An analytical model of required returns to equity under taxation with imperfect loss offset

1. An analytical model of required returns to equity under taxation with imperfect loss offset Diderik Lund University of Oslo, Norway December 27, 2004 Presentation IAEE conference, Taipei, June 2005 http://folk.uio.no/dilund/capcost/

2. Required expected rates of return according to Capital Asset Pricing Model (CAPM) • How does taxation of companies affect β? • Traditional focus: Effect on β of leverage • Then tax has effect via interest deduction • Here: Risk of tax cash flow (and after-tax cash flow) differs from risk of pre-tax cash flow • No focus on leverage; simplify: all equity

3. Relevance for energy sector? • Energy companies operate internationally • Many different tax systems • Several countries attempt to capture natural resource rent through high tax rates • Result from this study: After-tax required expected rates of return depend negatively on tax rates, even without leverage • Message: Companies should adjust their practices

4. Valuation by element better? sufficient? • Valuation by element (Adjusted present value, APV) recommended by Myers (1974) • No need to consider risk of net after-tax cash flow • Method in this paper fully consistent with APV • However: Most companies still apply one single required expected rate of return (WACC) • Interesting to find how this should vary with tax rate, depreciation rate, price volatility, etc.

5. Model and main intuition • Single firm, one project, one production period • Consider first a pure cash flow tax with payout when the tax base becomes negative: • After-tax cash flow is percentage of pre-tax cash flow, and thus has same β • Compared with this: Typical profits (or rent) tax has depreciation deductions instead of expensing • These postponed deductions are less risky than pre-tax cash flow; risk-wise opposite of leverage

6. Relation to Lund (2002), International Tax and Public Finance • Lund (2002) has similar model, but considers mainly a situation with risk free tax deductions • If a large, diversified company considers a new project: Project’s pre-tax cash flow, PQ, is risky • Future period’s tax is typically t PQ – t c I • c depreciation rate, I investment (neglect op. cost) • Value of deprec. deduction, t c I, almost risk free • Here: What if tax deduction somewhat risky?

7. Analytical model of decreasing returns to scale and imperfect loss offset • Marginal project added to more profitable projects • Modeled as analytical production function, Iα • Scale elasticity, α, is in (0,1]; α = 1 means CRS • Riskiness of tax cash flow depends on probability of whole company being in tax position • If CRS: All projects equally risky; marginal project easily falls out of tax position • Lower α implies more rent, less risky tax position

8. After-tax β as function of scale elasticity, αFirst CRS; Risk free versus risky deductions Deductions are risky Deductions are risk free, company always in tax position

9. After-tax β as function of scale elasticity, αIncreasing curve: Marginal β, deductions risky

10. After-tax β as function of scale elasticity, αSolid curve: Average β in risky case

11. After-tax β as function of scale elasticity, αTax 70% (note vertical axis), previously 30%

12. Conclusions • Under decreasing returns, many tax system will create difference between marginal and average β • Under DRS and risky tax position: • After-tax β is decreasing in tax rate • After-tax β is increasing in output price volatility • Marginal after-tax β is increasing in scale elasticity, while average is non-monotone, convex • Potential contribution to empirical CAPM • Important for analysis of tax reforms