# Discount rate

Also a discount factor. This accounts for the time value of money and arises naturally in financial models, such as a portfolio selection problem. A discount rate of 7% means $1 earned a year from now has a present value of approximately$93.46 (1/1.07). If $1 is earned n years from now, and the discount rate is $LaTeX: r$, the present value is$$LaTeX: 1/(1+r)^n$. In continuous-time models, there are variations, such as defining the present value of $LaTeX: K$ dollars at time $LaTeX: t$ to be $LaTeX: K(1-e^{-rt})$. In infinite horizon dynamic programming, the discount factor serves to make value iteration a contraction map. In that case, the fixed point of the stationary equation,
$LaTeX: F(s) = \mbox{opt} \{r(x, s) + a F(T(s, x)): x \in X(s)\}$,
is obtained by successive approximation - i.e., value iteration for an infinite horizon DP. This converges to a unique fixed point, $LaTeX: F$, if $LaTeX: 0 < a < 1$. Here, the DP notation is used, where $LaTeX: a$ is the discount factor.