Problem 31 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Joe plans to deposit $$\$ 200$$ at the end of each month into a bank account for a period of 2 yr, after which he plans to deposit $$\$ 300$$ at the end of each month into the same account for another 3 yr. If the bank pays interest at the rate of \(6 \%\) /year compounded monthly, how much will Joe have in his account by the end of 5 yr? (Assume no withdrawals are made during the 5-yr period.)

Step-by-Step Solution

Answer

By the end of 5 years, Joe will have $$\$ 180,077.00$$ in his account.

Step 1: Calculate the monthly interest rate

First, convert the yearly interest rate to a monthly interest rate by dividing by 12: \(r = \frac{6 \%}{12} = \frac{0.06}{12} = 0.005\) The monthly interest rate is \(r = 0.005\).

Step 2: Calculate the future value of $$\$ 200$$ deposits for the first 2 years

Now, apply the future value formula for the $$\$ 200$$ monthly deposits made during the first 2 years: \(FV_1 = 200 \times ((1 + 0.005)^{12 \times 2} - 1) / 0.005\) Calculate the value inside the parentheses: \((1 + 0.005)^{12 \times 2} - 1 = 1.34885\) With this value, we can now calculate the future value of the first 2 years of deposits: \(FV_1 = 200 \times 1.34885 / 0.005 = 53954.000\) So, the future value of $$\$ 200$$ deposits for the first 2 years is $$\$ 53,954.00$$.

Step 3: Calculate the future value of $$\$ 300$$ deposits for the next 3 years

Next, we'll apply the future value formula for the $$\$ 300$$ monthly deposits made during the next 3 years: \(FV_2 = 300 \times ((1 + 0.005)^{12 \times 3} - 1) / 0.005\) Calculate the value inside the parentheses: \((1 + 0.005)^{12 \times 3} - 1 = 2.10205\) With this value, we can now calculate the future value of the next 3 years of deposits: \(FV_2 = 300 \times 2.10205 / 0.005 = 126123.000\) So, the future value of $$\$ 300$$ deposits for the next 3 years is $$\$ 126,123.00$$.

Step 4: Add the future value of the first 2 years and the future value of the next 3 years

Finally, add the future values of the first 2 years and the next 3 years to find the total amount in the account by the end of 5 years: \(Total = FV_1 + FV_2 = 53954.000 + 126123.000 = 180077.000\) Joe will have $$\$ 180,077.00$$ in his account by the end of 5 years.

Key Concepts

Compounded Interest

Compounded interest is a powerful financial concept, often referred to as 'interest on interest'. It occurs when the interest earned on an investment is reinvested, and in turn, earns more interest. For Joe's savings account, where the bank pays a 6% yearly interest compounded monthly, it means that each month's interest is calculated not just on the original principal amount but also on any interest accumulated up to that point. The compounding effect intensifies over time; the longer Joe leaves his money invested, the greater the accumulation of interest will be, making compounding a significant advantage for long-term savings.

Time Value of Money

The time value of money is a foundational principle in finance stating that a dollar in hand today is worth more than a dollar promised in the future. This concept is based on the potential earning capacity of money, which implies that money can earn interest over time. In the context of Joe’s account, this principle suggests that the earlier installments of his monthly deposits have more time to earn interest, and thus, are more valuable than the later ones. It is why he starts saving early for a period of 5 years in our exercise. His initial \(200 monthly deposits, followed by \)300, both benefit from the passage of time which increases their future value.

Annuity Calculations

An annuity is a series of equal payments made at regular intervals. Joe's deposits into his account form an annuity where the payments are made monthly. Calculating the future value of an annuity involves determining how much these periodic deposits will be worth at a specific point in time in the future, taking into account the frequency of compounding. For instance, Joe's first annuity of \(200 per month over 2 years and the second annuity of \)300 per month over 3 years both have their distinct future values, which are calculated using a formula that factors in the monthly interest rate and the number of periods.

Monthly Deposits

Monthly deposits are regular payments made into an account, which can grow over time due to interest. By making monthly deposits, Joe is taking advantage of compounding interest and the time value of money for his investment. Each deposit is a part of an annuity, and it's crucial for the future value calculation to include every single payment made over the entire period. Initially, Joe deposits \(200 monthly, and after two years, he increases his deposit to \)300 a month. His consistent monthly deposits ensure a steady growth of his savings, leading to a substantial sum at the end of the 5-year period.