Question

Jasper want to venture into the food stall. She plans to create a fund by making deposits of 2,000 in a bank that gives 4% interest compounded quarterly, how much money will be in the fund after 4 years?*228,892.31*101,891.58*228,805.31*103,891.58

Answer 1

We have to calculate the future value of making monthly deposits of $2000 in a bank that gives 4% interest compounded quarterly.

As the frequencies between the deposits and the compounding are different, we have to calculate a equivalent rate that compounds at the same frequency as the deposits (monthly) that keeps the same effective interest rate.

We have a nominal annual rate of 4% that compounds quarterly (m = 3). We can calculate the equivalent nominal annual rate as:

`i=q\cdot\lbrack(1+(r)/(m))^{(m)/(q)}-1\rbrack`

where m = 3 is the current compounding subperiod, q = 12 is the new compounding subperiod and r = 0.04 is the current annual rate.

We replace the values and calculate:

`\begin{gathered} i=12\cdot\lbrack(1+(0.04)/(3))^{(3)/(12)}-1\rbrack \\ i\approx12\cdot(1.0133^{(1)/(4)}-1) \\ i\approx12\cdot(1.003316795-1) \\ i\approx12\cdot0.003316795 \\ i\approx0.0398 \end{gathered}`

We can now use the interest rate i = 0.0398 compounded monthly as the equivalent rate.

We can calculate the future value of the annuity as:

`FV=(PMT)/((i)/(q))\lbrack(1+(i)/(q))^(n\cdot q)-1\rbrack`

Where PMT = 2000, i = 0.0398, q = 12 and n = 4.

We can replace with the values and calculate:

`\begin{gathered} FV=(2000)/((0.0398)/(12))\cdot\lbrack(1+(0.0398)/(12))^(4\cdot12)-1\rbrack \\ FV\approx603015.075\cdot\lbrack(1+0.003317)^(48)-1\rbrack \\ FV\approx603015.075\cdot\lbrack1.1722636-1\rbrack \\ FV\approx603015.075\cdot0.1722636 \\ FV\approx103877.55 \end{gathered}`

We get a future value of the annuity of $103,877.55.

We have some differences corresponding to the roundings made in the calculation, but this value correspond to the option $103,891.58.

Answer: $103,891.58