Problem 5 - Applied Calculus

Question

Twenty annual payments of \(\$ 5000\) each, with the first payment one year from now, are to be made from an account earning \(10 \%\) per year, compounded annually. How much must be deposited now to cover the payments?

Step-by-Step Solution

Answer

Deposit approximately $42,568 now.

Step 1: Understanding the Problem

We need to find out how much money needs to be deposited right now (the present value) to ensure that there are sufficient funds to make 20 annual payments of $5000 starting a year from now, at an interest rate of 10% compounded annually.

Step 2: Identifying the Formula

The problem requires us to calculate the present value of an annuity because there are multiple future payments of the same amount. The formula for the present value of an annuity is: \[PV = P \times \left(1 - (1 + r)^{-n} \right) / r\]where \(PV\) is the present value, \(P\) is the payment amount, \(r\) is the interest rate per period, and \(n\) is the number of periods.

Step 3: Substituting Values into the Formula

We know that \(P = 5000\), \(r = 0.10\), and \(n = 20\). Substituting these values into the formula gives us:\[PV = 5000 \times \left(1 - (1 + 0.10)^{-20} \right) / 0.10\]

Step 4: Calculate Using the Formula

First, calculate \((1 + 0.10)^{-20}\):\[(1.10)^{-20} \approx 0.14864\]Then calculate the expression:\[PV = 5000 \times \left(1 - 0.14864 \right) / 0.10\]\[PV = 5000 \times 0.85136 / 0.10\]\[PV = 5000 \times 8.5136\]Finally, calculate the present value:\[PV \approx 42568\]

Step 5: Conclusion

The amount that must be deposited now to cover the 20 annual payments of $5000, given a 10% annual interest rate, is approximately $42,568.

Key Concepts

Annuity Formula

When planning financial decisions involving recurring payments, like our exercise with annual payments, the annuity formula is crucial. It allows us to determine the present value, or how much we need today to satisfy future payments, considering a specific interest rate. An annuity refers to a series of equal payments made at equal intervals. This could include things like monthly car payments or annual insurance payments. In our case, it’s annual payments.
The annuity formula used to calculate the present value (PV) is:

• \[PV = P \times \left(1 - (1 + r)^{-n} \right) / r\]

Here:

• \(P\) is the payment amount per period,
• \(r\) is the interest rate per period, and
• \(n\) is the number of periods.

By substituting the values into the formula, you can find out how much needs to be deposited today to fund these future payments. In financial terms, this allows us to evaluate whether we can afford a specific series of payments or if changes are needed to our financial strategy.

Compound Interest

Compound interest is a powerful concept in financial mathematics that can dramatically affect the outcome of investments and loans. Unlike simple interest which is calculated on the principal alone, compound interest calculates on both the initial principal and the accumulated interest from previous periods. It creates a situation where your investment grows at an increasing rate over time.
In our exercise, compound interest is compounded annually, which means that all interest is calculated once per year. The rate used is \(10\%\). When using the annuity formula, it's important to apply compound interest correctly because it influences how much needs to be deposited now, by considering how the investment will grow over the specified time (20 years).

• To simplify, consider compound interest as earning interest on your interest, which boosts your growth exponentially.
• This is why the amount you need now to meet future payments can often be less than the total future payouts.

Financial Mathematics

Financial mathematics is a branch of applied mathematics concerned with finance and investment. It provides the tools and formulas to evaluate potential financial decision outcomes. Understanding these tools, like the annuity formula and compound interest, can lead to better financial planning and management.
It combines knowledge of mathematics, economics, and finance and is crucial for anyone dealing with annuities, investments, or any financial planning.

• Financial mathematics helps assess the time value of money, which is the concept of how the value of money changes over time due to potential earning capacity.
• In our example, the annuity formula is used to calculate the present value of the annuity given a specific interest rate and determine how much to invest today for future returns.
• Being versed in financial mathematics can empower individuals to make informed choices about savings, investments, and expenditures.

By mastering financial mathematics, students can develop skills in logical thinking, problem-solving, and risk assessment, which are valuable in personal finance and professional financial roles.