Simultaneity of Value and Non-Cash Expenses in Small Business Valuation
指出在估值中，当折旧摊销依赖于价值时，现金流折现法存在价值与费用相互依赖的问题，需用联立方程求解最优价值。
SIMULTANEITY OF VALUE AND NON-CASH EXPENSES IN SMALL BUSINESS VALUATION There are many approaches to valuing a small business, ranging from simple rules of thumb, such as gross revenue multipliers, to more sophisticated techniques such as discounted cash flow (DCF) analysis (Pratt 1981). Between these extremes lie a host of other techniques such as the use of comparables, price/earnings approaches, and replacement value methods. In the valuation of businesses with high growth potential, though, DCF analysis is particularly encouraged due to its emphasis on future or anticipated benefits (cash flow) as opposed to historical performance. However, in situations where depreciation and/or amortization (D/A) occurs on more than just the value of the physical assets but is in fact dependent upon the value (price), a problem arises with the DCF approach not heretofore recognized: that value cannot be determined without estimating D/A, and D/A cannot be determined without value. Therefore, a simultaneous-equation method must be used in order to solve for the optimal value, or at least the maximum value for the business, granted all other assumptions and inputs. There is no mention of this relationship in the literature, with the exception of an oblique reference to the problem in a capital budgeting application (Mayer 1978). The purpose of this discussion, therefore, is to point out the problem and to show how it can be addressed. The importance of this problem is twofold: first, ignoring the interdependency undercuts the DCF method and essentially leads to guesstimates of business value; secondly, there is no financial way to incorporate the interdependency into other value approaches such as those mentioned above. The next section develops a general model based on the simultaneity of value and tax shields. An application of the model to small business valuation is then provided, followed by a summary. THE MODEL The general DCF valuation model assumes that the value of an asset is the value today of its expected cash flows (Carland and White 1980). All assets (e.g., bonds, stocks, capital projects, small businesses) are essentially valued in the same way: (1) the cash flows are estimated over the life of the asset; (2) the required rate of return is estimated for the cash flows; and (3) each cash flow is discounted at the required rate of return and summed to find the value. Equation 1 formalizes this process: V = [CF.sub.1]/[(1+k).sup.1] + [CF.sub.2]/(1 + [k).sup.2] +... + [CF.sub.n]/(1 + [k).sup.n], where: V = the present value of the asset, [CF.sub.t] = the expected cash flow in period t, k = the required rate of return, n = the life of the asset. In the case of small business valuation, [CF.sub.t] includes all sources of cash flow, including but not limited to sales, cash expenses, tax shields of non-cash expenses, changes in long-term debt, gross capital expenditures, salvage values, and the change in net working capital (Lloyd and Hand 1982). To focus on the simultaneity problem, however, let [CF*.sub.t] represent the net cash flow in period t from all sources except tax shields of non-cash expenses. That particular source of cash flow and value can then be isolated by rewriting the general model as follows: [Mathematical Expression Omitted] where [NCE.sub.t] = total non-cash expenses providing tax shields in period t, T is the marginal tax rate, and all else is as previously defined. Let [NCE.sub.t] = [d.sub.t]V, where [d.sub.t] is the D/A rate in period t and V is the value or price as previously defined. Substituting into equation (2) above then gives: [Mathematical Expression Omitted] In this form, the simultaneity between value and cash flow is apparent. When the V terms are isolated on the left hand side and the tax rate T is held constant, the model reduces to: [Mathematical Expression Omitted] which further simplifies to: [Mathematical Expression Omitted] The value in the numerator is simply the present value of the net cash flow (exclusive of tax shields) for each period summed over the life of the asset. …