Final answer:
To withdraw $10,000 at the end of year 3 from an account with varying interest rates of 5%, 7%, and 10% in successive years, calculate the present value for each year starting from year 3 and discounting back to the present. Determine the present value for year 3 using the 10% rate, then discount that value at 7% for year 2, and finally discount that result at 5% for year 1 to find the initial deposit required.
Step-by-step explanation:
To determine the amount of money you need to deposit today to withdraw $10,000 at the end of year 3 with varying interest rates each year, you need to calculate the present value of that amount considering the sequential individual interest rates for each year. Since the account pays different interest rates in each year, the calculation should be made by discounting the final amount back to the present value for each year separately.
Starting with the third year, where the interest rate is 10%, you would calculate the present value for the third year as follows:
Present Value in Year 2 = Future Value at the end of Year 3 / (1 + Interest Rate in Year 3)
Present Value in Year 2 = $10,000 / (1 + 0.10)
Next, use the calculated present value for the second year, and apply the second year's interest rate of 7%:
Present Value in Year 1 = Present Value in Year 2 / (1 + Interest Rate in Year 2)
Present Value in Year 1 = Result from Year 2 / (1 + 0.07)
Finally, discount back the resultant present value in Year 1 using the first year's interest rate of 5% to find the initial deposit needed:
Initial Deposit = Present Value in Year 1 / (1 + Interest Rate in Year 1)
Initial Deposit = Result from Year 1 / (1 + 0.05)
Through this process, you can solve for the initial deposit amount, and then round it to the nearest dollar to determine how much you need to deposit initially. Remember that every step you are calculating the present value, which is the amount that, if deposited today at the specified interest rates, would grow to your desired future value.