Annuity is defined as: the same cash flows happen at the same interval for n periods. A few examples are shown below. Case #1 c c c c c [19 equal cash flows] |----|----|----|----|... |----| 0 1 2 3 4 19 20 Case #2: A general one c c c c |----|----|----|... |----| ... |----| 0 1 2 3 k k+1 k+n-2 k+n-1

Case #3, the 1st cash flow happens at the end of the 1st period. c c c c c |-------|---------|---------| .........|---------| 1 2 3 (n-1) n

For case #3, we have the following two equivalent equations. fv_annuity= c(1+R)^(n-1) +c(1+R)^(n-2) + .... + c(1+R)^1+c(1+R)^0 (1) c = --- *[ (1+R)^n -1 ] (2) R What is the future value, if a person saves $500 at the end of each year for 40 years? Assume that the annual discount rate of 5% and the first cash flow happens at the end of the first year. 1) Write an R function for Equation (2). 2) Use at least two types of loops based on Equation (1). 3) What is the future value if the first cash flow happens at the end of year 5?