 If we accrue simple interest at a rate s per unit of time, the accumulation function is a of t equals 1 plus s t, where the future value of a dollar increases by a fixed amount s per unit of time. The analog to simple interest is simple discount with discount function 1 minus s of t. And like the accumulation function, this is based on one unit of currency. In this case, it corresponds to the amount you'd receive now to pay back a dollar at time t. So for example, suppose you agree to pay back $1,200 to be discounted by 10% per month. How much will you receive now if you agree to pay back $1,200 in six months? So our discount function will be, so if t equals 6, the discount will be 40%. So we find 40% of $1,200, and so you'd receive $1,200 minus $480, or 720 now, and pay back $1,200 in six months. Now simple discount does introduce some peculiarities. If we use simple discount, then if t is large enough, d of t will be 0. And what this means is you'd get 0 now, but pay $1 at time t. And if t is even larger, d of t will be negative. And we might interpret this as this means you pay now and pay an additional amount in the future. Both situations are unrealistic. So remember a of t represents the amount you'd pay back at time t if you received $1 now. So our discount function, the reciprocal, represents the amount you'd receive now to pay back $1 at time t. So given the accumulation function a of t, we can find the discount function and rate. Now it's useful to keep in mind the distinction between the two. The discount rate is the amount off. It's how much less you'd receive if you got the money now. The discount function, on the other hand, gives you the actual amount of money received. For example, suppose you agreed to pay back $1,000 in five years. How much will you receive now if the loan accrues interest at a rate of 8% per year compounded? Since the interest is 8% per year compounded, our accumulation function will be 1.08 to the t's power. So the discount function will be, so if t equals 5, via 5 will be, which is the amount you'd receive now for a dollar paid back in five years. So if you agreed to pay back $1,005 you'd receive, or about $680.58. So how do discount and interest rates relate? An important concept, percentages don't add. This is sometimes called the percentage paradox. For example, suppose the value of an investment drops by 50%, then increases by 50%. How does the new value compare to the old? And here it's useful to keep in mind it never hurts to create a concrete example. So suppose our initial value was $100. If it dropped by 50%, it would decrease by 50% of $100 or $50. So the new value would be $50. Now if we then increased by 50%, it would increase by 50% of $50. That's the current value, or $25. And so the new value would be $75. And so even though the percentage drop and percentage increase were equal, the overall change was a loss. And this generally happens. Now let's apply some mathematics. Suppose the effective interest rate over a time period t is i, while the effective discount rate is d. Since the discount function is the reciprocal of our accumulation function, then you'd pay back $1 at time t to receive v of t now. So what you'd receive now for $1 would be 1 divided by 1 plus i. Alternatively, that $1 would be discounted at a rate d to 1 minus d. So remember the discount rate is the amount off while the discount function is the actual amount received. So since we received 1 divided by 1 plus i, or we paid 1 minus d for that amount, those two are equal, and so we have a relationship between i and d. Now while this is a relation between i and d, it would be nice to have a formula for it. But remember, don't memorize formulas. Understand concepts. So given our relationship, we can solve for the discount rate d, which will be. Or we can solve for the interest rate i, which will be