Question

Suppose you are investing $856 per year in an account that is compounded every quarter. The quoted annual rate is 5% per year. What is the value of this account after 4 years?

How would one do this without excel?

Answer 1

Here, since the deposits are annual and interest is compounded quarterly, so we need to find the effective annual rate as per below:

Effective annual rate = (1 + r/m)^m - 1

where, r = rate of interest = 5%, m = number of compounding periods

putting these values in the above formula, we get,
Effective annual rate = (1 + 5% / 4)⁴ - 1

Effective annual rate = (1 + 0.0125)⁴ - 1

Effective annual rate = (1.0125)⁴ - 1

Effective annual rate = 1.050945 -1

Effective annual rate = 0.050945 or 5.0945%

Now, for calculating the value of the account after 4 years, we will use the below formula of future value of annuity:

FVA = P * ((1 + r)ⁿ - 1 / r)rmula of

where, FVA is future value of annuity, P is the periodical amount = $856, r is the rate of interest = 5.0945% compounded annually and n is the time period = 4

Now, putting these values in the above formula, we get,

FVA = $856 * ((1 + 5.0945%)⁴ - 1 / 5.0945%)

FVA = $856 * ((1 + 0.050945)⁴ - 1 / 0.050945)

FVA = $856 * ((1.050945)⁴ - 1 / 0.050945)

FVA = $856 * ((1.21988798341- 1 / 0.050945)

FVA = $856 * (0.21988798341/ 0.050945)

FVA = $856 * 4.3161837944

FVA = $3695

So, value of the account after 4 years will be $3695