 Sometimes, you have the inconvenience of receiving an annuity that doesn't begin paying out for some time. This is a deferred annuity. While it's terribly inconvenient to have a wealthy benefactor, the mathematics is the same. For example, on your 11th birthday, you inherit an annuity paying 1,000 on each of your 18th through 26th birthdays. Assuming the annuity earns 7% annual interest, find the value on your 11th birthday. So one approach is to find the present value of the payment starting at t equals 7. So our present value will be, and we can compute, if we remove a factor of v to the 7th we get, and we can apply our geometric series summation formula, and at 7% annual interest, v is the reciprocal 1 over 1.07, and so the present value will be, which we round down. As an alternative, imagine the annuity being paid starting on your 12th birthday, and continuing until your 26th birthday, but... Someone steals the payments made from your 12th birthday, t equals 1, until your 17th, t equals 6. So again, using the same value for v, our present value will be the total value of all 15 payments, minus the payments that were stolen. We can compute these values separately and find our total present value. Or perhaps you want to set up an annuity, say, for your favorite online math teacher, hint, hint. The annuity should pay, oh, 10,000 a year for 20 years, beginning 10 years from now. Assuming an interest rate of 5%, how much will it cost to set up the annuity? So if we use the stolen payments concept, it doesn't matter if we begin at t equals 0 or t equals 1, but the formulas are a little easier if we start at t equals 0. So if we imagine the annuity being paid at t equals 0, 1, 2, and so on, up to 29, but the first payments from t equals 0 to t equals 9 are stolen, then the present value of the annuity, using our discount function, will be 10,000 a double dot angle 30 at 5%, that's all the payments, minus 10,000 a double dot angle 10 at 5%, those are the stolen payments. And we can compute these values separately. To ensure there will be enough money, we should round this up to the nearest half million, to the nearest penny.