Problem 86 - College Algebra and Calculus: An Applied Approach

Question

Find the amount of an annuity with income function \(c(t)\), interest rate \(r\), and term \(T\). $$ c(t)=\$ 500, \quad r=7 \%, \quad T=4 \text { years } $$

Step-by-Step Solution

Answer

The total amount (present value) of the annuity is approximately $1715.17.

Step 1: Identify Parameters

From the problem, we can identify the necessary parameters to calculate the amount of the annuity. They are \(c(t) = $500\), which is the regular payment, \(r = 7\%\) or 0.07 is the annual interest rate, and \(T = 4\) years, which is the length of time the annuity is active.

Step 2: Apply The Formula

For an annuity where the payments are made at the end of the periods, we use the formula for the present value of an ordinary annuity:\[ A = c(t) \times \left(\frac{1 - (1 + r)^{-T}}{r}\right) \]This formula calculates the total value of an annuity at present time, taking into consideration that money received in the future is worth less than the same amount of money today.

Step 3: Calculation

Substitute the given parameters into the formula: \[ A = 500 \times \left(\frac{1 - (1 + 0.07)^{-4}}{0.07}\right) \]. Now perform the operations within the parentheses first, then multiplication.

Step 4: Final Result

After calculating, the annuity value \(A\) comes out to be approximately \$1715.17.

Key Concepts

Interest Rate

Interest rate is a crucial concept in financial mathematics, particularly when dealing with annuities. It represents the cost of borrowing money or the gain from investing money, expressed as a percentage of the principal. In the context of annuities, the interest rate affects how future payments are valued in present terms.
When dealing with annuities, the interest rate helps determine the discount factor, which is used to calculate the present value of future payments. A higher interest rate usually means a lower present value of future cash flows, as money today is considered more valuable due to its potential earning capacity.

• The rate is typically expressed annually as a percentage.
• An ordinary annuity assumes payments are made at the end of each period.
• The interest rate is used in the formula to calculate the total present value of the annuity.

Overall, understanding the interest rate is essential for accurately calculating the value of annuities and making informed investment decisions.

Present Value

Present value is a fundamental concept in financial mathematics. It represents today's value of money that is expected to be received or paid in the future. When calculating annuities, understanding present value is critical because it helps us evaluate how future cash flows are compared to their value if received today.
An ordinary annuity, like the one in the exercise, involves regular payments or receipts that occur at the end of each period. The method to calculate present value allows us to sum these future annuity payments into one lump sum value in today's terms.

• The present value formula used for ordinary annuities is: \[ PV = c(t) \times \left( \frac{1 - (1 + r)^{-T}}{r} \right) \]
• The present value formula assumes a constant interest rate over the annuity's term.
• This calculation helps in personal finance decisions, like deciding between a lump sum versus periodic payments.

By converting future payments into present value, individuals and businesses can assess investment opportunities effectively.

Ordinary Annuity

An ordinary annuity is a series of equal payments made at regular intervals at the end of each period. It is a common financial product encountered in savings and loans, making it a significant aspect of financial mathematics.
Ordinary annuities are used to calculate scenarios like mortgage payments, retirement savings, and investment profits, where the timing of payments impacts the total financial outcome.

• In an ordinary annuity, payments happen at the end of each period.
• The formula used considers the timing of these payments, adjusting for interest rate and number of periods to calculate present value.
• They are different from annuities due at the beginning of the period, known as annuity-due.

Understanding ordinary annuities is key for evaluating how cash flows will affect personal wealth over time, allowing better financial planning and decision-making.

Financial Mathematics

Financial mathematics is an essential field that involves the application of mathematical methods to solve problems in finance. This discipline encompasses various concepts including present value, interest rate, and annuities, all of which are vital in making sound financial decisions.
Through financial mathematics, individuals can model cash flows and determine fair pricing for financial instruments like bonds and loans. Mathematical tools help in valuing annuities, which offer structured payments over time, providing stability and predictability for financial planning.

• It aids in the accurate calculation of annuities by using specific formulas.
• Helps in understanding how interest rates affect investment outcomes.
• Facilitates the valuation of future cash flows in present terms.

By mastering financial mathematics, individuals can enhance their financial literacy, enabling them to manage assets and liabilities better and optimize investment strategies.