Not Quite the Dullest Post I Ever Wrote

A few weeks back I hurled some unintelligible mismash on corporate debt:  I said that debt on a leveraged balance sheet ought to be cheaper than pure equity--but that as I penciled it out, I kept coming to the conclusion that debt on a leveraged balance sheet was more expensive than pure equity,.  This seemed wrong to me and I wondered whether I had led myself into a rookie error.  I tried the piece on a couple of smart people who had the grace to pretend that they never got the message, and on a colleague who agreed that yes, I must be wrong, but he wasn't sure why.  On a rethink, I've come to the conclusion that (a) no, I was not wrong; but (b) it's actually fairly trivial.  I was muddling one important premise, this regarding the allocation of risk.  Assume the unleveraged balance sheet bears some risk (Doesn't it always?  I mean, nothing is certain, right?).  Add leverage; normally we would expect (leveraged) equity to bear the enterprise risk, and for debt to be risk free.

But what if debt is not risk-free?  What if debt  bears some of the enterprise risk.  Let's go to the numbers.

LittleCo has assets worth $1.000. There are 10 shares which implies a share value of $100 (each).   Suppose LittleCo also throws off $100 in earnings/dividends every year forever.  This looks like a straight-out 10 percent return.  Everything parses.

Note, I'm not saying that LittleCo is risk-free.  That $100 represents the weighted sum of a spectrum of probabilities.  I mean, suppose that the value of LittleCo, starting at T=1 might be $1,200 or might be $800, with a 50 percent chance of each.  Then: 0.5(1200)+0.5(800)=$1,000, and everything still parses.

Now, suppose we decide to replace half our equity with debt.  Suppose debt costs eight percent (plucked out of the air--no way to derive it).  That seems to suggest that we can get $500 worth of debt for $40 a year ((500x0.08, yes?).  That seems to leave $60 for equity.

Equity costs 10 percent.  60/0.1=600=$600.  Woo hah.  Looks like we have just created $100 in value for equity.   Can this be right?   It cannot be right.  Rather, we have dumped all the enterprise risk on equity, so equity will demand a higher return. The formula is re=ra+(D/E)(ra-rd) (sorry, I can't do subscripts).  So 0.1+(500/500)(0.1-0.08)=0.12=12 percent.  Then 60/0.12)=500=$500 and we are back to ground zero.  What you lose on the swings you  make up on the roundabouts.

So far this is baby steps. But take one step further.  Suppose the risk spectrum is not just 800-1,200.  Suppose it is: 50 percent chance of zero and 50 percent chance of $2,000..  The probability weighted sum value for the enterprise is still $1,000. Now, again replace $500 equity with debt.  At what rate?  Again I don't know--but I also know that this time debt is not risk free.  The probability weighted sum of the returns to debt are: 0.5(500)+0.5 (zero) = 250=$250.  And just to close the circle, the potential payoff to equity if 0.5(2000-500)+0.5(0)= 750=$750.  So enterprise value is 250+750=1,000=$1,000 and everything parses again.

In words, debt will be more risky than pure equity when it bears some of the enterprise risk.  My only remaining cavil is with the idea that debt is "risk free."  Finance types are always assuming that debt is "risk free," whereas truly risk-free debt, if it exists (but how could it), must be the exceptional case.

This looks obvious enough once I spell it out that I probably should be embarrassed even to post it.  I guess I was focusing on the simple examples in the Modigliani-Miller chapter of the coursebook where debt is always risk free. FWIW, my beloved colleague John Hunt pointed out that Robert K. Merton clearly understood and accounted for this result in his seminal paper laying the math foundations for Black-Scholes.