Sackur–Tetrode equation

In finance, leverage (sometimes referred to as gearing in the United Kingdom and Australia) is a general term for any technique to multiply gains and losses. Most often this involves buying more of an asset by using borrowed funds. The belief is that the income from the asset will more than pay for the cost of borrowing. Almost always this involves the risk that borrowing costs will be larger than the income from the asset - causing a reduction in profits.

Where leverage is found

• Individuals lever their savings when buying a home by financing a portion of the purchase price with mortgage debt.
• Individuals lever their exposure to financial investments by borrowing from their broker.
• Securities like options and futures contracts are bets between parties where the principal is implicitly borrowed/lent at very short t-bill rates.
• Equity owners of businesses leverage their investment by having the business borrow a portion of its needed financing. The more it borrows, the less equity it needs, so any profits or losses are shared among a smaller base and are proportionately larger as a result.
• Businesses leverage their operations by using fixed cost inputs when revenues are expected to be variable. An increase in revenue will result in a larger increase in operating income.
• Hedge funds may leverage their assets by financing a portion of their portfolios with the cash proceeds from the short sale of other positions.

Risk

While leverage magnifies profits when the returns from the asset more than offset the costs of borrowing, losses are magnified when the opposite is true. A corporation that borrows too much money might face bankruptcy or default during a business downturn, while a less-levered corporation might survive. An investor who buys a stock on 50% margin will lose 40% of his money if the stock declines 20%.

Risk may be attributed to a loss in value of collateral assets. Brokers may require the addition of funds when the value of securities hold declines. Banks may fail to renew mortgages when the value of real estate declines below the debt's principal. Even if cash flows and profits are sufficient to maintain the ongoing borrowing costs, loans may be called.

This may happen exactly when there is little market liquidity and sales by others are depressing prices. It means that as things get bad, leverage goes up, multiplying losses as things continue to go down. This can lead to rapid ruin, even if the underlying asset value decline is mild or temporary. The risk can be mitigated by negotiating the terms of leverage, by maintaining unused room for additional borrowing, and by leveraging only liquid assets.

On the other hand, the extreme level of leverage afforded in forex trading presents relatively low risk per unit due to its relative stability when compared with other markets. A standard unit of measurement known as a pip equals just .0001 USD. Compared with other trading markets, forex traders must trade a much higher volume of units in order to make any considerable profit. For example, many brokers offer 100:1 leverage for investors, meaning that someone bringing $1,000 can control $100,000 while taking responsibility for any losses or gains their investments incur. This intense level of leverage presents equal parts risk and reward.

There is an implicit assumption in that account, however, which is that the underlying levered asset is the same as the unlevered one. If a company borrows money to modernize, or add to its product line, or expand internationally, the additional diversification might more than offset the additional risk from leverage. Or if an investor uses a fraction of his or her portfolio to margin stock index futures and puts the rest in a money market fund, he or she might have the same volatility and expected return as an investor in an unlevered equity index fund, with a limited downside. Or if both long and short positions are held by a pairs-trading stock strategy the matching and off-setting economic leverage may lower overall risk levels.

So while adding leverage to a given asset always adds risk, it is not the case that a levered company or investment is always riskier than an unlevered one. In fact, many highly-levered hedge funds have less return volatility than unlevered bond funds, and public utilities with lots of debt are usually less risky stocks than unlevered technology companies.

Worked example

Calculate equity return given:
5% Projected Return on Investment
4% Cost of Debt
8:1 Leverage Debt:Equity = Investment in Asset of 9

LONG-FORM MATH

Profit from investment = Size of asset * Return on investment = 9 * 5% = 0.45
less Interest on debt = Size of debt * Cost of debt = 8 * 4% = 0.32
Equals equity's profit = 0.45 - 0.32 = 0.13
Return on Equity = Profit divided by Equity portion = (0.13 / 1) = 13%

SHORT-FORM GENERIC CALCULATION

Interest rate differential = Return on investment less Cost of debt = 5% - 4% = 1%
Debt to equity multiple = 8 divided by 1 = 8
Multiply first two lines = 1% * 8 = 8%
Add return on investment = 5% Equals return on equity = 8% + 5% = 13%

Measuring leverage

A good deal of confusion arises in discussions among people who use different definitions of leverage. The term is used differently in investments and corporate finance, and has multiple definitions in each field.

Investments

Accounting leverage is total assets divided by the total assets minus total liabilities. Notional leverage is total notional amount of assets plus total notional amount of liabilities divided by equity. Economic leverage is volatility of equity divided by volatility of an unlevered investment in the same assets. To understand the differences, consider the following positions, all funded with $100 of cash equity:

• Buy $100 of crude oil with money out of pocket. Assets are $100 ($100 of oil), there are no liabilities, and assets minus liabilities equals owners' equity. Accounting leverage is 1 to 1. The notional amount is $100 ($100 of oil), there are no liabilities, and there is $100 of equity, so notional leverage is 1 to 1. The volatility of the equity is equal to the volatility of oil, since oil is the only asset and you own the same amount as your equity, so economic leverage is 1 to 1.
• Borrow $100 and buy $200 of crude oil. Assets are $200, liabilities are $100 so accounting leverage is 2 to 1. The notional amount is $200 and equity is $100, so notional leverage is 2 to 1. The volatility of the position is twice the volatility of an unlevered position in the same assets, so economic leverage is 2 to 1.
• Buy $100 of crude oil, borrow $100 worth of gasoline, and sell the gasoline for $100. You now have $100 cash, $100 of crude oil, and owe $100 worth of gasoline. Your assets are $200, and liabilities are $100, so accounting leverage is 2 to 1. You have $200 in notional assets plus $100 in notional liabilities, with $100 of equity, so your notional leverage is 3 to 1. The volatility of your position might be half the volatility of an unlevered investment in the same assets, since the price of oil and the price of gasoline are positively correlated, so your economic leverage might be 0.5 to 1.
• Buy $100 of a 10-year fixed-rate treasury bond, and enter into a fixed-for-floating 10-year interest rate swap to convert the payments to floating rate. The derivative is off-balance sheet, so it is ignored for accounting leverage. Accounting leverage is therefore 1 to 1. The notional amount of the swap does count for notional leverage, so notional leverage is 2 to 1. The swap removes most of the economic risk of the treasury bond, so economic leverage is near zero.

Abbrevations

• EBIT means Earnings before interest and taxes.
• DOL is Degree of Operating Leverage
• DFL is Degree of Financial Leverage
• DCL is Degree of Combined Leverage
• ROE is Return on Equity
• ROA is Return on assets

Corporate finance

$\mathrm{D}\mathrm{O}\mathrm{L}=\frac{\mathrm{E}\mathrm{B}\mathrm{I}\mathrm{T}+\mathrm{F}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}}{\mathrm{E}\mathrm{B}\mathrm{I}\mathrm{T}}$

$\mathrm{D}\mathrm{F}\mathrm{L}=\frac{\mathrm{E}\mathrm{B}\mathrm{I}\mathrm{T}}{\mathrm{E}\mathrm{B}\mathrm{I}\mathrm{T}\mathrm{-}\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}}$

$\mathrm{D}\mathrm{C}\mathrm{L}=\mathrm{D}\mathrm{O}\mathrm{L}*\mathrm{D}\mathrm{F}\mathrm{L}=\frac{\mathrm{E}\mathrm{B}\mathrm{I}\mathrm{T}+\mathrm{F}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}\mathrm{s}}{\mathrm{E}\mathrm{B}\mathrm{I}\mathrm{T}\mathrm{-}\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{I}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}\mathrm{E}\mathrm{x}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{e}}$

Accounting leverage has the same definition as in investments. There are several ways to define operating leverage, the most common. is:

$\mathrm{O}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}=\frac{\mathrm{R}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{e}-\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}}{\mathrm{R}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{e}-\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}-\mathrm{F}\mathrm{i}\mathrm{x}\mathrm{e}\mathrm{d}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}}=\frac{\mathrm{R}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{e}-\mathrm{V}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{C}\mathrm{o}\mathrm{s}\mathrm{t}}{\mathrm{O}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{I}\mathrm{n}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{e}}$

Financial leverage is usually defined as:

$\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}=\frac{\mathrm{T}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}\mathrm{A}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{t}\mathrm{s}}{\mathrm{S}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{e}\mathrm{r}{\mathrm{s}}^{\prime }\mathrm{E}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{t}\mathrm{y}}$

or:

$\mathrm{F}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{g}\mathrm{e}=\frac{\mathrm{R}\mathrm{O}\mathrm{E}}{\mathrm{R}\mathrm{O}\mathrm{A}}$

Operating leverage is an attempt to estimate the percentage change in operating income for a one-percent change in revenue.
Financial leverage tries to estimate the percentage change in net income for a one-percent change in operating income.
The product of the two is called Total leverage, and estimates the percentage change in net income for a one-percent change in revenue.
There are several variants of each of these definitions, and the financial statements are usually adjusted before the values are computed. Moreover, there are industry-specific conventions that differ somewhat from the treatment above.

Bank regulation concerning leverage

Before the 1980s, quantitative limits on bank leverage were rare. Banks in most countries had a reserve requirement, a fraction of deposits that was required to be held in liquid form, generally precious metals or government notes or deposits. This does not limit leverage. A capital requirement is a fraction of assets that is required to be funded in the form of equity or equity-like securities. Although these two are often confused, they are in fact opposite. A reserve requirement is a fraction of certain liabilities (from the right hand side of the balance sheet) that must be held as a certain kind of asset (from the left hand side of the balance sheet). A capital requirement is a fraction of assets (from the left hand side of the balance sheet) that must be held as a certain kind of liability or equity (from the right hand side of the balance sheet). Before the 1980s, regulators typically imposed judgmental capital requirements, a bank was supposed to be "adequately capitalized," but not objective rules.

National regulators began imposing formal capital requirements in the 1980s, and by 1988 most large multinational banks were held to the Basel I standard. Basel I categorized assets into five risk buckets, and mandated minimum capital requirements for each. This limits accounting leverage. If a bank is required to hold 8% capital against an asset, that is the same as an accounting leverage limit of 1/.08 or 12.5 to 1.

While Basel I is generally credited with improving bank risk management it suffered from two main defects. It did not require capital for all off-balance sheet risks (there was a clumsy provisions for derivatives, but not for certain other off-balance sheet exposures) and it encouraged banks to pick the riskiest assets in each bucket (for example, the capital requirement was the same for all corporate loans, whether to solid companies or ones near bankruptcy, and the requirement for government loans was zero).

Work on Basel II began in the early 1990s and it was implemented in stages beginning in 2005. Basel II attempted to limit economic leverage rather than accounting leverage. It required advanced banks to estimate the risk of their positions and allocate capital accordingly. While this is much more rational in theory, it is more subject to estimation error, both honest and opportunitistic. The poor performance of many banks during the financial crisis of 2007-2009 led to calls to reimpose leverage limits, by which most people meant accounting leverage limits, if they understood the distinction at all. However, in view of the problems with Basel I, it seems likely that some hybrid of accounting and notional leverage will be used, and the leverage limits will be imposed in addition to, not instead of, Basel II economic leverage limits.

The financial crisis of 2007–2009

The financial crisis of 2007–2009, like many previous financial crises, was blamed in part on "excessive leverage."

• Consumers in the United States and many other developed countries had high levels of debt relative to their wages, and relative to the value of collateral assets. When home prices fell, and debt interest rates reset higher, and business laid off employees, borrowers could no longer afford debt payments, and lenders could not recover their principal by selling collateral.
• Financial institutions were highly levered. Lehman Brothers, for example, in its last annual financial statements, showed accounting leverage of 30.7 times ($691 billion in assets divided by $22 billion in stockholders’ equity). Bankruptcy examiner Anton R. Valukas determined that the true accounting leverage was higher: it had been understated due to dubious accounting treatments including the so-called repo 105 (Allowed by Ernst & Young).
• Banks' notional leverage was more than twice as high, due to off-balance sheet transactions. At the end of 2007, Lehman had $738 billion of notional derivatives in addition to the assets above, plus significant off-balance sheet exposures to special purpose entities, structured investment vehicles and conduits, plus various lending commitments, contractual payments and contingent obligations.
• On the other hand, almost half of Lehman’s balance sheet consisted of closely offsetting positions and very-low-risk assets, such as regulatory deposits. The company emphasized "net leverage", which excluded these assets. On that basis, Lehman held $373 billion of "net assets" and a "net leverage ratio" of 16.1. This is not a standardized computation, but it probably corresponds more closely to what most people think of when they hear a leverage ratio.

Use of language

Levering has come to be known as "leveraging", in financial communities; this may have originally been a slang adaptation, since leverage was a noun. However, modern dictionaries (such as Random House Dictionary and Merriam-Webster's Dictionary of Law) refer to its use as a verb, as well. It was first adopted for use as a verb in American English in 1957.

See also

• Leveraged buyout
• Repurchase agreement
• Homemade Leverage

References

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