ABSTRACT

Note that if we set the dirty surplus variable ε(t) to be identically equal to zero then we will have the clean surplus identity as developed in §5-10 of Chapter 5. Now, it will be recalled here (as in §9-2) that the recursion value of equity η(t) is the present value of the stream of future dividends the firm expects to pay computed under the assumption that the firm’s existing investment opportunity set will remain in force indefinitely, or

η(t) = Et ⎡⎣ ∞∫

e−r(s−t)D(s)ds

⎤⎦ where Et[·] is the expectation operator taken at time t and r is the cost of capital (per unit time) applicable to equity. Moreover, we can use the dirty surplus equation in conjunction with the above expression for the recursion value of equity to link the expected present value of the firm’s dividends to its bookkeeping and other information variables. To do this, we first note that the dirty surplus equation given earlier will mean that that the dividend payment at time s can be expressed as

D(s)ds = [x(s)+ ε(s)]ds− db(s) Substituting this result into the expression for the recursion value of equity η(t) given above then gives

η(t) = Et ⎡⎣ ∞∫

e−r(s−t)[x(s)+ ε(s)]ds ⎤⎦−Et

⎡⎣ ∞∫ t

e−r(s−t)db(s)

⎤⎦ Now, ifwe apply integration by parts to the final termon the right-hand side of this expression, it follows that

e−r(s−t)db(s) = [e−r(s−t)b(s)]∞ t

+ r ∞∫ t

e−r(s−t)b(s)ds

We can then ensure that the recursion value of equity will remain finite for all t by imposing the following transversality requirement (as in §9-4):

lim s→∞e

−r(s−t)Et[b(s)] = 0

This will mean that the final term in the above expression for the recursion value of equity can be evaluated as

Et

⎡⎣ ∞∫ t

e−r(s−t)db(s)

⎤⎦= −b(t)+ rEt ⎡⎣ ∞∫

e−r(s−t)b(s)ds

⎤⎦ Moreover, substitution then shows that the recursion value of the firm’s equity can be restated as

η(t) = Et ⎡⎣ ∞∫

e−r(s−t)[x(s)+ ε(s)]ds ⎤⎦+ b(t)− rEt

⎡⎣ ∞∫ t

e−r(s−t)b(s)ds

⎤⎦ or equivalently

η(t) = b(t)+Et ⎡⎣ ∞∫

e−r(s−t)a(s)ds

⎤⎦+Et ⎡⎣ ∞∫

e−r(s−t)ε(s)ds

⎤⎦ where a(t) = x(t)− rb(t) is the abnormal earnings (per unit time) attributable to the firm’s equity (as in §5-10). This shows that under dirty surplus accounting the expected present value of the future stream of dividends to be paid by the firm is equivalent to the book value of the firm’s equity plus the expected present value of the stream of abnormal earnings and the expected present value of the stream of dirty surplus adjustments, where in both cases the expectation is taken at time t. Note how this result exhibits one crucial difference when comparedwith the ‘equivalent’ formulation based on the clean surplus identity as summarized in §5-10 and §9-5. This is that under the clean surplus identity there can, by definition, be no dirty surplus adjustments and so the recursion value of equity will be based on only the first two terms on the right-hand side of the above expression for the recursion value of equity. That is, when the clean surplus identity holds, the expected present value of the stream of dirty surplus adjustments, Et

[∫∞ t e

−i(s−t)ε(s)ds ] , will be identically equal to zero. The dirty

surplus integral given here has some important implications for the valuation of a firm’s equity, and so we now develop them in further detail.