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I'm calculating the historical and forward quarterly WACC for a security. For the historical WACC, should I use the quarter's tax rate (which in some cases is very negative) or a TTM tax rate or something else? The wide quarterly variances throw off the calculation considerably. The purpose of this would be to look for historical trends.

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    $\begingroup$ Is this for academic research or to fulfill a corporate mandate? $\endgroup$ – Dave Harris Nov 8 '17 at 4:30
  • $\begingroup$ Academic research. $\endgroup$ – American Luke Nov 8 '17 at 4:41
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    $\begingroup$ Just from common sense some kind of averaging would seem to be required. Using the trailing twelve months seems reasonable. $\endgroup$ – noob2 Nov 8 '17 at 20:45
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The various forms of the Capital Asset Pricing Model, as well as Black-Scholes, are built on top of the axioms of Frequentist Decision Theory. An assumption in the model is that the parameters are known with a probability of one. Generally speaking, this assumption is harmless. There are many models where the result with perfect knowledge is also the result without such knowledge. Unfortunately, finance cannot be such a case.

To demonstrate this case, consider the simple one period profit function of $w_{t+1}=Rw_t+\epsilon_{t+1}$. If $\epsilon$ is drawn from a normal distribution and $R$ is known, then all of the properties of mean-variance finance will hold. In fact, $w_{t+1}$ will converge to a normal distribution on repeated sampling if you rescale to 1 at every sample. However, if $R$ is unknown and $R>1$, which it must be in finance, then two papers show that this class of problem has no non-Bayesian solution.

The first paper was by Mann and Wald in 1943 and the second by White in 1958. Mann and Wald solved that the estimator for the coefficient was the least squares estimator, a form of the sample mean. White showed that the distribution of $\hat{R}-R$ was the Cauchy distribution. This is where it gets weird. The Cauchy distribution lacks a population mean or variance. Hence no useful estimator exists. Therefore WACC cannot be estimated unless you actually know the means, variances, and covariances of every security with perfect knowledge.

I wrote a paper solving the distribution of returns for all asset and liability classes. It notes something else that most people have ignored. The first, above, is that no one actually checked the properties of estimators created by these models. The second though is interesting as well. Returns cannot be data, except bank or insurance company guaranteed investments.

Returns are not data as they are not observed. Prices are observed. Volumes are observed. Cash flows are observed. Returns are calculated from either prices or cash flows. They are a function of the data. The definition of a statistic is that it is a function of the data. Returns are not data. They are statistics.

As such, they inherit the properties of the underlying data via the mathematical transformation that turns them into returns. There are four methods that economists convert prices to returns discussed in the article. A fifth, using the internal rate of return method, was not thought about by me when I wrote the article although it is a form of regression and so should be covered.

In the simple case where a return is a future value divided by the present value minus one is the ratio of two random variates. Under some simple, real-world assumptions, the distribution of returns of going concerns that are traded on an exchange will converge to the truncated Cauchy distribution with a very slight skew due to the intertemporal budget constraint. For assets sold at other types of auctions, such as Sotheby’s, it will be the ratio of two Gumbel distributions because of the different rules.

The short version of this is that WACC cannot be measured if it exists. Any attempt at measurement would have zero relative asymptotic efficiency to a median based measurement. This would violate the theory. It isn’t median-interquartile range finance.

The reason you are getting wide sample variances that do not settle down is that the truncated Cauchy distribution lacks a variance. It looks like heteroskedasticity, but it isn’t. The Cauchy distribution is askedastic, and so no measurement of a traditional understanding of volatility exists. This is not to say it cannot be measured. The interpretation is very different because there is no such thing as a variance or covariance.

Bayesian methods are admissible here, but they don’t work like it is generally taught in finance. Indeed, since they do not view data as random, but instead view parameters as uncertain the differences can be rather large. Before going down that path, I recommend picking up an introductory text on Bayesian methods.

If you look at a mean or variance as a property of a distribution, then you may have an easier time thinking about it. A nose is a property that vertebrates share. Trees do not. If you attempt to measure the number of noses on a tree, you will fail no matter what method you use. On the other hand, trees do pass gasses with the environment. They just don’t do it through the structure we call a nose. All living parts of a tree are engaged in respiration. If you widen your concept, the lack of a nose is not a problem. Not having a variance is a similar type of problem. The idea of volatility works differently quite a bit in the same sense that trees lack noses but respirate through an entirely different model.

The correct solution is to look at the marginal cost of capital on a firm by firm basis. For firms with lines of credit, this is easy. For firms that would not qualify for commercial loans then you would have to look at the discount on initial public offerings. I doubt a data source exists to provide you with your requested information. The existence of mean-variance finance implied that it was not necessary information. Someone really needs to go back and fill in that gap.

For my paper, you can find it at http://www.scirp.org/Journal/PaperInformation.aspx?PaperID=78849

For White’s paper, you can find it at: White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197.

I don’t have the Mann and Wald citation where I am at, but I think you can get to it via the White paper in its bibliography.

I do apologize, but your problem cannot be solved by anyone. No such methodology exists within null hypothesis based methods.

Just for completeness, it does not matter what tax rate you use as you cannot solve the necessary other components. IF you used the marginal cost of funds, then you should use the actual quarterly marginal tax rate, even if it is negative to be mathematically consistent with the idea of marginal thinking.

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    $\begingroup$ I believe the variance in the tax rate of the firm I'm looking at have more to do with accounting method and tax loss carry-forward and when the company elects to use them. Great answer on the quant side though. $\endgroup$ – American Luke Nov 11 '17 at 22:44
  • $\begingroup$ The tax must have finite variance, returns won't though. $\endgroup$ – Dave Harris Nov 12 '17 at 0:36
  • $\begingroup$ Yes, but the tax rate is not a random variable. It is semi-dependent on earnings before tax and presumably the rest of the variance is tax credits and such. The most significant variances occur when the denominator accounting income before taxes is near zero and there is still a sizable tax income base left because of differences in tax accounting. This leads to extraordinary tax rates that are not marginal. Does that make any sense? $\endgroup$ – American Luke Nov 14 '17 at 21:34
  • $\begingroup$ Yes, and the rate is random, though not necessarily a continuous variable. Your tax rate may be ill-behaved, however. Your rate is a ratio of two random variates, but are probably not well described by any simple function. Your tax rate may not have a mean or a variance, depending on the calculation process in the numerator and the denominator. Have you considered median based methods such as Theil's regression or median based statistical tests? $\endgroup$ – Dave Harris Nov 16 '17 at 0:30
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If you are trying to determine the quarterly WACC your tax rate needs to be quarterly as well. Personally, I would take a look at the change in income tax expense. You can also try experimenting with change in (income tax expense / ebit) or adding in income tax expense + deferred tax liability.

However, I would say use the quarterly tax rate even if it is negative. It is negative due to corporate tax subsidies, deferments, low quarterly earnings etc. Point being that these, in so far as they are eventually translated in to capital losses or gains should impact the weighted average cost of capital. Example: One of the ways the government will try and encourage businesses to expand is lowering the tax rate. So that they will get more bang for their buck, so to speak.

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