It is possible to consider the future value of annuities. If you are investing cashflow C at the end of each period for t periods, with a per period discount rate of r%, we define the future value of this annuity to be

a) Lisa plans on investing a portion of her monthly check into a mutual fund at the end of each month. If the fund promises a yearly return of 6.5%, with monthly compounding, and Lisa has a goal of $50,000 in her mutual fund by the end of 3 years, how much must she invest each month?

b) Suppose the mutual fund also has a scheme that returns 7% a year, but with only semi-annual compounding. How much must Lisa invest a month now? Would she prefer this scheme to a)?

Hint 1: If an investment returns 7% a year, but compounding only happens halfway through the year, Lisas deposits will not grow any interest those first five months. Instead, the money will accumulate, interest (what is the interest rate?) will occur after 6 months. Continue this pattern through the rest of the year.

Hint 2: If Lisa contributes the same amount every month, she must be contributing the same amount every six months.

c) Lisa needs to spend 30% of her net income on rent, and another 20% on necessary expenses. If she earns (net) $8,500 a month, will she have enough left over to reach her goal of $50,000? Assume she is investing in the scheme found in part a).

d) Now, lets say Lisa gets a bit more ambitious with her investment goals. She wants to grow her monthly principal investment amount by .4% every month. Her first months investment will be equal to the value found in part a), and assuming the compounding scheme in part a), Lisa would like to know how much her mutual fund will have after 2 years.

Thankfully she knows she there is a growing, annuity future value formula, where g% is the per period growth rate of future cash flow,

What is the amount Lisa will find to be in her mutual fund after 2 years? Is this sufficient time to meet her investment goal of $50,000?

e) Suppose Lisa decides to go ahead with the growing investment scheme detailed in part d). In what year will her growing monthly investment principal be more than her disposable income after rent and necessary expenses, found in part c)? Round to the nearest whole year.