 Civilized countries have health care systems that ensure retirees aren't bankrupted by medical costs. US citizens have the freedom to choose between saving for retirement medical costs or risking bankruptcy and homelessness. Typically, workers contribute to a retirement fund during their careers and the fund disperses payments after the worker retires. The US health care system helps by ensuring retirees die before they run out of money. Suppose you make n deposit of $1 starting at t equals 0. To find the value at t equal n, we'll have to consider what happens to all of these deposits. The first dollar deposited accrues interest from t equals 0 to t equals n, or n minus 0 n times. So at t equals n, it will be worth a of n. The second dollar deposited accrues interest from t equals 1 to t equals n, or n minus 1 times, so its value will be a of n minus 1. The third dollar accrues interest n minus 2 times, so its value will be a of n minus 2, and so on. And so the value at t equals n will be where we read our notation as s double dot angle n. If we're assuming a compound interest rate of i per time unit this becomes where a is 1 plus i and a notation is read as s double dot angle n at interest rate i. For example, to save for retirement, Professor Jeff deposits $100 each month into an account earning 3% annually convertible monthly. How much will his account be worth after 20 years? Since the deposits are made monthly and the interest is convertible monthly, we'll let our interest rate be 312%, and 20 by 12, 240 deposits are made at t equals 0, 1, 2, and so on. So the value at t equals 240 will be 100 s double dot angle 240 at 312%. If we let a be 1 plus 312, then our accumulation function is a to power t, and our value is going to be 100 s double dot 240 at 312%, which will be, notice that if we remove a factor of a and reverse the order of our summation, we can apply the geometric series summation formula and find, and so the value at t equals 240 will be, or about $32,912.28 to last him the rest of his life. So for retirement planning, we'll introduce one more idea. The value of a angle n at interest rate i assumes the first payment is made at t equal to 1. In an annuity due, the first payment is made at the beginning at t equal to 0. Mathematically, there isn't that much difference. An annuity due makes payments of $1 at t equal to 0, 1, 2, 3, and so on up to n, where the discount function is v of t. Then the present value will be, as usual, the sum of the discount functions evaluated at the payment points, and our notation here is read as a double dot angle n. If our accumulation function is based on compound interest at rate i per unit, then our present value can be expressed as the corresponding powers of v, where v is 1 over 1 plus i, and our notation is read as a double dot angle n at interest rate i. So for example, suppose Professor Jeff begins receiving an annuity of $1,000 per year for the next 10 years. Find the present value of the annuity, assuming a 5% annual interest rate. So there are payments of $1,000 at t equal to 0, 1, 2, and so on up to 9, and so we want to find $1,000 a double dot angle 10 at 5%. So we'll compute these values. If the interest rate is 5%, then v is 1 divided by 1.05. So a double dot angle 10 at 5% is, and so 1,000 a double dot angle 10 at 5% is, which we can round to 2 decimal places. Remember the present value is in some sense what the financial instrument is worth right now. So imagine that Professor Jeff's retirement account has 8,187.82 earning 5% interest. He wants to be able to withdraw the same amount of money R each year for the next 20 years. How much can he withdraw? In other words, we've gone from 1,000 every year for 10 years to some other amount for 20 years. Note there will be disbursements at t equal to 0, 1, 2, and so on up to 19, so we want the amount 8,107.82 to be the present value of this new annuity. This gives us the equation 8,107.82 is R a double dot angle 20 at 5% finding our values we get, and since v is 1 divided by 1.05 we find, and we can solve for R, so R is $619.61.