CFA Level 1 Study Notes
Rayllionaire (http://blog.5m10y.com/)
CFA Level 1 Candidate June 2009
The latest version of this document is always located at http://blog.5m10y.com/. This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License.
Version History
Version | Last Update | Comments |
0.1 | 26-Oct-08 | Completed notes for Study Session 2 Reading 5, 6, 7 |
0.2 | 16-Nov-08 | Completed notes for Study Session 1 Reading 1, 2 |
0.3 | Completed notes for SS1, Reading 3, 4 |
Ethical and Professional Standards 5
Reading 1: Code of Ethics and Standards of Professional Conduct 5
Reading 2: “Guidance” for Standards I-VII 6
Reading 3: Introduction To The Global Investment Performance Standards (GIPS®) 16
Reading 4: Global Investment Performance Standards (GIPS®) 17
Study Session 2: Basic Concepts 22
Reading 5: Time Value of Money 22
Reading 6: Discounted Cash Flow Applications 25
Reading 7: Statistical Concepts and Market Returns 29
Reading 8: Probability Concepts 35
Reading 14: Efficiency and Equity 39
Reading 15: Markets in Action 39
Reading 16: Organizing Production 39
Reading 17: Output and Costs 39
Welcome to my CFA Level 1 study notes! Originally, I’m creating this document for myself, mainly to make sure that the concepts and ideas I get from reading the books are really sticking to my head, and for my personal review later on nearer to the exam day.
For my preparation, I’m using strictly the materials from the CFA Institute itself. I’m not planning to buy any notes or tapes or videos or whatever from Schweser, Kaplan, Stalla, or whatever. Not because I have any doubts about their quality, I’m sure they’re awesome, even if expensive. But because:
I should have enough time to prepare for the June 2009 exam, even with my crazy work schedule, my family, and the fact that everybody says that there are just too many freakin pages in the 6 books of CFAI (3000+ of them I heard, although I haven’t really counted).
I’m in this not only for the CFA designation alone. I’m in this for the knowledge. I love this stuff. If CFAI books don’t only tell you the things that you need to know to pass the exam, hey, I’ll be happy to learn about them too!
This note will be updated based on the order I’m progressing with the plan, which is posted here: http://blog.5m10y.com/my-cfa-level-1-exam-june-2009-plan/. As you can see I’m starting with Study Session 2 first. I’m planning to read about Ethics later.After finding out that Ethics is a very big part of the exam (36 out of 240 questions), I’ve decided to tackle Ethics early. It seems like such a low hanging fruit!
It’s a bit tricky to study this session because it’s all about applications of these code and standards.
Useful for this particular topic are:
Reading the freely downloadable “CFA Institute Code of Ethics and Standards of Professional Conduct” is helpful—it’s actually a lot less boring than I thought it’d be.
Take the Ethics free self-examination.
The links to both can be found among my CFA bookmarks here: http://delicious.com/rayllionaire/cfa
Learning Outcomes
describe the structure of the CFA Institute Professional Conduct Program and the process for the enforcement of the Code and Standards;
CFAI Professional Conduct Program (PCP):
All members and candidates are required to comply with the Code and Standards.
Rules of Procedure form the basic structure of enforcing the Code and Standards. These Rules of Procedure are based on two principles: (a) fair process to the member and candidate (b) confidentiality of proceedings.
The oversight and responsibility are maintained by CFAI Board of Governors, through the Disciplinary Review Committee, which is responsible for the enforcement.
Professional Conduct (PC) Staff under the direction of CFAI Designated Officer conducts the professional conduct inquiries. Several things can prompt such inquiries:
Members and candidates must self-disclose on the annual Professional Conduct Statement all matters questioning their professional conduct
Written complain to PC Staff
CFAI may become aware of questionable conduct through the media or other public source
CFA exam proctors can submit violation report about any candidate suspected to have compromised the exam.
Inquiry steps:
Can include requesting written explanation, interview of member/candidate, complaining parties, collecting supporting document and records.
Then the Designated Officer can conclude in one of these ways:
No disciplinary sanction
Issue of cautionary letter
Continue proceedings to discipline the member or candidate
The member/candidate can reject the proposed sanction (which will be proposed by the Designated Officer if he/she finds that indeed a violation did happen). Then this matter can be brought to a hearing by a panel of CFAI members.
state the six components of the Code of Ethics and the seven Standards of Professional Conduct;
explain the ethical responsibilities required by the Code and Standards, including the multiple sub-sections of each Standard.
(Explained further below under reading 2.)
Learning Outcomes
demonstrate a thorough knowledge of the Code of Ethics and Standards of Professional Conduct by applying the Code and Standards to situations involving issues of professional integrity;
distinguish between conduct that conforms to the Code and Standards and conduct that violates the Code and Standards;
recommend practices and procedures designed to prevent violations of the Code of Ethics and Standards of Professional Conduct.
Code of Ethics summary:
Act with integrity and ethics with everyone
Place profession and clients above yourself
Use care and independent judgment in your activities
Practice professionalism and ethics, encourage others to do the same
Promote integrity and uphold rules of capital markets
Maintain and improve your own competence and others’
Standards of Professional Conduct (mnemonic: Parties In Denmark Don’t Include Corny Racecars):
Professionalism
Knowledge of the Law
Must know and comply with all applicable laws
In conflict, choose the stricter one! If there’s no law, use the Code and Standards.
Must not knowingly participate in any violation—knowingly means it applies when you know or should know.
If imminent or ongoing client or employer are doing illegal or unethical activities, you MUST dissociate yourself from the activities. The steps:
Attempt to stop by bringing up to supervisor or compliance dept.
Consider directly confronting the person
If still not successful, you MUST dissociate, such as removing name from written reports, asking different assignment, or in extreme cases, leave the employer. Remember:
Inaction + continuing association == participation or assistance!!!
Code and Standards do NOT require that you report violations to government or regulatory organization, EXCEPT if required by applicable laws. Code and Standards also do NOT require you to report to CFAI potential violations by others.
Recommendations:
For members & candidates:
Stay informed
Review procedures regularly
Maintain current files
And for firms:
Develop/adopt code of ethics. This thing must start from the top.
Make available and distribute to employees pertinent info about laws and regulations
Establish written protocols for reporting suspected violations
Independence and Objectivity
Must maintain independence and objectivity
Must not offer, solicit, or accept anything that can compromise the two above
Modest gifts and entertainment are acceptable, but the BEST practice dictates that you MUST reject ANY offer of gift or entertainment that could be expected to threaten your independence and objectivity.
Gifts from clients are different from gifts from entities seeking to influence you to the detriment of other clients. In client relationship, the compensation arrangement is there, so a gift can be considered supplementary compensation. So bonuses/gifts from clients are alright, as long as these are disclosed to your employer.
There are a few possible source of pressure on an analyst:
Investment banking(IB) and research have symbiotic relationship. It’s important to make sure that analysts are protected from any undue pressure from the IB side. This is the firm’s responsibility, along with public disclosure of actual conflicts of interest to investors.
Public companies management don’t like unfavorable opinions on their companies. Research analysts may justifiably fear retaliatory acts such as limiting “negative” analysts’ access to information, or even bringing legal action against them. CFAI has Best Practice Guidelines Covering Analyst/Corporate Issuer Relations for this.
Buy-side clients may get adversely affected if an analyst downgrade a stock in their portfolio. Conversely, they’d like to inflate rating as well.
Issuer-Paid Research companies hire analysts to produce research reports. This is obviously fraught with potential conflicts. Best practice in this would be to accept a flat fee, prior to writing the report, regardless of their conclusions.
If your employer is unwilling to publish negative opinions about their client, you should encourage them to remove the firm from research, and put it on a restricted list.
Recommendations:
Members & candidates should follow certain practices and encourage their firms to:
Protect integrity of opinions (compensation, reporting structure, etc.)
Restricted list if firm is unwilling to publish negative opinions about clients
Restrict special cost arrangement. Meetings with the issuer shouldn’t always be hosted by the issuer. The issuer shouldn’t pay for air transportation. Corporate aircraft shouldn’t be used unless commercial transport is not available.
Limit gifts.
Restrict investments (equity or equity-related IPOs, private placements)
Review procedures for compliance with personal investment activities policies
Formal written policy on independence and objectivity of research.
Misrepresentation
Misrepresentation is any untrue statement, omission of a fact, or any statement that is otherwise false or misleading.
This covers all forms of communication, be it verbal, advertising, electronic, or written. Web pages, chat rooms, blogs, and emails are included too.
Misrepresentations can be in the form of:
Misrepresentation of qualifications/credentials, qualifications or services provided by firms, or characteristics of an investment.
Guarantee specific returns on investments that are volatile (unless it’s a REAL guarantee—e.g.: the firm will pay the client the difference if the return is not met)
Plagiarism, which can be in the form of:
Using material without acknowledging the form
Not specific in reference (e.g.: “leading analyst”)
Presenting figures, etc. without including the caveats that may have been used
Charts and graphs without credits
Computer spreadsheets or algorithms without authorization from the author
Firms can distribute third-party reports etc. as long as sources are identified and credited.
Recommendations:
Employees should know the limit of the firms and individual capabilities.
Firms should make clear of available services and qualifications.
Firms can appoint employees who are authorized to speak on behalf of the firm.
For plagiarism:
Maintain copies of all sources used
Attribute quotations, tables, statistics, etc.
Attribute summaries
In case of intermediary source, best practice is: obtain complete study from the original source and cite only that source, or use the intermediary, and cite both the intermediary and original sources.
Misconduct
Members & candidates must not engage in any fraud, deception, or dishonest conducts, or conducts that damage trustworthiness or competence even if they are not illegal. Only covers professional life—however remember that personal conducts can affect professional life as well.
Generally this standard is not meant to cover legal transgressions from civil disobedience supporting one’s personal beliefs (e.g.: getting arrested for participating in non-violent protests).
Recommendations
Develop code of ethics for all employees
Disseminate list of potential violations and sanctions
Check references of potential employees
Integrity of Capital Markets
Material Nonpublic Information
Material means it can impact the price of the security--something you\'d want to know before deciding on it. Examples:
Earnings
Mergers, acquisitions, changes in management, significant legal disputes
Changes in assets, bankruptcies
New licenses, patents, trademarks, rejections, acquisition or loss of contracts, etc.
Orders for large trades before execution
Material depends on how reliable it is. Speculations are not material.
Nonpublic means it hasn\'t been disseminated or made available to the marketplace in general. Just because it’s been disclosed to a select group of investors/analysts, doesn’t mean it is public. Even if the info is in the form of guidance/interpretation of publicly available info, that guidance can be nonpublic info.
Mosaic theory states that analysts are free to use public + NONmaterial nonpublic info to come up with conclusions, even if those conclusions would have been material, were they company insiders.
A well-known analyst recommendation can otherwise be considered “material” since it can move prices. But since he’s not a company insider, this is not material, and he doesn’t need to make it public before giving it to his clients.
Recommendations:
If a member or candidate determines that information is material, he/she should make reasonable efforts to achieve public dissemination. If not possible, he/she must NOT take investment action based on that, and he/she MUST communicate this only to designated supervisory and compliance personnel within the firm.
Members and candidates should encourage firms to have compliance procedures, and
Disclosure policies to make sure information disseminated to the market place in an equitable manner, between small/big firms, buyside/sellside analysts, and analysts who have given negative reports in the past.
If material nonpublic info is released to a group of analysts, company should promptly issue a press release as well.
“Firewalls” to segregate info flow within firm—an employee can only be on one side of the wall at any given time.
Firm should maintain written records of communications between various departments.
Risk-arbitrage trading is even more sensitive
Market Manipulation
Market manipulation is about distorting security prices or trading volume with the intent to mislead other participants—this can be in the form of:
Transaction-based manipulations such as artificially distorting prices or volume, or securing a controlling dominant position). Intent is critical--if the security is thinly traded, your legitimate order can move the price significantly anyway.
Dissemination of false or misleading info to induce trading by others.
Duties to Clients
Loyalty, Prudence, and Care
This emphasizes again that client is #1—a member/candidate owes duty of loyalty and duty to exercise reasonable care.
Prudence means caution and discretion—e.g.: in managing portfolio this means following investment parameters set by client, and balancing risk and return.
The steps for members/candidates in fulfilling their duty are:
Find out to whom exactly the duty of loyalty is owed. (This may not be obvious, e.g.: if you’re managing a portfolios of pension plans or trusts, the client is not the person or entity hiring you, but the ultimate beneficiaries of the plan or trust. The duty of loyalty is owed to these beneficiaries in this case.)
Be aware of whether you have custody or effective control of client assets. If yes, the level of responsibility increases. Management of these assets under control MUST be in accordance with the terms of governing documents. Anything violating these documents is also a violation of the standard.
Endeavor to avoid all real or potential conflicts of interest, and put the clients before yourself.
A member/candidate’s duty is satisfied with a particular investment if they have thoroughly considered the investment’s place in the overall portfolio.
This duty of loyalty can apply in other situations too, such as voting proxies in a responsible and informed manner. “Soft dollars” or “soft commissions” has to be treated carefully as well. If a manager pays higher commission (a form of “soft dollars”) in exchange for services that don’t benefit the client, he violates the duty of loyalty.
“Directed brokerage”, where client directs the manager to use a particular brokerage, is OK. The manager however, is still duty-bound to:
Seek best price and execution with that brokerage
Be assured by the client that goods and services purchased from that brokerage will benefit the account beneficiaries
Disclose to client that they may not be getting best execution
Recommendations:
Submit to each client at least quarterly, itemized statement containing funds, securities, debits, credits, transactions, and separate client’s assets from any other party’s assets.
Review investments periodically to ensure compliance with governing documents
Establish policies/procedures for proxy voting and use of client brokerage (including soft dollars)
If in doubt on certain matters, ask yourself what you would expect or demand if you were the client. If in doubt, disclose in writing, and obtain client’s approval
Address and encourage firms to address the topics when drafting manuals for responsibilities towards clients:
Follow applicable rules and laws
Establish investment objectives of clients
Diversify
Deal fairly
Disclose conflicts of interest
Disclose compensation arrangements
Vote proxies
Confidentiality
Seek best execution
Place client interests first
Fair Dealing
Fair dealing is about treating all clients fairly in providing investment analysis, recommendations, taking action, or other activities.
Must treat all clients fairly--not discriminate any clients in terms of information
Still takes into account clients needs and criteria--not everybody needs to know everything
Offering premium services (more personal, in-depth service) are OK, provided: (a) different levels don\'t negatively affect clients (b) different service levels should be disclosed and offered to all clients.
Covers investment recommendations and actions
For investment recommendations, you should:
Ensure that information is disseminated in such a way that ALL clients have a fair opportunity to act on every recommendation. In practice, this can be tricky because not everybody can be contacted the same way, etc. Material changes are even more important than the initial recommendations. This should be communicated to all current clients, if possible before any orders contrary to the changes have been executed.
For investment actions:
If issue is oversubscribed, then the issue should be pro-rated to all subscribers on a round-lot basis. Members should even forgo any sales to themselves or immediate families so clients can get more.
Members should treat all clients fairly, including those who don’t have multiple relationships.
Disclose to clients and prospects the written allocation procedures and how it’ll affect the client or prospect. Note that our duty of fairness and loyalty to clients cannot be overridden by their consent to patently unfair allocation procedures.
Members must not take advantage of their position in the industry to the detriment of the clients.
Recommendations:
Members should recommend their firms to establish procedures to ensure fairness, which should have these things considered:
Limit number of people who know that a recommendation is going to be disseminated
Shorten timeframe between decision and dissemination. Sometimes detailed recommendations may take weeks to finish. Analysts can do a flash update, a brief summary that goes out before the detailed version.
Guidelines on predissemination—no action or discussion on pending recommendation
Simultaneous dissemination—for instance by trading restriction for the firm’s employees until recommendation is widely distributed to all relevant clients.
Maintain a list of clients and their holdings—to make sure all who needs to know can be informed fairly.
Written Trade Allocation procedure.
Disclose this trade allocation procedure
Establish systematic review to make sure no preferential treatment.
Disclose level of service
Suitability
Reasonable inquiry to find out clients\' investment profiles (i.e.: age, occupation, experience, risk/return objectives, and constraints), and update this regularly
Make sure investment is suitable for the clients.
This part is mainly for members/candidates who are in advisory relationships with clients. They should gather investment-related info about the client, which should be incorporated into a written investment policy statement (IPS). This should be repeated at least annually and prior to material changes to recommendations/decisions.
Suitability is effective only if client fully discloses the complete financial portfolio, including those not managed by member/candidate. If client withholds this information, suitability analysis must be done based on the information provided, which may not be complete. Members can be responsible only for assessing suitability of an investment given the information and criteria provided by the clients.
Also, if a member/candidate manages a fund with a stated mandate, his responsibility is to follow the mandate, NOT to assess the suitability of that fund for individual investors (unless those individual investors happen to be his clients).
Recommendations:
In formulating the written IPS for the client, a member/candidate should:
Identify clients’ type and nature, existence of separate beneficiaries, and approximate portion of total client assets.
Understand investor’s objectives, in terms of risk and return.
Understand constraints (e.g.: liquidity, expected cash flows, investable funds, time horizon, tax, legal, preferences, proxy-voting, etc.)
Performance measurements benchmarks.
This should be reviewed periodically.
Performance Presentation
Members/candidates must make reasonable efforts to make sure presentation and measurement of candidate\'s performance is fair, accurate, and complete.
Investment performance communication to clients:
Must be fair and complete.
Must not state or imply that clients will get historical rate of return.
If presentation is brief, more details must be provided upon requests
Recommendations:
Compliance with GIPS is the best way.
Consider audience knowledge and sophistication
Present the performance of weighted composite (instead of using a single representative account!)
Include terminated accounts as well as part of performance history
Include disclosures to explain fully the result (e.g.: if it’s simulated, or that it’s gross fees, net fees, after tax, etc.)
Maintain data and records used to calculate the performance.
Preservation of Confidentiality
Members/candidates must keep information of past, present, and future clients confidential unless:
It concerns illegal activities
Law requires otherwise (following applicable law comes first!)
The clients expressly permit/authorize it.
Needed for cooperating with CFA Institute Professional Conduct Program (PCP).
Recommendations:
Not much, just don’t do it. In certain instances that you want to disclose information received from clients that is outside the scope of the confidential relationship and not illegal, ask yourself two things:
Is the information relevant to work?
Will disclosing this result in better service to the clients?
Duties to Employers
Loyalty
Basically can be summarized as:
Next to the integrity of the capital markets and the clients, is the employer. It does not, however, mean that employer should come ahead in all personal matters. Balance and dialogue are the keys.
Abstain from independent competitive activities that can conflict with employer\'s interests.
Must leave in good will, no hijacking of clients, trade secrets, and other confidential stuff. Once one has left, though, the knowledge and experience she gained from ex-employer is no longer confidential unless deemed such by law.
Employee can be a whistleblower if it\'s justified, e.g.: when employer is engaged in an illegal or unethical activity)
As independent contractors, the duties are governed by oral or written agreement.
Additional Compensation Arrangement
Summary:
Must not accept any compensation that competes or maybe conflict with employer\' interest.
Unless there\'s written consent from all parties involved.
Written means any form of communication that can be documented (e.g.: emails).
Recommendations:
Members/candidates should make an immediate written report to employers detailing any compensation they propose in addition to the compensation from their employers. This report should contain terms of any agreement under which this additional compensation will be received, such as:
Nature of compensation
Approximate amount
Duration of the agreement
Responsibilities of Supervisors
Must make reasonable effort to detect and prevent violations of applicable laws, rules, regulations, and Code and Standards by anybody we supervise.
Delegating doesn\'t relieve us of the responsibility
Must understand compliance procedures.
Investment Analysis, Recommendations, and Action
Diligence and Reasonable Basis
Must ensure due diligence
Have reasonable bases supported by research and investigation, for any investment actions.
Communication with Clients and Prospective Clients
Must inform clients the basic format and general principles of the investment process to analyze, select securities, construct portfolios--including what happens if the assumptions are wrong.
Must promptly inform clients of any changes that will materially affect those processes.
Use reasonable judgment to decide which factors are important.
Distinguish facts and opinions.
Methods of communication not limited to written--but brief comms must be supported by background reports/data available upon request.
Record Retention
Must maintain records of to back up reasons for actions and conclusions.
Records can be in hard copies or electronic form.
Property of employer, employee cannot take it to the new firm unless expressly allowed by former employer.
Without recreating these supporting records, member cannot use historical recommendations or research created at the previous firm. (Unless the previous firm expressly consents to him bringing these records to the new firm).
Following the local laws may satisfy the standard—but member must explicitly determine if it does.
In absence of regulatory guidance, CFAI recommends maintenance of at least SEVEN YEARS.
Conflicts of Interest
Disclosure of Conflicts
Must make full and fair disclosure of all potential conflict of interests to current and prospective clients, and employer.
Disclosure must be prominent, in plain language, communicated effectively.
Priority of Transactions
No front-running! Clients, and then employer, come first.
Referral Fees
Must disclose to clients, prospective clients, employer, any compensation/benefit received or paid for recommendation of products or services.
They (clients & employer) then should be able to evaluate (a) partiality (b) full cost.
Responsibilities as a CFA Institute Member or CFA Candidate
Conduct as Members and Candidates in the CFA Program
Basically, don\'t cheat in the exam. Don\'t try to "outsmart" the security measures of the exam, leak the exam questions, etc.
This applies to everyone, including exam graders.
Don\'t use affiliation with CFA program to inappropriately further personal or professional goals.
Reference to CFA Institute, the CFA Designation, and the CFA Program
CFA must always be used as adjectives (not: "he is a CFA", but: "he\'s a CFA charterholder")
No partial designation. If you haven\'t passed all 3 levels and not awarded the charter, you are not a charterholder.
No over-promising competency and/or future performance result
Learning Outcomes
Explain why GIPS standards were created, what parties the GIPS standards apply to and who is served by the standards;
GIPS standards were created to people can compare apples to apples (or, investments to investments, anyway). It ensures fair representation and full disclosure of performance information, and removes misleading practices such as “representative accounts” (selecting those that perform well), “survivorship bias” (excluding terminated accounts from average performance), or “varying time periods” (fixing a time period during which the performance is the strongest).
With regards to compliance:
Compliance with the GIPS standards is NOT required by any law or regulations—it’s all voluntary.
Only investment management firms that actually manage assets can claim compliance with GIPS. That’s it. Software that does GIPS calculation cannot be said to be GIPS-compliant. Plan sponsors and consultants cannot claim GIPS compliance, although they can say they endorse it.
It’s an all-or-nothing thing. Either (a) the whole firm (as opposed to just a single product or composite) fully comply with all (not one, not two, ALL) requirements of the GIPS standards and claim compliance through the GIPS Compliance Statement, or (b), they’re not compliant. That’s it.
GIPS benefits investment management firms and prospective clients.
Explain the construction and purpose of composites in performance reporting;
A composite is an aggregation of all actual, fee-paying, discretionary portfolios into a single group representing a particular investment objective or strategy.
The decision of which portfolios to include should be done ex-ante, not after the fact. (So that firms cannot select just the best performing portfolios to include in the composite.)
Explain the requirements for verification of compliance with GIPS standards.
Firms are responsible for their claim of compliance and maintenance of that compliance. They can, however, hire a third-party to verify their claim of compliance. The verification basically tests for two things:
Whether composite construction are GIPS-compliant on a firm-wide basis, and
Whether the calculation and presentation methods are compliant with the GIPS standards.
Unlike claim and maintenance, verification can NOT be done by the firm itself—it has to be an independent third party.
Learning Outcomes
describe the key characteristics of the GIPS standards and the fundamentals of compliance;
Eleven key characteristics of GIPS:
For GIPS, FIRM must be defined as an investment firm, subsidiary, or division held out to (potential) clients as a DISTINCT BUSINESS ENTITY.
GIPS are ethical standards to ensure fair representation and full disclosure of a FIRM’s performance.
GIPS require firms to:
Include ALL actual fee-paying, discretionary portfolios in composites.
Initially show GIPS-compliant history for min 5 years (or since inception, if FIRM is in existence in less than 5 years). After presenting 5 years of compliant history, the firm must add annual performance each year, up to 10 years minimum.
GIPS require firms to use certain calculation & presentation methods, and make certain disclosures.
GIPS rely on integrity and accuracy of input data. (e.g. benchmarks and composites should be created before the fact)
GIPS require and recommend. Firms are encouraged to adopt the recommended provisions, and they MUST meet the requirements.
GIPS must be applied with the goal of full disclosure and fair representation. This will likely require more than just complying with GIPS’s minimum requirements. Some situations are not addressed or open for interpretation. In this, firms are encouraged to present all relevant additional and supplemental info.
All requirements, clarifications, updated info, and guidance, MUST be adhered to when determining a claim of compliance.
In case of GIPS conflict with local law or regulation, firms MUST comply with the local law or regulation, and make full disclosure of the conflict.
Certain GIPS recommendations may become REQUIRED in the future.
GIPS include supplemental provisions for PRIVATE EQUITY and REAL ESTATE that MUST be applied to these asset classes.
For GIPS Fundamentals of Compliance, see further below.
describe the scope of the GIPS standards with respect to an investment firm’s definition and historical performance record;
FIRMS from any country may come into compliance with the GIPS standards.
Historical performance record, FIRMS:
MUST present at minimum 5 years of GIPS-compliant annual investment performance. If firm or composite has been there for less than 5 years, then since inception.
MUST present additional annual performance up to a total of 10 years minimum.
MAY link non-GIPS-compliant record, as long as:
No non-compliance is presented after Jan 1, 2000
Periods of non-compliance are disclosed and explained.
If previously compliant with CVG (Country version of GIPS), FIRMS can claim compliance with GIPS for records prior to Jan 1, 2006. The FIRM MUST continue to show the historical CVG-compliant trace record up to 10 years.
explain how the GIPS standards are implemented in countries with existing standards for performance reporting and describe the appropriate response when the GIPS standards and local regulations conflict;
IPC (Investment Performance Council) encourages countries without investment performance standards to adopt GIPS.
If discrepancy arises between translations of the Standards, the English version is the ultimate authority.
FIRMS are strongly encouraged to comply with GIPS IN ADDITION to local laws. Local laws take precedence, but firms MUST disclose any conflicts of local law with GIPS.
characterize the eight major sections of the GIPS standards.
Eight major sections/provisions of GIPS:
Fundamentals of Compliance
Definition of Firm:
GIPS MUST be on a firm-wide basis. Either a firm is fully compliant, or not compliant.
FIRM MUST be defined as an investment firm, subsidiary, or division held out as a distinct business entity.
TOTAL FIRM ASSETS must be the aggregate of the MARKET VALUE (as opposed to cost basis or book value) of all discretionary AND nondiscretionary, fee-paying AND non-fee-paying assets under management within the firm.
FIRMS MUST include performance of assets assigned to sub-advisor in a COMPOSITE provided the FIRM has discretion in selecting the sub-advisor.
Changes in a FIRM’s organization must NOT lead to alteration of historical composite results.
FIRMS are ENCOURAGED to adopt the broadest, most meaningful definition of firm.
Document Policies and Procedures:
FIRMS MUST document in writing all the policies and procedures used for maintaining compliance with GIPS.
Claim of Compliance:
Once FIRM has met all the REQUIRED elements, the compliance statement must be like this:
“<FIRM> has prepared and presented this report in compliance with the Global Investment Performance Standards (GIPS®).”
There’s no “partial” compliance. If some of the requirements are not met, FIRMS MUST NOT say “We’re GIPS compliant except for…”
No statements about calculation methodology being GIPS compliant are to be made.
No statements about the performance of a single existing client as being done according to GIPS are to be made. EXCEPT when a GIPS-compliant firm reports the performance of an individual account to an existing client.
Firm Fundamental Responsibilities:
FIRMS MUST make every reasonable effort to provide ALL prospective clients with GIPS compliant presentation. They cannot selectively choose whom to be presented with a compliant performance.
FIRMS MUST provide a COMPOSITE list and COMPOSITE DESCRIPTION to any prospective client that requests them. FIRMS MUST list discontinued COMPOSITES for at least 5 years after discontinuation.
The list and description mentioned above MUST be compliant.
In joint-marketing effort, it MUST be made clear which FIRM claims GIPS-compliance, and this firm MUST be sure that it is clearly defined and separate relative to other FIRMS not claiming compliance.
FIRMS are ENCOURAGED to comply with recommendations and MUST comply with all requirements, including any updates, reports, guidance statements, interpretations, and clarifications published by CFAI and IPC, available via the website and the GIPS Handbook.
Verification:
FIRMS are ENCOURAGED to undertake the verification process (can only be done by a third party)
FIRMS are ENCOURAGED to add disclosure to presentations or advertisements that the firm has been verified. The verification disclosure should read:
“<FIRM> has been verified for the periods <dates> by <verifier>. A copy of the verification report is available upon request.”
Input Data: must be consistent.
Calculation Methodology: uniformity of method is required, and mandated by the standard.
Composite Construction: Composite is an aggregation of one or more portfolios that represents a particular investment objective of strategy. COMPOSITE RETURN is asset-weighted average of the performance results of all the portfolios.
Disclosures: allow firms to elaborate and give proper context to understand performance results. Some disclosures are required of all firms, some are not. No negative assurance language is required for those that are not required.
Presentation and Reporting: When appropriate, firms have the responsibility to include other information not necessarily covered by the standard.
Real Estate: apply to ALL investments where returns are from holding, trading, development or management. Apply regardless of level of control firm has over the management of investment. Apply whether the assets are producing revenue.
Private Equity: apply to all private equity other than open-end or evergreen funds (which must comply with main GIPS).
Like its title suggests, SS2 is really about the basic of quantitative analysis that will be used throughout the remainder of the curriculum. In particular, SS2 is mainly about:
Time value of money, and
Statistics and probability theory
It’s the perfect time to start familiarizing yourself with the financial calculator as well!
Learning Outcomes
Interpret interest rates as required rate of return, discount rate, or opportunity cost.
An interest rate, denoted r, is a rate of return that reflects the relationship between differently dated cash flows. They can be thought of in three ways:
Required rates of return: that is, the minimum rate of return that an investor must get to be interested enough in the investment.
Discount rates: the rate at which we discount a future amount to find its present value.
Opportunity cost: the value that investors forgo by choosing a particular course of action. Let’s say the interest rate is 5% per annum, and you have $10k. If you put this in the bank, 1 year from now you’ll have $10,500. But if you spend it, you’re forgoing the opportunity to earn 5% on your money. We say that 5% is the opportunity cost of current consumption.
Explain an interest rate as the sum of a real risk-free rate, expected inflation, and premiums that compensate investors for distinct types of risk.
Let’s say 2 guys come to you asking you to lend them money. Guy A has a reputation of being trustworthy, and has a lot of valuable assets. Guy B... well, you’ve never even heard of guy B. Which one would you charge with higher interest? Obviously B, because it’s riskier to lend your money to him. For all you know, tomorrow B might run away with your money. So to compensate you for taking this risk, you’re charging him with (a lot) higher interest on the money you’re lending him.
So in short, interest is the premium you pay to compensate lenders/investors for lending you their money and taking all the risks that come with it. In other words, interest rate is composed of a real risk-free interest rate, plus a set of four premiums for compensating lenders/investors for bearing 4 distinct types of risk.
The real risk-free interest rate is the single-period interest rate for a completely risk-free security, if no inflation were expected. It reflects the time preferences of individuals for current vs. future real consumption.
The inflation premium compensates investors for the expected inflation rate (which reduces the purchasing power of money) over the maturity of the debt. In #1 above, we’re talking about the real risk-free interest rate, which doesn’t take inflation into account. When we take inflation into account, we get the nominal risk-free interest rate:
(1 + nominal rate) = (1 + real rate)(1 + inflation rate)
(The CFAI book says that the nominal rate is the sum of real rate and inflation premium rate, however this is just an approximation—as explained in the footnote.)
The default risk premium compensates investors for the risk that the borrower won’t be able to pay at the contracted time the contracted amount.
The liquidity premium compensates investors for the risk of loss because the investment cannot be converted into cash easily, a.k.a. illiquid, in which case she may need to accept a price that is below the fair value because there are few buyers.
The maturity premium compensates investors for the increased sensitivity of the debt value to a change in market interest rate. You noticed how banks give higher interest rates for longer-term fixed deposit? That’s the maturity premium. The longer the term of the debt is, generally the higher this premium will be too.
Calculate and interpret the effective annual rate, given the stated annual interest rate and the frequency of compounding, and solve time value of money problems when compounding periods are other than annual.
The time value of money concept is one of the main principles of financial valuation. Basically it’s about the relationship between the following factors:
Present Value (PV): the value of money now.
Future Value (FV): the value of money at some point in the future.
Interest rate per period (r): interest rate per compounding period, which is not necessarily a year. So if the interest is 12% annually, but it’s compounded every month, the effective annual rate is not 12%, but (1 + 0.12/12) ^ 12 – 1 = 12.68%
The number of compounding periods (n): not necessarily annual, but the interest rate per period has to match this!
Intermediate cash flows (PMT)
These are especially easy to compute using the financial calculator.
Interest rate is usually stated or quoted annually—which is why we have stated annual interest rate or quoted interest rate. This is a quoting convention, and is different from the effective interest rate if the compounding period is less than one year.
Important equations:
Calculating future value from present value, with stated annual interest rate (rs), number of compounding periods per year (m), and the number of years (N):
FV–N=PV1+rsmmN
Continuous compounding: we see that the more compounding periods we have, the higher the effective interest rate is. What if each compounding period is infinitesimally small? That is, we have continuous compounding? In that case the equation becomes like this:
FVN= PVersN
Now, I don’t know about you, but the first time I asked myself when I heard of continuous compounding is: what is the point of it? I mean, it’s not like there’s any bank that would give us continuously compounding savings account, right? Sure it’s cool theoretically, but what’s the use?
Effective rates:
EAR=1+Periodic interest ratem- 1
For continuous compounding, the EAR is defined as:
EAR=ers- 1
Of course, given all these equations, it’s easy to calculate PV for a given FV—just rearrange them a bit! PV and FV are reciprocals.
Calculate and interpret the future value (FV) and present value (PV) of a single sum of money, an ordinary annuity, an annuity due, a perpetuity (PV only), and a series of unequal cash flows.
When we’re talking about series of cash flows, there are a few terminologies:
An annuity is a finite set of level, sequential cash flows (level means each cash flow is of the same amount).
An ordinary annuity has itsfirst cash flow occurs one period from now (indexed t = 1). Also known as payments in arrears. Mortgage is an example of this.
The FV for ordinary annuity is:
FVN=A1+rN-1r
And the PV for ordinary annuity is:
PV=A[1-11+rNr]
An annuity duehas the first cash flow that occurs immediately, NOW (indexed t = 0). A.k.a. payments in advance. Lease is an example of this.
A perpetuity is perpetual annuity, the first cash flow occurring one period from now, but it never ends, it keeps paying you forever (How cool is that?) The present value of a perpetuity is not infinite though—the further the cash flow is in the future, the smaller the present value is—the present value of $100 a thousand years from now is almost zero, so the present value of perpetuity actually converges to:
PV=Ar
(Obviously it doesn’t make sense to calculate the FV of a perpetuity—by definition if you take a point in the future for calculating FV, then you’re dealing with an annuity since you know when it ends.)
Draw a time line, and solve time value of money applications (for example, mortgages and savings for college tuition or retirement).
Remember that other than just FV and PV, there are other factors in time value of money problems, namely, rates, number of periods, and size of annuity payments. Luckily they can be easily calculated by simple rearranging our basic equation FVN=PV1+rN:
Growth rate/interest rate:
g=FVNPV1N-1
Number of periods:
N=lnFVNPVln1+r
Size of annuity payments is also easy to calculate. Given the formula for FV and PV of ordinary annuity, and the r, N, and PV or FV, we can calculate A (the size of annuity payments) easily.
Cash flow additivity principle states that amounts of money indexed at the same point in time are additive.
Learning Outcomes
Calculate and interpret the Net Present Value (NPV) and the Internal Rate of Return (IRR) of an investment, contrast the NPV rule to the IRR rule, and identify problems associated with the IRR rule.
Net Present Value of an investment is the PV of its cash inflows minus the PV of its cash outflows (that’s why it’s called “net”). The steps to calculate NPV of an investment:
Identify all cash inflows and outflows
Determine discount rate/opportunity cost r. Opportunity cost is the alternative return we will be forgoing if we undertake this investment.
Use the discount rate to find the PV of all cash flows
Sum them all, inflows are positive, outflows are negative
NPV rule: if the net PV is negative, obviously you shouldn’t take the investment. If you have 2 or more, pick the one with the highest NPV.
Here’s the formula for finding NPV:
NPV=t=0NCFt1+rt
It’s basically a summary of the 5 steps above—with CFt being the expected cash flow at time t, discounted accordingly and summed.
Internal Rate of Return is the discount rate that makes net present value equal to zero. It equates the PV of outflows and the PV of inflows (so the total sum is 0). The rate is “internal” because unlike NPV (for which we need to determine the opportunity cost), the rate for IRR really only depends on the cash flows of the investment itself—no external data are needed.
Because it is internal though, it means that we can get an actual rate that is lower (or higher) than the IRR—because we may not be able to reinvest the cash generated by the project at exactly the IRR rate. Also, IRR can have multiple solutions!
The IRR rule: “Accept projects for which the IRR is greater than the opportunity cost”.
Sometimes the NPV and the IRR rules can tell you different things—the rule here is to pick the one chosen using the NPV rule, because:
IRR assumes that cash is reinvested at the IRR rate—which may not be the realistic rate one can get.
IRR can cause us to miss the “big picture”—it can give us misleading results if we compare projects of different size (e.g.: $300 investment generating $600 in one year gives higher IRR than $1000 investment generating $1500 in one year. But the latter earns the shareholders more money).
IRR can give us misleading result based on the timing of the cash flows. If the projects being compared have different timings, the IRR can be misleading as well.
Define, calculate, and interpret a holding period return (total return).
When you invest your money, obviously you want to assess how successful your investment is. This involves performance measurement (calculating the returns), and performance appraisal/evaluation.
A fundamental concept in measuring performance is holding period return (HPR): the return that an investor earns over the holding period. For investment with just one cash payment at the end of holding period, the formula is straightforward—we just subtract the original investment from the total amount of money at the end, and divide it by the original investment:
HPR=P1- P0+ D1P0
If the measurement is over many periods, during which there may be withdrawals and additions, it’s more complicated (see below).
Calculate, interpret, and distinguish between the money-weighted and time-weighted rates of return of a portfolio and appraise the performance of portfolios based on these measures.
IRR is the money-weighted return of a portfolio. What it means is that it is significantly influenced by when the money is given and how much money—the total return will be swayed towards the return of the period where more money is invested.
This is no good for evaluating investment managers because usually they don’t have control over when the money is given to them and how much—that is the decision of their clients. We should be able to evaluate an investment manager based on her own actions, or things that are under her control.
The time-weighted rate of return works better in this regard because it is not sensitive to the additions and withdrawals of funds—time-weighted here means that returns are averaged over time.
The steps to get the time-weighted return:
Price the portfolio, break the overall periods into subperiods, based on the dates of cash inflows and outflows.
Calculate the HPR for each subperiod.
Link or compound the HPRs to obtain the annual rate of return for that year. For multi-year returns, take the geometric mean of the annual rate of returns.
Now, in practice, step 1 can be costly. So instead, we approximate it by valuing the portfolio in regular intervals—daily valuation is common. In that case, the steps are:
Calculate the daily return (MVB/MVE = Market Value Begin/End):
rt=MVEt- MVBtMVBt
We get the annual return for that year by linking the daily return like this:
r=1+r11+r21+r3…1+r365- 1
Once we’ve got the annual return for each year, we can get the time-weighted return as the geometric mean like this:
rTW=N1+r11+r2…1+rN- 1
Note that step 3 is different from step 2. In step 3 we’re looking for the mean. In step 2 we’re compounding the daily return to get the annual return.
Calculate and interpret the bank discount yield, holding period yield, effective annual yield, and money market yield for a U.S. Treasury bill; and convert among holding period yields, money market yields, effective annual yields, and bond equivalent yields.
The money market is the market for short-term debt instruments (maturity <= 1 yr). Some will repay you the amount + interest, whereas some are pure discount instruments, meaning that you simply buy it at a price lower than the face value, and you’ll get the face value at maturity. A U.S. Treasury bill (T-bill) is an example of this.
The market for each of these instruments has its own quoting conventions. For T-bills, they are quoted on a bank discount basis—a convention that annualizes the discount as a percentage of face value, based on a 360-day year. So here’s the formula for bank discount yield:
rBD=DF360t
Bank discount yield is no good for measuring returns because it’s:
Based off the face value, not your original investment
Annualized based on 360 instead of 365-day year
Doesn’t take into account compound interest
So we have 3 alternative yield measures. They are:
Holding Period Yield (HPY): this is just another name for our good ol’ HPR above. For interest bearing instrument (like coupon-bearing bonds), then the accrued interest must be considered.
Effective annual yield: it takes the HPY, compounds it for one 365-day year, and then subtract one:
EAY=1+HPY365t- 1
Money market yield/CD equivalent yield: annualized HPY based on 360-day year—not compounded! It’s like bank discount yield, but computed based on the purchase price (as opposed to face value):
rMM=HPY360t=360rBD360-trBD
Bond-equivalent yield is the way of calculating yield to maturity by ignoring compounding. E.g.: if a bond has the semi annual yield of 4%, the effective annual yield is (1 + 0.04)^2 – 1. But the bond-equivalent yield is simply to double it.
Learning Outcomes
Differentiate between descriptive statistics and inferential statistics, between a population and a sample, and among the types of measurement scales.
Statistical methods include descriptive stats and inferential stats. Descriptive statistics is about making sense out of a mass of numerical details—it turns data into information. Statistical inference (inferential statistics) is about making predictions or judgment about a larger group from the smaller group actually observed.
There are four measurement scales (mnemonic: NOIR), ranging from weakest to strongest:
Nominal scales: categorize data but do not rank them.
Ordinal scales: categorize data and rank them, but we don’t know the difference between scale values.
Interval scales: categorize data, rank them, and make the difference between scale values equal. So the difference between “1” and “3”, and the difference between “5” and “7” are the same. Has no true/natural zero.
Ratio scales: Interval scales with true zero. This way, we can meaningfully compute ratios. E.g.: money—somebody with twice the money has twice the purchasing power. Zero money means no money.
Explain a parameter, a sample statistic, and a frequency distribution.
A population is all members of a specified group. Any descriptive measure of a population characteristic is called a parameter. Examples of parameters in investment analysis are the mean value, the range of investment returns, and the variance.
It’s usually impractical to observe all the members of a population. So usually we take a sample, which is a subset of the population, with the hope that it is characteristic of the population. A sample statistic (statistic for short) is a descriptive measure of a sample characteristic.
A frequency distribution is data summarized into a relatively small number of non-overlapping intervals. For instance, if we tabulate the number of times the annual total return of the S&P 500 falls under certain intervals (e.g.: -44% <= observation < -42%, -42% <= observation < -40%... and so on), then we have a frequency distribution.
Calculate and interpret relative frequencies and cumulative relative frequencies, given a frequency distribution, and describe the properties of a dataset presented as a histogram or a frequency polygon.
The relative frequency is (Absolute frequency of interval)Total number of observations. Cumulative relative frequency is, well, cumulative. So it adds up as we go from the first to the last interval (so in the last interval the cumulative relative frequency is always 100%).
Histogram is the graphical representation of a frequency distribution. Say, if we have a frequency distribution table, and we draw an Excel bar chart with frequency as the y-axis and intervals as the x-axis, we have a histogram.
If we then take the midpoint of each of the interval along the x-axis, and put a dot whose y value represents the absolute frequency for that interval, then connect those dots, we have a frequency polygon.
Define, calculate, and interpret measures of central tendency, including the population mean, sample mean, arithmetic mean, weighted average or mean (including a portfolio return viewed as a weighted mean), geometric mean, harmonic mean, median, and mode.
Measures of central tendency are just that: they specify where the data are centered. There are many measures of central tendency:
Arithmetic mean—we can compute this for both populations and samples.
Population mean μ=i=1NXiN
Sample mean average X=i=1nXin
Means computed for individual units are called cross-sectional means. Those computed over time are called time-series mean. The problems with means are that they are sensitive to extreme values (e.g.: 1, 2, 3, 4, 10000). Another approach is to use the median.
Median: the value of the middle item of a sorted set. When there are odd n samples, the median is the one occupying the n+12 position (e.g.: for 5 items, the median is the 3rd). If n is even, then we take the mean of the two items occupying the n2 and the n+22 position (e.g.: for 6 items, the median is the mean of 3rd and 4th items). Unlike the mean, the median is not affected by extreme values.
Mode: the most frequently occurring value in a distribution. A distribution can have one mode (unimodal), two (bimodal), or even three (trimodal). If all the values in the data set are different, then the distribution has no mode. Continuous distributions may not have a modal outcome, although when we divide them into intervals, we can have modal interval(s).Mode is the only measure that can be used with nominal data (where categories are not ranked).
Weighted mean: sometimes we need to find the mean of a set of observations that have different weights.
Weighted mean Xw=i=1nwiXi
Where the sum of the weights equals 1. The arithmetic mean can be seen as a special case of weighted mean with all the weights equal. In portfolio, asset held long has positive weight, and asset held short negative. When we take this for forward-looking data, it’s called expected value.
Geometric mean: the arithmetic mean of n values is the value that, when you multiply it by n, equals the sum of all n values. Geometric mean is similar to that—the difference is that we’re talking about multiplication. The geometric mean of n values is the value that, when you raise it to the nth power, equals the product of all n values.
G=nX1X2X3…Xn
For portfolio returns, the geometric mean of returns is the compound return. Note the difference between the geometric and the arithmetic mean:
Geometric mean <= arithmetic mean.
Geometric mean return represents the growth/compound rate of return. The arithmetic mean focuses on the average single-period performance. (So if you double your money in year 1, and lose all that profit in year 2, the geometric mean return is 0—because you haven’t gained anything. But the arithmetic mean is 25%, because in year 1 you got 100% return and in year 2 -50%.)
Harmonic mean: is a special type of weighted mean in which an observation’s weight is inversely proportional to its magnitude.
XH=ni=1n1Xi
The harmonic mean is used in dollar cost averaging, to find out the average price paid for the security over time (note that when the security price goes down, the number of units that we buy go up and vice versa). The weighted mean can be used too in this scenario.
Harmonic mean <= geometric mean <= arithmetic mean. The equality only happens when all observations have the same value.
Describe, calculate, and interpret quartiles, quintiles, deciles, and percentiles.
Other than measures of central tendency, we have other measures of location: quantiles. The term quantile or fractile is the most general term of a value at or below which a stated fraction of the data lies.
Quartiles divide the distribution into quarters, quintiles fifths, deciles tenths, and percentiles hundredths. The first quartile Q1 divides a distribution such that 25% of the observations lie at or below it—so it’s also the 25th percentile.
The formula for the position of the yth percentile in a sorted ascending array of n entries is:
Ly=n+1(y100)
When Ly is not a whole number, we can do linear interpolation to estimate the value. So let’s say L25 is 4.25 (so we have 16 entries—4.25 is (16 + 1) * (25 / 100)). Let’s say the 4th item is 100 and the 5th item is 200—4.25 means L25 lies a quarter of the distance between the 4th and the 5th item, in this case
P25=X4+4.25-4X5-X4=100+0.25*100=125
In investment practice, quantiles are used in portfolio performance evaluation and investment strategy R&D. Analysts use quantiles to rank performance—for instance, Morningstar mutual fund star rankings associate the number of stars with percentiles of performance relative to similar-style mutual funds.
In investment research, dividing data into quantiles based on some characteristic allows analysts to evaluate the impact of that characteristic on a quantity of interest.
Define, calculate, and interpret 1) a range and a mean absolute deviation, and 2) the variance and standard deviation of a population and of a sample.
The mean return of an investment tells us where the returns are centered. To better understand an investment, though, we also need to know the dispersion around that center. Dispersion, then, is the variability around the central tendency.
The most common measures of (absolute, as in without comparison to any benchmark) dispersion are:
Range: quite simply, the difference between the maximum and minimum values in a data set. Alternatively it can report the values themselves, which gives more information than just the difference.
Mean Absolute Deviation (MAD): the deviations around the mean always sum to 0. So in MAD we take the absolute values:
MAD=i=1nXi-X n
MAD uses all the observations in the sample, so it’s better than Range. But we cannot take the derivative of MAD thanks to the absolute values—that’s why we use variance.
Variance and Standard Deviation: variance is the average of the squared deviations around the mean. Standard deviation is the positive square root of variance. If we know every member of a population, then we can compute the population variance and std dev (remember the population std dev σ is just the square root of the variance σ2):
σ2=i=1NXi-μ2N
However in investment management, a subset or sample of the population is all that we can observe. So we have another formula for sample variance and std dev:
s2=i=1nXi-X2n-1
Why is n-1 used instead of n as the divisor? Because using n-1 makes the sample variance an unbiased estimator—meaning the sample variance will be close to the population variance.
Semivariance and Semideviation: when we’re concerned only about a certain group of observations, e.g.: the variance for returns below the mean. Or the variance of returns below 10%. The formula for calculating the semivariance is the same as of that for variance, except that we discard observations that we’re not interested in. Semivariance and semideviation are not often used.
Calculate and interpret the proportion of observations falling within a specified number of standard deviations of the mean, using Chebyshev’s inequality.
Chebyshev developed an inequality using std dev as a measure of dispersion.
Chebyshev’s Inequality: the proportion of observations within k standard deviations of the arithmetic mean is at least
1-1/k2
For all k > 1, no matter how the data are distributed.
Define, calculate, and interpret the coefficient of variation and the Sharpe ratio.
Coefficient of variation(CV) is a measure of relative dispersion, defined as:
CV=s/X
The idea is that standard deviation alone might be difficult to interpret if we’re using it to compare data sets that have markedly different mean. For instance, a standard deviation of $20 million may mean a high degree of variability if the mean is only $70 million. But if the mean is $1 billion, then a standard deviation of $20 million is nothing—the variability is actually very small.
The inverse of CV reveals something about return per unit of risk. The Sharpe Ratio recognizes this, plus the existence of a risk-free return (e.g.: the mean U.S. T-bill return). The formula is defined as:
Sh=Rp-RFsp
The numerator is the mean return of portfolio – mean return of risk-free asset over the sample period. So the Sharpe ratio measures the reward (that is, the excess return the investor gets for the added risk he’s taking by choosing this portfolio instead of the risk-free asset) per unit of the portfolio’s risk. So the larger the Sharpe ratio is, the larger the excess reward for each unit of risk.
Two cautions concerning Sharpe Ratio:
When it’s negative, increased risk actually results in a number that is closer to 0 (numerically larger). For instance doubling the risk may increase the Sharpe ratio from -2 to -1. Obviously this doesn’t mean that the latter has better risk-adjusted performance.
The ratio only considers the standard deviation of return, which is most appropriate for approximately symmetric return distributions. So Sharpe Ratio may not adequately describe the risk of portfolio strategies with asymmetric returns such as those with option elements.
Define and interpret skewness, explain the meaning of positively or negatively skewed return distribution, and describe the relative locations of the mean, median, and mode for a nonsymmetrical distribution.
In normal distribution, the rule of thumb is: 68% observations lie between ±1 std dev, 95% observations lie between ±2 std dev, 99% observations lie between ±3 std dev.
Mean and variance may not adequately describe an investment’s distribution of returns. We don’t know whether large deviations are likely to be positive or negative, for example. In a symmetrical distribution of returns, equal loss and gain intervals have about the same frequency.
Some distributions are skewed, however. A positively skewed distribution has a long tail to the right (frequent small losses, few extreme gains); negatively skewed one has a long tail to the left (so frequent small gains, few extreme losses).
For positive skew, mean > median > mode. For negative skew, mean < median < mode. (Remember mean, median, mode, is in alphabetical order).
Skewness isthe statistical measure of a skew. (A symmetric distribution has, not surprisingly, a skewness of 0.) The sample skewness formula is defined as:
SK=nn-1n-2i=1nXi- X3s3
Some researchers believe that investors should prefer positive skewness, all else being equal.
Define and interpret measures of sample skewness and kurtosis.
Other than skew, a return distribution may differ from normal distribution by having more returns closely clustered around the mean (more peaked), and more returns with large deviations from the mean (fatter tails). (Most investors would perceive the fatter tails bit as increased risk.)
Kurtosis is the statistical measure for this “peakiness”. A distribution that is more peaked than normal is called leptokurtic (lepto = slender), one that’s less peaked is called platykurtic (platy = broad, but I always think of a platypus’s beak), the one that is identical to normal distribution is called mesokurtic (meso = middle).
The image below (taken from the Wikipedia article: http://en.wikipedia.org/wiki/Kurtosis) shows these distributions, starting from the red one with the highest excess kurtosis, to the purple one, with the lowest (negative) excess kurtosis.
Note the difference between excess kurtosis and kurtosis. The former tells you the difference between a particular distribution’s kurtosis and the normal distribution’s kurtosis (3).
KE=nn+1n-1n-2n-3i=1nXi-X4s4-3n-12n-2n-3
Learning Outcomes
Define a random variable, an outcome, an event, mutually exclusive events, and exhaustive events
A random variable is a quantity whose outcomes (possible values) are uncertain. An event is a specified set of outcomes.
Explain the two defining properties of probability, and distinguish among empirical, subjective, and a priori probabilities
Probability P(E) is a number between 0 ≤ P(E) ≤ 1 that measures the chance that a stated event E will occur. The sum of the probabilities of any set of mutually exclusive (only one event can occur at a time) and exhaustive (the events cover all possible outcomes) events is 1.
There are 3 broad approaches to estimating probability:
Empirical probability: estimating the probability of an event as the relative frequency of historical occurrence.
Subjective probability: sometimes we may not have historical records, so we may need to draw on personal, that is, subjective, judgments.
A priori probability: sometimes we can deduce probability through reasoning—so we base is on logical analysis instead of observation or personal judgment.
Empirical and a priori probability don’t vary from person to person, so they are called objective probabilities.
State the probability of an event in terms of odds for or against the event
Odds for E happening are 1 to 9, that means P(E) = 1/(1+9) = 0.1.
Odds against E happening are 17 to 1, means that P(E) = 1/(1+17) = 0.056
Distinguish between unconditional and conditional probabilities
Conditional probability P(A | B) is the answer to the question: what is the probability of A, given that B has happened?
The definition of conditional probability:
PA B=PABPB, P(B)≠0
With P(AB) being the joint probability of A and B.
Note the difference between P(AB) and P(A | B): P(AB) is the probability of A and B, considering all possible events, whereas P(A | B) is the probability of P(AB) occurring, given P(B).
Calculate and interpret 1) the joint probability of two events, 2) the probability that at least one of two events will occur, given the probability of each and the joint probability of the two events, and 3) a joint probability of any number of independent events
The joint probability is an “and” probability—meaning the probability of both A and B occurring. Sometimes we’re interested in “(inclusive) or”, meaning the probability of either A happens, or B happens, or both A and B happen. This is defined as:
PA or B=PA+ PB- P(AB)
Of course if A and B are mutually exclusive, P(A or B) will simply be P(A) + P(B).
Distinguish between dependent and independent events
Two events are independent if the occurrence of one event doesn’t affect the probability of the other event. More formally, A and B are independent if and only if P(A | B) = P(A), or P(B | A) = P(B).
Note that independent events are different from mutually exclusive events! This becomes more obvious when we consider the joint probability of two independent events A and B:
PAB=PAP(B)
Whereas mutually exclusive events cannot happen at the same time—therefore P(AB) = 0.
Calculate and interpret, using the total probability rule, an unconditional probability
The total probability rule:
PA=PAS1+PAS2+PAS3+…+P(ASn)
Where S1… Sn are mutually exclusive and exhaustive scenarios/events. In other words, we can calculate the unconditional probability P(A), if we know the joint probabilities of A and mutually exclusive and exhaustive events.
Explain the use of conditional expectation in investment applications
Expected value E(X) of a random variable X is the probability-weighted average of the possible outcomes of the random variable.
EX=i=1nPXiXi
(For continuous random variables the corresponding operation is integration.)
The variance of a random variable is the expected value of squared deviations from the random variable’s expected value (yes, it’s an expected value of expected value kinda thing):
σ2X=EX-EX2= i=1nPXiXi-EX2
We can have conditional expected value E(X | S) as well, which is the expected value of random variable X, given S.
EX | S=i=1nPXi | SXi
The total probability rule applies as well for expected value, given S1...Sn, where they are mutually exclusive and exhaustive.
Diagram an investment problem, using a tree diagram
Calculate and interpret covariance and correlation
Calculate and interpret the expected value, variance, and standard deviation of a random variable and of returns on a portfolio
Calculate and interpret covariance given a joint probability function
Calculate and interpret an updated probability, using Bayes’ formula
Bayes formula is easier to remember like this:
PABPB=PBAP(A)
Substituting A with Event and B with Information, it’s just a matter of rearranging the equation to get to the form we see in the CFAI book:
PEventInformation=PInformationEventPEventPInformation
Bayes theorem is about how our beliefs about observing the “Event” are updated by having observed the new “Information”.
Identify the most appropriate method to solve a particular counting problem, and solve counting problems using the factorial, combination, and permutation notations
Calculate and interpret the elasticities of demand (price elasticity, cross elasticity, income elasticity) and the elasticity of supply, and discuss the factors that influence each measure;
Calculate elasticities on a straight-line demand curve, differentiate among elastic, inelastic, and unit elastic demand, and describe the relation between price elasticity of demand and total revenue.