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I've read the so-called "leverage-effect" for stocks models the fact that if a company is leveraged, its volatility should increase as the stock price moves lower and closer to the level of debt.

Can someone please explain this to me?

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Explain what? What are you looking for? Google shows tons of results for "leverage effect". From this paper's abstract: A standard explanation ties the phenomenon to the effect a change in market valuation of a firm's equity has on the degree of leverage in its capital structure, with an increase in leverage producing an increase in stock volatility. – chrisaycock Jan 9 '13 at 4:08
I guess I'm looking for a mathematical model – Steve Lorimer Jan 9 '13 at 6:17
up vote 10 down vote accepted

The key to this is to think about the enterprise value of a business separately from how it is financed.

For simplicity sake, consider a business that comprises a sole gold bar (no workers, no extraction costs, etc). The value of the business is clearly just the value of the gold bar. If it were a listed company, with no debt, then the equity capitalization would be the value of the gold bar, and the volatility of the share price would be equal to the volatility of the gold price.

Now consider the same company financed with $50\%$ debt (at zero interest) and $50\%$ equity. The enterprise value of the geared company remains the same as before, but the equity capitalization is half as much (since the debt holders are owed the other half). However, whereas the claims of debt holders is fixed in nominal dollars, the equity holders get the benefit/cost of a higher/lower gold price.

E.g. If the gold bar is initially worth $\$100$ (financed with $\$50$ equity and $\$50$ debt), but then rises to $\$110$, then the value of equity becomes $\$60$, while the value of debt remains at $\$50$. Equity holders enjoy a $20\%$ increase ($=\frac{10}{50}$) in share value, against $10\%$ ($=\frac{10}{100}$) in the unlevered case. In moving from $0\%$ gearing to $50\%$ gearing, the volatility of equity value has doubled.

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For a mathematical model you can have a look at this paper:

The Valuation of Compound Options by Robert Geske

where after equation (17) it is shown that $\partial \sigma_s/\partial S<0$.

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