 If we consider level annuity is certain, then there are three variables. The regular payment amounts, the interest rate, the present value, and the number of payments. Careful attention to details and some algebra will allow us to solve for any one of these given the others. So, for example, suppose you're to receive 500 a month for the next 10 years as part of a settlement against for-profit universities starting next month, which will take to be t equal to 1. You're offered $30,000 instead. Determine whether you should take the one-time payment. So the value of annuity paying 500 a month for 10 years at monthly interest rate i is 500 a angle 120 i. If the one-time payment is equal to the present value, then 30,000 is 500 a angle 120 at interest rate i. So, what's the interest rate? Well, let's find out. We have removing a factor of v and applying a geometric series summation formula gives us, and we can solve this equation. Again, as a general rule, equations of this type cannot be solved algebraically, and we'll need to resort to some sort of computing device to solve them. Now, we do have two solutions. Since v is the reciprocal of 1 plus i, and in this case, we're assuming that i is greater than 0, only the positive solution is meaningful. And so the interest rate i satisfies. So i works out to be about 1.32%. At this point, it's important to remember no computer was ever fired for confusing a client. While the interest rate is 1.32%, remember we're measuring time in months. So this gives us an interest rate of 1.32% per month. We're probably more interested in the effective annual rate, so we'll use our more accurate unrounded value of i, and find that in one year, a dollar will grow too. So this $30,000 offer at the start of the annuity is equivalent to earning an annual rate of 17.07%. It's worth noting that the rule of thumb for stock market returns is about 10% per year, and bank accounts usually pay far less. So earning 17.07% interest, this would be a very good deal for you, and a very bad deal for whoever offers it. And remember, no computer was ever fired for losing money. Let's consider a different case. Suppose you charge $2000 on a credit card that charges 12% annual interest compounded monthly, and want to pay it off over a year. What will your monthly payments be? So note that the $2000, since you've received it now, is the present value. Since the payments are monthly, we'll use months as our time units, and our payments will be at t equals 1, 2, 3, and so on up to 12. Meanwhile, our interest rate will be 1212% or 1%. So the present value of 2000 should equal the present value of 12 monthly payments of Q, which gives us the equation 2000 equals Q, A, Angle 12, at 1%, which we can then solve for Q. That is, once we find A Angle 12 at 1%. So A Angle 12 at 1% is the sum of V, V squared, and so on up to V to the 12th. Removing a factor of V, we can now use our geometric series summation formula. V is our discount rate, which is the reciprocal of 1.01, and so we find. So we find that Q is approximately 175.94, so you'd need to pay back 175.94 each month for 12 months. Or let's consider student loans. Suppose a student loan of 10,000 accrues 8% annual interest convertible monthly. If you can make payments of $70 per month, how long will it take to pay off the loan? Since our time unit is months, we use an interest rate of 812% per month. So we have the present value of 10,000, and that should equal the present value of 70 payments at A Angle N at interest rate 812%. So that's our sum V plus V squared and so on up to V to the N where we don't know what N is, but we can still apply the geometric series summation formula. We'll factor out V, then apply our formula. So with a present value of 10,000 and payments of $70 per month at an interest rate of 812% per month, then N satisfies the equation. Remember V is the reciprocal of the accumulation function, so it's a known value, and we're trying to solve for N, so we can rearrange. We'll hit both sides with a log and simplify, and letting V equal the reciprocal of our accumulation function, we find that we require about 458.2 payments, are a little over 38 years. Now reducing the interest rate means the bank would make less, and this would cause the world's economy to collapse in an unprecedented disaster. What if we did? So let's drop that interest rate to 7.5% and run those same computations. So with the present value of 10,000 and payments of $70 per month at an interest rate of 7.512% per month, our present value 10,000 is 70, a angle N at 7.512%. And if we solve this, we find that N is approximately 358.5 or about 8 years less. Reducing that interest rate by half a percent cut out a significant number of payments. Unfortunately, this means the banker won't make as much money, and somebody has to look out for the interest of these poor Wall Street billionaires.