Question

In Chapter 1 of the text we looked at calculating a monthly payment for a loan. A related formula is to calculate the amount accruing when regular payments are made into an interest bearing account - often called the Savings Plan formula. (A is the accrued amount after t years of making regular payments, PMT, into an account at interest rate, r%, compounded ntimes each year.) A(t) = PMT·((1 + r/N)N·t - 1)/(r/N) = PMT*((1 + r/N)^(N*t) - 1)/(r/N) The second version is essentially in the form used in Excel Suppose you want to buy a car and have decided that you can save $100 a month. Using information from an internet source, determine the current interest rate on savings accounts and use the information to answer the following: How much money will you have saved in two year’s time? How much will be interest? Why wouldn’t a linear model work here? Can you put the answer in excel?

Answer 1

The given formula for accrued amount = A(t) = PMT*((1 + r/N)^(N*t) - 1)/(r/N)

This formula for an accrued amount after n periods of making regular payments P into an account at the interest rate of r% per period compounded after each period is

P{(1+r)ⁿ - 1)∕r}

Where P = periodic payments, r = rate of interest per period and n – number of periods.

Given P = 100

Periods n , the no. of months = 2 years = 24 months

We calculate the amount for three rates of interests

Interest rate per annum = 4%. 5%. 6%, 10% and 12%

Corresponding interest rates per period(month) = Interest rate p a /12

Converted to decimals.

0.003

0.004

0.005

0.008

0.010

Calculate the accumulated amount using the formula P{(1+r)ⁿ - 1)∕r}

P=	100		n=	24
Interest Rate p a	0.04	0.05	0.06	0.10	0.12
r	0.00	0.00	0.01	0.01	0.01
1+r	1.00	1.00	1.01	1.01	1.01
(1+r)24	1.08	1.10	1.13	1.22	1.27
(1+r)24 - 1	0.08	0.10	0.13	0.22	0.27
((1+r)24 - 1)/r	24.94	25.19	25.43	26.45	26.97
P*((1+r)24 - 1)/r	2494.29	2518.59	2543.20	2644.69	2697.35

Thus $ 100 deposited every month into a savings bank account for 24 months ( two years) will get accumulated to $ 2644.69 at the rate of interest 10% per annum.

The following table gives the total amount saved for 2 years at different rates of interest and the amount of interest.

rate of interest p a	total amount	Interest amount
4.00%	2494.29	94.29
5.00%	2518.59	118.59
6.00%	2543.20	143.20
10.00%	2644.69	244.69
12.00%	2697.35	297.35

The interest for every month is added to the principal and then interest for the current month is calculated.

If we had calculated simple interest then we can work with a linear model.