Engineering Economy

Question

Starting seven years from today, a mining company plans to withdraw $300 000 at the beginning of every 3-month interval from its exploration fund to carry out exploration over a 4year period (there are 16 withdrawals). The company now has $5 million in its exploration fund that pays an annual interest rate of 6 percent compounded quarterly. What annual amount must be deposited in the fund, starting today and ending 6 years from today, such that there remains a $10 million balance in the fund immediately after the company makes its last $300 000 withdrawal? Kindly, provide an explained answer with the cash flow diagram.

Answer 1

Hi Khaled,

Certainly, let's summarize the entire solution in a format that should be easy to copy and paste onto Wyzant.com.

Step 1: Understanding Cash Flow Activities

1. The mining company starts withdrawing $300,000 every 3 months for 4 years, totaling 16 withdrawals.
2. The fund has an annual interest rate of 6% compounded quarterly.
3. Deposits are to be made starting today and ending 6 years from today.
4. The objective is to have a $10 million balance in the fund after the last withdrawal.

Step 2: Cash Flow Diagram

Year 0 to Year 6: Annual deposits (D) Year 7 to Year 10: Quarterly withdrawals of $300,000 (W) Year 11: Remaining balance of $10 million

Step 3: Interest Rate and Compounding Frequency

1. Annual interest rate: 6%
2. Quarterly interest rate: 6% / 4 = 1.5%

Step 4: Future Value Calculation for Withdrawals

Using the formula for an ordinary annuity due, the Future Value of withdrawals is calculated as follows:

=300000×(1+0.015)16−10.015×(1+0.015)

FVwithdrawals​=300000×0.015(1+0.015)16−1​×(1+0.015)

This comes out to be approximately $5,460,407.

Step 5: Future Value Calculation for Desired Fund Balance

The Future Value needed in the fund after 6 years is: =10,000,000+5,460,407=15,460,407

FVneeded​=10,000,000+5,460,407=15,460,407

Step 6: Future Value of Current Fund Balance

The Future Value of the current $5 million fund balance after 6 years at a 6% annual rate compounded quarterly is:

=5,000,000×(1+0.015)(4×6)

FVcurrentbalance​=5,000,000×(1+0.015)(4×6)

This comes out to be approximately $7,147,514.

Step 7: Annual Deposit Calculation

The annual deposit needed is calculated using the formula for an ordinary annuity:

(1+0.015)24−10.015D=0.015(1+0.015)24−1​FVneeded​−FVcurrentbalance

​​

This gives an annual deposit of approximately $290,320, rounded to the nearest dollar.

Summary:

To ensure a $10 million balance in the fund after the last $300,000 withdrawal, an annual deposit of approximately $290,320 must be made from today until 6 years from today.

Thank you,

Benjamin M.

p.s. If this was helpful, I would greatly appreciate your feedback as I am new to this platform and your feedback is incredibly helpful. :)