Discount factors

What is a discount factor? Why does it exist?

In introductory finance courses, we learn that money today is worth more than money tomorrow due to the time value of money. The discount factor is the quantification of this concept.

Suppose we have a utility of consumption,

$$U(c_0, c_1, \dots, c_T)$$

where $c_t$ is consumption at time $t$. Given an investment opportunity that pays $x$ at time $T$ and costs $p$ at time $t\leq T$; buying $n$ units of this investment will give us a payoff of $nx$ at time $T$ and cost us $np$ at time $t$. We will buy the investment if it increases our utility, i.e. if

$$U(c_0, \dots, c_t - np, \dots, c_T + nx) > U(c_0, \dots, c_t, \dots, c_T).$$

Adding and subtracting $U(c_0, \dots, c_t, \dots, c_T)$

$$U(c_0, \dots, c_t - np, \dots, c_T + nx) - U(c_0, \dots, c_t - np, \dots, c_T) > U(c_0, \dots, c_t, \dots, c_T) - U(c_0, \dots, c_t - np, \dots, c_T).$$

if and only if

$$x\frac{U(c_0, \dots, c_t - np, \dots, c_T + nx) - U(c_0, \dots, c_t - np, \dots, c_T)}{nx} > p\frac{U(c_0, \dots, c_t, \dots, c_T) - U(c_0, \dots, c_t - np, \dots, c_T)}{np}.moremoremoremoremoremo$$

Taking the limit as $n\to 0$,

$$x\frac{\partial U(c_0, \dots, c_t - np, \dots, c_T + nx)}{\partial c_T} > p\frac{\partial U(c_0, \dots, c_t - np, \dots, c_T)}{\partial c_t}.$$

In fact, this holds with equality as well, since we can always find a small enough $n$ such that the inequality holds. Rearranging,

$$p = x\frac{\partial U(c_0, \dots, c_t - np, \dots, c_T + nx)}{\partial c_T} \bigg/ \frac{\partial U(c_0, \dots, c_t - np, \dots, c_T)}{\partial c_t}.$$