Given the choice between money today and money tomorrow, one would prefer the money today. This is true because money today can be hoarded until tomorrow, or alternately used for other purposes before tomorrow – “money tomorrow” is just one out of many options available to someone with “money today” and hence “money today” must be of greater than or equal value to “money tomorrow”. Similarly, a greater sum of money is always preferable to a lesser sum, because I have the option of discarding a part of the greater sum to get the smaller sum. It is then plausible that there be a greater sum of money tomorrow (s2,t2) which is of equal value to a smaller sum today (s1,t1).

What is the relationship between (s2,t2) and (s1,t1)? So far we have found that (s2-s1)(t2-t1) >= 0. Only by assuming that investment opportunities are of much shorter duration and of much smaller size than the quantities under consideration, and also available equally throughout time (i.e. having multiple instantiations at (t1+dt,t2+dt) for all dt), do we get the conventional discounting rule, where (s2/s1) = r^(t2-t1) for some r > 0.

To go from the inequality to the equality requires work. This specific equality only emerges because of the additional assumptions made. Given access to a pool (defined by our assumptions of time-invariance and scale-invariance) of investments with discount rate r, we can now arbitrage any cash flow to its present value. The net present value (NPV) is this concept, and when positive it represents situations in which choosing the investment over choosing the pool results in a cash gain.

So far, I have only dealt with the opportunity cost of money, as represented by participation in the pool of investments. I will discuss the risk of the investment itself next.

Looking forward to the risk discussion!