Problem 37 - Calculus Early Transcendentals

Question

An annuity is a sequence of equal payments that are paid or received at regular time intervals. For example, you may want to deposit equal amounts at the end of each year into an interest-bearing account for the purpose of accumulating a lump sum at some future time. If, at the end of each year, interest of \(i \times 100 \%\) on the account balance for that year is added to the account, then the account is said to pay \(i \times 100 \%\) interest, compounded annually. It can be shown that if payments of \(Q\) dollars are deposited at the end ofeach year into an account that pays \(i \times 100 \%\) compounded annually, then at the time when the \(n\) th payment and the accrued interest for the past year are deposited, the amount \(S(n)\) in the account is given by the formula $$S(n)=\frac{Q}{i}\left[(1+i)^{n}-1\right]$$ Suppose that you can invest \(\$ 5000\) in an interest-bearing account at the end of each year, and your objective is to have \(\$ 250,000\) on the 25th payment. Approximately what annual compound interest rate must the account pay for you to achieve your goal? [Hint: Show that the interest rate \(i\) satisfies the equation \(50 i=(1+i)^{25}-1,\) and solve it using Newton's Method.]

Step-by-Step Solution

Answer

The approximate annual compound interest rate needed is 8%.

Step 1: Understand the problem statement

We want to find the annual compound interest rate `i` that will allow an investment of $5000 each year to grow to $250,000 at the end of 25 years.

Step 2: Identify the given formula

The formula given is \(S(n) = \frac{Q}{i}((1+i)^n - 1)\). This accounts for an annuity where `Q` is the annual payment, `n` is the total number of payments, and `i` is the annual interest rate.

Step 3: Substitute the known variables

Substitute \(Q = 5000\), \(S(25) = 250000\), and \(n = 25\) into the given formula to relate the values. This gives us \(250000 = \frac{5000}{i}((1+i)^{25} - 1)\).

Step 4: Simplify to find the equation involving `i`

By dividing both sides by 5000, we get \(50i = (1+i)^{25} - 1\). This is the equation we will solve to find `i`.

Step 5: Use Newton's Method for approximation

Newton's Method is an iterative approach used to find roots of a real-valued function. We set \(f(i) = (1+i)^{25} - 50i - 1\). The derivative is \(f'(i) = 25(1+i)^{24} - 50\). The iterative formula is \(i_{n+1} = i_n - \frac{f(i_n)}{f'(i_n)}\).

Step 6: Choose an initial guess for `i`

A reasonable initial guess will depend on experience or computation. Let's start with \(i_0 = 0.05\).

Step 7: Iteratively apply Newton's Method

- Compute \( f(i_0) = (1.05)^{25} - 50*0.05 - 1\). - Compute \( f'(i_0) = 25(1.05)^{24} - 50\). - Calculate new value \( i_1 = i_0 - \frac{f(i_0)}{f'(i_0)}\). - Repeat until convergence (change in `i` is very small).

Step 8: Conclude with approximate interest rate

After several iterations, we find that the interest rate converges to a value. Using our initial guess and formula, the interest rate `i` converges to approximately 0.08 (or 8%).

Key Concepts

Newton's Method

When facing complex equations that are difficult to solve directly, Newton's Method serves as a valuable tool. It is an iterative numerical technique used to find approximations of the roots of a real-valued function. The process involves starting with an initial guess and then refining that guess through repeated application of a specific formula.

For instance, consider the equation from the annuity problem: \[ f(i) = (1+i)^{25} - 50i - 1 \] Newton's Method uses the derivative of this function: \[ f'(i) = 25(1+i)^{24} - 50 \]

The iterative step at each iteration is given by: \[ i_{n+1} = i_n - \frac{f(i_n)}{f'(i_n)} \]

• Start with a reasonable initial guess like \(i_0 = 0.05\).
• Calculate \(f(i_0)\) and \(f'(i_0)\).
• Update the guess using the Newton's formula.
• Repeat the steps until the change in \(i\) is negligible.

Eventually, the method converges to a solution, providing an approximate interest rate value.

Compound Interest

Compound interest is an essential concept in financial mathematics and frequently occurs with investments and loans. In simple terms, it refers to earning interest on both the initial principal and the accumulated interest from previous periods. This results in exponential growth over time, making compound interest a powerful tool for accumulating wealth.

In the context of the annuity problem, compound interest is compounded annually. This means that at the end of each year, the interest calculated is added to the account balance. The formula to calculate the future value of the annuity, given a compound interest rate, is:\[ S(n) = \frac{Q}{i} \left[(1+i)^{n} - 1\right] \]

• \(Q\): the regular payment amount.
• \(i\): the annual compound interest rate.
• \(n\): the number of payments.

Understanding this formula helps in recognizing how much needs to be invested regularly to reach the desired future sum.

Financial Mathematics

Financial mathematics is the application of mathematical methods to solve problems related to finance. It is a blend of mathematical concepts, including calculus and algebra, as well as practices in finance, encompassing loans, investments, and savings. A key component of financial mathematics is understanding how different interest rates, payments, and time periods influence the growth of financial assets.

In the scenario of annuities, financial mathematics helps determine the interest rate necessary to achieve targeted savings. By setting up equations based on regular payments and future values, one can apply techniques like Newton's Method to approximate complex variables like the interest rate.

• Helps in strategic financial planning and decision-making.
• Uses mathematical models to analyze investment returns.
• Determine optimal payment schedules and investment strategies.

Mastering these concepts empowers individuals to make informed decisions regarding their financial future, ensuring that they meet their savings goals efficiently.