# Buy firm, leverage and sell, seems like arbitrage strategy. What's wrong with this argument?

I recently came across with someone valuing a firm with different D/E ratios. My question is that this looks a bit spurious in that its mechanical. For instance if one wants to buy an unlevered firm and finds a value $V_U$ for its equity (there is not debt at this point). If they take debt then the firm value changes to $V_L=V_U + Dt$ and in particular equity value changes to $V_U- D(1-t)$. Then they sell the equity. The cash flow is

$$-V_U + D + (V_U- D(1-t))= Dt$$

where the first term is the money spent to buy unlevered equity, second is the cash gotten by taking debt and the third is the money obtained by selling off the equity.

This looks like arbitrage. What's wrong in this strategy though?

• First you buy the equity, then the firm issues debt to X, then you sell THE EQUITY (right? not the firm). X also has a claim on the firm, you are no longer the only claimant, value of your claim is the same as before Feb 1 '18 at 15:06
• The firm value does not change after taking on debt (cash increases by $Dt$, debt increases by $Dt$, net does not change) Feb 1 '18 at 15:12
• @noob2 You are right. I am correcting the question wording to reflect that. The argument still stands though. Feb 1 '18 at 15:35
• @AntoineConze The firm value increase by Dt where D is debt and t is tax rate. It even has a name -- tax shield. For a derivation you can see my answer in quant.stackexchange.com/questions/33384/… for instance. Feb 1 '18 at 15:40
• If I'm correct you assume that the firm takes on perpetual debt and that interest paid is tax deductible hence the $+Dt$ present value assuming the firm lives forever. However the cash raised would have to be invested over the life of the firm, and the return on that investment would be taxed, resulting in a $-Dt$ present value if the rate of return is equal to the interest rate on the debt. So the firm value only increases if the rate of return is on average larger than the interest rate, which would not hold under a risk neutral probability. I don't think this would qualify as arbitrage. Feb 1 '18 at 16:09

This sounds like a classic, introductory corporate finance question.

If interest payments on debt are tax deductible, increasing debt lowers corporate income taxes. Taking total firm cash flows as given, increasing debt effectively redirects cash flows from the government to equity holders. With no counteracting forces, firm owners will choose an all debt firm.

Why not do this? There are numerous objections you can raise to assumptions behind the above argument. For example:

1. The cost of bankruptcy isn't zero.
2. Excessive debt can cause agency problems (eg. underinvestment due to debt overhang) and reduce firm cash flows.
3. There are numerous other ways to avoid taxes.

The list goes on... Anyway, these issues will be discussed in any corporate finance textbook or intro corporate finance class.

Rather than give some simplistic survey of that theory here, I'd recommend the reader start with any introductory corporate finance text or class. The only thing else I'd mention is that the theory of firm capital structure is not a solved problem. Many existing theories are difficult to test, and to the extent that we can build empirical models to predict leverage ratios from theory, the $R^2$ you get aren't overwhelming.