Diderik Lund, Ph.D., Department of Economics, University of Oslo, P.O.Box 1095, Blindern, Oslo, NO-0317, Norway
Lund (Int Tax and Public Finance, 9(4), 483–503, 2002) showed in a CAPM-type model how tax depreciation schedules affect required expected returns after taxes. Even without leverage higher tax rates imply lower betas when tax deductions are risk free. This has important implications for the capital budgeting practice of firms operating under more than one tax system, and applying the commonly used weighted-average cost of capital (WACC). The equity part of the WACC should depend on the tax system when the cash flow has systematic risk, because taxes and tax deductions affect the after-tax systematic risk.
The present paper is again purely theoretical. It extends the model by, more realistically, allowing tax deductions to be risky. In this case total risk, not only systematic risk, matters. Marginal investment is taxed together with inframarginal in an analytical model of decreasing returns to scale. With imperfect loss offset, tax claims are analogous to call options, and a modified Black-Scholes formula is used in the valuation. This is necessary even though the investment decision is not supposed to include any real option. The model has no value of waiting to invest.
The paper shows how the required expected return after taxes depends on the exogenous output price volatility and on the endogenous riskiness in the firms’ tax positions. The latter depends on the amount of inframarginal investment, which depends on the elasticity of the production function. The uncertainty of the tax position modifies the effect in Lund (2002), which relied on tax deductions being risk free. But there is still a substantial difference between the betas before and after tax for reasonable values of the output price volatility. The beta of equity is still decreasing in the tax rate, but increasing in the volatility.
Modelling the riskiness of the tax position leads to a distinction between marginal and inframarginal investment. This in turn leads to a distinction between marginal and average beta. Both of these after-tax betas are determined in the model for some given beta before tax. The average beta will be relevant for the firm as a whole, i.e., for betas observed in the stock market. Thus it is also relevant for inferences about asset betas and their relation to stock betas. The marginal beta is, however, relevant for investment decisions in an established firm. This shows the complexity of finding the correct after-tax discount rate under taxation with imperfect loss offset.
The marginal beta is increasing as the scale elasticity approaches unity from below, i.e., as one goes from decreasing towards constant returns to scale. This reflects higher risk, since the inframarginal investment, which would reduce the riskiness of the tax position, vanishes as one approaches constant returns to scale.
The model also gives results for the cost of capital before tax, which is the traditional measure of distortionary effects of taxation. In contrast with the after-tax results, these before-tax results make precise what has more or less been known previously. Imperfect loss offset creates additional distortions.
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www.oekonomi.uio.no/memo/memopdf/memo1305.pdf