leverage

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The first syllable is pronounced leev in British English and lev in American English.

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Various statistics that give information about the reliability of the estimates of the multiple regression modelE(Y)=Xβ,where Y is an n×1 vector of independent and identically distributed response variables, β is a p×1 vector of unknown parameters, and X is an n×p matrix. If β is replaced by its least squares estimate, β̂, the estimated column vector of fitted values, ŷ, is given by

where the n×n matrix H, the hat matrix, is given by
X′ is the transpose of X, (X′X)⁻¹ is the inverse of the matrix X′X, and y is the column vector of observed values. Denote the element in the jth row and kth column of H by h_jk. The fitted value, ŷ_j, for the jth observation, y_j, is given by


Thus there is a direct link between the fitted and observed values in the form of h_jj. This is the leverage: a large value (e.g.>2_p/n) indicates an observation having a large influence on the form of the fitted model.

The most obvious guide to the fit of a model are the residuals, e₁, e₂,..., where e_j is given by
.If the random variables have common variance σ² and if s² is an unbiased estimate of σ², then the standardized residual is sometimes defined by e_j/s. However, an unbiased estimate of the variance of e_j is not s² but s²(1−h_jj) and a more appropriate residual (having unit variance if the model is correct) is given by r_j, where


This is sometimes called the standardized residual and sometimes the Studentized residual.

The deletion residual is given by
where ŷ_j,-j is the fitted value for observation j based on the fit of the model to all the observations except the observation y_j. Dividing the deletion residual by its estimated standard error, we get the Studentized deletion residual which can be written as


where s²_-j is the unbiased estimate of σ² obtained when observation j is omitted. Confusingly, this may also be called the Studentized residual. See also Anscombe residual; deviance residual.

A related influence statistic is DFFITS, which is an abbreviation for difference in fits. For observation j, DFFITS_j is


The influence statistic DFBETA (difference in beta values) applies the idea embodied in DFFITS to the parameter estimates rather than the fitted values. For β_k, DFBETA_k,−j is


where β̂_k is the estimate of β_k from the complete data, β̂_k,-j is the estimate when observation j is omitted, and m_kk is the corresponding diagonal element of the p×p matrix (X′X)⁻¹.

A statistic that usefully combines information about leverage and influence is Cook's statistic, D_j, given by


This statistic can also be interpreted as measuring the effect on the parameter estimates of omitting the jth observation. Large values point to possible outliers.

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Operating leverage: extent to which a company’s costs of operating are fixed (rent, insurance, executive salaries) as opposed to variable (materials, direct labor). In a totally automated company, whose costs are virtually all fixed, every dollar of increase in sales is a dollar of increase in operating income once the breakeven point has been reached, because costs remain the same at every level of production. In contrast, a company whose costs are largely variable would show relatively little increase in operating income when production and sales increased because costs and production would rise together. The leverage comes in because a small change in sales has a magnified percentage effect on operating income and losses. The degree of operating leverage— the ratio of the percentage change in operating income to the percentage change in sales or units sold—measures the sensitivity of a firm’s profits to changes in sales volume. A firm using a high degree of operating leverage has a breakeven point at a relatively high sales level.


Financial leverage: debt in relation to equity in a firm’s capital structure—its long-term debt (usually bonds), preferred stock, and shareholders’ equity—measured by the debt-to-equity ratio. The more long-term debt there is, the greater the financial leverage. Shareholders benefit from financial leverage to the extent that return on the borrowed money exceeds the interest costs and the market value of their shares rises. For this reason, financial leverage is popularly called trading on the equity. Because leverage also means required interest and principal payments and thus ultimately the risk of default, how much leverage is desirable is largely a question of stability of earnings. As a rule of thumb, an industrial company with a debt to equity ratio of more than 30% is highly leveraged, exceptions being firms with dependable earnings and cash flow, such as electric utilities.
Since long-term debt interest is a fixed cost, financial leverage tends to take over where operating leverage leaves off, further magnifying the effects on earnings per share of changes in sales levels. In general, high operating leverage should accompany low financial leverage, and vice versa.


Investments: means of enhancing return or value without increasing investment. Buying securities on margin is an example of leverage with borrowed money, and extra leverage may be possible if the leveraged security is convertible into common stock. rights, warrants, and Option contracts provide leverage, not involving borrowings but offering the prospect of high return for little or no investment.
See also deleverage.

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Use of borrowed funds to increase purchasing power and, ideally, to increase the profitability of an investment.
See FINANCIAL LEVERAGE, OPERATING LEVERAGE, POSITIVE LEVERAGE, REVERSE LEVERAGE .


Example: Collins wishes to invest in real estate. The property costs $100,000 and produces net operating income of $10,000 per year. If purchased with all cash, Collins’s annual rate of return is 10% ($10,000 ÷ $100,000). If she leverages the investment by borrowing $75,000, her return on equity may be higher. If the debt cost is 8% ($6,000) annually, the leverage results in a return of 16% ($4,000 ÷ $25,000). However, if the debt cost is 12% ($9,000), the leverage is negative because it reduces the return on equity to 4% ($1,000 ÷ $25,000).

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noun

The power to produce an effect by indirect means: influence, sway, weight. Informal clout. Slang pull. See affect/ineffectiveness.

This entry contains information applicable to United States law only.

A method of financing an investment by which an investor pays only a small percentage of the purchase price in cash, with the balance supplemented by borrowed funds, in order to generate a greater rate of return than would be produced by paying primarily cash for the investment; the economic benefit gained by such financing.

Real estate syndicates and promoters commonly use leverage financing. A leveraged investor builds up equity or ownership in the investment by making payments on the amount of principal borrowed from a third person. The money allotted to the repayment of interest charged on the borrowed principal is treated typically as a deduction that reducestaxable income. The greater the amount of principal borrowed, the larger the interest payments and the resulting deductions. Obviously, a taxpayer who pays cash is not entitled to deductions for interest payments. In many cases, deductions for the depreciation of the capital asset constituting the investment are also permitted.

Any investor receives an anticipated rate of return from the investment although the rate may fluctuate depending upon the economic climate and the management of the investment. Because of the favorable tax treatment enjoyed as a result of this method of financing, the leveraged investor keeps more of the income generated by the investment than an investor who financed the investment mainly through cash. There is, however, risk involved in leverage financing. If the income generated by the investment decreases, there might not be adequate funds available to meet payment of the outstanding principal and interest, leading to substantial losses for the investor.

The amount in which a purchase is paid for in borrowed money. The greater the leverage, the greater the possible gain or potential loss.

1. The use of various financial instruments or borrowed capital, such as margin, to increase the potential return of an investment.

2. The amount of debt used to finance a firm's assets. A firm with significantly more debt than equity is considered to be highly leveraged.

Leverage is most commonly used in real estate transactions through the use of mortgages to purchase a home.

Investopedia Says:
1. Leverage can be created through options, futures, margin and other financial instruments. For example, say you have $1,000 to invest. This amount could be invested in 10 shares of Microsoft stock, but to increase leverage, you could invest the $1,000 in five options contracts. You would then control 500 shares instead of just 10.

2. Most companies use debt to finance operations. By doing so, a company increases its leverage because it can invest in business operations without increasing its equity. For example, if a company formed with an investment of $5 million from investors, the equity in the company is $5 million - this is the money the company uses to operate. If the company uses debt financing by borrowing $20 million, the company now has $25 million to invest in business operations and more opportunity to increase value for shareholders.

Leverage helps both the investor and the firm to invest or operate. However, it comes with greater risk. If an investor uses leverage to make an investment and the investment moves against the investor, his or her loss is much greater than it would've been if the investment had not been leveraged - leverage magnifies both gains and losses. In the business world, a company can use leverage to try to generate shareholder wealth, but if it fails to do so, the interest expense and credit risk of default destroys shareholder value.

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For a list of words related to leverage, see:

• Securities - leverage: use of credit in business activity, esp. to enhance speculative capacity; borrowed capital used in business

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See words rhyming with "leverage."

Dansk (Danish)
n. - vægtstangsforhold, vægtstangssystem, vægtstangsbevægelse, magt, indflydelse, låntagning med marginalforrentning for øje
v. tr. - spekulere med lånte penge i forventning om at profitten vil overstige lånerenten

Nederlands (Dutch)
macht, invloed, kracht/ werking van hefboom, hefboomstelsel, kredietspeculatie, van speculatiekrediet voorzien, met krediet speculeren

Français (French)
n. - prise, influence, (Écon, Pol) force d'appui, (Fin) effet de levier, (Phys) puissance de levier
v. tr. - exercer une influence

Deutsch (German)
n. - Macht, Einfluss, Hebelkraft, Hebelwirkung
v. - beeinflussen, Geld investieren

Ελληνική (Greek)
n. - ενέργεια/δύναμη μοχλού, (μτφ.) επιρροή, μόχλευση

Italiano (Italian)
potenza, influenza

Português (Portuguese)
n. - ação (f) de uma alavanca, influência (f), poder (m)

Русский (Russian)
действие рычага, система рычагов, средство для достижения цели, использование кредита для биржевой игры, использовать кредит для биржевой игры

Español (Spanish)
n. - apalancamiento, fuerza de la palanca, influencia, ventaja
v. tr. - ejercer poder, influenciar sobre

Svenska (Swedish)
n. - hävstångsverkan, hävstångskraft, hävstångsanordning, makt, inflytande

中文（简体）(Chinese (Simplified))
杠杆作用, 杠杆率, 杠杆装置, 使举债经营

中文（繁體）(Chinese (Traditional))
n. - 槓桿作用, 槓桿率, 槓桿裝置
v. tr. - 使舉債經營

한국어 (Korean)
n. - 지레 장치, 행동력
v. tr. - 차입금으로 투기하게 하다

日本語 (Japanese)
n. - てこの作用, 手段, 影響力, 権力

العربيه (Arabic)
‏(الاسم) فعل الرافعه أو المزل, فعاليه, قوة, نفوذ‏

עברית (Hebrew)
n. - ‮הנפה, היתרון המכני המושג בשימוש במנוף, אמצעים להשגת מטרה: כוח, השפעה, תנופה, יחס כמות הלוואות במבנה הון וחישובי רווח, מערכת מנופים‬
v. tr. - ‮הפעיל השפעה על, השקיע‬

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