Question

Consider the case of the following annuities, and the need to compute either their expected rate of return or duration Ethan inherited an annuity worth $4,391.74 from his uncle. The annuity will pay him six equal payments of $950 at the end of each year. The annuity fund is offering a return of Ethan's friend, Zach, wants to go to business school. While his father will share some of the expenses, Zach still needs to put in the rest on his own. But Zach has no money saved for it yet. According to his calculations, it will cost him $26,506 to complete the business program, including tuition, cost of living, and other expenses. He has decided to deposit $3,800 at the end of every year in a mutual fund, from which he expects to earn a fixed 6% rate of return It will take approximately years for Zach to save enough money to go to business school.

Answer 1

To calculate the duration for Zach to save for business school, the future value of an annuity formula is used, considering a fixed 6% rate of return on annual deposits of $3,800. The importance and impact of compound interest on long-term savings are illustrated by examples where small changes in rates or fees significantly affect the final amount.

Step-by-step explanation:

The scenario given involves calculating the duration it will take for Zach to save enough money to attend business school. Given that Zach plans to deposit $3,800 each year in a mutual fund with an expected fixed 6% rate of return, we can determine how many years it will take for Zach to save the $26,506 needed. We use the future value of an annuity formula FV = P * [((1 + r)^n - 1) / r], where FV is the future value of the annuity, P is the annual deposit, r is the rate of return per period, and n is the number of periods. To solve for n, we need to re-arrange the formula. In this case, we must use a method such as trial and error combined with a financial calculator or spreadsheet, as there is no direct algebraic way to solve for n when it is an exponent in this equation.

If we look at the provided examples, we can learn about the power of compound interest and its significant impact over long periods. For instance, saving $3,000 at a 7% real annual rate of return would grow to $44,923 in 40 years. Similarly, if Yelberton saves $100,000 at a 9% rate of return, it would grow to $1,327,000 in 30 years. In the case of Alexx and Spenser, administrative fees make a difference to the final amount accumulated. Alexx's direct investment at 5% annual return results in more money after 30 years than Spenser's investment at 4.75% due to the 0.25% fee. This example highlights the long-term effects of even small differences in return rates or fees.