Financial ratio

Accounting Dictionary: Financial Ratio

Mathematical relationship between one quantity and another. There are many categories of ratios such as those that evaluate a business entity's liquidity, solvency, return on investment, operating performance, asset utilization, and market measures. An example of a ratio is the earnings yield that equals dividends per share divided by market price per share. While the computation of a ratio is a basic arithmetical operation, its analytical interpretation is more complex. A financial ratio should be computed only if the relationship between accounts or categories has significance. The financial ratio may provide the accountant with clues and symptoms of underlying financial condition. To be meaningful, a given financial ratio of a company for a given year must be compared to (1) prior years to examine the trend, (2) industry norm, and (3) competing companies. See also Financial Statement Analysis.

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Small Business Encyclopedia: Financial Ratios

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Financial ratios illustrate relationships between different aspects of a small business's operations. They involve the comparison of elements from a balance sheet or income statement, and are crafted with particular points of focus in mind. Financial ratios can provide small business owners and managers with a valuable tool to measure their progress against predetermined internal goals, a certain competitor, or the overall industry. In addition, tracking various ratios over time is a powerful way to identify trends as they develop. Ratios are also used by bankers, investors, and business analysts to assess various attributes of a company's financial strength or operating results.

Ratios are determined by dividing one number by another, and are usually expressed as a percentage. They enable business owners to examine the relationships between seemingly unrelated items and thus gain useful information for decision-making. "They are simple to calculate, easy to use, and provide a wealth of information that cannot be gotten anywhere else," James O. Gill noted in his book Financial Basics of Small Business Success. But, he added, "Ratios are aids to judgment and cannot take the place of experience. They will not replace good management, but they will make a good manager better. They help to pinpoint areas that need investigation and assist in developing an operating strategy for the future."

Virtually any financial statistics can be compared using a ratio. In reality, however, small business owners and managers only need to be concerned with a small set of ratios in order to identify where improvements are needed. "As you run your business you juggle dozens of different variables," David H. Bangs, Jr. wrote in his book Managing by the Numbers. "Ratio analysis is designed to help you identify those variables which are out of balance."

It is important to keep in mind that financial ratios are time sensitive; they can only present a picture of the business at the time that the underlying figures were prepared. For example, a retailer calculating ratios before and after the Christmas season would get very different results. In addition, ratios can be misleading when taken singly, though they can be quite valuable when a small business tracks them over time or uses them as a basis for comparison against company goals or industry standards. As a result, business owners should compute a variety of applicable ratios and attempt to discern a pattern, rather than relying on the information provided by only one or two ratios. Gill also noted that small business owners should be certain to view ratios objectively, rather than using them to confirm a particular strategy or point of view.

Perhaps the best way for small business owners to use financial ratios is to conduct a formal ratio analysis on a regular basis. The raw data used to compute the ratios should be recorded on a special form monthly. Then the relevant ratios should be computed, reviewed, and saved for future comparisons. Determining which ratios to compute depends on the type of business, the age of the business, the point in the business cycle, and any specific information sought. For example, if a small business depends on a large number of fixed assets, ratios that measure how efficiently these assets are being used may be the most significant. In general, financial ratios can be broken down into four main categories—profitability or return on investment, liquidity, leverage, and operating or efficiency—with several specific ratio calculations prescribed within each.

Profitability or Return on Investment Ratios

Profitability ratios provide information about management's performance in using the resources of the small business. As Gill noted, most entrepreneurs decide to start their own businesses in order to earn a better return on their money than would be available through a bank or other low-risk investments. If profitability ratios demonstrate that this is not occurring—particularly once a small business has moved beyond the start-up phase—then the entrepreneur should consider selling the business and reinvesting his or her money elsewhere. However, it is important to note that many factors can influence profitability ratios, including changes in price, volume, or expenses, as well the purchase of assets or the borrowing of money. Some specific profitability ratios follow, along with the means of calculating them and their meaning to a small business owner or manager.

Gross profitability: Gross Profits / Net Sales—measures the margin on sales the company is achieving. It can be an indication of manufacturing efficiency or marketing effectiveness.

Net profitability: Net Income / Net Sales—measures the overall profitability of the company, or how much is being brought to the bottom line. Strong gross profitability combined with weak net profitability may indicate a problem with indirect operating expenses or non-operating items, such as interest expense. In general terms, net profitability shows the effectiveness of management. Though the optimal level depends on the type of business, the ratios can be compared for firms in the same industry.

Return on assets: Net Income / Total Assets—indicates how effectively the company is deploying its assets. A very low ROA usually indicates inefficient management, whereas a high ROA means efficient management. However, this ratio can be distorted by depreciation or any unusual expenses.

Return on investment 1: Net Income / Owners' Equity—indicates how well the company is utilizing its equity investment. Due to leverage, this measure will generally be higher than return on assets. ROI is considered to be one of the best indicators of profitability. It is also a good figure to compare against competitors or an industry average. Experts suggest that companies usually need at least 10-14 percent ROI in order to fund future growth. If this ratio is too low, it can indicate poor management performance or a highly conservative business approach. On the other hand, a high ROI can mean that management is doing a good job, or that the firm is undercapitalized.

Return on investment 2: Dividends / Stock Price Change / Stock Price Paid—from the investor's point of view, this calculation of ROI measures the gain (or loss) achieved by placing an investment over a period of time.

Earnings per share: Net Income / Number of Shares Outstanding—states a corporation's profits on a per share basis. It can be helpful in further comparison to the market price of the stock.

Investment turnover: Net Sales / Total Assets—measures a company's ability to use assets to generate sales. Although the ideal level for this ratio varies greatly, a very low figure may mean that the company maintains too many assets or has not deployed its assets well, whereas a high figure means that the assets have been used to produce good sales numbers.

Sales per employee: Total Sales / Number of Employees—can provide a measure of productivity, though a high figure can indicate either good personnel management or good equipment.

Liquidity Ratios

Liquidity ratios demonstrate a company's ability to pay its current obligations. In other words, they relate to the availability of cash and other assets to cover accounts payable, short-term debt, and other liabilities. All small businesses require a certain degree of liquidity in order to pay their bills on time, though start-up and very young companies are often not very liquid. In mature companies, low levels of liquidity can indicate poor management or a need for additional capital. Any company's liquidity may vary due to seasonality, the timing of sales, and the state of the economy. But liquidity ratios can provide small business owners with useful limits to help them regulate borrowing and spending. Some of the best-known measures of a company's liquidity include:

Current ratio: Current Assets / Current Liabilities—measures the ability of an entity to pay its near-term obligations. "Current" usually is defined as within one year. Though the ideal current ratio depends to some extent on the type of business, a general rule of thumb is that it should be at least 2:1. A lower current ratio means that the company may not be able to pay its bills on time, while a higher ratio means that the company has money in cash or safe investments that could be put to better use in the business.

Quick ratio (or "acid test"): Quick Assets (cash, marketable securities, and receivables) / Current Liabilities—provides a stricter definition of the company's ability to make payments on current obligations. Ideally, this ratio should be 1:1. If it is higher, the company may keep too much cash on hand or have a poor collection program for accounts receivable. If it is lower, it may indicate that the company relies too heavily on inventory to meet its obligations.

Cash to total assets: Cash / Total Assets—measures the portion of a company's assets held in cash or marketable securities. Although a high ratio may indicate some degree of safety from a creditor's viewpoint, excess amounts of cash may be viewed as inefficient.

Sales to receivables (or turnover ratio): Net Sales / Accounts Receivable—measures the annual turnover of accounts receivable. A high number reflects a short lapse of time between sales and the collection of cash, while a low number means collections take longer. It is best to use average accounts receivable to avoid seasonality effects.

Days' receivables ratio: 365 / Sales to receivables ratio—measures the average number of days that accounts receivable are outstanding. This number should be the same or lower than the company's expressed credit terms. Other ratios can also be converted to days, such as the cost of sales to payables ratio.

Cost of sales to payables: Cost of Sales / Trade Payables—measures the annual turnover of accounts payable. Lower numbers tend to indicate good performance, though the ratio should be close to the industry standard.

Cash turnover: Net Sales / Net Working Capital (current assets less current liabilities)—reflects the company's ability to finance current operations, the efficiency of its working capital employment, and the margin of protection for its creditors. A high cash turnover ratio may leave the company vulnerable to creditors, while a low ratio may indicate an inefficient use of working capital. In general, sales five to six times greater than working capital are needed to maintain a positive cash flow and finance sales.

Leverage Ratios

Leverage ratios look at the extent that a company has depended upon borrowing to finance its operations. As a result, these ratios are reviewed closely by bankers and investors. Most leverage ratios compare assets or net worth with liabilities. A high leverage ratio may increase a company's exposure to risk and business downturns, but along with this higher risk also comes the potential for higher returns. Some of the major measurements of leverage include:

Debt to equity ratio: Debt / Owners' Equity—indicates the relative mix of the company's investor-supplied capital. A company is generally considered safer if it has a low debt to equity ratio—that is, a higher proportion of owner-supplied capital—though a very low ratio can indicate excessive caution. In general, debt should be between 50 and 80 percent of equity.

Debt ratio: Debt / Total Assets—measures the portion of a company's capital that is provided by borrowing. A debt ratio greater than 1.0 means the company has negative net worth, and is technically bankrupt. This ratio is similar, and can easily be converted to, the debt to equity ratio.

Fixed to worth ratio: Net Fixed Assets / Tangible Net Worth—indicates how much of the owner's equity has been invested in fixed assets, i.e., plant and equipment. It is important to note that only tangible assets are included in the calculation, and that they are valued less depreciation. Creditors usually like to see this ratio very low, but the large-scale leasing of assets can artificially lower it.

Interest coverage: Earnings before Interest and Taxes / Interest Expense—indicates how comfortably the company can handle its interest payments. In general, a higher interest coverage ratio means that the small business is able to take on additional debt. This ratio is closely examined by bankers and other creditors.

Efficiency Ratios

By assessing a company's use of credit, inventory, and assets, efficiency ratios can help small business owners and managers conduct business better. These ratios can show how quickly the company is collecting money for its credit sales or how many times inventory turns over in a given time period. This information can help management decide whether the company's credit terms are appropriate and whether its purchasing efforts are handled in an efficient manner. The following are some of the main indicators of efficiency:

Annual inventory turnover: Cost of Goods Sold for the Year / Average Inventory—shows how efficiently the company is managing its production, ware-housing, and distribution of product, considering its volume of sales. Higher ratios—over six or seven times per year—are generally thought to be better, although extremely high inventory turnover may indicate a narrow selection and possibly lost sales. A low inventory turnover rate, on the other hand, means that the company is paying to keep a large inventory, and may be overstocking or carrying obsolete items.

Inventory holding period: 365 / Annual Inventory Turnover—calculates the number of days, on average, that elapse between finished goods production and sale of product.

Inventory to assets ratio: Inventory / Total Assets—shows the portion of assets tied up in inventory. Generally, a lower ratio is considered better.

Accounts receivable turnover: Net (credit) Sales / Average Accounts Receivable—gives a measure of how quickly credit sales are turned into cash. Alternatively, the reciprocal of this ratio indicates the portion of a year's credit sales that are outstanding at a particular point in time.

Collection period: 365 / Accounts Receivable Turnover—measures the average number of days the company's receivables are outstanding, between the date of credit sale and collection of cash.

Summary

Although they may seem intimidating at first glance, all of the aforementioned financial ratios can be derived by simply comparing numbers that appear on a small business's income statement and balance sheet. As Gill noted, small business owners would be well-served to "think of ratios as one of your best friends when scrutinizing your business."

Financial ratios can be an important tool for small business owners and managers to measure their progress toward reaching company goals, as well as toward competing with larger companies within an industry. Ratio analysis, when performed regularly over time, can also give help small businesses recognize and adapt to trends affecting their operations. Yet another reason small business owners need to understand financial ratios is that they provide one of the main measures of a company's success from the perspective of bankers, investors, and business analysts. Often, a small business's ability to obtain debt or equity financing will depend on the company's financial ratios.

Despite all the positive uses of financial ratios, however, small business managers are still encouraged to know the limitations of ratios and approach ratio analysis with a degree of caution. As Gill explained, "[Ratios] do not make decisions for you, but will provide information from which decisions may be made."

Sources of data for financial ratios

Values used in calculating financial ratios are taken from the balance sheet, income statement, statement of cash flows or (sometimes) the statement of retained earnings. These comprise the firm's "accounting statements" or financial statements. The statements' data is based on the accounting method and accounting standards used by the organization.

Purpose and types of ratios

Financial ratios quantify many aspects of a business and are an integral part of financial statement analysis. Financial ratios are categorized according to the financial aspect of the business which the ratio measures. Liquidity ratios measure the availability of cash to pay debt.^[2] Activity ratios measure how quickly a firm converts non-cash assets to cash assets.^[3] Debt ratios measure the firm's ability to repay long-term debt.^[4] Profitability ratios measure the firm's use of its assets and control of its expenses to generate an acceptable rate of return.^[5] Market ratios measure investor response to owning a company's stock and also the cost of issuing stock.^[6]

Financial ratios allow for comparisons

between companies
between industries
between different time periods for one company
between a single company and its industry average

Ratios generally hold no meaning unless they are benchmarked against something else, like past performance or another company. Thus, the ratios of firms in different industries, which face different risks, capital requirements, and competition are usually hard to compare.

Accounting methods and principles

Financial ratios may not be directly comparable between companies that use different accounting methods or follow various standard accounting practices. Most public companies are required by law to use generally accepted accounting principles for their home countries, but private companies, partnerships and sole proprietorships may not use accrual basis accounting. Large multi-national corporations may use International Financial Reporting Standards to produce their financial statements, or they may use the generally accepted accounting principles of their home country.

There is no international standard for calculating the summary data presented in all financial statements, and the terminology is not always consistent between companies, industries, countries and time periods.

Abbreviations and terminology

Various abbreviations may be used in financial statements, especially financial statements summarized on the Internet. Sales reported by a firm are usually net sales, which deduct returns, allowances, and early payment discounts from the charge on an invoice. Net income is always the amount after taxes, depreciation, amortization, and interest, unless otherwise stated. Otherwise, the amount would be EBIT, or EBITDA (see below).

Companies that are primarily involved in providing services with labour do not generally report "Sales" based on hours. These companies tend to report "revenue" based on the monetary value of income that the services provide.

Note that Shareholder's Equity and Owner's Equity are not the same thing, Shareholder's Equity represents the total number of shares in the company multiplied by each share's book value; Owner's Equity represents the total number of shares that an individual shareholder owns (usually the owner with controlling interest), multiplied by each share's book value. It is important to make this distinction when calculating ratios.

Other abbreviations

(Note: These are not ratios, but values in currency.)

COGS = Cost of goods sold, or cost of sales.
EBIT = Earnings before interest and taxes
EBITDA = Earnings before interest, taxes, depreciation, and amortization
EPS = Earnings per share

Ratios

Profitability ratios

Profitability ratios measure the firm's use of its assets and control of its expenses to generate an acceptable rate of return.

Gross margin, Gross profit margin or Gross Profit Rate^[7]^[8]

$\frac{\mbox{Gross Profit}}{\mbox{Net Sales}}$

$\frac{\mbox{Net Sales -- COGS}}{\mbox{Net Sales}}$

Operating margin, Operating Income Margin, Operating profit margin or Return on sales (ROS)^[9]^[8]

$\frac{\mbox{Operating Income}}{\mbox{Net Sales}}$

Note: Operating income is the difference between operating revenues and operating expenses, but it is also sometimes used as a synonym for EBIT and operating profit.^[10] This is true if the firm has no non-operating income. (Earnings before interest and taxes / Sales^[11]^[12])

Profit margin, net margin or net profit margin^[13] ^[14]

$\frac{\mbox{Net Profit}}{\mbox{Net Sales}}$

Return on equity (ROE) ^[14]

$\frac{\mbox{Net Income}}{\mbox{Average Shareholders Equity}}$

Return on investment (ROI ratio or Du Pont ratio)^[6]

$\frac{\mbox{Net Income}}{\mbox{Average Owners Equity}}$

Return on assets (ROA)^[15]

$\frac{\mbox{Net Income}}{\mbox{Total Assets}}$

Return on assets Du Pont (ROA Du Pont)^[16]

$\left(\frac{\mbox{Net Income}}{\mbox{Net Sales}}\right)\left(\frac{\mbox{Net Sales}}{\mbox{Total Assets}}\right)$

Return on Equity Du Pont (ROE Du Pont)

$\left(\frac{\mbox{Net Income}}{\mbox{Net Sales}}\right)\left(\frac{\mbox{Net Sales}}{\mbox{Average Assets}}\right)\left(\frac{\mbox{Average Assets}}{\mbox{Average Equity}}\right)$

Return on net assets (RONA)

$\frac{\mbox{Net Income}}{\mbox{Fixed Assets + Working Capital}}$

Return on capital (ROC)

$\frac{\mbox{EBIT(1 − Tax Rate)}}{\mbox{Invested Capital}}$

Risk adjusted return on capital (RAROC)

$\frac{\mbox{Expected Return}}{\mbox{Economic Capital}}$

$\frac{\mbox{Expected Return}}{\mbox{Value at Risk}}$

Return on capital employed (ROCE)

$\frac{\mbox{EBIT}}{\mbox{Capital Employed}}$

Note: this is somewhat similar to (ROI), which calculates Net Income per Owner's Equity

Cash flow return on investment (CFROI)

$\frac{\mbox{Cash Flow}}{\mbox{Market Recapitalisation}}$

Efficiency ratio

$\frac{\mbox{Non-Interest expense}}{\mbox{Revenue}}$

Net gearing

$\frac{\mbox{Net debt}}{\mbox{Equity}}$

Basic Earnings Power Ratio^[17]

$\frac{\mbox{EBIT}}{\mbox{Total Assets}}$

Liquidity ratios

Liquidity ratios measure the availability of cash to pay debt.

Current ratio^[18]

$\frac{\mbox{Current Assets}}{\mbox{Current Liabilities}}$

Acid-test ratio (Quick ratio)^[18]

$\frac{\mbox{Current Assets -- (Inventories + Prepayments)}}{\mbox{Current Liabilities}}$

Operation cash flow ratio

$\frac{\mbox{Operation Cash Flow}}{\mbox{Total Debts}}$

Activity ratios (Efficiency Ratios)

Activity ratios measure the effectiveness of the firms use of resources.

Average collection period^[3]

$\frac{\mbox{Accounts Receivable}}{\mbox{Annual Credit Sales ÷ 365 Days}}$

Degree of Operating Leverage (DOL)

$\frac{\mbox{Percent Change in Net Operating Income}}{\mbox{Percent Change in Sales}}$

DSO Ratio^[19]

$\frac{\mbox{Accounts Receivable}}{\mbox{Total Annual Sales ÷ 365 Days}}$

Average payment period^[3]

$\frac{\mbox{Accounts Payable}}{\mbox{Annual Credit Purchases ÷ 365 Days}}$

Asset turnover^[20]

$\frac{\mbox{Net Sales}}{\mbox{Total Assets}}$

Stock turnover ratio^[21]^[22]

$\frac{\mbox{COGS}}{\mbox{Average Inventory}}$

Receivables Turnover Ratio^[23]

$\frac{\mbox{Net Credit Sales}}{\mbox{Average Net Receivables}}$

Inventory conversion ratio^[4]

$\frac{\mbox{365 Days}}{\mbox{Inventory Turnover}}$

Inventory conversion period

$\left (\frac{\mbox{Inventory}}{\mbox{COGS}}\right)\mbox{365 Days}$

Receivables conversion period

$\left (\frac{\mbox{Receivables}}{\mbox{Net Sales}}\right)\mbox{365 Days}$

Payables conversion period

$\left (\frac{\mbox{Purchases}}{\mbox{Accounts Payable}}\right)\mbox{365 Days}$

Cash Conversion Cycle

Inventory Conversion Period + Receivables Conversion Period - Payables Conversion Period

Debt ratios (leveraging ratios)

Debt ratios measure the firm's ability to repay long-term debt. Debt ratios measure financial leverage.

Debt ratio^[24]

$\frac{\mbox{Total Liabilities}}{\mbox{Total Assets}}$

Debt to equity ratio^[25]

$\frac{\mbox{Long-term Debt + Value of Leases}}{\mbox{Average Shareholders Equity}}$

Long-term Debt to equity (LT Debt to Equity)^[25]

$\frac{\mbox{Long-term Debt}}{\mbox{Total Assets}}$

Times interest-earned ratio^[25]

$\frac{\mbox{EBIT}}{\mbox{Annual Interest Expense}}$

$\frac{\mbox{Net Income}}{\mbox{Annual Interest Expense}}$

Debt service coverage ratio

$\frac{\mbox{Net Operating Income}}{\mbox{Total Debt Service}}$

Market ratios

Market ratios measure investor response to owning a company's stock and also the cost of issuing stock.

Earnings per share (EPS)^[26]

$\frac{\mbox{Net Earnings}}{\mbox{Number of Shares}}$

Payout ratio^[26]^[27]

$\frac{\mbox{Dividends}}{\mbox{Earnings}}$

$\frac{\mbox{Dividends}}{\mbox{EPS}}$

Dividend cover (the inverse of Payout Ratio)

$\frac{\mbox{Earnings per Share}}{\mbox{Dividend per Share}}$

P/E ratio

$\frac{\mbox{Market Price per Share}}{\mbox{Diluted EPS}}$

Dividend yield

$\frac{\mbox{Dividend}}{\mbox{Current Market Price}}$

Cash flow ratio or Price/cash flow ratio^[28]

$\frac{\mbox{Market Price per Share}}{\mbox{Present Value of Cash Flow per Share}}$

Price to book value ratio (P/B or PBV)^[28]

$\frac{\mbox{Market Price per Share}}{\mbox{Balance Sheet Price per Share}}$

Price/sales ratio

$\frac{\mbox{Market Price per Share}}{\mbox{Gross Sales}}$

PEG ratio

$\frac{\mbox{Price per Earnings}}{\mbox{Annual EPS Growth}}$

Other Market Ratios

EV/EBITDA

$\frac{\mbox{Enterprise Value}}{\mbox{EBITDA}}$

EV/Sales

$\frac{\mbox{Enterprise Value}}{\mbox{Net Sales}}$

Cost/Income ratio

Sector-specific ratios

EV/capacity

EV/output

Capital Budgeting Ratios

Main article: Capital budgeting

In addition to assisting management and owners in diagnosing the financial health of their company, ratios can also help managers make decisions about investments or projects that the company is considering to take, such as acquisitions, or expansion.

Many formal methods are used in capital budgeting, including the techniques such as

References

^ Groppelli, Angelico A.; Ehsan Nikbakht (2000). Finance, 4th ed. Barron's Educational Series, Inc.. pp. 433. ISBN 0764112759.
^ Groppelli, p. 434.
^ ^a ^b ^c Groppelli, p. 436.
^ ^a ^b Groppelli, p. 439.
^ Groppelli, p. 442.
^ ^a ^b Groppelli, p. 445.
^ Williams, P. 265.
^ ^a ^b Williams, p. 1094.
^ Williams, Jan R.; Susan F. Haka, Mark S. Bettner, Joseph V. Carcello (2008). Financial & Managerial Accounting. McGraw-Hill Irwin. pp. 266. ISBN 9780072996500.
^ http://www.investorwords.com/3460/operating_income.html Operating income definition
^ Groppelli, p. 443.
^ Bodie, Zane; Alex Kane and Alan J. Marcus (2004). Essentials of Investments, 5th ed. McGraw-Hill Irwin. pp. 459. ISBN 0072510773.
^ Professor Cram. "Ratios of Profitability: Profit Margin" College-Cram.com. 14 May 2008 <http://www.college-cram.com/study/finance/presentations/104>
^ ^a ^b Groppelli, p. 444.
^ Professor Cram. "Ratios of Profitability: Return on Assets" College-Cram.com. 14 May 2008 <http://www.college-cram.com/study/finance/presentations/107>
^ Professor Cram. "Ratios of Profitability: Return on Assets Du Pont" College-Cram.com. 14 May 2008 <http://www.college-cram.com/study/finance/presentations/112>
^ Weston, J. (1990). Essentials of Managerial Finance. Hinsdale: Dryden Press. p. 295. ISBN 0030307333.
^ ^a ^b Groppelli, p. 435.
^ Professor Cram. "Ratios of Asset Management Study Sheet" College-Cram.com. 14 May 2008 <http://www.college-cram.com/study/finance/presentations/275>
^ Bodie, p. 459.
^ Groppelli, p. 438.
^ Weygandt, J. J., Kieso, D. E., & Kell, W. G. (1996). Accounting Principles (4th ed.). New York, Chichester, Brisbane, Toronto, Singapore: John Wiley & Sons, Inc. p. 801-802.
^ Weygandt, J. J., Kieso, D. E., & Kell, W. G. (1996). Accounting Principles (4th ed.). New York, Chichester, Brisbane, Toronto, Singapore: John Wiley & Sons, Inc. p. 800.
^ Groppelli, p. 440; Williams, p. 640.
^ ^a ^b ^c Groppelli, p. 441.
^ ^a ^b Groppelli, p. 446.
^ Groppelli, p. 449.
^ ^a ^b Groppelli, p. 447.

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