# Future value of a "future" investment

Consider the following well-known simple exercise :

You have a loan outstanding. It requires making three annual payments at the end of the next three years of \$1,000 each. Your bank has offered to allow you to skip making the next two payments in lieu of making one large payment at the end of the loan’s term in three years. If the interest rate on the loan is 5%, what final payment will the bank require you to make so that it is indifferent between the two forms of payment?

We just need to compute the future value of the three payments that is : $$FV=1000\sum_{i=0}^2 (1.05)^i=\\\3,152.21.$$

What if the proposition was a bit further in time ? For example suppose we had to make 6 payments of $$\\\$$1,000 annually over the next 6 years and suppose that at the end of year three (after the 3rd payment), you can renegotiate the loan and make the large payment at the end. Should we still compute the future value starting year 3, meaning that the answer remains the same or should we compute it regarding today's date , i.e $$FV=1000\sum_{i=4}^6 (1.05)^i=\\\3,831.88.$$ ?

Also, you might find formula for annuity: $$FV= \frac{C}{r}\left(1-\frac{1}{(1+r)^t}\right)$$ more convenient than laboriously evaluating those sums.