## U g

where FCFF0 is the current after-tax operating cash flow to the firm, pu is the unlevered cost of equity and g is the expected growth rate. In the more general case, you can value the firm using any set of growth assumptions you believe are reasonable for the firm.

The second step in this approach is the calculation of the expected tax benefit from a given level of debt. This tax benefit is a function of the tax rate of the firm and is discounted at the cost of debt to reflect the riskiness of this cash flow. If the tax savings are viewed as a perpetuity,

(Tax Rate)(Cost of Debt )(Debt) Cost of Debt Value of Tax Benefits = (Tax Rate)Debt)

The tax rate used here is the firm's marginal tax rate and it is assumed to stay constant over time. If we anticipate the tax rate changing over time, we can still compute the present value of tax benefits over time, but we cannot use the perpetual growth equation cited above.

The third step is to evaluate the effect of the given level of debt on the default risk of the firm and on expected bankruptcy costs. In theory, at least, this requires the estimation of the probability of default with the additional debt and the direct and indirect cost of bankruptcy. If na is the probability of default after the additional debt and BC is the present value of the bankruptcy cost, the present value of expected bankruptcy cost can be estimated.

= (Probability of Bankruptcy )PV of Bankruptcy Cost)

PV of Expected Bankruptcy cost

This step of the adjusted present value approach poses the most significant estimation problem, since neither the probability of bankruptcy nor the bankruptcy cost can be estimated directly.

In theory, the APV approach and the cost of capital approach will yield the same values for a firm if consistent assumptions are made about financial leverage. The difficulties associated with estimating the expected bankruptcy cost, though, often lead many to use an abbreviated version of the APV model, where the tax benefits are added to the unlevered firm value and bankruptcy costs are ignored. This approach will over value firms.

All of the principles that we have developed for valuation apply to private companies as well. In other words, the value of a private company is the present value of the expected cashflows that you would expect that company to generate over time, discounted back at a rate that reflects the riskiness of the cashflows. The differences that exist are primarily in the estimation of the cashflows and the discount rates: - When estimating cashflows, we should keep in mind that while accounting standards may not be adhered to consistently in publicly traded firms, they can diverge dramatically in private firms. In small, private businesses, we should reconstruct financial statements rather than trust the earnings numbers that are reported. There are also two common problems that arise in private firm accounting that we have to correct for. The first is the failure on the part of many owners to attach a cost to the time that they spend running their businesses. Thus, the owner of a store who spends most of every day stocking the store shelves, manning the cash register and completing the accounting will often not show a salary associated with these activities in her income statement, resulting in overstated earnings. The second is the intermingling of personal and business expenses that is endemic in many private businesses. When re-estimating earnings, we have to strip the personal expenses out of the analysis.

- When estimating discount rates for publicly traded firms, we hewed to two basic principles. With equity, we argued that the only risk that matters is the risk that cannot be diversified away by marginal investors, who we assumed were well diversified. With debt, the cost of debt was based upon a bond rating and the default spread associated with that rating. With private firms, both these assumptions will come under assault. First, the owner of a private business is almost never diversified and has his or her entire wealth often tied up in the firm's assets. That is why we developed the concept of a total beta for private firms in chapter 4, where we scaled the beta of the firm up to reflect all risk and not just non-diversifiable risk. Second, private businesses usually have to borrow from the local bank and do not have the luxury of accessing the bond market. Consequently, they may well find themselves facing a higher cost of debt than otherwise similar publicly traded firms.

- The final issue relates back to terminal value. With publicly traded firms, we assume that firms have infinite lives and use this assumption, in conjunction with stable growth, to estimate a terminal value. Private businesses, especially smaller ones, often have finite lives since they are much more dependent upon the owner/founder for their existence.

With more conservative estimates of cashflows, higher discount rates to reflect the exposure to total risk and finite life assumptions, it should come as no surprise that the values we attach to private firms are lower than those that we would attach to otherwise similar publicly traded firms. This also suggests that private firms that have the option of becoming publicly traded will generally succumb to that temptation even though the owners might not like the oversight and loss of control that comes with this transition.

Illustration 12.12: Valuing a Private Business: Bookscape

To value Bookscape, we will use the operating income of \$2 million that the firm had in its most recent year as a starting point. Adjusting for the operating lease commitments that the firm has, we arrive at an adjusted operating income of \$2,368.88 million.33 To estimate the cost of capital, we draw on the estimates of total beta and the assumption that the firm's debt to capital ratio would resemble the industry average of 16.90% that we made in chapter 4 (see illustration 4.16): Cost of capital = Cost of equity (D/(D+E)) + After-tax cost of debt (D/(D+E))

= 13.93% (.831) + 5.50% (1-.4) (.169) = .1214 or 12.14% The total beta for Bookscape is 2.06 and we will continue to use the 40% tax rate for the firm.

In chapter 5, we estimated a return on capital for Bookscape of 12.68% and we will assume that the firm will continue to generate this return on capital for the next 40 years, while growing its earnings at 4% a year. The resulting reinvestment rate is 31.54%: Reinvestment rate = Growth rate/ Return on capital = 4%/12.68% = 31.54% The present value of the cashflows, assuming perpetual growth, can be computed as follows:

Value of operating assets = 2,368.88 (1-.40)(1-.3154)/(.1214-.04) = \$12.483 million Value of cash holdings = \$ 2.500 million

Value of the firm = \$ 14.939 million

- Value of debt (operating leases) = \$ 6.707 million

Value of equity = \$ 8.231 million

If we wanted to be conservative, and assume that the cashflows will continue for only 40 years, the value of the operating assets drops marginally to \$12.3 million.

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