Discount rate

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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.

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