Question

Find the function values. \(A(P, r, t)=P\left[\left(1+\frac{r}{12}\right)^{12 t}-1\right]\left(1+\frac{12}{r}\right)\) (a) \(A(100,0.10,10)\) (b) \(A(275,0.0925,40)\)

Short answer

The value of \(A(100, 0.10, 10)\) is approximately 1071.51 and the value of \(A(275, 0.0925, 40)\) is approximately 169034.91

Step-by-step solution

Plug in values for \(A(100,0.10,10)\)

Begin by replacing \(P\), \(r\), and \(t\) in the equation with the corresponding values to get the equation \(A = 100\left[\left(1+\frac{0.10}{12}\right)^{12*10}-1\right]\left(1+\frac{12}{0.10}\right)\)

Perform calculations

Next, calculate the value for \(A\) in the equation previously generated. First, simplify inside the parentheses, then compute the power, subtract 1, multiply by 100 and finally multiply by \(\left(1+\frac{12}{0.10}\right)\), this gives us \(A \approx 1071.51\)

Repeat steps 1 and 2 for \(A(275,0.0925,40)\)

Now replug the new set of \(P\), \(r\), and \(t\) in the given equation which gives us \(A = 275\left[\left(1+\frac{0.0925}{12}\right)^{12*40}-1\right]\left(1+\frac{12}{0.0925}\right)\). Next, calculate the value for \(A\) in the equation previously generated. This gives us \(A \approx 169034.91\)

Key concepts

Financial Mathematics

Financial mathematics is a branch of applied mathematics that studies financial markets and the mathematical models that describe them. It includes topics such as interest rates, investments, and savings. One fundamental concept in financial mathematics is compound interest, which is the interest calculated on the initial principal as well as on the accumulated interest from previous periods. This concept is key to understanding how investments grow over time.

In financial terms, we often think about how the money we invest or save will grow over the years. The compound interest formula, exemplified above as the function A(P, r, t), is a powerful tool that gives us the figure we can expect our investment to grow to under certain conditions of rate and time periods. It considers not just a single initial investment (P) along with an annual interest rate (r), but it also includes the frequency of compounding - in the case of our function, compounding occurs monthly. The formula presented in the exercise exhibits how exponentially wealth can accumulate over time.

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. These functions, written in the form f(x) = b^x, describe situations where growth or decay is proportional to the current amount, such as populations, radioactive decay, and critically, compound interest in finance.

The function provided, A(P, r, t), shows exponential growth as money is repeatedly multiplied by a factor that depends on the interest rate. Each compounding period increases the amount not linearly, but exponentially, meaning the increments become larger as the investment grows. This is the key characteristic of exponential functions—the rate of growth becomes faster as time passes. Understanding the mechanics behind these functions helps in grasping how the value of an investment escalates much more dramatically over long periods due to the compound interest effect.

Algebraic Calculations

Algebraic calculations are the process of manipulating algebraic expressions and equations to resolve unknown values. In the context of financial mathematics, they are utilized to determine future values of investments based on variables such as principal, rate, and time.

The step by step solution provided is a practical example of algebraic calculations at play. Initially, we replace the variables with their corresponding values, a process known as 'substitution'. Next, we simplify the expression inside the parentheses and calculate powers, a type of algebraic operation. After simplifying the expression, the resulting product provides the final answer. Such algebraic calculations are fundamental because they allow for the customization of the compound interest formula to suit different financial scenarios, enabling individuals to effectively plan their finances based on theoretical projections.