## Important

• The discount rates on the tax obligations and on the tax shelter are usually not exact, but just reasonable and convenient approximations. The value consequences of reasonable errors are minor.

• It is common and usually reasonable to value tax liabilities at a discount rate equal to the firm's overall cost of capital (E(rFM)).

• For the tax shelter due to interest payments, assuming that the firm will grow over time, it is common and usually reasonable to

- use the debt cost of capital (E(roT)) if the firm plans on a decreasing debt ratio;

- use the firm cost of capital (E(rFM)) if the firm plans on a constant debt ratio;

- use the equity cost of capital (E(rEQ)) if the firm plans on an increasing debt ratio.

Taxes are important. You are only fudging the divisor, not the numerator!

Do not forget that the entire discussion—that you can allow yourself some latitude on errors—was only about the discount factor. Importantly, the (expected) amount of the tax shelter itself is not unimportant. This also applies to the idiosyncratic risk in the expected tax shelter, a quantity that figures into the present value numerator of the tax shelter, not the denominator (the discount rate). For example, an R&D project may not generate any tax shelter half the time—in which case, the expected tax shelter (in the PV numerator) to be discounted would be something like

/Tax Shelter If R&D is Successful^ Expected Tax Shelter = 50% ■ = Tax-Rate Times Interest Paid + 50% ■ \$0 (18.34)

Figure 18.2: Thinking About Proper Discount Rates For The Tax Shelter D/V Target

Time Time

The background of the other three graphs: The typical firm value grows over time.

This firm plans to reduce its debt ratio over time, perhaps to keep its dollar debt and its interest payments constant.

Use E (roT) to discount the tax shelter.

D/V Target

D/V Target

Firm R&D Time Time

This firm plans to keep its debt ratio constant.

Use E (rFM) to discount the tax shelter.

These two firms, called "R&D" and "Inc" plan to raise their debt ratios over time. Firm R&D wants to sharply increase its debt ratio only after it will have higher tax-deductible income.

Use E (teq) to discount the tax shelter.

V is the firm's value. D is the firm's debt. D/V is the firm's debt ratio.

These scenarios illustrate cases in which the firm's debt ratio changes over time, which in turn influences the discount rate that should be applied to the tax shelter. For example, if the firm wants to keep a constant debt ratio over the years, then it will have more debt and therefore a higher debt tax shelter if the firm experiences good times in the first year. This means that the value of the future tax shelter covaries positively with the firm value in the first year. It is therefore not almost risk-free (as it was in our example in which the firm existed only for one year), but more risky (in fact, almost as risky as the firm is in its first year).

Fortunately, although it would be a first-order error to compute the wrong tax shelter, it is often a second-order error to use the wrong discount factor on the tax shelter. Yes, you should try to get it right anyway, but realize that getting other quantities right is often more important than agonizing whether you should use E(rFM) or E(rDr).

CHAPTER 19

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