CFA Level 1 - A1 - Formulas

Permutation

An ordered listing is known as a permutation, and the formula that counts the number of permutations is known as the permutation formula. The number of ways that we can choose r objects from a total of n objects, where the order in which the r objects is listed does matter, is:

Annual Sales Based On Average Account Receivables

Annual Sales: = (365/(Receivables Turnover))*(Average Account Receivable) = (Days Sales Outstanding) * (Average Account Receivable)

Asset Turnover Ratio

Asset turnover ratio measures the value of a company's sales or revenues generated relative to the value of its assets. The Asset Turnover ratio can often be used as an indicator of the efficiency with which a company is deploying its assets in generating revenue. Asset Turnover = Sales / Average Total Assets BREAKING DOWN 'Asset Turnover Ratio' Asset turnover ratio is typically calculated over an annual basis using either the fiscal or calendar year. The total assets number used in the denominator can be calculated by taking the average of assets held by a company at the beginning of the year and at the year's end. Generally speaking, the higher the asset turnover ratio, the better the company is performing, since higher ratios imply that the company is generating more revenue per dollar of assets. The asset turnover ratio tends to be higher for companies in certain sectors than in others. Retail and consumer staples, for example, have relatively small asset bases but have high sales volume and, thus, often yield the highest asset turnover ratio. Conversely, firms in sectors, such as utilities and telecommunications, which have large asset bases will have lower asset turnover. Since this ratio can vary widely from one industry to the next, considering the asset turnover ratios of a retail company and a telecommunications company will not make for an accurate comparison. Comparisons are only meaningful when they are made for different companies within the same sector. Practical Example of the Asset Turnover Ratio Let's calculate the asset turnover ratio for four companies in the retail and telecommunications sector - Wal-Mart Stores Inc., Target Corporation, AT&T Inc., and Verizon Communications - for the fiscal year ended 2016. Asset Turnover Ratio Wal-Mart Stores - 2.30 Target Corp. - 1.79 AT&T Inc. - 0.41 Verizon - 0.52 For every dollar in assets, Wal-Mart generated $2.30 in sales, while Target generated $1.79. Target's turnover may indicate that the retail company is experiencing sluggish sales or holding obsolete inventory. Furthermore, its low turnover may also mean that the company has lax collection methods. The firm's collection period may be too long, leading to a higher accounts receivable. Target could also not be using its assets efficiently - for example, fixed assets such as property or equipment could be sitting idle or not being utilized to their full capacity. AT&T and Verizon have asset turnover ratios of less than 1, which is typical for firms in the telecommunications sector. Since these companies have large asset bases, it is expected that they would slowly turnover their assets through sales. From the table, it is clear that Verizon turns over its assets at a faster rate than AT&T. Clearly, it would not make much sense to compare the asset turnover ratios for Wal-Mart and AT&T, since they operate in very different industries. But comparing the asset turnover ratios for AT&T and Verizon Communications Inc. (VZ), for instance, may provide a clearer picture of asset use efficiency for these telecom companies. Limitations of the Asset Turnover Ratio: While the asset turnover ratio should be used to compare apples to apples and oranges to oranges, this kind of comparison does not necessarily paint the clearest possible picture. It is possible that a company's asset turnover ratio in any single year differs substantially from previous or subsequent years. For any specific company, then, one would do well to review the trend in the asset turnover ratio over a period of time to check whether asset usage is improving or deteriorating. The asset turnover ratio may be artificially deflated when a company makes large asset purchases in anticipation of a higher growth. Likewise, selling off assets to prepare for declining growth will artificially inflate the ratio.

Present Value Formula

In order to receive a single future cash flow N years from now, you must make an investment today in the following amount: Notice that the future cash flow is discounted back to the present. Therefore, the interest rate is called the discount rate. You should be able to calculate PVs and FVs using your calculator. * N = number of periods * I/Year = yield in market place or the required rate of return * PV = present value * PMT = payment amount per period * FV = the future value of the investment

GDP Equations

GDP = C + I + G + (X - M) Saving and Investment Saving can be absorbed in three different ways. Firstly, it can be absorbed into Investment spending (I). Secondly, it can be absorbed into financing government deficits (G - T). Lastly, it can be absorbed into building up financial claims for or against other economies. For a positive trade balance, (X - M) > 0. However, for a trade deficit, (X - M) < 0. This implies that domestic saving can be supplemented by inflowing foreign saving; hence overseas economies build up financial claims against the domestic economy. Therefore, the savings equation can be written as: S = I + (G -T) + (X - M) Fiscal Policy Rearranging the saving function gives us an equation for the fiscal deficit: G - T = (S - I) - (X - M) A fiscal deficit shows that the private sector requires increasing its saving and reducing investment. In this case, (S - I) > 0. Alternatively, the country might resolve to run a trade deficit with respect to corresponding inflow in foreign saving, such that (X - M) < 0.

Growth Rate

Growth Rate: =(Return on Equity) * (Retention Rate)

Future Value Formula

FV = future value at time n PV = present value r = interest rate per period N = number of years

Test Statistic For the Mean of a Distribution

For example, a test statistic for the mean of a distribution (such as the mean monthly return for a stock index) often follows a standard normal distribution. In such a case, the test statistic requires use of the z-test, P(Z <= test statistic = z). Where: X-bar = sample mean μ = hypothesized value σ = sample standard deviation n = sample size

Combination Formula AKA the Binomial Formula

For example, if you select two of the ten stocks you are analyzing, how many ways can you select the stocks? = 10! / [(10 - 2)! x 2!] = 45.

Continuous Probability Distribution.

For example, the possible outcomes are the integers 1 to 8 (inclusive), and the probability that the random variable takes on any of those possible values is the same for all outcomes (i.e., it is uniform). If a continuous random variable is equally likely to fall at any point between its maximum and minimum values, it is a continuous uniform random variable, and its probability distribution is a continuous probability distribution. The probability density function is a horizontal line with a height of 1/(b-a) over a range of values from a to b. The cumulative density function is a sloped line with a height of 0 to 1 over a range of values from a to b, and is a horizontal line with a height of 1 when the value of the variable equals or exceeds b.

Continuous Uniform Random Variable

For example, the possible outcomes are the integers 1 to 8 (inclusive), and the probability that the random variable takes on any of those possible values is the same for all outcomes (i.e., it is uniform). If a continuous random variable is equally likely to fall at any point between its maximum and minimum values, it is a continuous uniform random variable, and its probability distribution is a continuous probability distribution. The probability density function is: f(x) = 1/(b - a) for a ≤ x ≤ b; or 0 otherwise.

Standard Normal Distribution Formula - Z-Score Formula

Normal distributions can be transformed to standard normal distributions by the formula: where X is a score from the original normal distribution, μ is the mean of the original normal distribution, and σ is the standard deviation of original normal distribution.

Cash Conversion Cycle (Net Operating Cycle)

The cash conversion cycle (CCC) is a metric that expresses the length of time, in days, that it takes for a company to convert resource inputs into cash flows. The cash conversion cycle attempts to measure the amount of time each net input dollar is tied up in the production and sales process before it is converted into cash through sales to customers. This metric looks at the amount of time needed to sell inventory, the amount of time needed to collect receivables, and the length of time the company is afforded to pay its bills without incurring penalties. The cash conversion cycle is calculated as: CCC = DIO + DSO - DPO where DIO = days inventory outstanding DSO = days sales outstanding DPO = days payable outstanding The CCC is also referred to as the cash cycle. BREAKING DOWN 'Cash Conversion Cycle - CCC' Usually a company acquires inventory on credit, which results in accounts payable. A company can also sell products on credit, resulting in accounts receivable (A/R). Cash, therefore, is not a factor until the company pays the accounts payable and collects the accounts receivable. The cash conversion cycle (CCC) measures the time between the cash outlay and the cash receipt. The cash conversion cycle cannot be observed directly in cash flows, which are affected by financing and investment activities as well; rather, the cycle refers to the time span between a firm's disbursement and collection of cash. Days Inventory Outstanding (DIO) refers to the number of days it takes to sell an entire inventory. A smaller DIO is preferred as it would mean the company is making sales quickly. Days Sales Outstanding (DSO) refers to the number of days needed to collect on sales, or accounts receivable. A smaller DSO is also preferred as it would mean that the company takes fewer days to collect on its A/R. Days Payable Outstanding (DPO) refers to the company's payment of its own bills, or accounts payable. By maximizing this number, the company holds onto cash longer, increasing its investment potential. Thus, a longer DPO is preferred. What It Means The cash conversion cycle is a metric used to gauge the effectiveness of a company's management and, consequently, the overall health of that company. The calculation measures how fast a company can convert cash on hand into inventory and accounts payable, through sales and accounts receivable, and then back into cash. By combining these activity ratios, the measurement indicates the efficiency of the management's ability to employ short-term assets and liabilities to generate cash for the company. The CCC entails the liquidity risk associated with growth by measuring the length of time that a firm will be deprived of cash if it increases its investment in resources in an effort to elevate sales. It can be especially useful for investors who wish to draw a comparison between close competitors, as a low CCC signifies a well-managed company, and thus can be used to help evaluate potential investments. The CCC should be combined with other fundamentals, such as the return on equity (ROE) and return on assets (ROA), as an indicator of management effectiveness and company viability. While the cash conversion cycle applies to companies in any industry, the cycle is extremely important for retailers and similar businesses, as their operations consist of buying inventories and selling them to customers. The metric does not apply to companies for which this is not the case, such as those in the software or insurance industries. The cash conversion cycle illustrates how quickly a company can convert its products into cash through sales. The shorter the cycle, the less time capital is tied up in the business process, and thus the better for the company's bottom line. An important distinction is that the cycle applies to firms that buy and sell on account, while cash-only firms only accommodate data from sales operations in the equation, as their disbursed cash is directly measurable as purchase of inventory, and their collected cash is measurable as sale of inventory. This direct ratio does not exist for firms that buy and sell on account. Changes in inventory occasion payables and receivables rather than cash flows, and increases and decreases in cash will discount these accounting vehicles from statements. Therefore, the CCC is calculated according to the cycle of cash through receivables, inventory, payables and, eventually, back to cash.

Standard Normal Distribution Graph

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is denoted as N(0,1). Below are some confidence intervals for the standard normal distribution:

z-Test Formula for a Single Population Mean (μ)

The z-test should be used if the population variance is normally distributed with known variance. For a z-test concerning a single population mean (μ), the test statistic to be used is the z-statistic with n degrees of freedom.

t-Test Formula for a Single Population Mean (μ)

This test should be used if the population variance is unknown and either of the following conditions holds: 1.) The sample size is large (in general, n ≥ 30). 2.) The sample size is small (n ≤ 30) but the population is normally distributed or approximately normally distributed. For a t-test concerning a single population mean (μ), the test statistic to be used is the t-statistic with n - 1 degrees of freedom.

Chi-square graph

Unlike t-graphs and z-graphs, a chi-square graph is positively skewed. It is also truncated at zero, and thus is not defined for negative values. Like the family of t-graphs, the shape of the graph varies; the graph becomes more symmetrical as the degrees of freedom increase.

Standardized Value Formula

X = σZ + μ

Test statistic for a test of the difference between two population means (normally distributed populations, unequal and unknown population variances):

where s12 = [(n - 1)(s1^2) + (n - 1)(s2^2)] /(n1 + n2 - 2)

Cumulative Distribution Function (CDF)

A cumulative distribution function (cdf) gives the probability that a random variable X is less than or equal to a particular value x, P(X≤x). In contrast, a probability function is used to find the probability of a specific outcome. To derive a cumulative distribution function F(x), simply sum the values of the probability function for all outcomes less than or equal to x. Consider a probability function: p(X) = X/6 for X = 1, 2, 3 and p(X) = 0 otherwise. In a previous example it was shown that p(1) = 1/6, p(2) = 2/6, and p(3) = 3/6. 1.) F(1) indicates the probability that has been accumulated up to and including the point X = 1. Clearly, 1/6 of probability has been accumulated up to this point, so F(1) = 1/6. 2.) F(2) indicates the probability that has been accumulated up to and including the point X = 2. When X = 2 is reached, the accumulation of 1/6 is taken from X = 1 and 2/6 from X = 2; in total accumulation is 1/6 + 2/6 = 3/6 or, of the probability, so F(2) = 3/6. 3.) F(3) indicates the probability that has been accumulated up to and including the point X = 3. By the time X = 3 is reached, all the probability has been accumulated: 1/6 from X = 1, 2/6 from X = 2 and 3/6 from X = 3. Thus, 1/6 + 2/6 + 3/6 = 1. Therefore, F(3) = 1. It is also possible to calculate F(X) for intermediate values. F(0) = 0, as no probability has been accumulated up to the point X = 0; F(1.5) = 1/6, as by the time X = 1.5 is reached, 1/6 of probability has been accumulated from X = 1; F(7) = 1, as by the time 7 is reached, all possible probability from X = 1, 2 and 3 has been collected.

Test Statistic Formula

A test statistic is simply a number, calculated from a sample, whose value, relative to its probability distribution, provides a degree of statistical evidence against the null hypothesis. In many cases, the test statistic will not provide evidence sufficient to justify rejecting the null hypothesis. However, sometimes the evidence will be strong enough so that the null hypothesis is rejected and the alternative hypothesis is accepted instead. The value of the test statistic is the focal point of assessing the validity of a hypothesis. Typically, the test statistic will be of the general form:

Working Capital Turnover Ratio

DEFINITION of 'Working Capital Turnover' Working capital turnover is a measurement comparing the depletion of working capital used to fund operations and purchase inventory, which is then converted into sales revenue for the company. The working capital turnover ratio is used to analyze the relationship between the money that funds operations and the sales generated from these operations. For example, a company with current assets of $10 million and current liabilities of $9 million has $1 million in working capital, which may be used in fundamental analysis. BREAKING DOWN 'Working Capital Turnover' The working capital turnover ratio measures how well a company is utilizing its working capital for supporting a given level of sales. Because working capital is current assets minus current liabilities, a high turnover ratio shows that management is being very efficient in using a company's short-term assets and liabilities for supporting sales. In contrast, a low ratio shows a business is investing in too many accounts receivable (AR) and inventory assets for supporting its sales. This may lead to an excessive amount of bad debts and obsolete inventory. Calculating Working Capital Turnover When calculating a company's working capital turnover ratio, the amount of net sales is divided by the amount of working capital. The calculation is typically made on an annual or trailing 12-month basis and uses the average working capital during that time period. For example, Company A has $12 million of net sales over the past 12 months. The average working capital during that time was $2 million. The calculation of its working capital turnover ratio is $12,000,000/$2,000,000 = 6. Pros and Cons of High Working Capital Turnover A high working capital turnover ratio shows a company is running smoothly and has limited need for additional funding. Money is coming in and flowing out on a regular basis, giving the business flexibility to spend capital on expansion or inventory. A high ratio may also give the business a competitive edge over similar companies. However, an extremely high ratio, typically over 80%, may indicate a business does not have enough capital supporting its sales growth. Therefore, the company may become insolvent in the near future. The indicator is especially strong when the accounts payable (AP) component is very high, indicating that management cannot pay its bills as they come due. For example, gold mining and silver mining have average working capital turnover ratios of approximately 82%. Gold and silver mining requires ongoing capital investment for replacing, modernizing and expanding equipment and facilities, as well as finding new reserves. An excessively high turnover ratio may be discovered by comparing the ratio for a specific business to ratios reported by other companies in the industry. Investopedia

Days Sales Outstanding

Days Sales Outstanding =365/(Receivables Turnover)

Net Operating Cycle (Cash Conversion Cycle)

Days of inventory in hand + Days of sales outstanding - Number of days of payables = Net operating cycle:

Days' Inventory on Hand Ratio

Days' Inventory on Hand Ratio Days' inventory on hand (also called days' sales in inventory or simply days of inventory) is an accounting ratio which measures the number of days a company takes to sell its average balance of inventory. It is also an estimate of the number of days for which the average balance of inventory will be sufficient. Days' sales in inventory ratio is very similar to inventory turnover ratio and both measure the efficiency of a business in managing its inventory. Formula Days' inventory on hand is usually calculated by dividing the number of days in a period by inventory turnover ratio for the period as shown in the following formula: Days of Inventory: = (Number of Days in the Period)/(Inventory Turnover for the Period) Thus, if we have inventory turnover ratio for the year, we can calculate days' inventory on hand by dividing number of days in a year i.e. 365 by inventory turnover. If we substitute inventory turnover as "cost of goods sold ÷ average inventory" in the above formula and simplify the equation, we get: Days of Inventory = (Average Inventory/Cost of Goods Sold) × Number of Days in the Period Analysis Since inventory carrying costs take significant investment, a business must try to reduce the level of inventory. Lower level of inventory will result in lower days' inventory on hand ratio. Therefore lower values of this ratio are generally favorable and higher values are unfavorable. However, inventory must be kept at safe level so that no sales are lost due to stock-outs. Thus low value of days of inventory ratio of a company which finds it difficult to satisfy demand is not favorable. Days' sales in inventory varies significantly between different industries. For example, business which sell perishable goods such as fruits and vegetables have very low values of days' sales in inventory whereas companies selling non-perishable goods such as cars have high values of days of inventory.

DuPont Analysis / Formula (Modified ROE)

DuPont analysis is a fundamental performance measurement framework popularized by the DuPont Corporation and is also referred to as the "DuPont identity." DuPont analysis is a useful technique used to decompose the different drivers of the return on equity (ROE). Decomposition of ROE allows investors to focus their research on the distinct company performance indicators otherwise cursory evaluation. BREAKING DOWN 'DuPont Analysis' According to DuPont analysis, there are three major financial metrics drive return on equity (ROE): operating efficiency, asset use efficiency and financial leverage. Operating efficiency is represented by net profit margin or net income divided by average shareholders' equity. Asset use efficiency is measured by total asset turnover or the asset turnover ratio. Finally, financial leverage is analyzed through observation of changes in the equity multiplier. DuPont Analysis Components DuPont analysis breaks ROE into its constituent components to determine which of these components is most responsible for changes in ROE. Net margin: Expressed as a percentage of the total revenue, net margin is the revenue that remains after subtracting all operating expenses, taxes, interest and preferred stock dividends from a company's total revenue. Asset turnover ratio: This ratio is an efficiency measurement used to determine how effectively a company uses its assets to generate revenue. The formula for calculating asset turnover ratio is total revenue divided by total assets. As a general rule, the higher the resulting number, the better the company is performing. Equity multiplier: This ratio measures financial leverage. By comparing total assets to total stockholders' equity, the equity multiplier indicates whether a company finances the purchase of assets primarily through debt or equity. The higher the equity multiplier, the more leveraged the company, or the more debt it has in relation to its total assets. DuPont analysis involves examining changes in these figures over time and matching them to corresponding changes in return on equity (ROE). By doing so, analysts can determine whether operating efficiency, asset use efficiency or leverage is most responsible for return on equity (ROE) variations.

Hypothesis Tests Concerning Differences between Means

In practice, analysts often want to know whether the means of two populations are equal or whether one is larger than the other. The test statistic to be used in this section is a t-value, but it varies based on the assumptions. The assumption has been made throughout that the population means are normally distributed. 1. Test statistic for a test of the difference between two population means (normally distributed populations, population variances unknown but assumed equal): where sp2 = [(n - 1)(s1^2) + (n - 1)(s2^2)] /(n1 + n2 - 2)

Return on Equity (ROE)

Return on equity (ROE) is the amount of net income returned as a percentage of shareholders equity. Return on equity measures a corporation's profitability by revealing how much profit a company generates with the money shareholders have invested. ROE is expressed as a percentage and calculated as: Return on Equity = Net Income/ Shareholder's Equity Net income is for the full fiscal year (before dividends paid to common stock holders but after dividends to preferred stock.) Shareholder's equity does not include preferred shares. Also known as "return on net worth" (RONW). The ROE is useful for comparing the profitability of a company to that of other firms in the same industry. It illustrates who effective the company is at turning the cash put into the business into greater gains and growth for the company and investors. The higher the return on equity, the more efficient the company's operations are making use of those funds. How Return on Equity is Determined There are several variations on the formula that investors may use: 1. Investors wishing to see the return on common equity may modify the formula above by subtracting preferred dividends from net income and subtracting preferred equity from shareholders' equity, giving the following: return on common equity (ROCE) = net income - preferred dividends / common equity. 2. Return on equity may also be calculated by dividing net income by average shareholders' equity. Average shareholders' equity is calculated by adding the shareholders' equity at the beginning of a period to the shareholders' equity at period's end and dividing the result by two. 3. Investors may also calculate the change in ROE for a period by first using the shareholders' equity figure from the beginning of a period as a denominator to determine the beginning ROE. Then, the end-of-period shareholders' equity can be used as the denominator to determine the ending ROE. Calculating both beginning and ending ROEs allows an investor to determine the change in profitability over the period. Things to Remember: 1.) If new shares are issued then use the weighted average of the number of shares throughout the year. 2.) For high growth companies you should expect a higher ROE. 3.) Averaging ROE over the past 5 to 10 years can give you a better idea of the historical growth.

z-Test versus t-Test Summary

Summary 1.) In practice, the population variance is typically unknown. 2.) The table below summarizes tests concerning the population mean when the population has unknown variance.

Chi-square Test For Variance or standard Deviation

Suppose an analyst is interested in testing whether the variance from a single population is statistically equal to some hypothesized value. Let σ^2 represent the variance and let σ0^2 represent the hypothesized value. The null and alternative hypotheses would be expressed as: H0 : σ^2 = σ0^2 , versus H1: σ^2 ≠ σ0^2 Also note that directional hypotheses could be made instead: H0: σ^2 ≤ (σ0)^2, versus H1: σ^2 > (σ0)^2, or H0: σ^2 ≥ (σ0)^2, versus H1: σ^2 < (σ0)^2 The test statistic to be used is a chi-square (χ^2) statistic with n-1 degrees of freedom.

Cash Reinvestment Ratio

The cash reinvestment ratio is used to estimate the amount of cash flow that management reinvests in a business. While a high cash reinvestment ratio might initially appear to indicate that management is committed to improving the business, it could also mean that an excessive amount of investment in fixed assets and working capital is required to run the operation. Thus, the measure can be misleading, unless coupled with other metrics to obtain a more complete picture of company operations. In particular, compare the company's ratio of fixed assets to revenues to those of well-run companies in the industry, as well as the ratio of working capital to revenues. If these ratios indicate better performance by the peer group, there is a strong likelihood that the subject company is investing more cash than necessary. The formula for the cash reinvestment ratio requires you to summarize all cash flows for the period, deduct dividends paid, and divide the result into the incremental increase during the period in fixed assets and working capital. Additional points regarding the formula are: Fixed asset sales. If any fixed assets are sold during the measurement period, factor out the impact of the sale. Working capital elimination. A variation on the formula is to exclude working capital changes from the numerator. Doing so focuses attention solely on new fixed asset additions.

Defensive Interval Ratio

The defensive interval ratio (DIR), also called the defensive interval period (DIP) or basic defense interval (BDI), is a financial metric that indicates the number of days that a company can operate without needing to access noncurrent assets, long-term assets whose full value cannot be obtained within the current accounting year, or additional outside financial resources. The DIR is sometimes viewed as a financial efficiency ratio, but is most commonly considered a liquidity ratio. BREAKING DOWN 'Defensive Interval Ratio' The formula for calculating the DIR is: DIR (expressed as number of days) = current assets / daily operational expenses Current assets = cash + marketable securities + net receivables Daily operational expenses = (annual operating expenses - noncash charges) / 365 The DIR is considered by some market analysts to be a more useful liquidity ratio than the standard quick ratio or current ratio due to the fact that it compares assets to expenses rather than comparing assets to liabilities. The DIR is commonly used as a supplementary financial analysis ratio, along with the current or quick ratio, to evaluate a company's financial health, since there can be substantially different DIR and quick or current ratio values if, for example, a company has a large amount of expenses but little or no debt. The DIR is called the defensive interval ratio because its calculation involves a company's current assets, which are also known as defensive assets. Defensive assets consist of cash, cash equivalents such as bonds or other investments, and other assets that can readily be converted to cash such as accounts receivables (AR). For example, if a company has $100,000 cash on hand, $50,000 worth of marketable securities, and $50,000 in accounts receivables, it has a total of $200,000 in defensive assets. If the company's daily operational expenses equal $5,000, the DIR value is 40 days - 200,000 / 5,000. Importance of the Defensive Interval Ratio The DIR is a helpful tool in evaluating a company's financial health because it provides the real world metric of how many days the company can operate in terms of meeting daily operational expenses without running into any financial difficulty that would likely require it to access additional funds through either new equity investment, a bank loan or the sale of long-term assets. In that respect, it can be considered a more useful liquidity measure to examine than the current ratio, which, while providing a clear comparison of a company's assets to its liabilities, does not give any definitive indication of how long a company can function financially without encountering significant problems in terms of simply operating day to day.

Discrete Uniform Distribution

The discrete uniform distribution is the simplest of all probability distributions. This distribution has a finite number of specified outcomes, and each outcome is equally likely. Mathematically, suppose that a discrete uniform random variable, X, has n possible outcomes: x1, x2, ..., xn-1, and xn. 1.) p(x ) = p(x ) = p(x ) = ... = p(x ) = p(x ) = p(x). That is, the probabilities for all possible outcomes are equal. 2.) F(x ) = kp(x ). That is, the cumulative distribution function for the k outcome is k times of the probability of the k outcome. 3.) If there are k possible outcomes in a particular range, the probability for that range of outcomes is kp(X).

Free Cash Flow To Equity (FCFE)

What is 'Free Cash Flow To Equity - (FCFE)' Free cash flow to equity (FCFE) is a measure of how much cash is available to the equity shareholders of a company after all expenses, reinvestment, and debt are paid. FCFE is a measure of equity capital usage. It is calculated as: FCFE = Net Income - Net Capital Expenditure - Change in Net Working Capital + New Debt - Debt Repayment. BREAKING DOWN 'Free Cash Flow To Equity - FCFE' Free cash flow to equity (FCFE) is often used by analysts in an attempt to determine the value of a company. This method of valuation gained popularity as an alternative to the dividend discount model (DDM), especially if a company does not pay a dividend. Although FCFE may calculate the amount available to shareholders, it does not necessarily equate to the amount paid out to shareholders. Components of the FCFE Specifically, free cash flow to equity is composed of net income, capital expenditures, working capital, and debt. Net income is located on the company income statement. Capital expenditures can be found within the cash flows from investing section on the cash flow statement. Working capital is also found on the cash flow statement; however, it is in the cash flows from operations section. In general, working capital represents the difference between the company's most current assets and liabilities. These are short-term capital requirements related to immediate operations. Net borrowings can also be found on the cash flow statement in the cash flows from financing section. It is important to remember that interest expense is already included in net income so you do not need to add back interest expense. How to Interpret the FCFE Analysts also use FCFE to determine if dividend payments and stock repurchases are paid for with free cash flow to equity or some other form of financing. Investors want to see a dividend payment and share repurchase that is fully paid by FCFE. If FCFE is less than the dividend payment and the cost to buy back shares, the company is funding with either debt or existing capital, or issuing new securities. Existing capital includes retained earnings made in previous periods. This is not what investors want to see in a current or prospective investment, even if interest rates are low. Some analysts argue that borrowing to pay for share repurchases when shares are trading at a discount and rates are historically low is a good investment. However, this is only the case if the company's share price goes up in the future. If the company's dividend payment funds is significantly less than the FCFE, then the firm is using the excess to increase its cash level or to invest in marketable securities. Finally, if the funds spent to buy back shares or pay dividends is approximately equal to the FCFE, then the firm is paying it all to its investors.

Debt to Assets Ratio

What is 'Total Debt to Total Assets' Total debt to total assets is a leverage ratio that defines the total amount of debt relative to assets. This metric enables comparisons of leverage to be made across different companies. The higher the ratio, the higher the degree of leverage (DoL) and, consequently, financial risk. The total debt to total assets is a broad ratio that includes long-term and short-term debt (borrowings maturing within one year), as well as all assets - tangible and intangible. BREAKING DOWN 'Total Debt to Total Assets' Total debt to total assets is a measure of the company's assets that are financed by debt, rather than equity. This leverage ratio shows how a company has grown and acquired its assets over time. Investors use the ratio to not only evaluate whether the company has enough funds to meet its current debt obligations, but to also assess whether the company can pay a return on their investment. Creditors use the ratio to see how much debt the company already has and if the company has the ability to repay its debt, which will determine whether additional loans will be extended to the firm. Interpreting the Total Debt to Total Assets Ratio A ratio greater than 1 shows that a considerable portion of debt is funded by assets. In other words, the company has more liabilities than assets. A high ratio also indicates that a company may be putting itself at a risk of default on its loans if interest rates were to rise suddenly. A ratio below 1 translates to the fact that a greater portion of a company's assets is funded by equity. From the example above, Sears has a much higher degree of of leverage than Disney and Chipotle, and therefore, a lower degree of financial flexibility. With more than $13 billion in total debt, there is a high chance of Sears declaring bankruptcy in the following months. Investors and creditors will consider Sears a risky company to invest in and loan to due to its very high leverage. This is because debt servicing payments have to be made under all circumstances, otherwise the company would breach debt covenants and run the risk of being forced into bankruptcy by creditors. While other liabilities, such as accounts payable and long-term leases, can be negotiated to some extent, there is very little "wiggle room" with debt covenants. Therefore, a company with a high degree of leverage may find it more difficult to stay afloat during a recession than one with low leverage. It should be noted that total debt measure does not include short-term liabilities such as accounts payable and long-term liabilities such as capital lease and pension plan obligations. One shortcoming of the total debt to total assets ratio is that it does not provide any indication of asset quality, since it lumps all tangible and intangible assets together. For example, assume Disney took on $50.8 billion of long-term debt to acquire a competitor, and booked $20 billion as goodwill for this acquisition. Let's say the acquisition does not perform as expected and results in all the goodwill being written off. In this case, the ratio of total debt to total assets (which would now be $95.8 billion - $20 billion = $75.8 billion) would be 0.67. Like all other ratios, the trend of the total debt to total assets should also be evaluated over time. This will help assess whether the company's financial risk profile is improving or deteriorating. For example, an increasing trend indicates that a business is unwilling or unable to pay down its debt, which could indicate a default at some point in the future.

Fixed-Charge Coverage Ratio

What is the 'Fixed-Charge Coverage Ratio' The fixed-charge coverage ratio (FCCR) measures a firm's ability to satisfy fixed charges, such as interest expense and lease expense. Since leases are a fixed charge, the calculation for determining a company's ability to cover fixed charges includes earnings before interest and taxes (EBIT), interest expense, lease expense and other fixed charges. It is calculated as: = [EBIT + Fixed Charge (Before Tax)]/ [Fixed Charge (Before Tax) + Interest] BREAKING DOWN 'Fixed-Charge Coverage Ratio' Also referred to as the solvency ratio, the fixed-charge ratio is commonly used by lenders attempting to analyze the amount of cash flow a company has available for debt repayment. A low ratio means a drop in earnings could be dire for the company, a situation lenders try to avoid. As a result, many lenders use coverage ratios, including the times-interest-earned ratio (TIE) and the fixed-charge ratio, to determine a company's ability to take on additional debt. A company that can cover its fixed charges at a faster rate than its peers is not only more efficient, but more profitable. This is a company that wants to borrow for growth rather than hardship, which is the kind of company in which most investors are looking to invest. Fixed vs. Variable Charges There are two main parts to any profit and loss statement: sales and the cost of sales. Some costs are variable and dependent on the volume of sales over a particular time period. Other costs are fixed and must be paid regardless of business activity. These are called fixed costs and include line items such as lease payments, insurance payments and preferred dividend payments. Interpretation and Usage The goal of the fixed-charge coverage ratio is to see how well earnings can cover fixed charges. This ratio is a lot like the TIE ratio, a popular ratio used to measure a company's ability to cover interest with EBIT. TIE is calculated by dividing EBIT by the total interest paid on debt. A TIE of five means the company's EBIT is able to cover the company's interest payment five times. The fixed-charge coverage ratio is a more conservative measure as it takes additional fixed charges, including lease expense, into consideration. The fixed-charge coverage ratio is slightly different from the TIE, though the same interpretation can be applied. The fixed-charge coverage ratio is calculated by adding EBIT to fixed charges before taxes, and then dividing by the total of interest expense and fixed charges before taxes. For example, say company A records EBIT of $300,000, fixed charges before taxes of $200,000 and $50,000 in interest expense. The calculation is $300,000 plus $200,000 divided by $50,000 plus $200,000, which is $500,000 divided by $250,000, or 2:1. The company's earnings are two times greater than its fixed costs, which is low. Like the TIE, the higher the ratio the better.

Interest Coverage Ratio

What is the 'Interest Coverage Ratio' The interest coverage ratio is a debt ratio and profitability ratio used to determine how easily a company can pay interest on its outstanding debt. The interest coverage ratio may be calculated by dividing a company's earnings before interest and taxes (EBIT) during a given period by the company's interest payments due within the same period. The method for calculating interest coverage ratio may be represented with the following formula: BREAKING DOWN 'Interest Coverage Ratio' Essentially, the interest coverage ratio measures how many times over a company could pay its current interest payment with its available earnings. In other words, it measures the margin of safety a company has for paying interest during a given period, which a company needs in order to survive future (and perhaps unforeseeable) any financial hardship that may arise. A company's ability to meet its interest obligations is an aspect of a company's solvency, and is thus a very important factor in the return for shareholders. To provide an example of how to calculate interest coverage ratio, suppose that a company's earnings during a given quarter are $625,000 and that it has debts upon which it is liable for payments of $30,000 every month. To calculate the interest coverage ratio here, one would need to convert the monthly interest payments into quarterly payments by multiplying them by three. The interest coverage ratio for the company is $625,000 / ($30,000 x 3) = $625,000 / $90,000 = 6.94. Staying above water with interest payments is a critical and ongoing concern for any company. As soon as a company struggles with this, it may have to borrow further or dip into its cash reserve, which is much better used to invest in capital assets or for emergencies. The lower a company's interest coverage ratio is, the more its debt expenses burden the company. When a company's interest coverage ratio is 1.5 or lower, its ability to meet interest expenses may be questionable. 1.5 is generally considered to be a bare minimum acceptable ratio for a company and the tipping point below which lenders will likely refuse to lend the company more money, as the company's risk for default may be perceived as too high. Moreover, an interest coverage ratio below 1 indicates the company is not generating sufficient revenues to satisfy its interest expenses. If a company's ratio is below 1, it will likely need to spend some of its cash reserves in order to meet the difference or borrow more, which will be difficult for reasons stated above. Otherwise, even if earnings are low for a single month, the company risks falling into bankruptcy. Generally, an interest coverage ratio of 2.5 is often considered to be a warning sign, indicating that the company should be careful not to dip further.

z-test and the Central Limit Theorem

When hypothesis testing a population mean, there are generally two options for the test statistic: The first statistic may be used if either the sample is large (n = 30 or greater) or, if n < 30, it may be used if the sample is at least approximately normally distributed. In most cases, this will be the statistic used, because in most practical problems, the population variance is not known with certainty. The second statistic is sometimes used with large sample sizes, because the central limit theorem implies that the distribution of a sample mean will be approximately normally distributed as the sample size increases. As can be seen above, when the degrees of freedom are low, the graph is fairly flat in the center and has long tails and a bigger standard deviation. As the degrees of freedom increase, the tails become narrower and flatter and the graph peaks in the center. Recall also that the area under any graph is 1, as this area represents a probability. So, as the degrees of freedom increase, the area that is "lost" in the flatter tails is "found" in the center of the graph, and that is why the graph becomes more peaked. However, 1.) There are differences between the t-test critical values and the z-test critical values (these can be significant but get smaller with large samples). 2.) The t-test is still the theoretically proper choice unless the population variance is known.