StarkNet的主要特点如下：

1. 高度可扩展：通过使用zk-STARKs，StarkNet可以处理大量的交易，显著提高以太坊的吞吐量。

2. 数据可用性：StarkNet将部分数据存储在链下，但保证链上数据可用性，以实现去中心化和安全性。

3. 安全性：StarkNet利用零知识证明技术确保交易的完整性和私密性，同时防止双重支付等攻击。

4. 兼容性：StarkNet与以太坊智能合约兼容，使得开发者可以轻松地将现有的dApps迁移到StarkNet平台。

5. 生态系统：StarkNet正在与其他Layer 2解决方案（如Optimism、zkSync等）合作，共同构建一个更加繁荣和可持续发展的去中心化生态系统。

总之，StarkNet作为一个基于zk-STARKs的Layer 2解决方案，旨在解决以太坊网络的拥堵问题，提高吞吐量和降低交易成本，为去中心化应用提供更好的基础设施。

A chief executive officer (CEO) is the highest-ranking executive in a company. Broadly speaking, a chief executive officer’s primary responsibilities include making major corporate decisions, managing the overall operations and resources of a company, acting as the main point of communication between the board of directors and corporate operations. In many cases, the chief executive officer serves as the public face of the company. 

The CEO is elected by the board and its shareholders. They report to the chair and the board, who are appointed by shareholders. 

KEY TAKEAWAYS

• The chief executive officer (CEO) is the highest-ranking person in a company.

• While every company differs, CEOs are often responsible for expanding the company, driving profitability, and in the case of public companies, improving share prices. CEOs manage the overall operations of a company.

• Across many companies, CEOs are elected by the board of directors.

• CEOs of the 350 largest companies in America earn on average $24 million, or 351 times more than an average employee.

• Studies suggest that 45% of company performance is influenced by the CEO, while others show that they affect 15% of the variance in profitability.

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What CEOs Actually Do

Understanding Chief Executive Officers (CEOs)

A CEO's role varies from one company to another depending on the company's size, culture, and corporate structure. In large corporations, CEOs typically deal only with very high-level strategic decisions and those that direct the company's overall growth. For example, CEOs may work on strategy, organization, and culture. Specifically, they may look at how capital is allocated across the firm, or how to build teams to succeed. 

In smaller companies, CEOs often are more hands-on and involved with day-to-day functions. 

One study from Harvard Business review analyzed how CEOs spend their time. They found that 72% of CEOs' time was spent in meetings versus 28% alone. Moreover, 25% was spent on relationships, 25% on business unit review and functional reviews, 21% on strategy, and 16% on culture and organization. Some food for thought: the study showed that just 1% of time was spent on crisis management and 3% was allocated to customer relations.1

Not only that, CEOs can set the tone, vision, and sometimes the culture of their organizations. 

CEO Pay and Notoriety

On average, CEOs of the 350 largest companies in the U.S. have earned $24 million in annual salaries. To look at it another way, that's 351 times the salary of a worker. Since the 1970s, CEO pay is estimated to have skyrocketed over 1,300%. By contrast, worker compensation has grown 18%.2 

Because of their frequent dealings with the public, sometimes the chief executive officers of large corporations achieve fame. As of Feb. 9, 2022, Elon Musk, founder of Tesla (TSLA) has over 73 million followers on Twitter.3 Similarly, Steve Jobs, founder and CEO of Apple (AAPL), became such a global icon that following his death in 2011, an explosion of both cinematic and documentary films about him emerged.

Related Chief Positions

Corporate America houses numerous titles of senior executives that begin with the letter C, for "chief." This group of top senior staffers has come to be called C-suite, or C-level in the corporate vernacular.

It's worth noting that for small organizations or those that are still in the startup or growth phases, for example, the CEO may also be serving as the CFO and the chief operating officer (COO), and so on. This can lead to a lack of clarity, not to mention an overworked executive. Assigning multiple titles to a single executive-level individual can wreak havoc on a business's continuity and ultimately may affect its long-term profitability negatively. In short, when it comes to executive-level positions within an organization, assigned titles and the functions associated with each can become muddled quickly. 

The Difference Between CEO and COB

The CEO directs the operational aspects of a company. Comparatively, the board of directors—led by the chair of the board (COB)—oversees the company as a whole. While the chair of the board does not have the power to overrule the board, the board has the power to overrule the CEO's decisions. Effectively, the chair is considered a peer with the other board members. In some cases, the CEO and the chair of the board can be the same person, but many companies split these roles between two people.

The Difference Between CEO and CFO

The CFO is the chief financial officer of a company. While CEOs manage general operations, CFOs focus specifically on financial matters. A CFO analyzes a company's financial strengths and makes recommendations to improve financial weaknesses. The CFO also tracks cash flow and oversees a company's financial planning, such as investments and capital structures. Like CEOs, the CFO seeks to deliver returns to shareholders through focusing on financial discipline and driving margin and revenue growth.

The 11th District Cost of Funds Index (COFI) is a monthly weighted average of the interest rates paid on checking and savings accounts offered by financial institutions operating in Arizona, California, and Nevada. It is one of many indices used by mortgage lenders to adjust the interest rate on adjustable rate mortgages (ARM) and was launched in 1981. With an ARM mortgage, the interest rate on a mortgage moves up and down along with some standard interest rate chosen by the lender, and COFI is one of the most popular indices in the western states.

Published on the last day of each month, the COFI represents the cost of funds for western savings institutions that are members of Federal Home Loan Bank of San Francisco, a self-regulatory agency, and satisfy the Bank's criteria for inclusion in the index.

Understanding the 11th District COFI

The 11th District Cost of Funds Index (COFI) is computed using several different factors, with interest paid on savings accounts comprising the largest weighting in the average. As a result, the index tends to have low volatility and follow market interest rate changes somewhat slowly; it is generally regarded as a two-month lagging indicator of market interest rates. The interest rate on a mortgage will not match the COFI, rather the ARM rate is typically 2% to 3% higher than COFI, depending on the borrower's credit history, the size and terms of the loan, the ability of the borrower to negotiate with the bank and many other factors.

Because it is computed using data from three western states, the COFI is primarily used in the western U.S., while the 1-year Treasury index is the measure of choice in the eastern region. On April 30, the Federal Home Loan Bank of San Francisco announced the COFI for March 2018 of 0.814%, slightly lower than February.

Joint and several liability favors the plaintiff suing for damages because it empowers them to pursue full payment, if necessary, from the party with the deepest pockets if the others named cannot pay.

If all of the parties involved are insolvent and uninsured, the plaintiff collects nothing.

Joint and several liability differs in law from comparative fault, in which multiple parties are assigned responsibility for a portion of the damages in relation to the degree of fault that they bear for the harm. In such cases, a plaintiff may be left in the position of seeking damages from the party that is least able to pay.

Comparative fault is sometimes called "pure several liability." It is more common in the U.S. than joint and several liability.

Most states in the U.S. have limited the use of joint and several liability, or have developed a hybrid approach. For example, a state might allow joint and several liability to apply only to parties who are found responsible for more than 50% of the damage done.

The hybrid approach was adopted as a means of reforming a system that appeared to encourage some plaintiffs to add a single party with deep pockets, such as a large corporation, to a suit in order to get an outsized award.

Criticism of Joint and Several Liability

As noted, joint and several liability tends to benefit the plaintiff, as it increases the chances that all of the damages awarded can be collected.

Most states in the U.S. limit the use of joint and several responsibility or employ a hybrid approach.

On the other hand, it may be considered unfair to a party who bears only a minor responsibility for an adverse event to bear an outsized financial loss because of it.

Example of a Joint and Several Liability Suit

A joint and several liability case could be launched on behalf of workers who became ill after working at multiple job sites where they were exposed to harmful materials. For example, they might be construction workers who suffer physical ailments that can be attributed to contact with a toxic substance that was present in materials used in all of their workplaces.

The workers might argue that inadequate precautions were taken by several employers who were responsible for worker safety at various sites at which they worked.

The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as "the nth root product of n numbers." The geometric mean must be used when working with percentages, which are derived from values, while the standard arithmetic mean works with the values themselves.

KEY TAKEAWAYS

• The geometric mean is the average rate of return of a set of values calculated using the products of the terms.

• Geometric mean is most appropriate for series that exhibit serial correlation—this is especially true for investment portfolios.

• Most returns in finance are correlated, including yields on bonds, stock returns, and market risk premiums.

• For volatile numbers, the geometric average provides a far more accurate measurement of the true return by taking into account year-over-year compounding that smooths the average.

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Geometric Mean

The Formula for Geometric Mean

\begin{aligned} &\mu _{\text{geometric}} = [(1+R _1)(1+R _2)\ldots(1+R _n)]^{1/n} - 1\\ &\textbf{where:}\\ &\bullet R_1\ldots R_n \text{ are the returns of an asset (or other}\\ &\text{observations for averaging)}. \end{aligned}​μgeometric​=[(1+R1​)(1+R2​)…(1+Rn​)]1/n−1where:∙R1​…Rn​ are the returns of an asset (or otherobservations for averaging).​

Understanding the Geometric Mean

The geometric mean, sometimes referred to as compounded annual growth rate or time-weighted rate of return, is the average rate of return of a set of values calculated using the products of the terms. What does that mean? Geometric mean takes several values and multiplies them together and sets them to the 1/nth power.

The geometric mean is an important tool for calculating portfolio performance for many reasons, but one of the most significant is it takes into account the effects of compounding.

For example, the geometric mean calculation can be easily understood with simple numbers, such as 2 and 8. If you multiply 2 and 8, then take the square root (the ½ power since there are only 2 numbers), the answer is 4. However, when there are many numbers, it is more difficult to calculate unless a calculator or computer program is used.

The longer the time horizon, the more critical compounding becomes, and the more appropriate the use of geometric mean.

The main benefit of using the geometric mean is the actual amounts invested do not need to be known; the calculation focuses entirely on the return figures themselves and presents an "apples-to-apples" comparison when looking at two investment options over more than one time period. Geometric means will always be slightly smaller than the arithmetic mean, which is a simple average.1

How to Calculate the Geometric Mean

To calculate compounding interest using the geometric mean of an investment's return, an investor needs to first calculate the interest in year one, which is $10,000 multiplied by 10%, or $1,000. In year two, the new principal amount is $11,000, and 10% of $11,000 is $1,100. The new principal amount is now $11,000 plus $1,100, or $12,100.

In year three, the new principal amount is $12,100, and 10% of $12,100 is $1,210. At the end of 25 years, the $10,000 turns into $108,347.06, which is $98,347.05 more than the original investment. The shortcut is to multiply the current principal by one plus the interest rate, and then raise the factor to the number of years compounded. The calculation is $10,000 × (1+0.1) 25 = $108,347.06.

Example of Geometric Mean

If you have $10,000 and get paid 10% interest on that $10,000 every year for 25 years, the amount of interest is $1,000 every year for 25 years, or $25,000. However, this does not take the interest into consideration. That is, the calculation assumes you only get paid interest on the original $10,000, not the $1,000 added to it every year. If the investor gets paid interest on the interest, it is referred to as compounding interest, which is calculated using the geometric mean.

Using the geometric mean allows analysts to calculate the return on an investment that gets paid interest on interest. This is one reason portfolio managers advise clients to reinvest dividends and earnings.

The geometric mean is also used for present value and future value cash flow formulas. The geometric mean return is specifically used for investments that offer a compounding return. Going back to the example above, instead of only making $25,000 on a simple interest investment, the investor makes $108,347.06 on a compounding interest investment.

Simple interest or return is represented by the arithmetic mean, while compounding interest or return is represented by the geometric mean.

How Do You Find the Geometric Mean in Excel?

The shortcut to calculating the geometric mean in Excel is “=GEOMEAN.” Specifically, enter the function into a cell and then list the numbers (or cells containing the numbers) that you would like to calculate the geometric mean for. 

Can You Calculate Geometric Mean With Negative Values?

You cannot—it is impossible to calculate a geometric mean that includes negative numbers.

How Do You Find the Geometric Mean Between Two Numbers?

To calculate the geometric mean of two numbers, you would multiply the numbers together and take the square root of the result.

The Herfindahl-Hirschman Index (HHI) is a common measure of market concentration and is used to determine market competitiveness, often pre- and post-merger & acquisition (M&A) transactions.

KEY TAKEAWAYS

• The Herfindahl-Hirschman Index (HHI) is used to determine market competitiveness.

• A market with an HHI of less than 1,500 is considered a competitive marketplace, an HHI of 1,500 to 2,500 is moderately concentrated, and an HHI of 2,500 or greater is highly concentrated.

• The primary disadvantage of the HHI stems from the fact that it is such a simple measure that it fails to take into account the complexities of various markets.

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Herfindahl-Hirschman Index (HHI)

Formula and Calculation of the Herfindahl-Hirschman Index (HHI)

The Herfindahl-Hirschman Index (HHI) is a commonly accepted measure of market concentration. It is calculated by squaring the market share of each firm competing in a market and then summing the resulting numbers. It can range from close to zero to 10,000. The U.S. Department of Justice uses the HHI for evaluating potential merger issues.1

\begin{aligned}&HHI=s^2_1+s^2_2+s^2_3+\dots s^2_n\\&\textbf{where:}\\&s_n=\text{the market share percentage of firm } n\\& \qquad\,\text{ expressed as a whole number, not a decimal}\end{aligned}​HHI=s12​+s22​+s32​+…sn2​where:sn​=the market share percentage of firm n expressed as a whole number, not a decimal​

What the Herfindahl-Hirschman Index (HHI) Can Tell You

The closer a market is to a monopoly, the higher the market's concentration (and the lower its competition). If, for example, there were only one firm in an industry, that firm would have 100% market share, and the Herfindahl-Hirschman Index (HHI) would equal 10,000, indicating a monopoly. If there were thousands of firms competing, each would have roughly 0% market share, and the HHI would be close to zero, indicating nearly perfect competition.

The U.S. Department of Justice considers a market with an HHI of less than 1,500 to be a competitive marketplace, an HHI of 1,500 to 2,500 to be a moderately concentrated marketplace, and an HHI of 2,500 or greater to be a highly concentrated marketplace. As a general rule, mergers that increase the HHI by more than 200 points in highly concentrated markets raise antitrust concerns, as they are assumed to enhance market power under section 5.3 of the Horizontal Merger Guidelines jointly issued by the department and the Federal Trade Commission (FTC).1

The primary advantage of the Herfindahl-Hirschman Index (HHI) is the simplicity of the calculation necessary to determine it and the small amount of data required for the calculation. The primary disadvantage of the HHI stems from the fact that it is such a simple measure that it fails to take into account the complexities of various markets in a way that allows for a genuinely accurate assessment of competitive or monopolistic market conditions.

Regulators use the HHI Index using the 50 largest companies in a particular industry to determine if that industry should be considered competitive or as close to being a monopoly.2

Example of the Herfindahl-Hirschman Index (HHI)

The HHI is calculated by taking the market share of each firm in the industry, squaring them, and summing the result, as depicted in the equation above. Consider the following hypothetical industry with four total firms:

1. Firm one market share = 40%

2. Firm two market share = 30%

3. Firm three market share = 15%

4. Firm four market share = 15%

The HHI is calculated as:

\begin{aligned}HHI &= 40^2 + 30^2 + 15^2 + 15^2\\ &= 1,600 + 900 + 225 + 225 = 2,950\end{aligned}HHI​=402+302+152+152=1,600+900+225+225=2,950​

This HHI value is considered a highly concentrated industry, as expected since there are only four firms. But the number of firms in an industry does not necessarily indicate anything about market concentration, which is why calculating the HHI is important.

For example, assume an industry has 20 firms. Firm one has a market share of 48.59% and each of the 19 remaining firms has a market share of 2.71% each. The HHI would be exactly 2,500, indicating a substantially highly concentrated market. If firm number one had a market share of 35.82% and each of the remaining firms had a 3.38% market share, the HHI would be exactly 1,500, indicating a competitive marketplace.

Limitations of the Herfindahl-Hirschman Index (HHI)

The basic simplicity of the HHI carries some inherent disadvantages, primarily in terms of failing to define the specific market that is being examined in a proper, realistic manner. For example, consider a situation in which the HHI is used to evaluate an industry determined to have 10 active companies, and each company has about a 10% market share. Using the basic HHI calculation, the industry would appear highly competitive.

However, within the marketplace, one company might have as much as 80% to 90% of the business for a specific segment of the market, such as the sale of one specific item. That firm would thus have nearly a total monopoly for the production and sale of that product.

Another problem in defining a market and considering market share can arise from geographic factors. This problem can occur when there are companies within an industry that have roughly equal market share, but they each operate only in specific areas of the country, so that each firm, in effect, has a monopoly within the specific marketplace in which it does business.

For example, while the merger of Sprint and T-Mobile will increase the HHI value for the entire country by several hundred points because market share is concentrated in certain geographical areas, the index value would increase by more than 1,000 points in many markets.3 For these reasons, for the HHI to be properly used, other factors must be taken into consideration, and markets must be very clearly defined.

What Does the Herfindahl-Hirschman Index (HHI) Mean?

The Herfindahl-Hirschman Index (HHI) is a common measure of market concentration and is used to determine market competitiveness, often pre- and post-M&A transactions. The closer a market is to a monopoly, the higher the market's concentration (and the lower its competition).

How Is Market Concentration Defined?

A market with an HHI of less than 1,500 is considered a competitive marketplace, an HHI of 1,500 to 2,500 is moderately concentrated, and an HHI of 2,500 or greater is highly concentrated. As a general rule, mergers that increase the HHI by more than 200 points in highly concentrated markets raise antitrust concerns, as they are assumed to enhance market power.

What Is the Main Advantage of the Herfindahl-Hirschman Index (HHI)?

The primary advantage of the Herfindahl-Hirschman Index (HHI) is the simplicity of the calculation and the small amount of data required for the calculation. Also, firms are weighted according to their size, which makes the HHI superior to other measures, like the concentration ratio.

A NINJA loan is a slang term for a loan extended to a borrower with little or no attempt by the lender to verify the applicant's ability to repay. It stands for "no income, no job, and no assets." Whereas most lenders require loan applicants to provide evidence of a stable stream of income or sufficient collateral, a NINJA loan ignores that verification process.

NINJA loans were more common prior to the 2008 financial crisis. In the aftermath of the crisis, the U.S. government issued new regulations to improve standard lending practices across the credit market, which included tightening the requirements for granting loans. At this point, NINJA loans are rare, if not extinct.

KEY TAKEAWAYS:

• A NINJA (no income, no job, and no assets) loan is a term describing a loan extended to a borrower who may have no ability to repay the loan.

• A NINJA loan is extended with no verification of a borrower's assets.

• NINJA loans largely disappeared after the U.S. government issued new regulations to improve standard lending practices after the 2008 financial crisis.

• Some NINJA loans offer attractive low interest rates that increase over time. They were popular because they could be obtained quickly and without the borrower having to provide documentation.

How a NINJA Loan Works

Financial institutions that offer NINJA loans base their decision on a borrower’s credit score with no verification of income or assets such as through income tax returns, pay stubs, or bank and brokerage statements. Borrowers must have a credit score over a certain threshold to qualify. Since NINJA loans are generally provided through subprime lenders, however, their credit score requirements may be lower than those of mainstream lenders, such as major banks.

NINJA loans are structured with varying terms. Some may offer an attractively low initial interest rate that increases over time. Borrowers are required to repay the debt according to a scheduled timeframe. Failing to make those payments can cause the lender to take legal action to collect the debt, resulting in a drop in the borrower's credit score and ability to obtain other loans in the future.

Pros and Cons of NINJA Loans

Because NINJA loans require so little paperwork compared, for example, with traditional home mortgages or business loans, an application is processed quickly. Their speedy delivery makes them appealing to some borrowers, particularly those who lack the customary documentation or don't wish to produce it.

The loans can, however, be very risky for both the lender and the borrower. Because NINJA loans require no evidence of collateral, they are not secured by any assets that a lender could seize if the borrower defaults on the loan.

NINJA loans can be extremely risky for borrower and lender alike.

NINJA loans are also risky for the borrower, unfettered as they are by the traditionally conservative bank underwriting practices that often keep both sides out of trouble. Borrowers may be encouraged to take out larger loans than they can reasonably expect to repay, particularly if they focus on a low introductory interest rate that will rise in the future.

After a high level of loan defaults helped trigger the 2008 financial crisis and a crash in real estate values in many parts of the country, the government imposed stricter rules on lenders, making loans more highly regulated than before, with mortgage loans seeing the greatest impact.

The 2010 Dodd–Frank Wall Street Reform and Consumer Protection Act created new standards for lending and loan applications. The new rules largely did away with NINJA loans, requiring lenders to obtain more comprehensive information about prospective borrowers, including their credit scores and documented evidence of their employment and other income sources.

Pro rata is a Latin term used to describe a proportionate allocation. It essentially translates to "in proportion," which means a process where whatever is being allocated will be distributed in equal portions.

If something is given out to people on a pro rata basis, it means assigning an amount to one person according to their share of the whole. While a pro rata calculation can be used to determine the appropriate portions of any given whole, it is often used in business finance.

KEY TAKEAWAYS

• If something is given out pro rata, it typically means everyone gets their fair share.

• Pro rata means proportionally, such as fees that rise pro rata with employee salaries.

• The practice of prorating can apply in many areas, from billing for services to paying out dividends or allocating business partnership income.

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Pro-Rata

Understanding Pro Rata

Pro rata typically means that each party or person receives their fair share in proportion to the whole. Pro rata calculations can be used in many areas, including determining dividend payments, which are cash payments by corporations paid to shareholders.

In insurance, pro rata is used to determine the amount of premium due for a policy that only covers a partial term. Allocating the appropriate portion of an annual interest rate to a shorter time frame can also be done via pro rata.

Pro rata is also used to determine how much of a distribution from a qualified retirement account—such as an IRA, SEP, or 401(k)—is taxable when the account contains before and after-tax dollars.1 For example, an account holder has a 401(k) funded with 20% pre-tax dollars and 80% post-tax dollars. As a result, withdrawals will consist of 20% taxable and 80% non-taxable money.

Pro Rata and Dividends per Shareholder

When a company pays dividends to its shareholders, each investor is paid according to its holdings. If a company has 100 shares outstanding, for example, and issues a dividend of $2 per share, the total amount of dividends paid will be $200. No matter how many shareholders there are, the total dividend payments cannot exceed this limit. In this case, $200 is the whole, and the pro rata calculation must be used to determine the appropriate portion of that whole due to each shareholder.

Assume there are only four shareholders who hold 50, 25, 15, and 10 shares, respectively. The amount due to each shareholder is their pro rata share. This is calculated by dividing the ownership of each person by the total number of shares and then multiplying the resulting fraction by the total amount of the dividend payment.

The majority shareholder's portion, therefore, is (50 / 100) x $200 = $100. This makes sense because the shareholder owns half of the shares and receives half of the total dividends. The remaining shareholders get $50, $30, and $20, respectively.

Pro Rata for Insurance Premiums

Another common use is to determine the amount due for a partial insurance policy term. Most insurance policies are based on a 12-month period, so if a policy is needed for a shorter term, the insurance company must prorate the annual premium to determine what is owed. To do this, divide the total premium by the number of days in a standard term, and multiply by the number of days covered by the truncated policy.

For example, assume an auto policy that typically covers a full year carries a premium of $1,000. If the insured only requires the policy for 270 days, then the company must reduce the premium accordingly. The pro rata premium due for this period is ($1,000 / 365) x 270 = $739.73.

Pro Rata for Interest Rates

Pro rata calculations are also used to determine the amount of interest that will be earned on an investment. If an investment earns an annual interest rate, then the pro rata amount earned for a shorter period is calculated by dividing the total amount of interest by the number of months in a year and multiplying by the number of months in the truncated period. The amount of interest earned in two months on an investment that yields 10% interest each year is (10% / 12) x 2 = 1.67%.

When it comes to bonds, payment on accrued interest is calculated on a pro rata basis. Accrued interest is the total interest that has accumulated on a bond since its last coupon payment. When the bondholder sells the bond before the next coupon date, they're still entitled to the interest that accrues up until the time the bond is sold. The bond buyer, not the issuer, is responsible for paying the bond seller the accrued interest, which is added to the market price.

The formula for accrued interest is as follows:

\begin{aligned} &\text{AI} = \text{Face Value of Bond} \times \text{Coupon Rate} \times \text{Time Factor} \\ &\textbf{where:} \\ &\text{AI} = \text{Accrued Interest} \\ &\text{Coupon Rate} = \frac{ \text{Annual Coupon Rate} }{ \text{Number of Periods Per Year} } \\ &\text{Time Factor} = \frac{ \text{Days Lapsed Since Last Payment} }{ \text{Days in Payment Period} } \\ \end{aligned}​AI=Face Value of Bond×Coupon Rate×Time Factorwhere:AI=Accrued InterestCoupon Rate=Number of Periods Per YearAnnual Coupon Rate​Time Factor=Days in Payment PeriodDays Lapsed Since Last Payment​​

The factor is calculated by dividing the length of time the bond was held after the last coupon payment by the time from one coupon payment to the next.

For example, consider a bondholder who sells their corporate bond on June 30. The bond has a face value of $1,000 and a 5% coupon rate, which pays semiannually on March 1 and Sept. 1. The buyer of the bond will pay the seller:

\begin{aligned}&\$1,000 \times \frac { 5\% } { 2 } \times \frac { 122 }{ 184 } = \$16.58 \\\end{aligned}​$1,000×25%​×184122​=$16.58​

How Do You Calculate Pro Rata?

Naturally, calculating the pro rata of different items varies since it calculates a proportion of a given whole. To calculate the prorated interest rate over six months, for instance, consider a company that charges 20% interest per year. Here, the prorated interest rate would be calculated as (20% / 12) x 6 = 10%. 

How Does Pro Rata Apply to Dividends Per Share?

When a company distributes dividends, typically it is executed on a pro rata basis. For example, consider a majority shareholder, such as a founder or key executive, who owns 50% of a company's total 1,000 shares and the company is issuing a $1 dividend. Of the $1,000 in dividends, the majority shareholder would receive $500 in value. The formula would be as follows: (50 / 100) x $1,000 = $500. 

What Is a Pro Rata Discount?

A pro rata discount is a type of discount a merchant offers a customer. Companies offer customers discounts for a variety of reasons. They may offer a discount as an incentive to a new customer to try a product or service. They may offer a discount if the customer makes a purchase during a specific time period or as a bonus to a returning customer.

The pro rata part of the discount varies depending on how the merchant has structured their offer. For example, a merchant may offer a new customer $20 off their first purchase of products if they spend $100 or more. If the customer buys four products, each item would receive a $5 discount.

A pro rata discount could also apply if a customer joins a monthly subscription service on any day other than the first of the month. Rather than charging the customer the full subscription price for the month, the merchant would apply a pro rata discount and only charge the customer for the number of days in the month they actually had the service.

Technical analysis is a trading discipline employed to evaluate investments and identify trading opportunities by analyzing statistical trends gathered from trading activity, such as price movement and volume. Unlike fundamental analysis, which attempts to evaluate a security's value based on business results such as sales and earnings, technical analysis focuses on the study of price and volume.

KEY TAKEAWAYS

• Technical analysis is a trading discipline employed to evaluate investments and identify trading opportunities in price trends and patterns seen on charts.

• Technical analysts believe past trading activity and price changes of a security can be valuable indicators of the security's future price movements.

• Technical analysis may be contrasted with fundamental analysis, which focuses on a company's financials rather than historical price patterns or stock trends.

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Understanding Fundamental Vs. Technical Analysis

Understanding Technical Analysis

Technical analysis tools are used to scrutinize the ways supply and demand for a security will affect changes in price, volume, and implied volatility. It operates from the assumption that past trading activity and price changes of a security can be valuable indicators of the security's future price movements when paired with appropriate investing or trading rules.

It is often used to generate short-term trading signals from various charting tools, but can also help improve the evaluation of a security's strength or weakness relative to the broader market or one of its sectors. This information helps analysts improve their overall valuation estimate.

Technical analysis as we know it today was first introduced by Charles Dow and the Dow Theory in the late 1800s.1 Several noteworthy researchers including William P. Hamilton, Robert Rhea, Edson Gould, and John Magee further contributed to Dow Theory concepts helping to form its basis. Nowadays technical analysis has evolved to include hundreds of patterns and signals developed through years of research.

Using Technical Analysis

Professional analysts often use technical analysis in conjunction with other forms of research. Retail traders may make decisions based solely on the price charts of a security and similar statistics, but practicing equity analysts rarely limit their research to fundamental or technical analysis alone.

Technical analysis can be applied to any security with historical trading data. This includes stocks, futures, commodities, fixed-income, currencies, and other securities. In fact, technical analysis is far more prevalent in commodities and forex markets where traders focus on short-term price movements.

Technical analysis attempts to forecast the price movement of virtually any tradable instrument that is generally subject to forces of supply and demand, including stocks, bonds, futures, and currency pairs. In fact, some view technical analysis as simply the study of supply and demand forces as reflected in the market price movements of a security.

Technical analysis most commonly applies to price changes, but some analysts track numbers other than just price, such as trading volume or open interest figures.

Technical Analysis Indicators

Across the industry, there are hundreds of patterns and signals that have been developed by researchers to support technical analysis trading. Technical analysts have also developed numerous types of trading systems to help them forecast and trade on price movements.

Some indicators are focused primarily on identifying the current market trend, including support and resistance areas, while others are focused on determining the strength of a trend and the likelihood of its continuation. Commonly used technical indicators and charting patterns include trendlines, channels, moving averages, and momentum indicators.

In general, technical analysts look at the following broad types of indicators:

• Price trends

• Chart patterns

• Volume and momentum indicators

• Oscillators

• Moving averages

• Support and resistance levels

Underlying Assumptions of Technical Analysis

There are two primary methods used to analyze securities and make investment decisions: fundamental analysis and technical analysis. Fundamental analysis involves analyzing a company’s financial statements to determine the fair value of the business, while technical analysis assumes that a security's price already reflects all publicly available information and instead focuses on the statistical analysis of price movements.

Technical analysis attempts to understand the market sentiment behind price trends by looking for patterns and trends rather than analyzing a security's fundamental attributes.

Charles Dow released a series of editorials discussing technical analysis theory. His writings included two basic assumptions that have continued to form the framework for technical analysis trading.

1. Markets are efficient with values representing factors that influence a security's price, but

2. Even random market price movements appear to move in identifiable patterns and trends that tend to repeat over time.2

Today the field of technical analysis builds on Dow's work. Professional analysts typically accept three general assumptions for the discipline:

1. The market discounts everything: Technical analysts believe that everything from a company's fundamentals to broad market factors to market psychology is already priced into the stock. This point of view is congruent with the Efficient Markets Hypothesis (EMH) which assumes a similar conclusion about prices. The only thing remaining is the analysis of price movements, which technical analysts view as the product of supply and demand for a particular stock in the market.

2. Price moves in trends: Technical analysts expect that prices, even in random market movements, will exhibit trends regardless of the time frame being observed. In other words, a stock price is more likely to continue a past trend than move erratically. Most technical trading strategies are based on this assumption.

3. History tends to repeat itself: Technical analysts believe that history tends to repeat itself. The repetitive nature of price movements is often attributed to market psychology, which tends to be very predictable based on emotions like fear or excitement. Technical analysis uses chart patterns to analyze these emotions and subsequent market movements to understand trends. While many forms of technical analysis have been used for more than 100 years, they are still believed to be relevant because they illustrate patterns in price movements that often repeat themselves.3

Technical Analysis vs. Fundamental Analysis

Fundamental analysis and technical analysis, the major schools of thought when it comes to approaching the markets, are at opposite ends of the spectrum. Both methods are used for researching and forecasting future trends in stock prices, and like any investment strategy or philosophy, both have their advocates and adversaries.

Fundamental analysis is a method of evaluating securities by attempting to measure the intrinsic value of a stock. Fundamental analysts study everything from the overall economy and industry conditions to the financial condition and management of companies. Earnings, expenses, assets, and liabilities are all important characteristics to fundamental analysts.

Technical analysis differs from fundamental analysis in that the stock's price and volume are the only inputs. The core assumption is that all known fundamentals are factored into price; thus, there is no need to pay close attention to them. Technical analysts do not attempt to measure a security's intrinsic value, but instead, use stock charts to identify patterns and trends that suggest what a stock will do in the future.

Limitations of Technical Analysis

Some analysts and academic researchers expect that the EMH demonstrates why they shouldn't expect any actionable information to be contained in historical price and volume data; however, by the same reasoning, neither should business fundamentals provide any actionable information. These points of view are known as the weak form and semi-strong form of the EMH.

Another criticism of technical analysis is that history does not repeat itself exactly, so price pattern study is of dubious importance and can be ignored. Prices seem to be better modeled by assuming a random walk.

A third criticism of technical analysis is that it works in some cases but only because it constitutes a self-fulfilling prophecy. For example, many technical traders will place a stop-loss order below the 200-day moving average of a certain company. If a large number of traders have done so and the stock reaches this price, there will be a large number of sell orders, which will push the stock down, confirming the movement traders anticipated.

Then, other traders will see the price decrease and also sell their positions, reinforcing the strength of the trend. This short-term selling pressure can be considered self-fulfilling, but it will have little bearing on where the asset's price will be weeks or months from now.

In sum, if enough people use the same signals, they could cause the movement foretold by the signal, but over the long run, this sole group of traders cannot drive the price.

Chartered Market Technician (CMT)

Among professional analysts, the CMT Association supports the largest collection of chartered or certified analysts using technical analysis professionally around the world. The association's Chartered Market Technician (CMT) designation can be obtained after three levels of exams that cover both a broad and deep look at technical analysis tools.4

The association now waives Level 1 of the CMT exam for those who are Certified Financial Analyst (CFA) charterholders. This demonstrates how well the two disciplines reinforce each other.5

What Assumptions Do Technical Analysts Make?

Professional technical analysts typically accept three general assumptions for the discipline. The first is that, similar to the efficient market hypothesis, the market discounts everything. Second, they expect that prices, even in random market movements, will exhibit trends regardless of the time frame being observed. Finally, they believe that history tends to repeat itself. The repetitive nature of price movements is often attributed to market psychology, which tends to be very predictable based on emotions like fear or excitement. 

What's the Difference Between Fundamental and Technical Analysis?

Fundamental analysis is a method of evaluating securities by attempting to measure the intrinsic value of a stock. The core assumption of technical analysis, on the other hand, is that all known fundamentals are factored into price; thus, there is no need to pay close attention to them. Technical analysts do not attempt to measure a security's intrinsic value, but instead, use stock charts to identify patterns and trends that might suggest what the security will do in the future.

How Can I Learn Technical Analysis?

There are a variety of ways to learn technical analysis. The first step is to learn the basics of investing, stocks, markets, and financials. This can all be done through books, online courses, online material, and classes. Once the basics are understood, from there you can use the same types of materials but those that focus specifically on technical analysis. Investopedia's course on technical analysis is one specific option.

Zombies are companies that earn just enough money to continue operating and service debt but are unable to pay off their debt. Such companies, given that they just scrape by meeting overheads (wages, rent, interest payments on debt, for example), have no excess capital to invest to spur growth. Zombie companies are typically subject to higher borrowing costs and may be one just event—market disruption or a poor quarter performance—away from insolvency or a bailout. Zombies are especially dependent on banks for financing, which is fundamentally their life support. Zombie companies are also known as the "living dead" or "zombie stocks."

KEY TAKEAWAYS:

• Zombies are companies that earn just enough money to continue operating and service their debt.

• Zombie companies have no excess capital to spur growth and are considered close to insolvency.

• In rare cases, a zombie company might stretch itself financially, produce a lucrative product, and reduce its liabilities.

• Zombies are high-risk investments, and not for the faint-hearted.

Understanding Zombies

Zombies often fail, falling victim to the high costs associated with debt or certain operations, such as research and development. They may lack the resources for capital investment, which would create growth. If a zombie company employed so many people that its failure would become a political issue, it may be deemed "too big to fail," as was the case with many financial institutions during the 2008 financial crisis. Given that many analysts expect that zombies will eventually be unable to meet their financial obligations, such companies are considered riskier investments and will, therefore, see their share prices suppressed.

Zombies were first spoken of in reference to companies in Japan during the country's "Lost Decade" of the 1990s following the bursting of its asset price bubble. During this period, companies were dependent on bank backing to remain in operation, even though they were bloated, inefficient, or failing. Economists argue that the economy would have been better served by allowing such poorly performing companies to fail. The term "zombies" was picked up again in 2008 in response to U.S. government bailouts that were part of the Troubled Asset Relief Program (TARP).

While the ranks of zombie companies are small, years of loose monetary policy highlighted by quantitative easing, high leverage and historically low interest rates, have contributed to their growth. Economists argue that such policies preserve inefficiencies while stifling productivity, growth, and innovation. When the market shifts, zombies will be the first to fall victim, unable to meet their basic obligations as rising interest rates make their debt more expensive to service. Meanwhile, successful companies, which are less able to build on their success because of tight credit, may feel any downturn more than they should.

While keeping zombies on life support may preserve jobs, economists note that using such resources is misguided because it impedes growth at successful firms and, therefore, inhibits job creation.

Special Considerations

Zombie Investors

Because a zombie's life expectancy tends to be highly unpredictable, zombie stocks are extremely risky and are not suitable for all investors. For example, a small biotech firm may stretch its funds extremely thin by concentrating its efforts in research and development in the hope of creating a blockbuster drug. If the drug fails, the company can go bankrupt within days of the announcement. On the other hand, if the drug is successful, the company could profit and reduce its liabilities. In most cases, however, zombie stocks are unable to overcome the financial burdens of their high burn rates and most eventually dissolve.

Given the lack of attention paid to this group, there are often interesting opportunities for investors who have a high risk tolerance and are seeking speculative opportunities.

Mutually exclusive is a statistical term describing two or more events that cannot happen simultaneously. It is commonly used to describe a situation where the occurrence of one outcome supersedes the other. For example, war and peace cannot coexist at the same time. This makes them mutually exclusive.

KEY TAKEAWAYS

• Events are considered to be mutually exclusive when they cannot happen at the same time.

• The concept often comes up in the business world in the assessment of budgeting and dealmaking.

• If considering mutually exclusive options, a company must weigh the opportunity cost, or what it would be giving up by choosing each option.

• The time value of money (TVM) is often considered when deciding between two mutually exclusive choices.

• Not mutually exclusive means that two instances or outcomes can occur simultaneously, and one outcome does not limit the other from being possible.

Understanding Mutually Exclusive

Mutually exclusive events are events that can't both happen, but should not be considered independent events. Independent events have no impact on the viability of other options. For a basic example, consider the rolling of dice. You cannot roll both a five and a three simultaneously on a single die. However, you absolutely can roll a five and a three on two dice. Rolling a five and three simultaneously means this outcome is mutually exclusive. Rolling a five on one and a three on the other means they are not mutually exclusive outcomes.

Opportunity Cost

When faced with a choice between mutually exclusive options, a company must consider the opportunity cost, which is what the company would be giving up to pursue each option. The concepts of opportunity cost and mutual exclusivity are inherently linked because each mutually exclusive option requires the sacrifice of whatever profits could have been generated by choosing the alternate option.

The time value of money (TVM) and other factors make mutually exclusive analysis a bit more complicated. For a more comprehensive comparison, companies use the net present value (NPV) and internal rate of return (IRR) formulas to mathematically determine which project is most beneficial when choosing between two or more mutually exclusive options.

Example of Mutually Exclusive

The concept of mutual exclusivity is often applied in capital budgeting. Companies may have to choose between multiple projects that will add value to the company upon completion. Some of these projects are mutually exclusive.

For example, assume a company has a budget of $50,000 for expansion projects. If available Projects A and B each cost $40,000 and Project C costs only $10,000, then Projects A and B are mutually exclusive. If the company pursues A, it cannot also afford to pursue B and vice versa. Project C may be considered independent. Regardless of which other project is pursued, the company can still afford to pursue C as well. The acceptance of either A or B does not impact the viability of C, and the acceptance of C does not impact the viability of either of the other projects.

Moreover, when looking at opportunity costs, consider the analysis of Projects A and B. Assume that Project A has a potential return of $100,000, while Project B will only return $80,000. Since A and B are mutually exclusive, the opportunity cost of choosing B is equal to the profit of the most lucrative option (in this case, A) minus the profits generated by the selected option (B); that is, $100,000 - $80,000 = $20,000. Because option A is the most lucrative option, the opportunity cost of going for option A is $0.

What Does It Mean If Projects Are Mutually Exclusive?

In business, managers and directors often need to plan resource allocation. If a company is building a bridge and a skyscraper, and both projects require an extremely specialized piece of equipment and only one exists in the world, it would mean that these projects are mutually exclusive, as that piece of equipment cannot be used by both projects at the same time. This idea can be extended to consider specialized professionals, software systems (which cannot run both Mac and Windows), and allocated budgets.

What Is the Difference Between Independent and Mutually Exclusive?

To illustrate the difference between what is independent and what is mutually exclusive, consider the war and peace example from earlier. There could be war in France and peace in Italy. These are two independent nations and therefore each one could be in its own state of peace. However, there cannot be war in France and peace in France. Since they cannot coexist, that makes them mutually exclusive.

What Does Mutually Exclusive Mean in Finance?

Typically, this involves budgeting and payments. If a company has $180 million to spend, it cannot spend that $180 million both by reinvesting in the business and offering bonuses to upper management. In this case, those two options are mutually exclusive. If the company can only retain licensing in a single country, that means they should not attempt to be licensed in two separate countries as they are mutually exclusive.

The Bottom Line

Things that are mutually exclusive are not able to occur simultaneously. In business, this is typically concerning the undertaking of projects or allocating a budget. If two things are not mutually exclusive, it means the existence and occurrence of one does not necessarily mean the other cannot coexist.