In the controversial practice of appraisal arbitrage, activist investors buy shares of a corporation to be acquired by merger so as to assert appraisal rights challenging the merger price – which may already have been approved by the target’s stockholders. The practice is controversial because the appraisal remedy is widely seen as intended to protect *existing* stockholders who are forced to sell their shares in the merger and *not* to afford hedge funds a way to extract extra returns from the deal. But the puzzle is why appraisal arbitrage is profitable, since the remedy seeks to determine fair prices using the same techniques used by financial professionals who structure such deals.

Scholars have argued that the profit may derive from (1) a free option to assert appraisal rights at any time until target shares are cancelled (and indeed for a short time thereafter), (2) the use of a too-low supply-side discount rate (resulting in too-high multipliers) in the valuation of shares, and (3) the award of pre-judgment interest at a too-generous legal rate of 5 percent over the Fed discount rate.

None of these explanations is persuasive, but the second one may be on the right track.

As for why the discount rate matters, the basic formula for the value of a perpetual stream of returns (such as from a business) is:

VALUE = RETURN / DISCOUNT RATE

In this formula, the discount rate is the rate of return demanded by investors (the market) given the risk inherent in the business. The lower the discount rate, the higher the value. While the supply-side discount rate is indeed a bit lower than the historical rate used by many financial professionals, it has also become almost standard practice to reduce the discount rate even more by the projected rate of inflation and general economic growth (as measured by GDP), which may skew awards to the high side, as explained more fully below.

First, regarding the free option: It is almost impossible for information revealed after a merger is announced to have any effect on fair price as found by an appraisal court. Under the prevailing discounted cash flow (DCF) approach, valuation is based on cash flow projections prepared in the normal course of business and not for litigation. Similarly, the discount rate determined under the Capital Asset Pricing Model (CAPM) is based on historical rates of return as adjusted for risk and size. These factors are largely cast in stone before the deal happens. But an option has value because of the possibility that the underlying stock will *increase* in value. So an option increases in value in proportion to its duration in time. There is no such possibility in the context of an appraisal proceeding. This supposed appraisal option is just as valuable if it lasts for a day or a year. Moreover (and more important), although arbs can postpone tying up their own money, that does not change the amount of cash tied up by stockholders in the aggregate.

Second, regarding the discount rate: The supply-side rate reflects the belief of many finance scholars that future equity returns are unlikely to be as generous as they have been in the past (or at least since 1925 – the period for which we have reliable data based on the S&P 500). Given that growth in business returns (and thus stock prices) is the ultimate source of economic growth, slower economic growth going forward presupposes slower growth in stock prices and therefore less total return. Other scholars, who seem to be agnostic about the prospects for economic growth, note that about 1 percent of the 12 percent raw historical average equity return is attributable to an increase in price/earnings (P/E) ratios that is unlikely ever to be repeated.

While the former view is largely a matter of opinion, the latter argument for reducing discount rates is well taken. Some of the growth in stock prices since 1925 can be attributed to investor diversification (through mutual funds) and resulting reduction in risk without a concomitant reduction in return. As the Nobelist father of portfolio theory Harry Markowitz said, diversification is the only free lunch in the market. Thus, stocks became more valuable during this period as investors found new ways to reduce risk. So the supply of returns going forward is likely to be lower than the average historical rate. In other words, the market has already eaten the free lunch of diversification.

Although using the supply-side rate results in a roughly1 percent reduction in the benchmark discount rate (from about 12 percent to about 11 percent), the courts often reduce this figure still further – by as much as 5.5 percent – to adjust for projected inflation and GDP growth in returns during the terminal period. (Pay no attention to the irony behind the curtain.)

If returns are expected to grow at a steady rate, the valuation formula can be modified to account for such growth by reducing the discount rate by the growth rate:

VALUE = RETURN / (DISCOUNT RATE – GROWTH RATE)

To be clear, there is no need to make any such adjustment in the first few years (typically five), for which cash flow is projected year-by-year. Growth – as well as the diversion of funds to finance it (plowback) – is reflected directly in returns during this projection period. But it is also a mistake to adjust the terminal period discount rate for inflation and general economic growth unless return is reduced to reflect plowback. In most cases, growth comes from (re)investment of cash at ordinary rates of return. So the increase in value from a lower discount rate is exactly offset by the diversion of returns to new investment. A business that wants to increase sales may need to buy more inventory, rent a bigger warehouse, and hire additional employees. Growth does not grow on trees. To be sure, if a business finds opportunities that generate more return than its cost of capital, its value will grow. But such opportunities are rare and fleeting. As the Delaware courts have noted, the ability to generate returns in excess of the cost of capital will quickly be dissipated by competition. So it is fair to presume that growth of this sort is unlikely to persist beyond the projection period in the absence of positive evidence to the contrary.

On the other hand, if projected average cash flow for the terminal period builds in an implicit plowback rate – say a plowback rate based on the last year of the projection period – the discount rate should be adjusted accordingly. But there is no reason to assume that the company-specific plowback rate in combination with the company-specific discount rate will just happen to generate the projected rate of GDP growth. It is true that GDP growth ultimately comes from growth in returns from business. (At least that is how we measure it.) But it does not follow that growth in returns from business are spontaneously generated *because* the economy grows. The point is that if we know the company-specific plowback rate and discount rate, there is no reason to use a projected average GDP growth rate. Indeed, average GDP growth (about 3.4 percent in real terms since 1931) is significantly higher than average real growth in stock prices as measured by the S&P 500 (about 2.5 percent, and some of which is a one-time event). So even if GDP growth does ultimately derive from growth in returns, it appears to come disproportionately from small business, thus undercutting the rationale for using the GDP growth rate in appraisal where discount rates are derived from the S&P 500.

Finally, regarding pre-judgment interest: If investors expect 11 percent average annual returns, it is arguable that the interest rate should be the same or at least the compound rate – about 9.5 percent supply-side. If anything, the legal rate unduly *discourages* appraisal arbitrage.

Still, the question remains: Do we need appraisal arbitrage? Why should we permit new investors to buy into the claims of old investors – especially if the rationale is to compensate unwilling sellers who are forced to give up their shares in a merger? The answer is that arbs have bought the stock they hold from legacy investors. If the rights of arbs are curtailed, the result will be a bigger discount for stockholders who choose to sell. And that will raise the price of deals. It is certainly understandable that acquirers would want to curtail appraisal arbitrage ex post, but it serves them well ex ante.

Bargaining happens in the shadow of the law. Where there is a robust appraisal remedy, bidders will be induced to pay a fair price up front. Thus, appraisal may ultimately redound to the benefit of bidders as a sort of bonding mechanism that reassures target stockholders as to fairness of price. Indeed, this may explain why bidders continue to use the merger method rather than alternatives.

In the end, appraisal works best if appraisal arbitrage is possible. Otherwise bidders may reckon that many potential dissenters will decline to exercise their appraisal rights. If the bidder is thus required to pay a fair price only to *some* dissenting stockholders, the bidder comes out ahead. But appraisal arbitrage fixes this market failure. Thus, the problem with appraisal arbitrage (if any) appears to lie with valuation practice.

*This post comes to us from Professor Richard A. Booth, the **Martin G. McGuinn Chair in Business Law at Villanova University Charles Widger School of Law. It is based on his recent article, “The Real Problem with Appraisal Arbitrage,” which is forthcoming in The Business Lawyer and available here.*

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I find this column nonsensical, although some of that may be due to differences between legal and financial usage.

(1) The option value of appraisal rights does not refer to the incorporation of company-specific or market information after the merger announcement, but to the fact that litigating shareholders can win a higher price, but cannot do worse than the original merger price. It’s not a free option as there are litigation costs, but it only takes a small rate of success to make it worth buying shares and contesting nearly every merger.

(2) Projecting cash flows and discounting them is theoretically correct, but useless in practice without calibration to market. If you calibrate to market, then you’re only going through the mathematical form of a DCF analysis, you’re really valuing the company as a multiple of other companies.

In ISN Software, the court opined that an eightfold difference in DCF valuations implied a “best scenario” that “one expert, at the least, is wildly mistaken.” I take that to mean that the most likely scenario is that both experts were hired gun frauds.

But consider the DCF value for even the best-studied and most liquid S&P500 stocks. Half the present value for these stocks comes from cash flows more than 35 years out, so you have to estimate growth rates and discount rates for many decades to get accurate valuations. An error of 0.1% in either growth rate or discount rate means a 5% error in valuation. It takes a difference of 1.75% to get an eight-fold difference between estimates (it’s not linear).

Let’s assume we know the discount rate for certain and the only issue is to estimate the cash flow growth rate over the next century (think about estimating the 100-year cash flow growth rate of big US corporations in 1916). Let’s compare it to a much easier problem, estimating the actual growth rate of the entire US economy (diversifying away idiosyncratic features of individual companies) over the quarter (not century) that just ended (past, not future). And let’s not compare our estimate to the truth, just to an estimate made by the same people, using the same methods, using extra data that trickles out more than a few days after quarter end. The average difference between all the government’s estimates of annualized US growth rate from initial to final is 1.8%, more than enough to support an eight-fold difference in company valuation.

And this is for the largest, best-studied companies; without even worrying about the problem of appropriate discount rate. We ignore crucial issues like future tax rates, inflation and regulation. The problems are much worse for typical merger targets. There’s simply no rational quantitative basis for assigning a DCF value. All we can do is compare to similar companies and either do a direct multiplication based on Price / Earnings or Market / Book or some other ratio; or extract growth and discount rates from the comparable and apply them to the target; which just an indirect way of doing the same thing as a direct multiplication.

(3) The appropriate discount rate for a legal award is unrelated to the reason for the award. If a court finds that you should have given me $100 a year ago, you should pay me now $100 plus the interest I would have had to pay to borrow the $100. That is the damage to me. It doesn’t matter whether the $100 was underpayment for a stock that has since doubled in price, or has since gone to zero. The harm was a year ago, my remedy was to borrow $100, and you now owe me the costs of that. Of course, that raises issues if I cannot borrow at any price, in which case you might (or might not, it’s arguable) owe me compensation for specific damages like me losing my house. But in appraisal situations for public companies, it’s likely that the appropriate rate is LIBOR or something close to it, certainly not an equity risk premium over the risk-free rate. You took $100 from me, you didn’t take from me the opportunity to invest in good stocks.