Question

You will invest $7,500 at the end of each of the next 6 years, and $15,000 in years 7 though 12 in a project. You will invest in it $25,000 today. If other investments of equal risk earn 9% annually, what is the future value?

Answer 1

The question involves calculating the future value of multiple investments using the future value of annuity formula for annual investments and the compound interest formula for a lump sum, growing at 9% annually.

Step-by-step explanation:

The student is asking about calculating the future value of a series of investments given a certain annual interest rate. The investments are made at different times ($7,500 annually for 6 years, $15,000 annually for the next 6 years, and a $25,000 lump sum investment today) and grow at an annual rate of 9%. To calculate the future value, we use the future value formula of annuities for the periodic investments and compound interest formula for the lump sum investment.

For the $7,500 annual investment for 6 years and the $15,000 annual investment for the following 6 years, the future value annuity formula FV = Pmt * (((1 + r)^n - 1) / r) where Pmt is the annuity payment, r is the interest rate per period, and n is the number of periods, is used to determine their future values separately and then summed.

The $25,000 lump sum investment is grown using the compound interest formula FV = PV * (1 + r)^n where PV is the present value, r is the interest rate, and n is the number of periods. The future value of this lump sum investment is then added to the future values of the annuities to obtain the total future value of all investments.

Answer 2

The student's question involves calculating the future value of a series of investments using the future value formula for annuities and compound interest. The investments include annual payments of $7,500 for the first 6 years, $15,000 for the next 6 years, and an initial investment of $25,000, all growing at an annual interest rate of 9%.

Step-by-step explanation:

The student is asking about the future value of a series of investments made annually at different amounts, with an initial lump sum investment, and needing to calculate the total after a certain number of years at a given interest rate. To find the future value of the investments, we have to use the formula for the future value of an ordinary annuity for the $7,500 payments for the first six years, another annuity calculation for the $15,000 payments for the years seven through twelve, and compound interest for the $25,000 invested today.

The future value FV of an ordinary annuity is calculated using the formula FV = Pmt * ((1 + r)n - 1) / r, where Pmt is the annuity payment, r is the interest rate per period, and n is the number of periods. Additionally, we need to calculate the compound interest for the initial $25,000 using the formula FV = PV * (1 + r)n, where PV is the present value or initial investment amount.

First, for the $7,500 annuity, we substitute the respective values into the annuity formula for the first six years, with r equal to 9% (or 0.09) and n equal to 6. Then, for the $15,000 annuity, we apply the annuity formula with the same rate but n equal to 12, subtracting the first six years since those payments began later. The compound interest for the $25,000 is calculated separately. Lastly, to get the total future value, we sum all the calculated future values.