 Suppose an amount is deposited for time t in an account with accumulation function a of t. If the amount at the end of the period is c, then the future amount has present value c divided by a of t. Alternatively, since the discount function is the reciprocal of the accumulation function, the present value of amount c received at time t will be c v of t. We'll use the following sign convention. Money received by the person of interest has a positive sign, while money paid out by the person of interest has a negative sign. Since there are generally two parties in any transaction, it is very important to identify the person of interest. And remember, if it's not written down, it didn't happen. However, since the signs are opposite for the opposite parties, it doesn't really matter whether we use the positive or negative as long as you're careful to interpret your final answers correctly. Because remember, no computer ever went to prison for misleading a client. So, for example, suppose we borrow $1,000 from a bank to be paid back in six months, determine the sign about, from your point of view, and from the bank's point of view. So at a 7% discount, the $1,000 would be given to you as $930 now. And since this is money to you, this would be recorded as $930 with an optional plus sign indicating positive. Meanwhile, in six months, you pay $1,000, and since this is money you're paying out, this would be recorded as a negative number minus $1,000. Now, from the bank's point of view, all of these transactions are opposite since the bank is giving out $930 and receiving $1,000. The signs from the bank's point of view would be opposite, negative $930 and positive $1,000. Now, suppose you receive payments over an extended period of time. So, we might receive amount r0 at time 0, r1 at time t1, r2 at time t2, and so on. The net present value, NPV, is the sum of all the present values. So, let's compute that. The amount that we received at time 0, r0, the present value of r0 is r0 v0. That's the amount we received times the discount function. The present value of r1 to be received at time t1 is r1 v of t1, the value times the discount function. And the present value of r2 to be received at time t2 is r2 vt2, and so on. And the net present value is the sum of all these terms, which we can write this way. Or, suppose you deposit $1,000 now and $1,000 in a year and receive $1,200 in five years and another $1,200 in the six years. Let's find the NPV assuming a compound interest rate of 5% per year and interpret our results. Note that the transactions occur at t equals 0, now, t equals 1, a year from now, t equals 5, and t equals 6. At t equals 0 and 1, you pay $1,000, so these values are negative. At t equals 5 and 6, you receive $1,200, so these values are positive. And since the interest rate is 5% per year, then our accumulation function is 1.05 to the t, and our discount function is 1 divided by 1.05 to the t. Filling in our values and computing gives us a negative NPV. Now, remember, we're using the convention that money you pay out gets a negative sign, so since a negative amount corresponds to something you pay out, this investment is a net loss for you. And again, one way we might interpret this is we might say that we simply made the deposits into an account earning 5%, we'd be able to get more than $1,200 in the fifth and sixth years. So if this raises the question of how much we should get out. So suppose we deposit $1,000 now and in a year into an account earning compound interest at a rate of 3% per year. Let's find a payout amount after two years that make the NPV positive. Since there are transactions made at 0, 1, and 2, we have. Now we pay out $1,000 now, so r0 is negative $1,000 and v0 is equal to 1. We also pay out $1,000 in year 1, so r1 is negative $1,000. And since the account is earning 3%, our discount function is 1 divided by 1.03 to the t, which gives the second term of the NPV equation. We don't know the payoff at t equals 2, so r2 is x, the unknown value. And at time 2, v of 2 is 1 divided by 1.03 squared, which gives us the last term of our NPV equation. Now we want our net present value to be positive, to be greater than 0, but it's hard to work with inequalities. So a useful strategy, ignore inequalities, but remember they exist. So we'll start with NPV equal to 0 and solve. And to maintain as much accuracy as possible, we'll solve without computing any of the intermediate values and write our answer as a single expression. And we note that this gives us final answer rounded 2,090.90. So note that x equals 2,090.90 makes the NPV exactly equal to 0. And since this is the final payout amount, any greater amount will make the NPV greater. And so any payout greater than 2,090.90 will result in a positive NPV.