To solve this compound interest problem, let's break down the information given and set up the inputs required for the TVM (Time Value of Money) Calculator.

Given Information:

1. Present Value (PV) = $13,000 (initial investment amount)
2. Future Value (FV) = $42,885.62 (account balance at the end of the term)
3. Number of Years (t) = 11 years
4. Compounded Monthly = 12 compounding periods per year
5. Annual Interest Rate (I%) = This is unknown and needs to be calculated.

Variables for the TVM Calculator:

• PV = -$13,000 (negative because it's an outflow, or money invested)
• FV = $42,885.62 (positive because it’s the amount we want to receive)
• PMT = 0 (no additional payments are being made during the investment period)
• N = Total number of compounding periods, calculated as $N = 11 \times 12 = 132$
• I% = $x$ (unknown; this is what we need to solve)
• P/Y and C/Y = 12 (since compounding is monthly)

Solving for the Annual Interest Rate

The TVM calculator or financial calculator can be used to solve for the annual interest rate $x$.

However, we can also approximate this using a formula for compound interest:

$FV = PV \times (1 + \frac{r}{n})^{n \times t}$

Where:

• $FV = 42,885.62$
• $PV = 13,000$
• $n = 12$ (compounded monthly)
• $t = 11$ years
• $r$ is the annual interest rate we’re solving for.

Plugging in the Values:

1. Rearrange the equation to solve for $r$: $\frac{FV}{PV} = (1 + \frac{r}{n})^{n \times t}$
2. Substitute in the values: $\frac{42885.62}{13000} = (1 + \frac{r}{12})^{132}$
3. Solve for $r$ by taking the 132nd root on both sides and isolating $r$.

Using a calculator or the TVM feature on a financial calculator will yield the exact annual interest rate.

Would you like me to go into detailed calculations or use a step-by-step approximation method?

Follow-up Questions:

1. What is the formula for continuous compounding interest?
2. How does compounding frequency affect the future value of an investment?
3. What is the difference between nominal and effective interest rates?
4. How can we use logarithms to solve exponential equations in finance?
5. How would the result change if the investment were compounded quarterly instead of monthly?

Tip:

For compounded interest problems, always remember to convert percentages to decimal form before applying them in equations.