A stochastic perpetuity takes the form D∞=∑
n=0
∞ exp(Y
1+⋯+Y
n
)B
n
, where Y
n
:n≥0) and (B
n
:n≥0) are two independent sequences of independent and identically distributed random variables (RVs). This is an expression for the stationary distribution of the Markov chain defined recursively by D
n+1=A
n
D
n
+B
n
, n≥0, where A
n
=e
Y
n
; D
∞ then satisfies the stochastic fixed-point equation D
∞D̳AD
∞+B, where A and B are independent copies of the A
n
and B
n
(and independent of D
∞ on the right-hand side). In our framework, the quantity B
n
, which represents a random reward at time n, is assumed to be positive, unbounded with EB
n
p
<∞ for some p>0, and have a suitably regular continuous positive density. The quantity Y
n
is assumed to be light tailed and represents a discount rate from time n to n-1. The RV D
∞ then represents the net present value, in a stochastic economic environment, of an infinite stream of stochastic rewards. We provide an exact simulation algorithm for generating samples of D
∞. Our method is a variation of dominated coupling from the past and it involves constructing a sequence of dominating processes.