Journal .f Business Finance & Accounfing, 17(5), Winter 1990 0306-686X $2.50

DISCOUNTING OF DEFERRED TAX LIABILITIES

BRETT COLLINS, JOHN RICKARD AND MICHAEL SELBY *

The issue of discounting deferred tax liabilities has generated a series of notes in this journal, see, for example, Brown and Lippitt (1987), Findlay and Williams (1981 and 1985), Williams and Findlay (1975), Wolk and Tearney (1980) and Wolk, Saubert and Tiernan (1984). A paper by Nurnberg (1972) in the Accounting Review used an intuitive exploration of the opportunity cost concept to incorrectly argue that these liabilities should be discounted at the pre-tax rate. Wolk et al. (1984) and Findlay and Williams (1985) extended the qualitative debate by evaluating a numerical example. Brown and Lippitt (1987) examined conceptual difficulties which can arise when discounting deferred taxes without resolving whether pre-tax or post-tax rates should be used. This note provides an explicit general proof that discounting at the after- tax rate is correct and, as a result, resolves this issue for once and for all.

It will be convenient to denote the deferred tax liability in period t by L(t), t = 1, 2, . . . , n, and let X be the present value of the tax liabilities. Suppose that we can invest this cash sum at a pre-tax interest rate of i per period and that tax is payable at the end of each period at rate t on interest earned during that period. We are therefore assuming that there is no lag in the payment of tax on earned interest.

The cash position at the end of the first period is given by

C(1) = X(l + i ) - tiX - L(1) = 4 1 +(1-t)i] - L(1),

where the first term in the initial expression represents the end of period value of the initial cash investment X, the second term represents the tax payable on the interest earned and the final term is the deferred tax payment due.

Similarly, the cash position at the end of the second period is given by

C(2) = C(1)(1 + z ] - tiC(1) - L(2) = C(l)[ l+( l - t ) i ] - L(2). (2)

(3)

On substituting for C(1) from (1) into (2), we obtain

C(2) = 4 1 +(l-t)i]Z - [1+(1--t)i] L(1) - L(2).

'The first and second authors are respectively, a Lecturer and a Professor in the Graduate School of Management, The University of Melbourne, Victoria, Australia; the third author is a Reader in Finance at the London School of Economics. They would like to thank Victor Kacala, Bob Officer and Ivan Woods for helpful comments on an earlier draft of this paper. (Paper rectived June 1988)

757

758 COLLINS, RICKARD AND SELBY

In the example discussed by Findlay and WilliAms (1981 and 1985) and Wolk et al. (1984), L(l) = L(2) = $1000, L(t) = 0 , t 2 3, i = 0.10 (10%) and t = 0.50 (50%). It follows that in this case C(2) = 0, since there are no further tax liabilities to fund. Hence, from (3) we have

x = 4 1 ) + L(2) 1 + ( l - t ) i [ l +(l-t)i]Z ’

or, using the above data,

j y = - + - - ‘Oo0 ‘Oo0 - $1859.41. 1.05 (1.05)*

(4)

(5)

Apart from the impact of rounding errors, equations (4) and (5) substantiate the view of Findlay and Williams, namely that the appropriate discount rate is the after tax rate of return. Equally important, equation (4) has been derived from first principles without recourse to somewhat imprecise philosophical arguments as employed in some of the earlier contributions.

The generalisation of equation (4) is readily obtained. It is immediately apparent from reasoning analogous to that used earlier that

C(J] = CO’-l)(l + i ) - tic0‘-1) - L(J) = C0‘-l)[l+(l-t) i] - L(J)

= 4 1 +(l-t)iY - [ l + ( l - t > i Y - k L ( k ) . A = I

If L(n) is the final deferred tax liability then C(n) = 0 and it follows from (6) that

x = c [l + (1 - t ) i ] -kL(k) , (7) k = I

which confirms that the after tax rate of return is appropriate in the general case.

REFERENCES

Brown, S. and J. Lippitt (1987), ‘Are Deferred Taxes Discountable?’, Journal of Business Finance

Findlay, M.C. and E.E. Williams (1981), ‘Discounting Deferred Tax Liabilities: A Reply’,Journal

- (1985), ‘Opportunity Cost, Discounting and Deferred Tax Liabilities: A Final Note’,

Nurnberg, ,J. (1972). ‘Discounting Deferred Tax Liabilities’, The Accounting Review (October 1972),

Williams, E.E. and M.C. Findlay (1975), ‘Discounting Deferred Tax Liabilities: Some Clarifying

Wolk, H. and M. Tearney (1980), ‘Discounting Deferred Tax Liabilities: Review and Analysis’,

-, L. Saubert and F. Tiernan (1984), ‘A Further Note on Discounting Deferred Taxes’,

&Accounting (Spring 1987), pp. 121-30.

of Business Finance & Accounting (Winter 1981), pp. 593-7.

Journal of Business Finance & Accounting (Summer 1985), pp. 183-4.

pp. 655-65.

Comments’, Journal of Business Finance & Accountinf (Spring 1975), pp. 121-33.

Journal of Business Finance & Accounting (Spring 1980), pp. 119-33.

Journal of Business Finance &Accounting (Summer 1984), pp. 253-5.