First, I agree that when tax shield is discounted at the same rate as asset, the formula used in the blog is correct as the tax shield has the same beta as asset.

Second, I agree that the value of tax shield can be lower than D*t due to the risk of profitability.

Without financial distress risk associated with the high leverage, adding more debt always increases the firm value by lessening the tax burden. So the WACC should monotonically decrease as leverage increases.

]]>Thank you for your comment. Let me answer your questions in reverse order.

Regarding the (“distorted”) conclusion of this article, it’s useful to keep the top-level picture in mind: it provides a useful cross-check of our algebraic results. The profits (EBIT) of a firm are divided among three claimants: the tax authority (government), lenders, and shareholders. The sum of the lender and shareholder claims equals the (levered) value of the firm. For the same EBIT risk and growth rate (that is, for no change in the operations of the firm), the value of firm increases if and only if the value of the government claim falls. As we increase leverage, the expected value of tax payments falls, and firm value increases. A direct algebraic implication is also that WACC falls. Any “smiley” WACC curve (that shows — anywhere — WACC increasing for an increase in debt) violates this fundamental framework.

Regarding the calculation of tax shield: it is certainly less than the value you suggest, t*D. That value derives from the annuitization of the annual reduction in tax payments ( = t * KD * D ) using a discount rate equal to the cost of debt ( KD ), leaving t*D. However a firm only enjoys a reduction in tax *if it pays tax*; i.e. if it is profitable. Profitability depends on equity (not total, or enterprise) flows, and thus is riskier than the cost of debt.

In the derivation in the post, we assume the risk of tax shield flows equals the asset risk of the firm. (Asset risk is greater than debt risk, but less than equity risk.) Both of these assumptions about the appropriate discount rate for tax shield flows (cost of debt, and cost of assets) are incorrect however. The actual calculation of tax shield is presented in A Reconsideration of Tax Shield Valuation (Arzac and Glosten, 2004). They derive the value of the tax shield as

t * KD * D * (1+KE) / [(KE – g)(1 + KD)]

..which doesn’t translate easily into a neat discount rate, even if we assume a zero growth rate. Note that this value is less than t*D whenever KD is less than KE; i.e., always.

So, finally, to your first comment about the calculation of equity beta. I believe your derivation assumes that the riskiness of the tax shield flows equals the riskiness of debt. If your formula leads to the conclusion that increasing leverage sometimes increases WACC (producing the WACC “smile”), then it must be incorrect.

]]>RoA = E / A * RoE + D / A * ( 1 – tax rate ) * Interest

Another caveat is the calculation of tax shield. It should be D * tax rate intead of D * tax rate * kd / ka.

These fallacies actually distort the conclusion of this article. The smiley curve of WACC is a result of the value depreciation due to the risk of financial distress when the leverage is high, rather than the mispricing of the equity.

]]>Thank you for your thoughtful question.

The value of the tax shield, labeled “S” in the above tables, accrues to equity investors. Thus

V = A + S = D + E

As the tax shield increases, E and V both increase by the change in S. Examples of this calculation are shown in the tables above. ]]>

There is something that is bothering me about your calculations, perhaps you could clarify it somehow: You say that Be = Ba(V/E) – Bd(D/E), where B stands for Beta. I think that would be correct only if E + D = A = V, but that only happens if tax = 0, that is, if there are no tax shields.

Am I missing something?

Regards

JP ]]>

Thank you for pointing out the broken link (regarding alternative calculations of debt beta).

Alas I cannot find a link to the original article (by Julian Franks at LBS). In its place I’ve inserted several other references. I hope this suffices. ]]>

The link is now broken.

Thanks ]]>

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2400101 ]]>

Thank you for your note. Two thoughts in response:

1) Whether or not a convex WACC curve (with a minimum point) is more familiar, it’s certainly incorrect. For it to be valid, we would have to abandon Modigliani & Miller’s capital structure irrelevance proposition.

2) CAPM theory, by supposition, applies to all investable assets, and it explains the observed yield on those assets. The returns of any risky asset certainly include nonsystematic (idiosyncratic) variations, but CAPM purports that investors ignore those variations, and only price in systematic risk. This applies to equities, bonds, derivatives, fine art collections, and so on. So if we disallow the use of CAPM for explaining bond yields, it seems inconsistent to allow it to explain equity yields. If you have a non-CAPM framework for explaining asset yields, I would be happy to discuss it! ]]>

Debt beta is a measure of the *systematic* risk of debt securities. The DRP in Kd would account for more than just systematic risk. So I believe your debt beta estimates (calculated via CAPM) are probably too high. If you correct for this and lower the debt betas then the more familiar WACC curve should return.

Tony

]]>Thank you for your post.

In the example above, “Assets” refers to the value of assets when all-equity funded. Note that this value is after-tax. Consistent with the example above, we may assume that periodic pre-tax earnings from Assets are $15.38; after a 35% tax they are $10. Capitalizing at the cost of assets (10%) produces a $100 value of assets when all-equity funded.

If we add leverage to the capital structure, total after-tax payments to investors (lenders and shareholders) increase above $10/period, since the company pays less tax. The value of these tax savings is labeled “Tax Shield” in the tables above. This benefit accrues to equity investors.

Your example points out that the same asset may be of different value to different investors, depending on their tax status. We see this today in the rush to wrap real assets in real estate investment trusts (REITs). In this case the same asset is worth more to investors when held by a REIT than when held by a C-corp.

]]>Thank you for your comment. Two points:

1. The post’s text originally said m=5%; it turns out the calculations and charts assume m=6%. I have update thie corresponding text, and apologize for the error. I have also added the note that risk-free rate = 4%.

2. Though the asset beta is assumed constant (at 1.0), the equity beta increases with leverage.

]]>the point is that if you buy the house with all equity then you have the lost opportunity cost and if you buy the house with all debt then you have your interest payment minus tax return. probably in the end without house price change the opportunity cost is bigger than (interest minus tax return) but the asset price is not dependent to tax shield.

you should add tax shield to equity and not to asset for comparison.

Regards,

H.

Per your assumption, m = 5%. Since all equity Betas are 1, based on the Asset return Rf = 5% as well.

If I then plug this into the Debt beta assumption, I get:

0 0.2 0.4 0.6 and 0.8 for each of the alternative capital structure scenarios.

What am I missing?

Thanks!

MS

]]>I see where I was missing something from your diagram. You only go up to 100% debt-to-equity. LBOs where debt can be several multiples of equity make the smiley face curve even taking into account debt-beta.

Thanks for the follow-up.

Steve

Thank you for the thoughtful challenge. A couple of points:

1) I don’t “think” that cost of capital decreases with debt — that’s a simple consequence of algebra. (Note that many wacc calculations do the algebra wrong — they ignore debt beta, and in the process violate M&M.) Furthermore, I didn’t create any models — what’s calculated above is pure WACC.

2) Cost of _debt_ rises with leverage, but WACC declines — for a taxpayer. (In the absence of taxes, we’re back to the M&M “capital structure irrelevance” world.)

3) It’s true that at some point additional leverage will reduce enterprise value — not because WACC increased, but because cash flows decrease. See https://www.quantcorpfin.com/cookbook/debt-capacity/optimal-capital-structure-with-business-disruption-costs/ for more.

]]>Maybe I missed the point of the post. What was your intent here? As a practical matter your treatment of debt-beta seems counter-productive.

]]>For others reading this: any spreadsheets linked to from this blog are available to registered users. (The content is freely available, but we’d like to know who’s using it!)

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