Stochastic Firm Valuation with Dependent Cash Flows: Analytical Results for Random Walk and ARMA Models

Introduction:
Discounted cash flow (DCF) valuation is a standard method for estimating firm value, yet many applications implicitly assume independent cash flows. In practice, however, financial cash flows frequently exhibit temporal dependence, which may substantially affect valuation risk.

Methods:
This study derives analytical expressions for the expectation and variance of discounted cash flows under three stochastic processes: white noise, ARMA(1,1), and random walk (ARIMA(0,1,0)). The formulas are obtained for finite and infinite planning horizons and also include limiting cases of discounting structures.

Results:
The results show that autocorrelation increases valuation uncertainty while leaving expected firm value unchanged. For ARMA(1,1) processes, valuation risk depends on both innovation variance and the induced covariance structure. In the random walk case, shocks are fully persistent, leading to a strong accumulation of uncertainty over time and a more than proportional growth of valuation variance.

Conclusions:
The analysis demonstrates that ignoring temporal dependence can lead to a substantial underestimation of valuation risk in DCF models. The derived closed-form expressions provide a tractable framework for incorporating dependence structures into firm valuation. Future research may extend the framework to long-memory ARFIMA processes, which allow for gradually decaying dependence between the short-memory ARMA structure and the fully persistent random walk case, offering a more flexible description of persistence in cash-flow dynamics.