 So far, we've viewed interest as the additional amount you must pay when you return money. If you borrow $1,000 and return $1,050, then you've paid $1,050 minus $1,050 in interest. Another way to view interest is that it's the difference between what you eventually pay and what you actually receive. Since you pay $1,050 for $1,000, you paid $50 extra. And this leads to the concept of the discount rate. You can think of this as the upfront cost of borrowing money. So let's throw down some mathematics. Suppose that borrowing amount K requires first paying an amount KD, where D is the discount rate. The easiest way to pay that amount is to take it from the amount you receive. So if you borrow K at a discount rate of D, you receive K, but you immediately pay KD to borrow the money. So you actually receive K minus KD, and meanwhile, you'll pay back K. So this is a lot of words and symbols, so let's throw down an actual example. So as we borrow $500 at a discount rate of 8%, how much do you receive? And if you pay back the money after one year, what is the effective rate of interest? Since the discount rate is 8%, then the fee to borrow the money will be 8% of 500, which will be $40, so you'd actually receive 500 minus 40 or $460. Since you'd pay back the full 500 for a loan of $460, the effective rate of interest is the difference between the amount you pay back and the amount you actually got, divided by the amount you actually received, which will be, which works out to be about 8.7%. And notice that this is greater than the discount rate, and this is generally true. So let's throw some more advanced mathematics onto this, so let's talk about the discount function. So remember the accumulation function, A of t, is the amount $1, whatever, we'll grow to after time t. So if you get $1 at time t, what's the discount rate? So suppose we pay k dollars now for $1 at time t. If we assume A k t is k at, that is the amount function, is equal to the amount times the accumulation function, remember that's our standard assumption, we have A k t equals k at, since we want to get $1 at time t, then A k t is equal to 1, and so our discount rate is 1 divided by A of t. Consequently, you pay back $1 at time t, but you initially received 1 divided by A of t. And this motivates the discount function, which is the reciprocal of the accumulation function. It's helpful to remember the discount function represents how much you'd get now to return $1 at time t. So analogous to the effective interest rate, we can compute an effective discount rate. Suppose you borrow the money over some time interval t1 to t2, you receive the amount at the beginning, A of t1, but you pay back the amount at the end, A of t2. So the effective discount rate over the interval will be, and as with the effective interest rate, we can talk about the effective discount rate over the nth time period as Note that the only difference between the effective interest rate and the effective discount rate is the denominator. With the discount rate, our denominator is the amount at the end of the transaction. So suppose the accumulation function for an investment is, well, this horrible thing, where t is measured in years, let's find the effective interest rate and the effective discount rate over the fifth year. So note that the fifth year is the period between t equals 4 and t equals 5. So during the fifth year, the effective interest rate will be, now we need to compute A of 5 and A of 4, so we can compute them. And for accuracy, we should store these values instead of trying to round and copy them down, or we can use them as written. And so we find our effective interest rate will be, where we can round to this last step as it's the value we'll report. And so we might say the effective interest rate is about 3.89%. Again, the difference between the interest and the discount rate is what the denominator is. The numerator is still that difference A of 5 minus A of 4. But for the discount rate, we are dividing by the amount at the end of the transaction period. And we can compute. And again we can round to the last step as it's the value we'll report.