 There is no mathematical difference between an annuity where you are paid by a financial institution and a loan where you make payments to a financial institution. The important thing to remember, the present value is the amount you receive now. For example, suppose you purchase a house for $150,000. You put 10% down, then finance the rest with a 30-year loan at 3.5% annual interest convertible monthly. What are your monthly payments? Since you put 10% down, then you are able to pay 10% of $150,000, that's $15,000. So that means we need to have $150,000 minus $15,000, $135,000 now, so this is the present value of the loan. There are 12 times 3360 payments of P at 3.512% interest per month, so we want that present value, 135,000, to be the present value of the annuity, P times A angle 360 at 3.512%, or we can solve for P. We'll compute the value of A angle 360 at 3.512% interest, which will be, and so we find our payment to be approximately 606.2103 and so on, so we'll round in our payments will be $606.21, or will they? Generally speaking, our solutions to amounts will be rounded to the nearest hundredth of a dollar a cent, however the actual numbers may involve more decimal places, and there are two ways that this can be handled. You can round payments down, but the last payment will be larger than the others, this is called a balloon payment, or you can round payments up and make the last payment smaller than the others, a drop payment. So let's say what happens, so that 135,000 at 3.5% annual interest convertible monthly, suppose we round as we should to 606.21 for 360 months, let's find the present value of those payments. So the payments will have a present value of 606.21 times A angle 360 at 3.512%. Note that this amount, the A angle 360 at 3.512%, we've already computed that value. We can then multiply by our payment, and since this is our final answer, we can round this to $134,999.93, now since you've actually received $135,000, this means you've stolen about 7 cents from the bank. This will cause the collapse of western civilization, plagues of locusts, the sun going dark, and a rain of fire and brimstone. Well we can't have the banks losing money, so let's round up, so suppose we make our monthly payment $606.22, so this time we compute, again we're still using A angle 360 at 3.512%, so we find, and since you actually received $135,000, this means that you've paid the bank about $2.15 more than you received. The banker would say that this is just a round off error, and we can ignore such a tiny discrepancy. I mean come on, it's $135,000, you're going to quibble over $2.15? Well, yeah, the thing to remember here is that the only way to resolve differences fairly is through mathematics. We can use the final payment to resolve any differences between the actual present value and the amount received. Suppose we make n payments all together. If the first n minus 1 payments are q and the last payment is r, then the present value of the n minus 1 payments at q will be qA angle n minus 1. The last payment at r has present value r, v of n, where v of n is our discount function, and so the present value of this payment structure will be, and we can use this to find the final payment. So, let's go back to our scenario where this time we rounded down to 606.21, so we have the loan amount $135,000 equals the present value of the 359 payments of 606.21 at 3.512% interest. So that's 606.21, A angle 360 at 3.512% plus our last payment r, which since it will be paid in 360 months will be discounted by v of 360. Since we're assuming compound interest, our discount function v of t is v to the t, where v is the reciprocal of the accumulation function, solving our equation for r, so our final payment 606.42. Notice that with equal payments of 606.21 the present value would have been 7 cents short, but the final payment had to be increased by 20 cents. That reflects the fact that the present value of 20 cents is the 7 cents. What if we rounded up? So in that case our payments would have been 606.22, and again notice that our annuity value would have been the same, so we'll keep that computation, and this time when we solve, we find that our final payment is, this time notice that with equal payments of 606.22 the present value would have been $2.15 over, but in fact our final payment was reduced by about $6.15. Now this doesn't really answer the question of whether we should round up or round down, but generally speaking balloon payments are objectionable. Thus the rule is always round up. Well, except when you round down. Actually, it's kind of complicated. We'll see examples where the decision to round up or down has nothing to do with the mathematical process of rounding, but rather with the financial aspect of the problem we're trying to solve.