Question

A deposit of \(\$ 1000\) is made once a year, starting today, into a bank account earning \(4 \%\) interest per year, compounded annually. If 20 deposits are made, what is the balance in the account on the day of the last deposit?

Short answer

Answer: The balance in the account on the day of the last deposit after 20 years is approximately $29,778.

Step-by-step solution

Identify the variables

P = 1000, r = 0.04, n = 20.

Calculate the interest multiplier

First, we calculate \((1 + r) = (1 + 0.04) = 1.04\).

Calculate the interest multiplier raised to the power of n

Next, we calculate \((1 + r)^n = 1.04^{20} \approx 2.19112\).

Subtract 1 from the result of Step 3

Now, subtract 1 from the result of Step 3: \(2.19112 - 1 = 1.19112\).

Divide the result of Step 4 by the interest rate

Divide the result of Step 4 by the interest rate: \(\frac{1.19112}{0.04} = 29.778\).

Multiply the result of Step 5 by the deposit amount

To find the future value, multiply the result of Step 5 by the deposit amount: \(29.778\times1000 = 29,778\). So, the balance in the account on the day of the last deposit is approximately \(\$29,778\).

Key concepts

Future Value

The term "Future Value" refers to the amount of money that an initial investment will grow to, after earning interest over a specific period. This concept is crucial in understanding how savings can grow over time. Future Value calculations often include factors like the principal amount, interest rate, and compounding periods.
In the given exercise, calculating the future value involves determining what the regular deposits of \(1000 each will amount to after 20 years at a 4% annual interest rate.
Using the formula for the future value of an annuity \[FV = P \times \left( \frac{(1 + r)^n - 1}{r} \right) \] we substitute:

• \(P = \\)1000\) (Annual Deposit),
• \(r = 0.04\) (Interest Rate),
• \(n = 20\) (Years or Number of Deposits).

This gives a future value of approximately \(\$29,778\). This figure represents the total amount in the account after the last deposit.

Annual Deposits

Annual Deposits refer to the regular contributions made to an account each year. These deposits significantly contribute to the total future value of the investment because each deposit earns interest over the specified period.
In this context, the $1000 deposited once every year grows at the compound interest rate of 4%. Understanding the impact of frequent deposits helps in realizing how consistently saving money can substantially grow one's wealth over time.
It's worth noting how annual deposits, when coupled with compounding, increase the total future value significantly compared to a single lump-sum investment.

Interest Rate

An interest rate, expressed as a percentage, is a crucial factor in calculating the future value of investments. It determines how much your investment will grow over time. In this exercise, the account grows by 4% each year.
Interest can accumulate in various ways, but here we focus on compound interest, which means that each year's interest is added to the principal, allowing the investment to generate earnings from previous interest as well in subsequent years.
The formula \[(1 + r)^n\] gives insight into how interest rates affect the value of an investment over time. Here, since interest is compounded annually, the higher the rate, the greater the growth of the account balance.

Financial Mathematics

Financial Mathematics is the backbone of understanding how money grows and is managed. It encompasses concepts such as interest rates, annuities, and compounding—each playing a vital role in personal and corporate financial planning.
By applying these mathematical principles, investors and savers can make informed decisions on how best to maximize their wealth. This exercise illustrates using mathematical formulas to determine future value, ensuring better financial outcomes by understanding the long-term impacts of consistent savings and interest accrual.
Becoming familiar with these principles helps anyone plan better, evaluate investment opportunities, and make strategic decisions for future financial security.