/n)nt * p

A formula which you can paste directly into Excel, and then change the variables to cell references is:
= (1 + APR/n)^(n*t) * p

Note that we will follow the book's convention of n = 365 for daily compounding.

When n is greater than 1, the actual gain is NOT the same as the annual percentage rate times the principal. For instance, in example 9.2 above, the actual gain in the account is $60.68, while 0.03 * 2000 = 60.

The Annual Percentage Yield, or APY, can be calculated by dividing the interest gain in one year by the principal. For example 9.2, (A - p)/p = 60.68 / 2000 = 0.03034, or 3.034%. Banks typically list both the APR and the APY.

 

Rather than calculate savings generated by compound interest on a lump sum, we next consider smaller, periodic, deposits.

A series of fixed, regular payments is called an annuity. There are two types of annuities. An ordinary annuity is one where the payments are required at the end of each time period, and an annuity due has payments required at the beginning of each time period.

Example 9.6
A student is saving for a new car and deposits $150 at the end of each month, in a plan which earns monthly interest at an APR of 6%. Create a table for the first 6 months.

First, note the monthly interest is APR/n = 0.06/12 = 0.005

End Month i

 

Previous Balance

 

 

Interest on Previous Balance

 

 

Additional Deposit

 

 

New Balance

 

1 0 0 150 150
2 150 0.005*150 = 0.75 150 300.75
3 300.75 0.005*300.75 = 1.50 150 452.25
4 452.25 0.005*452.25 = 2.26 150 604.51
5 604.51 0.005*604.51 = 3.02 150 757.53
6 757.53 0.005*757.53 = 3.79 150 911.32

 

This leads to the accumulated savings formula, where:
A = Amount in account at a future data, sometimes called future value, or FV
PMT= regular payment into account
n= number of payment and compounding periods per year
t=number of years

A = PMT * ( (1 + APR/n)nt -1 )
                        (APR/n)

A formula which you can paste directly into Excel, and then change the variables to cell references is:
= PMT * ( (1 + APR/n)^(n*t)-1 ) /(APR/n)