Math 143 Quantitative Literacy

Question

Find the savings plan balance after 6 months with an APR of 6​% and monthly payments of $150.

Answer 1

There are formulas and online calculators that will provide an answer, but since there are only 6 compounding periods, a clear way to understand the process is to do it directly, 1 month at a time.

Here, I am assuming that payments are made at the beginning of each period. If they are made at the end of each period, the results are slightly different.

At beginning of 1st month

Balance is 0

Deposit +$150 yields $150

At end of first month, interest is .06/12 * 150 = $0.75

(monthly interest rate is annual percent rate of .06 / 12 months per year)

At beginning of 2nd month

Balance is $150 + 0.75 = 150.75

Deposit +$150 yields 300.75

At end of second month, interest is .06/12 * 300.75 = $1.50

At beginning of 3rd month

Balance is $300.75 + 1.50 = 302.25

Deposit +$150 yields 452.25

At end of third month, interest is .06/12 * 452.25 = $2.26

At beginning of 4th month

Balance is 452.25 + 2.26 = 454.51

Deposit +$150 yields 604.51

At end of 4th month, interest is .06/12 * 604.51 = 3.02

At beginning of 5th month

Balance is 604.51 + 3.02 = 607.53

Deposit +$150 yields 757.53

At end of 5th month, interest is .06/12 * 757.53 = 3.79

At beginning of 6th month

Balance is 757.53 + 3.79 = 761.32

Deposit +$150 = 911.32

At end of 6th month, interest is .06/12 * 911.32 = 4.56

So, at the end of 6th month, balance is 911.32 + 4.56 = $915.88

To check your answer for these kinds of problems, there is a good online calculator at

https://www.calculator.net/finance-calculator.html

You would fill in N = number of periods = 6

Start Principal = 0

I/Y % (yearly interest rate per period as a percent) = .5%

PMT Annuity Payment = +150

To solve this using a formula, this problem is the Future Value of an Annuity Due (where payment is made at the beginning of each month)

The formula is

S = R( (1 + i)ⁿ⁺¹ - 1) / i ) - R

where S is the future value

R is the periodic payment (here R = +$150)

i is the interest rate per period (here i = .06/12 = .005)

n is the number of periods (here n = 6)

This also gives a result of $915.88

If the payment is made at the end of each month, the problem is the Future Value of an Ordinary Annuity. Then the formula is

S = R( (1 + i)ⁿ - 1) / i )

The value of the variables is the same as above, and the answer is $911.33.