 So let's consider, suppose you will have amount c at time t, then the present value will be c times the reciprocal of the accumulation function. Remember that if the accumulation function is a of t, the present value of a dollar at time t is this reciprocal. So the value at time tau will be this present value times the accumulation function at tau. And we can simplify this expression. So, for example, suppose you inherit a fund that will pay out $500 in 8 years based on simple, not compound, annual interest of 4%, but you want to sell it. What's a fair price for the investment? For simple interest of 4% annually, the accumulation function is, so the present value of $500 will be, so a fair price to sell your inheritance today would be $378.79. Now here's a very important variation on this problem, same situation, but this time you plan to sell it in 3 years. What would be a fair price for the investment in 3 years? Well, we might reason as follows. The fund pays out $500 in 8 years, so 3 years from now will be 5 years from a payout of $500, so the value of the inheritance then will be the present value of $500 paid out in 5 years. And we could do this, but we would be wrong. And it's not obvious why we're wrong, so let's think about that. Remember we calculated the present value of the fund was $378.79. So suppose you deposited that $378.79 into an account earning simple interest of 4% annually. After 3 years, the account would have a balance of... So if someone offered 4.1667 for the account, you wouldn't want to sell it for that amount. Instead, the correct computation of the value in 3 years will be the present value of the account, that's 500 times the reciprocal of the accumulation function, times the accumulation function evaluated after 3 years. So a fair price in 3 years would be $424.24. Now, we could have any function at all for our accumulation function, so let's consider another example. Suppose we've received 507 years from now from an account with our accumulation function, which is a product of simple and compound interest functions. How much should you sell the account for in 2 years? And to really get at the difference, let's calculate the incorrect answer and the correct answer, and then explain why the correct answer is the correct one. So first, we'll do the problems the wrong way. In 2 years, the payoff of $500 will be 5 years away, and so the present value of $500 2 years from now will be... Again, this is the incorrect answer, $411.70 is not correct. So let's compute the actual value. So remember the value at time tau of an investment worth C at time t is C times this ratio at tau at, where at is the accumulation function. And so we compute the actual value for $1314. Now, if you're cynical, you'll say that the larger value must be correct because that's the way that banks can get more money from people. But in this particular case, there's a reason why this is also mathematically correct. So to explain why this is correct, let's take that expression apart. And this first factor, C times the reciprocal of A of 7, represents the present value of our payoff C. And again, if we deposit that amount into an account with the accumulation function A of t, then in 2 years the value will be... which is the expression we used. Of course, that's just a bunch of mathematical symbols thrown onto a page. And again, if you're cynical, you might suspect that there's some sort of subterfuge going on. So remember, never hesitate to use a concrete example. So the present value of the $500 to be received in 7 years is... and if you deposit this amount, or the rounded amount 38182, into an account with the same accumulation function, then in 2 years that amount will be... and again, if it's worth this much, you wouldn't want to sell it for a lesser price.