 The important question about saving for retirement... How much to save now to receive payments later? Let's set this up. Suppose you deposit Q for N time period starting at T equals 0 and ending at T equals N minus 1. The balance at T equals N will be Q s double dot angle N. We can view this as the present value of an annuity that will pay out R for M time periods. So for convenience, we'll reset the clock so this last has present value R a double dot angle M. And we want Q s double dot angle N to be R a double dot angle M. Essentially, we're using the accumulated holdings to buy the annuity. If we assume compound interest, then Q s double dot angle N at interest rate I should equal R a double dot angle M at interest rate J, and we have six variables. The contributions Q, the number of contributions N, the interest rate I during the accumulation period, the disbursements R, the number of disbursements M, and the interest rate J during the disbursement period. And so given any five, we should be able to find the sixth. So for example, Professor Jeff deposits 2000 each year into an account earning 6% annual interest starting at age 30 and ending at age 64. He then withdraws W each year until age 80. How much can he withdraw each year assuming the balance is exhausted by then? If we let age 30 correspond to T equals 0, then 35 annual contributions of 2000 are made at T equals 0, 1, 2, and so on. So the value of the account at T equals 35 will be 2000 s double dot angle 35 at 6%. Now we'll reset the clock and let T equals 0 correspond to age 65. Then Professor Jeff is hoping for payments of W at T equals 0, 1, 2, and so on up to 15. So at age 65 the account should have present value W a double dot angle 16 at 6%. And we want the value of the retirement account to be the present value of the annuity. At 6%, if we let A equal 1 plus 0.06, then our accumulation function A of T is A to power T. And so 2000 s double dot angle 35 at 6% will be... And there's a number of ways we can find this sum. Let V equal 1 over A and removing a factor of A to power 35 gives us... And a geometric series summation formula can be used and we can substitute in the values of A and V to find... Similarly, letting V equal the reciprocal of 1.06 gives A double dot angle 16 at 6% equal to... Which will be... Which we can use to find the value of W. So Professor Jeff can withdraw 22,053 and 42 cents every year until age 80 at which point? Let's consider a different scenario. Suppose Professor Jeff wants to retire in 10 years with an annuity paying 10,000 every year for the next 10. How much will he need to deposit each year into an account earning 4% annual interest? If he deposits Q annually at T equals 0, 1, 2, and so on, the value at T equal 10 will be Q s double dot angle 10 at 4%. This should equally present, that's actually the 10 years from now value of an annuity paying 10,000 each year for the next 10 years. That's 10,000 A double dot angle 10 at 4%. We can solve this for Q. We'll evaluate all these terms at 4% interest. We have A equal 1 plus 0.04, so A of T is A to power T, and V of T is the reciprocal of A of T. We find... So Professor Jeff needs to deposit 67,55,65 each year where we round up to ensure that we have enough in the account to pay out 10,000 each year. Another scenario is how long will the money last? So suppose Professor Jeff deposits 5,000 each year for the next 10 into an account earning 3% interest. He wants to be able to withdraw 10,000 each year. For how many years can this be done? The deposits are made at T equals 0, 1, 2, and so on, so the total value at T equals 10 will be 5,000 s double dot angle 10 at 3%. This should be the present value of an annuity paying 10,000 at T equals 0, 1, 2, up to n, where we don't know what n is, but we can at least write the notation 10,000 A double dot angle n at 3%. And we want these two values to be equal. Now the left-hand side has nothing we don't know, so we can express it exactly. The right-hand side is a geometric series where we don't actually know what that last term is, but we can still apply the geometric series summation formula. Now at an interest rate of 3%, we know that A is 1.03 and V is the reciprocal, so we can solve for n. So let's isolate that n term and we can calculate this to be about 6.18.