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The same global methodology can serve for modeling the standalone credit risk of a single firm. Consider a firm whose global value is the present value of its free cash flow at its cost of capital under the framework of valuation defined by Modigliani and Miller [53]. There is no need here to adopt a more general framework, because the actual credit risk models will be expanded in detail in the sections dedicated to standalone credit risk and to credit risk for portfolios (Sections 11 and 12).

The Firm Value

The value of the firm is random because its future cash flows are random. In their simplest setup, Modigliani and Miller assume a static no-growth and no-decay world without tax. Relaxing such assumptions has been done and is found in textbooks of corporate finance. The free cash flow of the firm is the cash flow available to shareholders after other claims from debt holders have been taken care of. Under this simple world, the free cash flow is constant. The firm is all equity funded or has a fraction of debt. This is not important at this stage, because it is shown that the weighted cost of capital (Wacc or w) of the firm depends only on operating risk and is constant whatever the leverage of the firm. If there is no debt, the firm cannot default and the cost of capital is the required return on equity, which depends only on the business risk of the firm. If there is debt, the firm defaults if the asset value falls below the contractual debt payment due at some horizon, which we assume is a given. The value of the firm is the present value of the constant cash flow F at the cost of capital, which collapses to Flw when the cash flow is constant and perpetual.

The firm's solvency is measured by its equity value. Suppose the random cash flow is driven by a single factor, which is the state of the economy. The firm asset value loss is the adverse deviation from its expected value. The firm has an expected cash flow such that its expected return on investment is E(r).

The cumulative yield on investment, from initial date 0 and final date 1, is random and has mean E(r) and standard deviation o(r). The yield is defined as the discount rate that makes the present value of the free cash flows equal to the initial investment, which is the book value of assets AQ. The cash flow is F — rAQ, with mean E(r)A . and standard deviation c(r)A0. As long as the yield of the investment is above the cost of capital, the firm creates value. The value is derived from the random value of the cash flow, and it is random. The loss point is when the firm destroys value. As long as F > wAQ, the firm creates value. When the yield on investment becomes equal to the cost of capital, the firm's creation of value is zero because the net present value1 of the investment is zero. If the yield on investment is lower than the cost of capital, the firm destroys value.

The yield on investment r is random. The firm's value at date 1 is A — Flw — rAJw. It has mean E(r)AJw and standard deviation c(r)A0/w. The loss point for r is w. As long as F > wA , the firm creates value, otherwise it destroys value. The value of the firm is random. The distribution of value is that of the cash flow, which is driven by r. The firm's solvency is impaired when it destroys value, which occurs when the final value moves below the initial investment. The lower the firm's value below that point, the more is the firm's value impaired, as well its equity value.

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