 Ideally, a retiree receives an annuity for the rest of their life. A basic level perpetuity due a double-dot angle infinity is an annuity that pays $1 at t equals 0, 1, 2, and so on forever. Likewise, a basic level perpetuity immediate, a angle infinity, is an annuity that pays $1 at t equals 1, 2, and so on forever. Suppose the annuity earns interest rate i, so v is 1 over a, where a is 1 plus i. Then, a double-dot angle infinity at interest rate i. That's going to be the infinite sum of the powers of v. And here we use the infinite geometric series summation formula if the absolute value of our common ratio is less than 1, then the sum 1 plus r plus i squared and so on is equal to 1 divided by 1 minus r. The important thing to remember is that this infinite series is only equal to this nice simple expression if our absolute value of that common ratio is strictly less than 1. So let's think about this. Since v is the reciprocal of 1 plus i, and generally speaking we'll assume that i is positive, then the absolute value of v will be less than 1. And so this infinite series will be, but we can simplify further. This reciprocal 1 over 1 minus v, remember v is 1 over 1 plus i, so we can rewrite. And let's simplify by multiplying numerator and denominator by 1 plus i, and that gives us. Now by essentially the same argument, a angle infinity at interest rate i, again that's the sum of the discount functions. We'll remove the factor of v. We can use our infinite geometric series summation formula. And this time we can simplify further. v times 1 over 1 minus v will be, note that since a double dot angle infinity at interest rate i includes a payment of 1 at t equal to 0, the present value of this annuity is 1 more than the present value of a angle infinity at interest rate i. For example suppose you deposit 1,000 every year for 30 years at the end of which you have a perpetuity due. Assuming an interest rate of 5%, what is the payment of the perpetuity due? So the value of the account at t equals 30 will be 1,000 s double dot angle 30 at 5%, while the value of the perpetuity paying p forever will be p a double dot angle infinity at 5%. So we want to solve the value of the perpetuity equals the value of the account. Now p a double dot angle infinity at 5%, that's the payment amount times the infinite geometric series of the discount functions. And we could use our geometric series summation formula to find that value. Similarly 1,000 s double dot angle 30 at 5% is going to be, which gives us an equation we can solve for the payment value. Solving our equation gives, or since a is 1.05 and v is the reciprocal we find. So the annuity can pay out $3,321.94 every year indefinitely where we round down to make sure that the annuity never runs out of money. Now it's worth having some method of verifying our answer. Remember no computer was ever fired for mismanaging a retirement fund. So we might check on this as follows. The accounts value at t equal to 30 is 1,000 s double dot angle 30 at 5% and we can find that to be. Now at that point let's suppose it pays out $3,321.94. Over the next year it will accrue interest equal to 5% of $6,977.79 minus the $3,321.94 payment and that amount will be. And so the payments equal the interest earned leaving the principal untouched. The previous problem leads to the following generalization. Suppose we fund a perpetuity due by making n payments of q into an account bearing interest at rate i. Then the perpetuity pays out p at interest rate i where p a double dot angle infinity at interest rate i is equal to q s double dot angle n at interest rate i. Now p a double dot angle infinity at interest rate i is equal to also q s double dot angle n at interest rate i is equal to. And because our interest rates are the same we can remove this common factor of 1 minus v. And we can go one step further since v is the reciprocal of a then a to the n times 1 minus v to the n is. And this gives us a nice relationship between the payment of a perpetuity and the contributions made over a time period. Now remember don't memorize and I'll forget it. So suppose the investment earns 8% interest compounded annually. How much would you need to contribute for 15 years to obtain a perpetuity due paying $10,000 each year? If you make n payments of q then a perpetuity due pays p where p is q times a to the n minus 1 where a is 1 plus i the interest rate. So we have a equals 1.08, p equals 10,000 and so we find. Now we want to make sure we've deposited enough so we'll round up to an annual contribution of 460370. Again we can verify this by noting that at t equal to 15 the account balance will be. At that point it pays out 10,000 and over the next year will generate a little bit over 10,000 in interest which will be available for the next payment.