Problem 51 - Technical Mathematics with Calculus

Question

If an amount \(a\) is invested at a compound interest rate \(n,\) it will be possible to withdraw a sum \(R\) at the end of every year for \(t\) years until the deposit is exhausted. The number of years is given by $$t=\frac{\log \left(\frac{a n}{R-a n}+1\right)}{\log (1+n)} \quad \text { (years) }$$ If \(\$ 200,000\) is invested at \(12 \%\) interest, for how many years can an annual withdrawal of \(\$ 30,000\) be made before the money is used up?

Step-by-Step Solution

Answer

Using the given formula and substituting the values for a, R and n, the number of years that the withdrawals can be made is calculated to be approximately 9.5708 years.

Step 1 - Convert percentage to decimal

The given interest rate is in percentage. Convert it into decimal form by dividing by 100. For an interest rate of 12%, the decimal form is 0.12.

Step 2 - Identify the parameters

Identify all the parameters given in the problem and plug them into the provided formula. We have the initial investment amount a = \(200,000, the annual withdrawal amount R = \)30,000, and the interest rate n in decimal form is 0.12.

Step 3 - Insert values into the formula

Insert the values of a, R, and n into the formula to find the number of years t. Therefore, the formula is now t = (log((200,000*0.12)/(30,000-200,000*0.12)+1))/(log(1+0.12)).

Step 4 - Compute the argument of the first logarithm

Evaluate the expression inside the first log. Compute (200,000*0.12)/(30,000-200,000*0.12) and then add 1 to this value.

Step 5 - Calculate the numerator

Use a calculator to determine the log of the value obtained in the previous step.

Step 6 - Calculate the denominator

Similarly, calculate the log of (1+0.12) to find the denominator of the formula.

Step 7 - Complete the calculation

Divide the value obtained in Step 5 by the value obtained in Step 6 to find t, the number of years the withdrawals can be made.

Key Concepts

Financial Mathematics

Understanding financial mathematics is crucial for accurately calculating transactions involving loans, investments, annuities, and budgets. Compound interest is one of the key elements in financial mathematics. It is the interest calculated on the initial principal, which also includes all of the accumulated interest from previous periods on a deposit or loan.

Think of compound interest as 'interest on interest'. It grows at a faster rate than simple interest, which is calculated only on the principal amount. The formula to calculate the number of years until an investment is exhausted through annual withdrawals incorporates compound interest concepts:
\[t = \frac{\log \left(\frac{a \cdot n}{R - a \cdot n} + 1\right)}{\log (1+n)}\]
Here, 'a' represents the initial investment, 'n' is the interest rate (in decimal form), and 'R' is the annual withdrawal amount. This formula provides a practical application of financial mathematics in personal financial planning.

Logarithmic Functions

Logarithmic functions play a critical role in various scientific, engineering, and financial calculations. In the context of the compound interest formula, logarithms help determine time periods related to growth rates, such as the number of years needed for an investment to reach a certain point.

Logarithms are the inverses of exponents. A logarithm answers the question: 'To what exponent must we raise a certain base to obtain a given number?' In the provided formula:

Function of Logarithms in the Formula

\[t = \frac{\log \left(\frac{a \cdot n}{R - a \cdot n} + 1\right)}{\log (1+n)}\]
The base of the logarithms is assumed to be 10 if not specified. The numerator computes the logarithm of the growth of the initial investment over the withdrawal amount and the denominator computes how quickly each invested unit grows by a rate of 'n'. These calculations are essential for solving problems involving the time value of money and compound interest.

Time Value of Money

The time value of money (TVM) is a core concept in finance that refers to the idea that money available at the present time is worth more than the same amount in the future due to its potential earning capacity. This principle underlies the concept of compound interest, where the value of an investment increases over time due to the interest earned on both the initial principal and the accumulated interest.

In the context of our compound interest formula:

Implementing TVM in Investment Decisions

\[t = \frac{\log \left(\frac{a \cdot n}{R - a \cdot n} + 1\right)}{\log (1+n)}\]
Time value of money is reflected in the formula, showing how long the invested money can last when withdrawals are made annually. By understanding TVM, investors can make informed decisions on their investments, considering factors like interest rates, inflation, and return on investment. This formula can help investors in determining the sustainability of their investment over a given period considering the withdrawal they need to make for expenses.