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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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    $\begingroup$ 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. $\endgroup$ – chrisaycock Jan 9 '13 at 4:08
  • $\begingroup$ I guess I'm looking for a mathematical model $\endgroup$ – Steve Lorimer Jan 9 '13 at 6:17
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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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Some levers can amplify the input force and give a larger output force. This function is called "leverage". The leverage principle in stock investment refers to the fact that investors use a portion of fixed interest rate funds to increase the return on investment of ordinary stocks, that is, the purchaser himself invests less, but may obtain high profits or large losses.

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Adding to Yugmorf’s excellent answer, the formal link between higher vol and lower (stock) price is the so-called Merton model of capital structure. Robert Merton being the one who (fairly) got Black’s Nobel for options pricing

Looking at any company’s balance sheet, its equity represents a long call option - all the upside, with limited liability. Its debt is a short put - fixed coupons, with all the ultimate potential loss.

The company itself is thus itself financed as long call plus short put equals just long...

So the (true nerd) answer to the original question is best answered by thinking about the options dynamics of its balance sheet financing.

Suppose the company does well. Its long-call equity goes ITM, its short-put debt goes OTM. Its gamma, ie its second-derivative change in price sensitivity to price changes (ie realised volatility) goes towards zero.

Suppose it goes badly. The long-call becomes worthless and insensitive; while the short-put debt becomes 100% sensitive to economic fundamentals, ie more volatile.

Hope this helps, DEM

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See Ronn and Verma 1986 Journal of Finance.

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