Final answer:
To purchase the beach house in 12 years, we calculate its future value with a 5% appreciation rate, resulting in $179,585.60. Then, using the annuity formula, we determine the monthly investment needed, considering a 7.50% annual return on the investment account, for the 10 years leading up to retirement.
Step-by-step explanation:
To calculate the amount needed to invest each month to buy a beach house in 12 years valued at $100,000 today which appreciates annually at 5%, considering a 7.50% annual rate of return on the Z-Trade investment account, we need to first find the future value of the house and then use the annuity formula to find the monthly investment.
The future value (FV) of the beach house after 12 years is calculated using the formula FV = PV(1 + r)n, where PV is the present value, r is the annual appreciation rate, and n is the number of years. Insert the values:
FV = $100,000(1 + 0.05)12 = $100,000(1.795856)
This gives us the future value of the house as $179,585.60.
Next, to find the monthly investment (PMT), we use the future value of an ordinary annuity formula, which is FV = PMT × ¶(× × × × ×), where PMT is the monthly payment, r is the monthly rate of return, and t is the total number of payments.
Since we are investing for 10 years, our monthly rate is (7.50% / 12) = 0.625%, and we have a total of 10*12 = 120 payments. Plugging into the annuity formula, we solve for PMT.
Now we calculate the monthly investment needed to reach the future value of the house after 10 years. After finding PMT, this will be the amount that needs to be invested at the end of each month for the 10 years leading up to the retirement to buy the dream home in Cabo San Lucas.
It's crucial to note that this calculation ignores taxes, fees, fluctuations in rates, and other real-world factors that can affect investment outcomes.