FTX3044F Finance IIA Chapter 1 Investment Ethics Unethical examples: • Bernard Madoff • Enron Section 1 Standard 1: Professionalism A: Knowledge of the Law B: Independence & Objectivity • Understand and comply will all CFA, government, etc laws • Take reasonable care to maintain independence and objectivity • Where in conflict, the strictest law must be adhered to • May not offer, solicit or accept gift, compensation, offer, etc which can compromise own or others’ independence and objectivity • May not knowingly violate or assist others in violating laws • Must dissociate from such violation - reporting of violations to governmental or regulatory bodies may be advisable, but it is not a requirement under CFA institute rules - recommend legal opinion be obtained • Companies should develop and adopt a code of ethics - put in place written procedures for the reporting of suspected violations • Benefits: small gifts permitted, shares from IPO not allowed, all gifts must be disclosed to employer, no gifts which can be perceived to possibly influence investment decisions • Research: must not be done because of pressure from companies or to attract potential clients, paid research must not influence the researcher, company paying for expenses on research visit acceptable if not excessive or out of the ordinary, must be no pressure to issue favorable reports • Integrity of opinions - protect, sate in reports, remunerate • Limit payment of expenses by outside parties • Restrict personal investment by investment professionals • Ensure supervision of activities of analysts and portfolio managers 2 C: Misrepresentation D: Misconduct • Must not knowingly make misrepresentation relating to investment analysis, recommendations, sources, actions, etc • May not engage in professional conduct involving dishonesty, fraud, deceit, etc • Give all info relevant to avoid possible misrep • May not commit act which reflects adversely on professional reputation, integrity, competence • Do not misrepresent investment performance, qualifications, abilities, track record or services that can be performed • Attempt to prevent distribution of misrepresentative literature • Plagiarism: citing quotations supposedly attributable to experts, etc without being specific about who they are; presenting opinions, data, etc of others with acknowledgement of source but without original caveats and qualifying statements • Understand limitations of firm or investment strategy • Maintenance of professional integrity, good reputation and competence • Not necessarily only illegal activities, but any that can affect above negatively • Do not commit acts that makes the profession look bad e.g. Drunkenness on the job • Avoid deceit, dishonesty and fraud • Be accurate and complete in presentations • Keep copies of all sources, background notes, etc relevant to or used in research reports • Attribute all quotations, projections, tables, statistics, models and ideas. Exceptions: when prepared by recognized financial or statistical reporting service e.g. Reuters 3 Section 2 Standard 2: Integrity of Capital Markets A: Material non-public information • Keep records of info as proof and basis and save all research • When in possession of material non-public information that could affect the value of an investment you may not use or cause others to act on this information • Influential analyst’s opinions may in itself affect security price, but not being a full insider the analyst does not need to make reports etc known to all • Material: can affect a reasonable investor’s investment decision or is likely to affect the value of an investment • If info is material, non-public encourage source company to disseminate it widely - if not possible, report to internal supervisory function and do not trade on it • Non-public: not widely disseminated to the broader public • Mosaic Theory: you are allowed to trade on a combination of public material information and non-public, non-material information • Material non-public info: earnings, tender offers, mergers and acquisitions, significant asset changes, management changes, patents, licences, significant supplier and customer changes, legal issues, financial position, cashflow issues, etc • Materiality determined by: substance and specificity of info, reliability of source, strength of link to price of security, time value of info • Do not knowingly induce company insiders to disclose material non-public info • Implement compliance procedures e.g. Monitoring of employee and proprietary trading and documentation • “Chinese walls” between relevant departments • Internal info barriers to ensure info only goes to those who need it • Companies should provide same info to all and not discriminate • Beware of selective disclosure by management of company insiders to analysts 4 B: Market Manipulation • May not engage in practices that distort prices, mislead markets or artificially inflate trading volumes • Artificially influencing prices of trading volumes on the market is prohibited • False rumours and misleading info • Legitimate transactions include trades for tax purposes (e.g. Selling and immediately re-buying a share) or affecting the price when trading in an illiquid share • Intent is the key. If activities are meant to mislead others, they are likely prohibited 5 Section 3 Standard 3: Duties to Clients A: Loyalty, prudence and care B: Fair Dealing • Duty of loyalty, must act with reasonable care and exercise prudent judgement • Must deal fairly and objectively with all clients in investment analysis recommendations, actions, etc • Must act for benefit of clients (above own and employer’s interests) • Objective and fair dealing with all clients - no discrimination • Must determine applicable fiduciary duty and comply herewith to persons/ interest to whom owed • Prudent Man Rule: exercise care and diligence a prudent person familiar with the investment situation/ needs would exercise • Investment decisions should be made with reference to the total portfolio • At least a quarterly transaction and holdings statement to clients with all info pertaining to assets held on their behalf • Periodically review investments to check compliance and mandates • Applies to dissemination of info or investment decisions/ actions • Investment recommendation should be communicated to all clients • Timing: dissemination to all at approximately same time • Allocate on an impartial basis • Limit no of people aware of pending recommendation and make a policy that people in the know cannot act • Written trade allocation procedures • Systematic account review to ensure no client treated preferentially • Diversify to minimize risk of loss • Treat all clients equally 6 C: Suitability D: Performance presentation • In advisory relationship: investigate and understand client’s investment experience, knowledge, risk-return objectives, financial needs, financial constraints, etc before making investment recommendations or taking investment actions • In communicating investment results, take care to ensure it is fair, accurate and complete • Determine that investment is suitable for client needs, constraint, written objectives and mandate before making investment recommendations or taking investment actions • When managing a portfolio or specific mandate to a mandate or given style: make only investment recommendations or take investment actions consistent with objectives, constraints and mandate • Investment Policy Statement (IPS): client description, investment objectives, investor constraints - liquidity, tax, cashflows, time horizons, legal requirements, preferences, etc - benchmarks for performance management --- base asset allocation on this and review regularly • GIPS (Global Investment Performance Standards) • No claims about ability to achieve future performance can be made • Do not selectively omit - include all relevant info • Keep records/ data • Consider level of understanding of clients • Include terminated accounts in presentation • Include other relevant disclosures to explain context of results obtained 7 E: Preservation of Confidentiality • Keep info on current, former and prospective clients confidential unless: • Info concerns illegal activities • Disclosure required by law • Client/ prospective client permits disclosure • Applies to current, past and prospective clients • Disclosure required by law overrides this standard • Disclosure to co-workers involved with client only on a need to know basis 8 Section 4 Standard 4: Duties to Employers A: Loyalty B: Additional compensation arrangements • Must act in interests of employer and not deprive employer of benefits of skills and efforts, divulge info or cause harm to employer • May not accept gifts, benefits, compensation, etc that competes (or may be reasonably expected to create a conflict of interest) with employer’s interest, unless written permission obtained from all parties involved • Confidentiality • Outside work for compensation: allowed if employer fully informed of all aspects and agrees to all aspects • Leaving employer: must act in employer’s best interests right up to point of leaving. No taking of confidential info, soliciting or clients prior to leaving, misappropriation of IP (models, reports, etc) and client databases • Includes monetary and non-monetary compensation • Compliance: written report to employer specifying nature of proposed outside service, parties involved, compensation, etc • Use of skills, experience and contacts gained at previous employer not prohibited • Personal and employer’s interests secondary to protection of clients interest and integrity of capital markets - “whistleblowing” allowed if for above purpose • A whistleblower is a person who tells the public or someone in authority about alleged dishonest or illegal activities (misconduct) occurring in a government department or private company or organization 9 C: Responsibilities of supervisors • Make reasonable efforts to detect and prevent violations of applicable rules, laws, regulations or Code of Standards by anyone under their authority or supervision • Must also take steps to detect this (investigate where suspicion) • Delegation of supervisory duties does not absolve supervisor from responsibility • Firm must have adequate compliance procedures, otherwise must decline supervisory responsibility in writing until in place • Members should take initiative and recommend that employer adopt code of ethics • Ideal is a clear code of ethics, supported by an efficient compliance system and processes • Identify situations in which violations are likely to occur and establish procedures that will address these 10 Section 5 Standard 5: Investment analysis, recommendations & Action A: Diligence and reasonable basis B: Communication with clients and prospective clients • Exercise diligence, independence and thoroughness in analyzing investments, making investment recommendations and taking investment actions • Disclose to clients format and basic principles of investment processes used to analyse investments, select securities and build portfolios • Have reasonable and adequate basis (supported by appropriate research and investigation) for investment analysis, recommendations and actions • Promptly disclose any changes that may materially affect these processes • Secondary and 3rd party research acceptable, but must be assessed thoroughly • Group research and decision making: if you are not in agreement, you may dissociate from decision, etc but do not have to if you believe the rest of the group used independent and informed judgement - document differences • Distinguish between fact and opinion in the presentation of investment analysis and recommendations • Ensure clients understand basic characteristics of relevant investments 11 C: Record retention • Develop and maintain appropriate records to support investment analysis, recommendations and actions, as well as investment-related communications with clients and prospective clients • Retain records - recommended for 7 years minimum if no other regulatory standards • Records are property of employer, not analyst, etc 12 Section 6 Standard 6: Conflicts of Interest A: Disclosure of conflicts B: Priority of transactions • Make full and fair disclosure of all matters that can reasonably be expected to impair independence or objectivity or interfere with duties to clients, prospective clients or employers • Investment transactions of clients must have priority over own transactions • Ensure that such disclosures are prominently delivered in plain language and effectively communicated • Own transactions only after clients and employer had opportunity to transact • Examples: direct or indirect beneficial ownership of shares • Own transactions include those for direct family or entity in which has a beneficiary interest • Relationships with company management, etc • Family members as beneficiaries: just another client • Financial interest or relationships between analyst or their company with companies, brokers, etc Compliance procedures: • Analysts within brokerage company or investment division of a bank where a negative research report on a current or potential client may compromise actual or potential income • IPOs: may create perception of competing with clients for limited opportunity, should declare any interests in IPOs and not benefit from client’s participation (front-running: you know that someone is going to buy a lot of shares so you buy before price rises) • Private placements: strict limitations for this regarding investment personnel, not often competing with clients but could be seen as a favor designed to influence analyst future judgements, clear incentive to recommend these deals to clients if self already a beneficiary 13 • Blackout/ restricted periods: period prior to trade for clients wherein those involved in investment process may not trade in security for themselves (avoid front-running), in large firms this could be a ban on private trading as the firm is nearly always trading in most securities on behalf of one or more clients • Reporting requirements: regular disclosure by investment personnel of private holdings, duplicate copies of private security transactions to be received by employer from personnel’s brokers, investment personnel required to receive preclearance for all private transactions C: Referral fees • Must disclose to employer, clients and prospective clients any compensation, consideration or benefit received or paid for recommendation of products or services • Acceptable but: must be disclosed to all relevant parties (whether paid in cash or not) so that can evaluate possible partiality in recommendations • Personal investments: acceptable provided does not disadvantage clients, investment professional does not benefit personally from trades done for clients, compliance with all regulatory requirements 14 Section 7 Results & Summary • Public loses confidence in investment industry - compromises sustainability • Need a well-functioning financial system - spurs economy growth • External controls e.g. Legislation and regulatory bodies • Internal controls e.g. Company rules, compliance officer and code of ethics • CFA Institute Code of Ethics • Investors get hurt - financially and otherwise • Costs to the economy of short-term selfish application of financial resources • Offenders may go to prison 15 Summary: 4. Duties to Employers 1. Professionalism • Loyalty • Knowledge of the law • Additional compensation arrangements • Independence and objectivity • Responsibilities of supervisors • Misrepresentation • Misconduct 5. Investment analysis, Recommendations and Action • Diligence and reasonable basis 2. Integrity of Capital Markets • Communication with clients and prospective clients • Material non-public information • Record retention • Market manipulation 6. Conflicts of Interest 3. Duties to Clients • Disclosure of conflicts • Loyalty, prudence and care • Priority of transactions • Fair dealing • Referral fees • Suitability • Performance presentation • Preservation of confidentiality 16 Chapter 2 The Investment Environment Investment: the current commitment of money or other resources in the expectation of reaping future benefits. Section 1 Real Assets vs Financial Assets Real Assets • Determine the productive capacity and net income of the economy • Something that produces income in the economy • E.g.: land, buildings, machines, knowledge used to produce goods and services • Financial assets are claims to the income generated by real assets • NOTE: patent = real asset 18 Section 2 Financial Assets Claims on real assets. • E.g. Dividends not paid in poor times Three types: 1. Fixed income or debt e.g. Bonds Derivatives: 2. Common stock or equity e.g. Shares • Value derives from prices of other securities, such as stocks and bonds 3. Derivative securities - derived from other assets • Used to transfer risk Fixed income: • Use them to gear your returns • Payments fixed or determined by a formula • Money market debt: short term (less than a year maturity), highly marketable, usually low credit risk • Capital market debt: long term bonds, can be safe or risky Common stock: • Equity or ownership in a corporation • Payments to stockholders are not fixed, but depend on the success of the firm 19 Section 3 Financial Markets and the Economy The Information Role of Financial Markets: Allocation of Risk: • Financial markets play a central role in the allocation of capital resources • Investors can select securities consistent with their tastes for risk • If a corporation seems to have good prospects for future profitability, investors will bid up its stock price • Capital therefore flows to companies with good prospects • Also benefits the firms that need to raise capital to finance their investments because each security is being sold for the best possible price because investors are selecting security types which best suit their risk-return preferences • Some companies may be “hot” for a short period of time, attract a large flow of investor capital, and then fail after only a few years Separation of Ownership and Management: • No-one knows with certainty what will happen - stock market encourages allocation of capital to those firms who appear at the time to have the best prospects • Gives a firm stability • Firm’s management should pursue strategies that enhance the value of their shares - can lead to agency problems Consumption Timing: • Agency problems: managers, who are hired as agents of the shareholders, may pursue their own interests instead • Individuals may earn more than they currently wish to spend • Ways to combat agency problems: • They can “store” their wealth in financial assets 1. Compensation plans tie the income of managers to the success of the firm • Investment in financial assets and postpone immediate consumption for future consumption 20 2. Boards of directors can force out management teams that are underperforming 3. Outsiders, such as security analysts, monitor the firm closely and make the life of poor performers at the least uncomfortable 4. Bad performers are subject to the threat of takeover Corporate Governance and Corporate Ethics: • For markets to effectively serve their purpose, there must be an acceptable level of transparency that allows investors to make well-informed decisions • Accounting scandals e.g. Enron, Rite Aid, HealthSouth • Auditors - watchdogs of the firms • Analyst scandals e.g. Arthur Andersen • Sarbanes-Oxley Act in America to tighten the rules of corporate governance • South Africa has the King III • E.g. Law that says you must have a certain number of independent directors 21 Section 4 The Investment Process Investors make two types of decisions in constructing their portfolios: 1. Asset Allocation • Choice among broad asset classes (top down) 2. Security Selection • Choice of which securities to hold within asset class • Security analysis to value securities and determine investment attractiveness 22 Section 5 Markets are Competitive The Risk-Return Trade-off: • Always risk associated with investments • High return = high risk Efficient markets: • Financial markets process all relevant information about securities quickly and efficiently ie the security price usually reflects all the information available to investors concerning its value If markets are efficient and prices reflect all relevant information - it is probably better to follow passive strategies instead of spending resources in a futile attempt to outguess your competitiors • Active Management: attempt to improve performance either by identifying misplaced securities or by timing the performance of broad assets classes - try beat the market • Passive Management: holding highly diversified portfolios without spending effort or other resources attempting to improve investment performance through security analysis - no attempt to find undervalued securities - don’t fight the market 23 Section 6 The Players • Business firms - net borrowers • Households - net savers • Government - can be both borrowers and savers • Investors trade previously issued securities among themselves in the secondary markets • Commercial Banking: • Take deposits and make loans Financial intermediaries: • Pool and invest funds • Bring lenders and borrows together • Banks, investment companies, insurance companies and credit unions Universal Bank Activities: • E.g. Standard Bank, ABSA, FNB, etc • Investment Banking: e.g. Investec : • Underwrite new stock and bond issues • Sell newly issued securities to public in the primary market 24 Section 7 Rise of Systematic Risk Systematic Risk: a potential breakdown of the financial system in which problems in one market spill over and disrupt others • One default may set off a chain of further defaults • Potential contagion from institution to institution, and from market to market • When firms are fully leveraged, losses on their portfolios ca force them to sell some of their assets to bring their leverage back into line • Wavs of selling from institutions that simultaneously need to “de-leverage” can drive down assets prices and exacerbate portfolio losses - forcing additional sales and further price declines in a downward spiral Policies that limit this risk: • Transparency - to allow traders and investors to assess the risk of their counterparties • Capital adequacy - to prevent trading participants from being brought down by potential losses, • Frequent settlement of gains or losses - to prevent losses from accumulating beyond an institution’s ability to bear them • Incentives - to discourage excessive risk taking • Accurate and unbiased risk assessment 25 Chapter 3 Asset Classes & Financial Instruments Money market instruments: shortterm debt Capital market instruments: bonds, long-term debt, equity securities, derivative securities Section 1 Equity Securities Common Stock as Ownership Shares American Depository Receipts: • Each share entitles its owner to one vote on any matters of corporate governance that are put to a vote at the corporation’s annual meeting • Way for a company to raise money from international investors without listing on a foreign stock exchange • Also entitles owner to a share in the financial benefits of ownership • Its not a share trading on the foreign stock exchange - instead a bank will give an investor exactly the same proceeds and rights as if they did have a share • Residual claim: to the profits of the company ie everyone else gets paid first and then the company only pays dividends • You have the same rights as a shareholder who owns the actual share • Limited liability: the most shareholders can lose in the event of failure of the corporation is their original investment Preferred Stock: perpetuity • Fixed dividends • Priority over common stock • Tax treatment - treated like a dividend 27 Section 2 Stock Market Indexes Dow Jones Industrial Average: Final value: $30 + $90 = $120 • Includes 30 large blue-chip corporations: tells you how their prices changed Percentage change in portfolio value = (120-125)/125 = -4% • Computed since 1896 • Price-weighted average: all 30 shares are treated equally, as if they have the same importance, but if the largest share went up - this should be more important than if the smallest share did so price-weighted Index: Initial index value: (25+100)/2 = 62.5 Final index value: (30 + 90)/2 = 60 Percentage change in index = (60-62.5)/62.5 = -4% NOTE: Standard & Poor’s Indexes: Index: “score” for total stock exchange. Measurement of how the shares in the stock exchange changed. E.g. Allshare 50 = score of top 50 shares. • S&P 500: broadly based index of 500 firms, market-valueweighted index • Investors can base their portfolios on an index: • Buy an index mutual fund Example: Price-Weighted Average • Buy exchange traded funds (ETFs) Portfolio: Initial value: $25 + $100 = $125 28 Other Indexes: Two company example: Yesterday: US Indexes: Company A: R11 x 20m = R220m cap • NYSE Composite Company B: R5 x 100m = R500m cap • NASDAQ Composite Total cap: R720m = 100 (base value) • Wilshire 5000 Today: Foreign Indexes: A: R15 x 20m = R300m cap • Nikkei (Japan) B: R4 x 100m = R400m cap • FTSE (UK) Total cap: R700m • DAX (Germany) • Hang Seng (China) 700/ 720 x 100 = 97 (new index) • TSX (Canada) FTSE/ JSE Allshare Index: • Market cap weighted but only the free float 29 Section 3 Debt, Preference Sahres & Ordinary Shares DEBT ORDINARY SHARES PREFS MATURITY Finite Perpetual Finite to perpetual SENIORITY OF CLAIM Contractual. Priority over ordinary shares and prefs Residual. Subordinate to debt and prefs Residual. Subordinate to debt and have priority over ordinary TAX FOR COMPANY Interest deductible Not deductible Not deductible TAX FOR INVESTOR Taxable Taxable from 1 April 2012 Taxable from 1 April 2012 Right to vote Limited voting rights - can only vote on issues that affect your rights VOICE IN MANAGEME NT No influence except in liquidation 30 Chapter 4 How Securities are Traded The first time a security trades is when it is issued to the public. Next, already-issued securities may be traded among investors. Section 1 How Firms Issue Securities Primary market: • Firms issue new securities through underwriter to the public • Private placement: issue that is usually sold to one or a few institutional investors and is generally held to maturity, cheaper than public offerings, not traded in secondary markets • Investors get new securities and the firm gets funding Markets for securities: Secondary market: • Investors trade previously issued securities among themselves 1. Institutional Security markets: • Formal, organized exchanges • Buyer and seller trading through broker Stocks: • IPO - Initial Public Offering: stocks issued by a formerly privately owned company that is going public for the first time • Seasoned offering: offered by companies that already have floated equity • Only members can trade (stock brokers) • Listing requirements • E.g. LSE, JSE, NYSE, etc 2. Over-the-Counter (OTC) markets: Bonds: • Decentralized market • Public offering: issue of bonds sold to the general investing public that can be traded on the secondary market • Unlisted shares • Network of dealers buy and sell shares 32 • E.g. NASDAQ, BJM OTC 3. Direct trading between two parties NOTE: - broker market: broker brings the buyer to the seller - dealer market: e.g. NASDAQ: the intermediary holds the share themself, intermediary brings the stock to the buyer Investment banking: • Underwriting: investment bank helps the firm to issue and market new securities • Prospectus: describes the issue and the prospects of the company - preliminary prospectus = red herring because it includes a statement printed in red stating that the company is not attempting to sell the security before the registration is approved • Road show: investment bankers and the company go and visit the people who they want money form - try to convince them to invest - investors also give an indication of what they think a fair price is • Firm commitment/ best effort agreement: investment bank purchases securities form the issuing company and then resells them to the public • Shelf registration: SEC Rule 415: allows firms to register securities and gradually sell them to the public for two years 33 Section 2 How Securities are Traded Types of Markets: Types of Offers: • Direct search markets • Market order - buyers and sellers seek each other out - executed immediately • Brokered markets - Trader receives current market price - brokers search out buyers and sellers • Price-contingent order • Dealer markets - traders specify buying or selling price - dealers have inventories of assets from which they buy and seek NOTE: a large order may be filled at multiple prices • Auction markets - traders converge to one place to trade Bid Vs Ask Prices: BID PRICE: • Bids are offers to buy • Price that investors are willing to pay PRICE BELOW THE LIMIT PRICE ABOVE THE LIMIT BUY Limit-Buy Order Stop-Buy Order SELL Stop-Loss Order Limit-Sell Order ASK PRICE: • Asks are offers to sell • Investors must buy at the ask/ offer price 34 Limit order vs option: • Limit order does not protect against sudden, large market movements • Option does, but more expensive • Can get the same protection of a limit order using an option option costs you money (premium) whereas dont have to put in money for a limit order • As the market price moves the computer looks and sees if the price is the same as the limit order - as soon as it goes above the limit it stops the order • But market might change in seconds and may not ‘touch’ every price so limit order doesn’t protect you if the price goes up by R5 suddenly and passes your limit Trading mechanisms: • Dealer markets • Electronic communication networks (ECNs): true trading systems that can automatically execute orders • Specialist markets: trading is managed by a specialist assigned responsibility for that security: maintain a “fair and orderly market” 35 Section 3 US Securities Markets NASDAQ: Electronic Communications Networks: • Lists about 3200 firms • ECNs: private computer networks that directly link buyers with sellers for automated order execution • Originally, NASDAQ was primarily a dealer market with a price quotation system • Today, NASDAQ’s Market Centre offers a sophisticated electronic trading platform with automatic trade execution • Large orders may still be negotiated through brokers and dealers New York Stock Exchange: • Major ECNs include NASDAQ’s Market Centre, arced, Direct Edge, BATS, and LavaFlow • “flash trading”: computer programs look for even the smallest misplacing opportunity and execute trades in tiny fractions of a second Bond Trading: • Lists about 2800 firms • Most bond trading takes place in the OTC market among bond dealers • Automatic electronic trading runs side-by-side with traditional broker/ specialist system • Market for many bond issues is “thin” • SuperDot: electronic order-routing system • DirectPlus: fully automated execution for small orders • NYSE is expanding its bond-trading system • NYSE bonds is the largest centralized bond market of any US exchange • Specialists: handle large orders and maintain orderly trading 36 Section 4 Market Structure in Other Countries London - predominately electronic trading Euronext - market formed by combination of the Paris, Amsterdam and Brussels exchanges, then merged with NYSE Tokyo Stock Exchange - switched to all-electronic trading Globalization and consolidation of stock markets - NYSE mergers and acquisitions: Archipelago (ECN), American Stock Exchange, Euronext - NASDAQ mergers and acquisitions: Instinet/ INET (ECN), Boston Stock Exchange - Chicago Mercantile exchange acquired: Chicago Board of Trade, New York Mercantile Exchange 37 Section 5 Trading Costs 1.Brokerage Commission: • Fee paid to broker for making the transaction • Explicit cost of trading • Full service (research team) vs discount brokerage (“no frills” service) 2. Spread • Difference between the bid and asked prices • Implicit cost of trading 38 Section 6 Buying on Margin • Buying part of the total purchase price of a position using a loan from a broker • Only available to you if your broker allows it • Investors contributes the remaining portion • Margin refers to the percentage or amount contributed by the investor Stock price falls to $70 per share New position: Stock $7000 - Borrowed $4000 ( ( ( Equity $3000 Margin % = $3000/ $7000 = 43% • You profit when the stock appreciates Example 2: How far can the stock price fall before a margin call? Let maintenance margin = 30% Example 1: Share price: $10 Equity = 100P - $4000 60% - Initial margin 40% - Maintenance margin 100 shares purchased Percentage margin = (100P - $4000)/ 100P So (100P - $4000)/ 100P = 0.3 P = $57.14 Initial position: Stock $10 000 - Borrowed $4000 ( ( ( Equity $6000 39 Section 7 Short Sales PURPOSE: to profit from a decline in the price of a stock or security MECHANICS: selling a share you do not have • Borrow stock through a dealer • Sell it and deposits proceeds and margin in an account • Closing out the position: buy the stock and return to the party from which is was borrowed Example 1: Stock owed: 1000 shares Stock falls to $70 per share Assets( ( ( ( $100 000 (sale proceeds)( ( Liabilities $70 000 (buy shares) $50 000 (initial margin)( ( ( ( ( ( ( ( Equity ( ( ( ( ( ( $80 000 Profit = ending equity - beginning equity = $80 000 - $50 000 = $30 000 1000 shares 50% - margin ( ( = decline in share price x number of shares sold short 30% - maintenance margin $100 - initial price Example 2: How much can the stock price rise before a margin call? Sales proceeds: $100 000 Margin & equity: $50 000 [($100 000 + $50 000) - 1000P]/ 1000P = 30% P = $115.38 40 Section 8 Regulation of Securities Markets Major regulations: Insider trading: • Securities Act of 1933 • Officers, directors and major stockholders must report all transactions in firm’s stock • Securities Act of 1934 • Securities Investor Protection Act of 1970 • Insider do exploit their knowledge - Jaffe study Self-regulation: • Financial Industry Regulatory Authority - Inside buyers > inside sellers = stock does well - Inside sellers > inside buyers = stock does poorly • CFA Institute standards of professional conduct • Sarbanes-Oxley Act - Public Company Accounting Oversight Board - Independent financial experts to serve on audit committees of boards of directors - CEOs and CFOs personally certify firms’ financial reports - Boards must have independent directors 41 Chapter 5 Macroeconomic & Industry Analysis Valuation: Fundamental Analysis: models a company’s value by assessing its current and future profitability. Purpose: to identify mispriced stocks relative to some measure of “true” value derived from financial data. Section 1 Introduction to Valuation Fundamental Analysis: Mispriced stocks: • Fundamental value > market value : BUY • Fundamental value < market value : SELL There are various Models of Equity Valuation as well as Valuation by Comparables. How then do you select a method? • Multiples incorporate current market conditions more so than free cash flows - so time period is important • Certain valuation methods suit certain industries better - gold mines valued with reference to their assets - comparable model. Banks use price to book ratio. Retail and industrial companies - free cash flow to firm Forecasting Earnings: Basics Top-Down Analysis: • CA approach Method selection: • Data availability - if given accounting information, cant do FCF • Minority share vs majority share - minority share: cannot influence dividend policy because don’t have control, best return you can expect is future dividends; if you can influence dividend policy: move away from dividend valuation to an earnings valuation 1. Analyse and forecast macroeconomic variables - which country is best? Which asset class is better ie debt or equity? 2. Analyse and forecast the industry - consider business cycles and which industry is better in which business cycle 3. Within industries with best expected future, select the firms with best future expectations relative to their current prices to invest in • Free cash flow models are theoretically better 43 Bottom-Up Analysis: Logic of Top-Down: 1. Analyse and forecast company-specific issues Safest way: diversify portfolio and aim to beat the market by a little. 2. Consider future of industry and macroeconomy as part of company specific issues 3. Invest in companies with the best expected future earnings relative to their current price Which is better? IT DEPENDS. Top-Down: don’t really expect share to go up or down that much = markets are efficient Bottom-Up: don’t expect market efficiency • Economic Analysis: determines asset class allocation bonds, equities, property, cash, etc • Industry Analysis: determines industry sector weightings in equity portfolio - financials, industrial, resources, etc Top-Down Analysis: • Global economic forecast • Company Analysis: determines company weightings per sector • Domestic macroeconomic forecast • Industry analysis • Analyse company positioning within the selected industries • Specific company valuations • Selection and investment decision 44 Section 2 The Global Economy & the Domestic Macroeconomy • Stock markets around the world responded in unison to the financial crisis of 2008 • Performance in countries and regions can be highly variable • It is harder for businesses to succeed in a contracting economy than in an expanding one • Political risk: the global environment may present much greater risks than normally found in US-based investments • Exchange rate risk: changes the prices of imports and exports • Stock prices rise with earnings • P/E ratios are normally in the range of 12-25 • The first step in forecasting the performance of the broad market is to assess the status of the economy as a whole • Key variables: GDP, unemployment rates, inflation, interest rates (increase in rates will suppress the global economy), budget deficit, consumer sentiment • Indebtedness (not just of the consumer but of the government sector and banking sectors as a whole) 45 Section 3 Demand & Supply Shocks Demand shock: Supply shock: • An event that affects demand for goods and services in the economy • An event that influences production capacity or production costs • Positive: reductions in tax rates, increases in the money supply, increases in government spending or increases in foreign export demand • Examples: changes in the price of imported oil, freezes, floods or droughts that might destroy large quantities of agricultural crops, changes in the educational level of an economy’s workforce, or changes in the wage rates at which the labour force is willing to work • Usually characterized by aggregate output moving in the same directions as interest rates and inflation • E.g. Increase in government spending - increase GDP - increase interest rates by increasing the demand for borrowed funds by the government as well as by businesses that might desire to borrow to finance new ventures - increase inflation rate if demand for goods and services is raised to a level at or beyond the total productive capacity of the economy • Usually characterized by aggregate output moving in the opposite direction of inflation and interest rates 46 Section 4 Federal Government Policy Demand-side policies: • Fiscal policy - government spending and tax • Monetary policy - manipulation of the money supply • Increasing the money supply lowers interest rates and stimulates the economy • Less immediate effect than fiscal policy • Tools of monetary policy include open market operations, discount rate and reserve requirements Fiscal policy: • Most direct way to stimulate or slow the economy • Formulation of fiscal policy is often a slow political process • To summaries the net effect of fiscal policy, look at the budget surplus or deficit: • Deficit stimulates the economy because: it increases the demand for goods (via spending) by more than it reduces the demand for goods (via taxes) Supply-side policies: • Goal: to create an environment in which workers and owners of capital have the maximum incentive and ability to produce and develop goods • Supply-siders focus on how tax policy can be used to improve incentives to work and invest Monetary policy: • Manipulation of the money supply to influence economic activity 47 Section 5 Business Cycles The transition points across cycles are called peaks and troughs. Defensive industries: • A peak is the transition from the end of an expansion to the start of a contraction • Little sensitivity to the business cycle • A trough occurs at the bottom of a recession just as the economy enters a recovery Cyclical industries: • Above-average sensitivity to the state of the economy • Examples: producers of consumer durables e.g. Autos and capital goods ie goods used by other firms to produce their own products • High betas • Examples: food producers and processors, pharmaceutical firms and public utilities • Low betas Economic indicators: • Leading indicators tend to rise and fall in advance of the economy • Coincident indicators move with the market • Lagging indicators change subsequent to market movements • If you forecast an upturn you want to be in a cyclical industry • If you forecast a slowdown you don’t want to be in a cyclical industry 48 Section 6 Industry Analysis General principle: company will perform better if the whole economy is doing well Defining an industry: • North American Industry Classification System (NAICS) codes • Firms with the same four-digit NAICS codes are commonly taken to be in the same industry Sensitivity to the Business Cycle: • Three factors determine how sensitive a firm’s earnings are to the business cycle 1. Sensitivity of sales: necessities vs discretionary goods. Items that are not sensitive to income levels e.g. Tobacco and movies vs items that are e.g. Machine tools, steel, autos 2. Operating leverage: the split between fixed and variable costs. Firms with low operating leverage (less fixed assets) are less sensitive to business conditions. Firms with high operating lev- erage (more fixed assets) are more sensitive to the business cycle 3. Financial leverage: the use of borrowing. Interest is a fixed cost that increases the sensitivity of profits to the business cycle BUSINESS CYCLE PICTURE **************** Sector Rotation: • Portfolio is shifted into industries or sectors that should outperform according to the stage of the business cycle • As the business cycle moves you put more emphasis on the sectors that are favored at that point in the business cycle - portfolio shift • Peaks - natural resource extraction firms • Contraction - defensive industries such as pharmaceuticals and food • Trough - capital goods industries • Expansion - cyclical industries e.g. Consumer durables 49 SECTOR ROTATION PICTURE ************ NOTE: not an “absolute” rotation - more like a shift of emphasis Industry Life Cycles: 4.Bargaining power of buyers: higher bargaining power = can demand price concessions 5. Bargaining power of suppliers: if a supplier has monopolistic control over the product it can demand higher prices for the good and squeeze profits • Start-Up Stage - rapid and increasing sales growth • Consolidation - stable sales growth • Maturity - slowing sales growth • Relative decline - minimal or negative sales growth Which life cycle is most attractive? IT DEPENDS. Industry structure and performance: To evaluate the competitiveness of an industry: use the Five Determinants of Competition: 1. Threat of entry: harder to enter = less competitive 2. Rivalry between existing competitors: more firms already in industry = more price competition 3. Pressure from substitute products: industry may face competition from firms in related industries 50 Chapter 6 Equity Valuation Models Models of Equity Valuation: • Balance sheet models • Dividend Discount Models • Price/ Earnings Ratios • Free Cash Flow Models Section 1 Valuation by Comparables • Compare valuation ratios of firm to industry averages • Ratios like price/ sales are useful for valuing start-ups that have yet to generate positive earnings Limitations of book value: • Book values are based on historical cost, not actual market values • It is possible, but uncommon, for market value to be less than book value • “floor” or minimum value is the liquidation value per share • Tobin’s q is the ratio of market price to replacement cost - in the long run this ratio will tend towards one (in theory) 52 Section 2 Intrinsic Value vs Market Price The return on a stock is composed of dividends and capital gains or losses. Required return: CAPM gives the required return, k: Is a stock attractively priced today given your forecast of next year’s price? K = rf + B[E(rM) - rf] The expected holding-period return: Expected HPR = E(r) = (E(D1) + [E(P1) - P0])/ P0 If the stock is priced correctly, k should equal expected return. K is the market capitalization rate. The expected HPR is the sum of the expected dividend yield, E(D1)/P0 and the expected rate of price appreciation = capital gains yield [E(P1) - P0]/ P0. The expected HPR may be more or less than the required return, based on the stock’s risk. The intrinsic value (IV) is the “true” value, according to a model. The market value (MV) is the consensus value of all market participants. IV > MV : BUY IV < MV : SELL or SHORT SELL IV = MV : HOLD or FAIRLY PRICED 53 Section 3 Dividend Discount Models (DDM) Vo = D1/ 1+k + D2/ (1+k)2 + D3/ (1+k)3 + … The DDM says the stock price should equal the present value of all expected future dividends into perpetuity Example: constant growth DDM A stock just paid an annual dividend of $3/ share. The dividend is expected to grow at 8% indefinitely and the market capitalization rate is 14%. V0 = $3(1+0.08)/0.14-0.08 Constant growth DDM: = $54 V0 = D0(1+g)/k-g = D1/k-g DDM Implications: Example: preferred stock No growth case. Value a preferred stock paying a fixed dividend of $2 per share when the discount rate is 8%. V0 = $2/0.08 - 0 = $25 The constant growth rate DDM implies that a stock’s value will be greater: 1. The larger its expected dividend per share 2. The lower the market capitalization rate, k 3. The higher the expected growth rate of dividends The stock is expected to grow at the same rate as the dividends. 54 Estimating Dividend Growth rates: g = ROE x b P0 = $2/0.15 -0.06 = $22.22 Where b = retention ratio = 1 - payout ratio PVGO = price per share - no growth value per share Present value of Growth Opportunities: = $22.22 - $5/0.15 The value of the firm equals: = $11.11 The value of the assets already in place, the no-growth value of the firm plus The NPV of its future investment, the present value of growth opportunities. Life Cycles and Multi-stage Growth Models: Expected dividends for Honda: 2010 - $0.50 P0 = E1/k + PVGO 2011 - $0.66 2012 - $0.83 Example: Firm reinvests 60% of its earnings in projects with ROE of 10%, capitalization rate is 15%. Expected year-end dividend is $2/ share, paid out of earnings of $5/share. 2013 - $1.00 Since the dividend payout ratio is 30% and ROE is 11%, the “steady-state” growth rate is 7.7% since: g = ROE x b = 0.11 x (1 - 0.30) = 7.7% g = ROE x b = 0.1 x 0.6 = 6% 55 Honda’s beta is 0.95 and the risk-free rate is 3.5%. If the market risk premium is 8% then k is: K = 3.5% + 0.95 x 8% = 11.1% So: P2013 = D2014/ k -g = D2013(1+g)/ k -g = $1(1+ 0.077)/0.111-0.077 = $31.68 Finally: V2009 = $0.50/1.111 + $0.66/1.1112 + $0.83/1.1113 + ($1+$31.68)/1.1114 In 2009, one share of Honda Motor Company Stock was worth $23.04. 56 Section 4 Price-Earnings Ratio & Growth The ratio of PVGO to E/k is the ratio of firm value due to growth opportunities to value due to assets already in place (ie the no-growth value of the firm) Wall Street rule of thumb: the growth rate is roughly equal to the P/E ratio. However, not true if you’re at extremes e.g. P/E = 25, not true that growth rate will be the same. P0/E1 = 1/k(1 + PVGO/(E/k)) When PVGO = 0, Po = E1/k. The stock is valued like a nongrowing perpetuity. P/E ratios and Stock Risk: When risk is higher, k is higher, so P/E is lower. P/E rises dramatically with PVGO ie high P/E = high growth opportunities. P/E = 1 - b/ k - g - higher growth rate = higher PE - Higher required return = lower PE P/E increases: • As ROE increases • As plowback increases, as long as ROE > k Pitfalls in P/E Analysis: Use of accounting earnings: P0/E1 = (1 - b)/(k - ROE x b) - forward looking because takes into account next period’s earnings • Influenced by accounting rules e.g. Historical cost of depreciation • Earnings management: practice of using flexibility in accounting tules to improve the apparent profitability of the firm • Choices on GAAP 57 • Inflation • Reported earnings fluctuate around the business cycle Other Comparative Valuation Ratios: • Price-to-book ratio: bank • Price-to-cash-flow ratio • Price-to-sales ratio: start up business 58 Section 5 Free Cash Flow Approach Value the firm by discounting free cash flow at WACC. Comparing the valuation models: Free cash flow to firm: attempts to value total firm, including parts of firm financed by long term debt. In practice: Free cash flow to equity: need to take the after tax cost of debt off and add back any net increase in long term debt. • Values from these models may differ • Analysts are always forced to make simplifying assumptions FCFF: After-tax EBIT + Non-cash flow items + Depreciation - Capital expenditures - Increase in net working capital 59 Section 6 The Aggregate Stock Market Explaining Past Behaviour: • Stock market is a leading economic indicator • Economic events and the anticipation of such events do have a substantial effect on stock prices • Two factors with greatest impact: interest rates and corporate profits Forecasting the Stock Market: • Earnings multiplier approach used to forecast 60 Chapter 7 Financial Statement Analysis Can be used to discover misplaced securities. Financial accounting data is widely available but, accounting earnings and economic earnings are not always the same thing. Section 1 The Major Financial Statements & Earnings Income Statement: Economic earnings: • Profitability over time • Sustainable cash flow that can be paid to stockholders without impairing productive capacity of the firm • Useful to distinguish four broad classes: cost of goods sold, general and administrative expenses, interest expense on the firm’s debt and taxes Accounting earnings: • Affected by conventions regarding the valuation of assets Balance Sheet: • Financial condition at a point in time • List of the firm’s assets and liabilities at that moment Statement of Cash Flows: • Tracks the cash implications of transactions 62 Section 2 Profitability Measures ROE: Financial Leverage and ROE: • Measures profitability for contributors of equity capital • ROE can differ from ROA because of leverage After-tax profit/ book value of equity • Leverage makes ROE more volatile • Let t = tax rate and r = interest rate then: ROA: • Measures profitability for all contributors of capital EBIT/ total assets • If there is no debt or ROA = r, ROE will simply equal ROA(1-t) Past vs Future ROE: • ROE is a key determinant of earnings growth • Past profitability does not guarantee future profitability • If ROA > r, the firm earns more than it pays out to creditors and ROE increases • If ROA < r, ROE will decline as a function of the debt-toequity ratio • Security values are based on future profits • Expectations of future dividends determine today’s stock value 63 Section 3 Ratio Analysis Decomposition of ROE: Du Pont Method Margin and turnover are unaffected by leverage. Where: ROA reflects soundness of firm’s operations, regardless of how they are financed. (1) = tax burden (2) = interest burden (3) = margin (4)= turnover Tax burden is not affected by leverage. Compound leverage factor = interest burden x leverage (5) = leverage Choosing a Benchmark: NOTE: Pretax = EBIT - Interest Expense Also: interest burden is similar to the times interest earned ratio: Interest coverage = EBIT/ Interest Expense • Compare the company’s ratios across time • Compare ratios of firms in the same industry • Cross-industry comparisons can be misleading NOTE: SEE RATIOS FROM FINANCE 1 64 Section 4 Accounting Fraud Categories: • Recording revenue too soon or of questionable quality - ZZZ, Enron • Boosting income with one-time gains • Shifting current expenses to later period • Failing to record liabilities • Shifting current revenue to later period • Shifting future expenses to the current period as special charge - provisions Signs: • Weak control environment • Extreme competitive pressure • Management known or suspected of having questionable character • Wary of fast-growing companies, basket case companies or newly listed companies 65 Chapter 8 EPS & Dilution EPS x P/E ratio = value Section 1 Earnings per Share Diluted Earnings per Share: • For complex capital structures - earnings to ordinary shareholders is adjusted to correct for potential dilution Subtract preferred dividends because do not accrue to ordinary shareholders. • Dilution - other parties have ability to convert other financial obligations/ instruments into ordinary shares, result: ordinary shareholder’s claim on company’s earnings reduced (diluted) Weighted average number of shares (WANOS): • Potentially dilutive securities - options, warrants, convertible debt • Number of common shares in issue during the year • Securities are only diluted if - exercising them reduces EPS • Weighted by the percentage of the year they were in issue • If not - securities are anti-dilutive ie should they be converted/ exercised they will increase the EPS for ordinary shareholders Alternatives: • Headline - JSE defined exclusions • Anti-dilutive securities are ignored when calculating diluted earnings per share (DEPS), as rational holders will not exercise them • Recurring headline - PSG defined exclusions • Attributable - excludes minority interest in profits • Diluted 67 Adjusted # of shares = WANOS + shares from conversion of preferred shares Where: + Shares from conversion of convertible debt Adjusted income available to common shares = + Shares issuable from share options earnings available to common shares - shares bought back with option proceeds + dividends on convertible preferred shares + after-tax interest on convertible debt Basic Rules: Diluted EPS Rule 1: If dilutive securities in issue throughout year: • Treat as though converted to ordinary shares at beginning of year If issued in year: Where: • Treat as though immediately converted on issue Adjusted earnings = (Net income - Preferred dividends) Rule 2: + convertible preferred dividends + Convertible debt interest x (1 - tax rate) • Exclude discontinued operations, extraordinary items and accounting changes from earnings • Only earnings from continued operations are relevant 68 Rule 3: Treasury Stock Method: Options • For diluted EPS: • Assumes funds hypothetically received from conversion of options used to repurchase shares in the market at the average market price • Denominator = basic EPS denominator adjusted for equivalent # of ordinary shares created on conversion of all outstanding dilutive instruments Rule 4: • Therefore, method reduces total number of shares that are created by hypothetical option exercise • Net shares created on exercise = shares created by option exercise - shares repurchased • Stock options and warrants: • Are dilutive only when: exercise price < market price Example: • No adjustment to net income in numerator Ordinary shares in issue for the year = 12000 • If dilutive, increase the # of shares using Treasury Share method Preferred shares (10%, par R100) in issue for the year = 1500 Convertible bonds in issue for the year (8%, par R1000) = 120 Rule 5: • If dilutive securities are present that will cause WANOS to change, numerator must be adjusted as follows: • If there are dilutive preferred shares, add preferred dividends back to income from continuing operations • If convertible bonds are dilutive, add after-tax interest expense back to earnings Options in issue for year (1 option = 10 shares @ R25 each) = 250 Tax rate = 30% Bond conversion factor: 1 bond = 10 ordinary shares Net income = R112650 Average market price of shares through the year = R33 Calculate both basic and diluted earnings. 69 1500 preferred shares in issue for the year, means preferred dividends of R15000 Net new total shares = 12000 + 2500 - 1894 = 12606 New earnings: = basic earnings - pref divs Next Step: test every potentially dilutive security for possible dilutive effect = 112650 - 15000 Convertible bonds - test for dilution: = R97650 Number of new shares created on conversion: # new shares created = 120 x 10 = 1200 New total shares = 12000 + 1200= 13200 Which is dilutive compared to 814 cents/share. New earnings: = basic earnings - pref divs + [conv debt int x (1 - tax rate)] Both convertible bonds and options are dilutive, so both are taken into account for DEPS calcs. = 112650 - 15000 + [120 x 1000 x 8% x (1 - 0.3)] = R104370 Options - test for dilution (Treasury Method): Number of new shares created on exercising: Shares bought back = (250 x 10 x R25)/ R33 = 1894 New earnings: Options do not affect earnings, but convertible bonds do, so new earnings = R104370. Both convertible bonds and options affect # of shares used for DEPS calcs. 70 New # of shares: Ordinary shares ( ( ( ( 12000 + shares from conversion of convertible bonds( ( +1200 + Shares issuable from share options( ( ( +2500 - shares bought back with share option proceeds( -1894 = adjusted # of shares( =13806 ( ( ( ( ( ( ( ( ( 71 Chapter 9 High Frequency Trading Section 1 HFT What is it? What information is used? Current happenings in Europe: Issue about STT in SA: 73 Chapter 10 Behaviour Finance & Technical Analysis Conventional Finance: prices are correct, resources are allocated efficiently, consistent with EMH. Behavioral Finance: what if investors don’t behave rationally? Section 1 The Behavioural Critique Information processing: Sample size neglect and representativeness: Forecasting errors: The notion of representativeness holds that people commonly do not take into account the size of a sample, acting as if a small sample is just as representative of a population as a large one. People give too much weight to recent experience compared to prior beliefs when making forecasts - memory bias. Tend to make forecasts that are too extreme given the uncertainty inherent in their information. Overconfidence: People tend to overestimate the precision of their beliefs or forecasts, and they tend to overestimate their abilities. Conservatism: A conservatism bias means that investors are too slow in updating their beliefs in response to new evidence. Such a bias would give rise to momentum in stock market returns. They may therefore infer a pattern too quickly based on a small sample and extrapolate apparent trends too far into the future. Behavioral biases: Framing: Decisions seem to be affected by how choices are framed. Individuals can act risk averse in terms of gains but risk seeking in terms of losses. Mental accounting: Mental accounting is a specific form of accounting in which people segregate certain decisions. 75 E.g. An investor may take a lot of risk with one investment account, but establish a conservative position with another account that is dedicated to her child’s education. The house money effect refers to gamblers’ greater willingness to accept new bets if they currently are ahead. Limits to Arbitrage: Fundamental risk: The fundamental risk incurred in exploiting apparent profit opportunities presumably will limit the activity of traders. Regret avoidance: E.g. Buying an underpriced security - risk is that the price will never converge to the intrinsic value. It has been found that individuals who make decisions that turn out badly have more regret when that decision was more unconventional. “Markets can remain irrational longer than you can remain solvent.” E.g. Buying a blue-chip share which fails is not as painful as buying an unknown share that fails. Blue-chip failure can be attributed to bad luck whereas unknown failure can be put to bad decision making. Prospect theory: Prospect theory modifies the analytic description of rational risk-averse investors. Intrinsic value and market value may take too long to converge. Implementation costs: Short-selling a security entails costs, short0-sellers may have to return the borrowed security on little notice, rendering the horizon of the short sale uncertain, other investors face limits on their abilities to short securities. Model risk: Conventional view: Utility depends on level of wealth. Behavioral view: Utility depends on changes in current wealth. There is a risk that an apparent profit opportunity is more apparent than real. 76 Section 2 Technical Analysis & Behavioural Finance Technical Analysis and Behavioral Finance: • Technical analysis attempts to exploit recurring and predictable patterns in stock prices • Prices adjust gradually to a new equilibrium • Market values and intrinsic values converge slowly • Disposition effect: the tendency of investors to hold on to losing investments • Demand for shares depends on price history • Can lead to momentum in stock prices Trends and Corrections: Dow Theory: 1. Primary trend: long-term movement in prices, lasting from several months to several years 2. Secondary or intermediate trend: short-term deviations of prices from the underlying trend line and are eliminated by corrections 3. Tertiary or minor trends: daily fluctuations of little importance Moving Averages: • The moving average is the average level of prices over a given interval of time • Bullish signal: market price though the moving average line from below - time to buy • Bearish signal: when prices fall below the moving average, it is time to sell Breadth: • Often measured as the spread between the number of stocks that advance and decline in price • A measure of the extent to which movement in a market index is reflected widely in the price movements of all the stocks in the market 77 Sentiment Indicators: Warning: Trin Statistic: • It is possible to perceive patterns that really don’t exist Common patterns: Ratios above 1.0 are bearish Confidence index: • The ratio of the average yield on 10 top-rated corporate bonds divided by the average yield on 10 intermediate-grade corporate bonds • Higher values are bullish Head-and-Shoulders • Three peaks close together but rising off a common base in a share price graph, with the middle one being the “head” and the other two either side of it the “shoulders” • Normally considered to be a very bearish (negative) pattern ie leading to a sharp decline in share price • Theory: if the share price persistently increases before falling back, but by the third time is unable to reach previous high, investors are likely losing confidence in the share Put/ call ratio: Cup-and-Handle • Calls are the right to buy - a way to bet on rising prices • Puts are the right to sell - a way to bet on falling prices • A rising ratio may signal investor pessimism and a coming market decline • Contrarian investors see a rising ratio as a buying opportunity • Consists of a “cup” in shape of U, followed by a period of relative price stability (the “handle”), which may have slight downward drift • This is a bullish pattern, considered to be the prelude to an important price breakout • Theory: as the share price rises from the low levels inside the “cup”, it starts to test old highs. People who had previously 78 bought at these highs sell their shares so as to break-even (the lows reached in the cup being fresh in their minds). This results in the share trading sideways for a while (increased supply temporarily halting the upward trend). Once all the excess supply is in the market (the above group having sold), the share is set to resume its upward trend (albeit at an accelerated pace the breakout). The Double Bottom: • Resembles a “W” • Considered to be very bullish, preceding a rise in share price • Idea is to buy when the price passes the highest point in the “handle” (the middle peak) on its way out of the double bottom • Theory: after two declines in the share price, all the uncertain investors have sold, and only the long-term investors are not still in the share. This means few potential sources of supply remain, resulting in a higher share price going forward. 79 Chapter 11 Risk and Return Section 1 Background to Portfolio Theory Markowitz’s Theory: Introduced diversification, the idea that risks are correlated in some way, a distinction between efficient and inefficient portfolios, the idea of risk-return trade offs on a portfolio and expected return and variance as criteria for portfolio selection Assumed only positive investments and said the investor will choose a portfolio from an efficient frontier Roy: Assumed positive and negative investments and said the investor will choose a specific portfolio Tobin: Separation Theorem: First select a portfolio of risky assets - subject to market risk. Decide on mix of this and the risk-free asset 81 Section 2 Introduction to Risk and Return Let: Given: R = nominal rate t = tax rate r = real rate R = nominal rate of interest i = inflation rate After-tax rate is R(1 - t) Real after-tax rate is R(1 - t) - i Excess returns and risk premiums: Also if: E(i) = current expectation of the inflation rate that will prevail over the coming period Excess return = the difference between the actual rate of return on a risky asset and the actual risk-free rate. Sharpe ratio: Measures the excess returns per unit of risk. 82 Chapter 12 Risky Aversion and Capital Allocation to Risky Assets Section 1 Risk Aversion and Capital Allocation to Risky Assets Speculator: assumes considerable risk in order to obtain commensurate gain (positive risk premium) Gamble: bet or wager on an uncertain outcome Risk averse investors are willing to consider only risk-free prospects or speculative prospects with positive risk premiums Risk-neutral investors look only at expected returns when making an investment decision. (e.g. Receive R100 @ 100% certainty vs. 50% chance of R200). Risk-neutral investors wouldn’t mind either way. Evidence of risk aversion: Insurance – protects you against the risk of a loss of an asset Preference of Bonds – people buy bonds with lower return rather than high risk equity with higher return. Are people completely risk loving or risk averse? Insurance The Basic assumption remains… Portfolios with higher expected returns and lower risk are considered more attractive and will therefore be ranked higher in terms of preference “Most investors devoting large sums of money to a portfolio are risk averse” If faced with portfolios of equal return, the risk averse investor will choose the portfolio with the lower risk…conversely….. 84 The$U&lity$Func&on$–$Differences$in$investor$risk$ aversion$ High%U'lity% • The$risk3return$trade3 off$is$one$in$which$ greater$risk$is$taken$if$ greater$returns$can$be$ expected,$resul&ng$in$ a$posi&ve$slope.$ • The$highest$ indifference$curve$(the$ one$in$the$most$ Lower%U'lity% northwestern$ posi&on)$offers$the$ greatest$u&lity.$$ Moderately%%Risk% Averse% Expected%Return% Expected%Return% The$U&lity$Func&on$ More%Risk% Averse% Less%%Risk%Averse% Risk%Loving% $ Same%measure%of% u'lity% Standard%Devia'on% (Risk)% Standard%Devia7on%(Risk)% Note:% • The$risk$lover$is$ prepared$to$accept$ lower$expected$ returns$as$risk$ increases.$ • By$contrast,$the$Risk$ Averse$investor$$ demand$significantly$ higher$expected$ returns$per$unit$of$ risk.$ $ The"U$lity"Func$on" Portfolio that is best for us will be the one that is tangent to the highest utility • To"rank"your"investment"por=olios"you"need"to"assign"them"a"level"" or"measure"of"u$lity"which"is"based"on"expected"return"and" variance.""This"is"defined"by"…" Where:& U""""""""""! ="U$lity"(Certainty"Equivalent"Rate"of"Return)" E"("r")!="expected"Return"on"the"asset"or"por=olio" A!="coefficient"of"risk"aversion" σ2"="variance"of"returns" Note"that"rates"of"return"are"expressed"as" decimals"and"not"integers"in"this"equa$on." 85 As can be seen from the equation, utility is enhanced by higher expected returns and lower risk. At the risk free level utility will be equal to the E(r) ….1/2Aδ2 will be equal to zero since risk (δ ) is equal to zero. Risk averse investors will have a higher value “A”, they tend to penalise risky investments severely. The utility score of a risky portfolio represents the certainty equivalent rate of return that the risk free asset must meet or beat if it is to be chosen. A portfolio will only be desired if the CERR is greater than that of a risk free alternative. !="($)−#/$ )*$( The&U3lity&Func3on& Example(1:( Assume&an&investor&with&the&following&u3lity&func3on:&& U(=(E(r)(0(3/2(σ2).( He&has&a&choice&between&2&assets&with&E&(&r)&and&σ&as&follows:& & P('n(P( Shoprite(C( & E((r()( 10%& 15%& & σ( 10%& 10%& & & Which&asset&(share)&should&he&invest&in?& Solution: Objective is to Maximize the investor’s utility, given their level of risk aversion (generally referred to as the co-efficient of risk aversion) !="($)−#/$ )*$( An investor that is risk neutral will have a risk index of zero. For them the portfolios CERR= E(r); risk is irrelevant. A risk lover will have a risk index <0. These investors are happy with “fair game” and “gambles” The"U9lity"Func9on" Example(2:( A"por&olio"has"an"expected"rate"of"return"of"0.15"and"a"standard" devia9on"of"0.15."The"risk<free"rate"is"6"percent."An"investor"has"the" following"u9lity"func9on:"U"="E(r)"<"(A/2)σ2.""Investor""X"has""coefficient" of"risk"aversion"of"4""and"investor"Y"has"a"coefficient"of"risk"aversion"of" 8."Which"investor"is"indifferent"between"the"risky"por&olio"and"the"risk" free"asset." " 86 Solution: Objective is to find the coefficient of risk aversion (A) at which the U given by U = E(r) - (A/2)σ2 is equal ton the utility of the risk free asset. We know that…U(Rf) = 6%. We can substitute the coefficient of risk aversion for each investor into the formula U = E(r) - (A/2)σ2 or.. greater than any portfolio within that quadrant. Similarly, portfolio “A” has a risk (SD) is equal to or lower than any portfolio that is located within quadrant IV. The criteria on which the choice of portfolio “A” is made is referred to as the mean-standard deviation or the mean-variance criterion. Under this criteria portfolio “A” will be superior to portfolio “B” if E(rA) >/= E(rB) and δA </= δB . We can solve for “A” given a Utility (U) of 6% The$Trade(off$Between$Risk$and$Returns$$ Mean-variance criterion. " E"(r")" II" I" E"(rp")" For this criterion to hold, if expected returns are equal [E(rA) = E(rB)], then the SD of portfolio “A” must be less than that of portfolio “B” [δA < δB ]. Alternatively, if the return on portfolio “A” id greater than that of portfolio “B’ [E(rA) > E(rB)], it is sufficient for the risk (SD) of the 2 portfolios to be equal [δA = δB ] for the mean-variance criterion to hold. Por,olio"“A”" III" IV" σp" Por,olio"“B”" Anything in quadrant 4 is worse than anything in quadrant 1 σ" For the investor the portfolio represented by ”A” has an expected return of E(rp) and a SD of δp. The risk averse investor will prefer this portfolio to any other portfolio in quadrant IV because portfolio “A” has an expected return that is at least equal to or 87 criterion. Under this criteria portfolio “A” will be superior to portfolio “B” if E(rA) >/= E(rB) and δA </= δB . The$Trade(off$Between$Risk$and$Returns$$ Mean-variance criterion conditions:. " Under$ this$ criteria$ por7olio$ “A”$ will$ be$ superior$ to$ por7olio$ “B”$ if$ E(rA)" >/=" E(rB)" and$δA"</="δB"."" $ $ If$expected$returns$are$equal$[E(rA)"="E(rB)],$ then$ $the$SD$of$por7olio$“A”$must$be$less$ than$that$of$por7olio$“B”$[δA"<"δB"]."" I" II" III" IV" $ AlternaEvely,$ if$ the$ return$ on$ por7olio$ “A”$ is$ greater$ than$ that$ of$ por7olio$ “B’$ [E(rA)" >" E(rB)]," it$ is$ sufficient$ for$ the$ risk$ (SD)$ of$ the$ 2$ por7olios$ to$ be$ equal$ [δA" =" δB" ]" for$ the$ mean(variance$ criterion$ to$ hold.$ For this criterion to hold, if expected returns are equal [E(rA) = E(rB)], then the SD of portfolio “A” must be less than that of portfolio “B” [δA < δB ]. Alternatively, if the return on portfolio “A” id greater than that of portfolio “B’ [E(rA) > E(rB)], it is sufficient for the risk (SD) of the 2 portfolios to be equal [δA = δB ] for the mean-variance criterion to hold. What about quadrants II and III?" NB: for exam purposes. Be about to explain the mean variance. The$Trade(off$Between$Risk$and$Returns$$ Mean-variance criterion. " E"(r")" Let’s turn our focus back to figure 6.1. For the investor the portfolio represented by ”A” has an expected return of E(rp) and a SD of δp. The risk averse investor will prefer this portfolio to any other portfolio in quadrant IV because portfolio “A” has an expected return that is at least equal to or greater than any portfolio within that quadrant. Similarly, portfolio “A” has a risk (SD) is equal to or lower than any portfolio that is located within quadrant IV. E(r) > δ I" II" Por,olio"“E”" A;C;D;E;F$ Dominates$B$ Por,olio"“C”" D;A;C$$($ Indifferent$ Por,olio"“F”" E"(rp")" Por,olio"“A”" E(r) < δ Por,olio"“D”" Por,olio"“B”" F;E$ Dominates$ D;A;C$;B$ IV" III" σp" σ" The criteria on which the choice of portfolio “A” is made is referred to as the mean-standard deviation or the mean-variance 88 When it comes to quadrant II and III we need to consider the trade off between risk and return. It is possible for the investor to identify other portfolios that are equally preferred to portfolio “A”. However this will call for some trade-off between risk and return. One such portfolio is portfolio “C”. The investor will be indifferent between portfolio C ,A and D, hence the indifference curve Capital'Alloca+on' Across&Risky&and&Risk.Free& Por2olios&& • Control'risk'' – Asset'alloca+on'choice' • Frac+on'of'the'por4olio'invested'in' Treasury'bills'or'other'safe'money' market'securi+es' Risk free asset Any portfolio to the left of the indifference curve is preferred to portfolios identified by the indifference curve because they have a higher return and lower risk (quadrant I) or a significantly higher return for marginal increase in risk (quadrant IV). Conditions that apply to quadrants 2 and 3: Would only accept investments in quadrant 2 if the increase in Expected Return is greater than the increase in standard deviation Would only accept investments in quadrant 3 if the decrease in Expected reurn is less than the decrease in standard deviation Risky asset Level of Risk You can control risk by simply re-allocating assets from risky to risk free assets. Incr. risk in risk free - risk goes down. Portfolios of one risky asset and one risk-free asset: It’s possible to split investment funds between safe and risky assets. Risk free asset: proxy; T-bills Risky asset: stock (or a portfolio) 89 Construc.on$of$a$Por(olio$of$1$Risky$Asset$$ and$1$Risk?Free$Asset$ The$por(olio$consis.ng$of$risk$free$and$risky$assets$in$known$as$a$complete( por*olio.(To$construct$the$complete$por(olio$we$will$need$the$following:$ Two versions of the formula. Can use either, depending on the information given and which would be best suited. Two sources of income for the portfolio. Return from risky asset and the return from risk-free asset. FOR(THE(RISK(FREE(ASSET:( The$rate$of$return,$known$as$the$risk$free$rate$of$return$$?$ r f( By$defini.on,$there$is$no$risk$aDached$to$the$risk$free$rate$of$return$therefore$ its$standard$devia.on$is$zero$$ σ(rf(=(0%( FOR(THE(RISKY(ASSET:( The$expected$return.$There$is$no$certainty$aDached$to$this$rate$of$return,$ hence$expected$$ E(rp)(( The$risk$aDached$to$the$expected$return$on$the$risky$asset$$ σp(( The$percentage$(of$funds$invested$)$in$the$risky$asset$–$ y" By$defini.on$therefore$the$percentage$(of$funds$invested$)$in$the$risk$free$$ asset$is$$$ (1%y)"" Por$olios(of(One(Risky(Asset(and(a(Risk4Free( Asset( Example:) Assump-ons:) • rf)=)7%) • E(rp))=)15%) • σp)=)22%) Expected)return)and)standard) devia-on)for)the)complete)por>olio:) E(rc))=))rf)+)y[E(rp)))–) rf]) ) E(rc))=))7)+)y[15C7]) ) E(rc))=))7)+)y8) σc)=)y)σ)p) ( σc)=)y)22) ( σc)=)22y) Interpreta-on:) We#can#now#define#the#expected#return#and#risk# for#the#complete#por1olio# In#terms#of#expecta/on#of#the#por1olio’s#rate#of#return…# • The(basic(rate(earned(on(this(por$olio(is(the(risk(free(rate(of(7%.( • The(por$olio(is(expected(to(earn(a(maximum(risk(premium(of(8%,)if(100%)is(invested( in(the(risky(asset((i.e.)y=1)(or(less)than)8%)depending(on(the(value)of)y) • The(total(expected(return(for(the(completed(por$olio(for(the(given(set(of(numbers(is( depends(on(the(amount)invested)in)the)risk)asset)(y))which(is(determined(by(the( investors’)level)of)risk)aversion)(will(touch(on(this(later)( E(rc)"=""yE(rp&)&+&(1*y)rf& & E(rc)"=""rf&+&y[E(rp&)&–&rf]" " Risk premium - [E(rp ) – rf] The#complete#por1olio’s#risk#is#given#by….# σc&=&y&σ&p& 90 Asset%Alloca*on% • Investment(Opportunity(Set( • • E'(r')' Capital'Alloca)on'Line'' (CAL)' • P E'(rp')'=' 15%' rf'='7%' M Slope'=' 8/22' E'(rp')'B'rf'='8%' σc'='y'σ'p' σp'='22%' σ E"(r")" Slope(of(the(CAL(is(o>en( referred(to(as(the(price(of( risk( 15%" The(slope(of(the(CAL( indicates(that(there((is( increase(an(expected( return(of(the(complete( porDolio(per(unit(of( addiEonal(risk( • Por,olio'P''is(100%( invested(in(the(risky(assets( (y=1)' • Por,olio'F'is(100%( invested(in(the(risk(free( asset((y=0)' • Por,olio'M'consists(of(a( mixture(of(risky(and(risk( free(assets((0<y<1)' F E(rc)'=''rf&+&y[E(rp&)&–&rf]' Slope&=([E(rp&)&–&rf]/&σp& 0.36& & This(is(the(Sharpe'Ra)o' Assump&on:*y=1* Return:* E(rc)"=""rf*+*y[E(rp*)*–*rf]* E(rc)!=%0.07%+%1%(0.15%4%0.07)%%=%0.15% * Risk:* σc*="yσp* σc!=%(1.0)%(0.22)%=%0.22% (CAL)" P M 11%" rf"="7%" Y"="1" Y"="0.5" F σp"="%" σp"="11%" Y"="0" σp"="22%" Assump&on*y=0* Return:* E(rc)"=""rf*+*y[E(rp*)*–*rf]* E(rc)!=%0.07%+%0%(0.15%x%0.07)%%=%0.07% σ Risk:* σc*="yσp* σc!=%(0.0)%(0.22)%=%0% Assump&on*y=*0.5* Return:* E(rc)"=""rf*+*y[E(rp*)*–*rf]* E(rc)!=%0.07%+%0.5%(0.15%4%0.07)%%=%0.11% * Risk:* σc*="yσp* σc!=%(0.5)%(0.22)%=%0.11% 91 E(rc)&=&&rf&+&y[E(rp&)&–&rf]& Asset%Alloca*on%–%Borrowing%@%rf% AssumpBon:& Borrow%at%the%RiskCFree%Rate%and%invest%in%the%%Risky% asset,%essen*ally,%Lending at 7% and Borrowing at 7% E&(r&)& rf&=&7%& Return:& E(rc)&=&&rf&+&y[E(rp&)&–&rf]& & E(rc)!=%0.07%+%(1.5%x%0.08)%=%0.19% Risk:& σc&=&yσp& P & σc!=%(1.5)%(.22)%=%0.33% E&(rp&)&/&rf&=& 8%& Slope&=& 8/22& Using&50%&Leverage:& Asset%Alloca5on%–%Borrowing%@%>%rf% AssumpAon:& Borrow%at%the%Risk.Free%Rate%and%invest%in%the%%Risky%asset,%essen5ally,%Lending at 7% and Borrowing at 9% E&(r&)& (CAL)& (CAL)& E&(rp&)&=& 15%& σc&=&y&σ&p& % Sharpe&ra9o& Slope&=&[E(rp&)&–&rf]/&σ&p& F P& E&(rp&)&=&15%& rB&=&9%& rf&=&7%& S&(y≤&1)&=&0.36& σ σp&=&22%& & Slope&=&0.36& Sharpe'ra(o' Slope'=&[E(rp')'–'rf]/'σ'p' ' F& Slope'=&(0.15'–'0.09)/'0.22' ' Slope'=&0.27' & Slope&=&[(0.15)&–&0.07]/&0.22& S&(y&<&&1)&=&0.27& σp&=&22%& σ E(rc)&=&&rf&+&y[E(rp&)&–&rf]& Asset%Alloca*on%–%Borrowing%@%rf% AssumpBon:& Borrow%at%the%Risk<Free%Rate%and%invest%in%the%%Risky% asset,%essen*ally,%Lending at 7% and Borrowing at 7% E&(r&)& (CAL)& P E&(rp&)&=& 15%& rf&=&7%& Slope&=& 8/22& E&(rp&)&/&rf&=& 8%& σc&=&y&σ&p& Using&50%&Leverage:& Alterna:ve% Return:& E(rc)&=&&Y[E(&Rp)]&+&(16y)&(Rf)& & E(rc)%=%1.5%(0.15)%+%(1<1.5)%(0.07)% % E(rc)%=%22.5%–%3.5% % E(rc)%=%0.19% F σp&=&22%& σ 92 E(rc)*=**rf&+&y[E(rp&)&–&rf]* * Asset%Alloca*on%–%Borrowing%@%>%rf% Assump&ons:* • rf%=%7%% • E(rp)%=%15%% • σp%=%22%% • Have%R100%to%invest% • Borrow%another% R100%@%9%%to%invest% in%Risky%assets% What*is*the…** • E%(rc)%% • σc% σc*=*y*σ*p* E(rc)*=**rBf&+&y[E(rp&)&–&rBf]* Investment(Opportunity(Set( Using&50%&Leverage:& E&(r&)& % E(rc)%=%%9%++1+(15%/9%)+++0.5(+15%/9%)+ + E(rc)%=%%9%+++6%+++3%+ + Lower%than%borrowing%@% E(rc)%=%%18%+ rf%rate….why?% + + Compare+the+return+to+that+under+the+ scenario+of++borrowing+at+the+risk+free+ rate…refer+to+graph+ & & σc*=*y*σ*p* % σc%=%1.5%(0.22)% Why%has%this%not%change?% % σc%=%0.33% & & (CAL)& P E&(rp&)&=& 15%& rB&=&9%& rf&=&7%& S&(y&<&&1)&=&0.27& Return:( E(rc)&=&&rf(+(y[E(rp()(–(rf]&& ( E(rc)!=(0.07(+((1.5(x(.06)( ( (0.15)(=(.16( Risk:( σc(=&yσp( S&(y≤&1)&=&0.36& F ( σc!=((1.5)((.22)(=(.33( σp&=&22%& σ Assump@on:& Borrow(at(the(RiskBFree(Rate(and(invest(in(the((Risky( asset,(essenFally,(Lending at 7% and Borrowing at 7% ( Sharpe(ra9o( Slope(=&[E(rp()(–(rf]/(σ(p( ( Slope(=&[(0.15)(–(0.07]/(0.22( ( Slope(=&0.36( * Effect of borrowing at a rate higher than the risk free rate, is a decreased expected return Higher levels of risk aversion lead to a larger proportion of your investment in the risk free asset (1.e. lower “y” values) Lower levels of risk aversion lead to larger a proportion of your investment in the portfolio of risky assets (i.e. higher “y” values) 93 Risk Tolerance and Asset Allocation: Example: Expected return of the complete portfolio is given by: rf = 7% E(rp) = 15% σp = 22% (risk in the portfolio) Variance is: Investor X has coefficient of risk aversion of 6 Investor Y has a coefficient of risk aversion of 2 Utility: Max U: Calculate the E(rc); σc and the reward-to- risk ration for each investor. Allocation to risky asset Use formula: Y* = (E(rp) – rf)/ Aσ2p Investor X: Y* = 0.28 Investor Y: Y* = 1.65 Expected return: Use Formula: E(rc) = rf + y[E(rp ) – rf] Investor X: 0.0924 94 E(rc)&=&&rf&+&y[E(rp&)&–&rf]& Investor Y: 0.202 σc&=&y&σ&p& Asset%Alloca*on%–%Borrowing%@%rf% Standard Deviation: E&(r&)& (CAL)& Investor&&y& E&(rc&)&=&20.2%& Use formula: σc = yσp Investor X: 0.0616 Investor Y: 0.363 Note: If the borrowing rate = risk free rate, then the capital allocation line will be straight. Investor&&X& E&(rc&)&=&9.24%& rf&=&7%& Risk-to-Reward Ratio: Slope = 0.36 P E&(rp&)&=&15%& F& σp&=&6.16%& σp&=&22%& σp&=&36.3%& σ Optimal portfolio in this particular example for the respective investors. Investor X is more risk averse than investor Y. Capital'Market'Line' E&(r&)& (CML)& Investor&&Z& E&(rc&)&=&20.2%& Evaluating the effect of a different coefficient of variance and its impact on expected return and risk of the portfolio. Shows low risk is in line with a lower return and a high risk with a higher expected return. P E&(rp&)&=&15%& Investor&&X& PorDolio&of&assets& E&(rc&)&=&9.24%& rf&=&7%& F& Reward-to-risk will be the same for both. σp&=&6.16%& σp&=&22%& σp&=&36.3%& σ 95 Chapter 13 Optimal Risky Portfolios Section 1 Optimal Risky Portfolios Assumptions: We are looking at a short term horizon Reduces the need to be concerned with skewness that is typical of investments over a long term horizon Normal distribution returns of a short term investment horizon therefore allows us to assume holding period return that are sufficiently accurate. Now our only concern is the mean and variance of the portfolio Systema*c-risk:-cannot-be-diversified-out-of-a-por>olio.- Diversifica*onPor$olio'Risk'as'a'Func0on'of'the'Number'of'Stocks'in'the'Por$olio' Risk (σ ) Stocks- may- have- different- risk– return-characteris*cs.-By-includingshares- with- different- risk–returncharacteris*cs- - in- a- por>olio- theinvestor- can- effec*vely- managehis-overall-por>olio-risk.- Diversifiable Risk a.k.a. Unique risk; Company-specific risk; Non-systematic risk Unique risk Non-diversifiable risk a.k.a. Market risk ; Systematic risk Note:'Systema*c-Risk-is-defined-by-the-variability-in-thereturn-on-all-risky-assets-cause-by-macroeconomic-variablesuch-as-growth-in-money-supply,-changes-in-interest-ratesetc.- Market risk Number of companies (n) Chalkboard: In the above diagram the risk return characteristics of the stock represented by the blue and red lines are “opposite” to each other i.e. they are negatively correlated, a concept which we may revisit a little later. If these 2 stocks made up the portfolio the risk would be represented by the blue line instead of the red or black line which would have been the case if the investor had only one of these 2 stocks in his portfolio. The risk-return charac97 teristics represented by the blue line(stock) is more stable than that represented any of the other lines (stock). This is the impact of diversification; it stabilises the risk-return characteristics of a portfolio. However, it goes beyond just stabilising the risk-return characteristics; it also reduces the risk in the overall portfolio. As can be seen in the diagram on the left, the more stock of different riskreturn characteristics your add to your portfolio, the lower the portfolio’s overall risk will be. But….not all risk is can be diversified, Market risk or systematic risk i.e. that which impacts the stock of all companies, cannot be diversified. An example of such risk is a slow down in the economy, all companies will be negatively impacted by this. It is only company specific risk, otherwise known as unique risk or non-systematic risk that can be diversified.. Two$Security-Por/olio:-Return!"="#$%$+#&%&! Linear-relaPonship- rp-=-Por/olio-ReturnwD-=-Bond-WeightrD-=-Bond-ReturnwE-=-Equity-WeightrE-=-Equity-Return- E(R) 13% Equity 8% E--(r-p)-=-wD!E!(rD!)!+-wE!E!(rE!)! The-por/olio-return-is-equal-tothe-sum-of-the-returns-derivedfrom-the-individual-stocks-thatmake-up-the-por/olio-i.e-WDrDand-WErE-.-! Similarly,!the!Expected! Returns…..! An-investor-may-invest-all-hismoney-in-debt-funds-at-a-return-of8%;-he-may-invest-all-his-money-inequity-at-a-return-of-13%,-or-hecan-invest-in-a-combinaPon-ofdebt-and-equity-and-end-up-with-apor/olio-return-that-will-be-theweighted-average-of-the-returnsbetween-debt-and-equity.-- Debt Fund (Corp. Bond) -0.5 0 1.0 1.5 1.0 0 Short Equity 2.0 W(Equity) -1.0 W(Bonds) =1-W(Equity) Short Debt We will now consider the construction of portfolio of 2 risky assets. One of these assets is a long term bond, denoted in the slide by the letter “D”, the other is Equity, denoted on the slide by the letter “E”. The weight of D and E in the portfolio will be denoted by WD and WE as indicated on the slide, and the return from D and E will be denoted by rD and rE, respectively. The return of the portfolio is denoted by rP. The equation for the portfolio return simply states that the portfolio return is equal to the sum of the returns derived from the individual stocks that make up the portfolio i.e WDrD and WErE . Similarly, the Expected returns of the portfolio will be defined by the summation of expected returns of the bonds and the expected returns of the equity. 98 Two-security Portfolio: Risk You will note that there is a linear relationship between the returns of the individual stocks and the portfolio. As indicated in the graph to the right of the slide, an investor may invest all his money in debt funds at a return of 8%; he may invest all his money in equity at a return of 13%, or he can invest in a combination of debt and equity stock and end up with a portfolio between the 2 extremes (debt and equity) at a return that will be the weighted average of the returns between debts and equity based on their weights in the portfolio. We have a fixed budget – can either go 100% debt and 100% equity or anything in between. But we can also borrow. If we invest 200% in our equity fund, it means we will go short debt and long equity, thus borrowing at the rate of debt.. Can borrow and lend in this portfolio. Short equity - borrowing at the rate of equity and if you short debt you are borrowing at the rate of debt. Covariance and correlation: Portfolio risk depends on the correlation between the (expected) returns of the assets in the portfolio Covariance and the correlation coefficient provide a measure of the way returns of two assets vary Correlation coefficient of +1: σP = WDσD + WE σE Correlation coefficient of -1: σP = WDσD - WE σE Examples: In these examples we want to see what the impact of changing the correlation coefficient is on the portfolios risk. Therefore we assume that the assets have the same expected return and standard deviation. Assume we have expected returns and standard deviations for debt and equity as follows: E(rD) = 0.20 E(rE) = 0.20 E(σD)=0.10 E(σE)=0.10 99 Assume further that you have a portfolio of debt and equity which is equally weighted. The$Trade(off$Between$Risk$and$Returns$–$ About$por8olio$dominance$$ Mean-variance criterion. " Examples (cont): Calculate the standard deviation for the portfolio of assets given the following correlation coefficients. E"(r")" E(r) > I" II" Por,olio"“E”" E"(rp")" Por,olio"“C”" D;A;C$$($ Indifferent$ Por,olio"“A”" E(r) < δ Por,olio"“D”" Por,olio"“B”" rD,E = 0.50 rD,E = -0.50 A;C;D;E;F$ Dominates$B$ Por,olio"“F”" rD,E = 1.00 rD,E = 0.00 δ F;E$ Dominates$ D;A;C$;B$ IV" III" σp" σ" rD,E = -1.00 Before we continue, we need to just re-cap the concept of portfolio dominance. If you recall, we discussed in a previous lecture the significance of the position of portfolio P on the risk – return plane. We said that any portfolio that lies within quadrant four i.e. to the south-east of portfolio P will be inferior to portfolio P. This is because the risk of such portfolio will be higher than that of portfolio P and the return will be lower. There for portfolio P in this example is said to dominate portfolio A. The conditions under which portfolio P will dominate Portfolio A is when the rP ≥ rA and the δP≤ δA ; Dominance requires at least one > condition to hold. 100 If we consider portfolio E relative to Portfolio P, we will see that we have an increase in risk, but that the increase in risk is less than proportional to the increase in return. Similarly, if we compare portfolio D with portfolio P, we can see that there is a decrease in return, but the decrease in return is less than proportional to the decrease in risk. As we have discovered in a previous lecture, with portfolios P, D and E being on the investor’s indifference curve, the investor will be equally satisfied any of these portfolios. We can therefore say that portfolio E is equivalent to portfolio P and portfolio P is equivalent to portfolio D, with portfolio D being equivalent to portfolio E. This is equivalent to saying that none of the portfolios on the IC dominates each other. Por$olio'Expected'Return'as'a'Func5on'of' Standard'Devia5on'' E Expected Return ρ"=##)1# ρ"=##0# rf C# ρ"=##0.30# B# C1# B1# A# ρ"=#1# A1# D Standard Deviation Any portfolio to the right of the IC will be dominated by a portfolio on the IC. Any portfolio on the IC will be dominated by a portfolio to the left of the IC. We have seen that… As you will see in the next slide, this principle is important in understanding the logic behind the shape of the efficient frontier of risky assets. The smaller the correlation, the greater the risk reduction potential and the greater the impact of diversification. We also saw that… If ρ = +1.0, no risk reduction is possible 101 Asset%alloca*on%including%a%risk%free%asset% Capital Allocation Line Expected Return P8 Indifference Curve P7 P6 P5 P4 P3 rf P2 P1 !"="$"(&")−&(/*" ! Standard Deviation Any asset to the north-west of portfolio 1, dominates portfolio 1. P3 – P8 = The efficient frontier Based on the mean-variance criterion and the concept of portfolio dominance we can see that portfolio P1 is not dominant, portfolios P2 to P 6 can offer higher returns at lower risk. As a result, portfolio P does not form part of the efficient frontier. We can extend this argument to portfolios P2 and P3 as well (portfolio P3 just about dominated by portfolio P4). Portfolios P2 and P3 therefore do not form part of the efficient frontier. However, portfolios P4 to P8 meet all the criteria of dominant portfolios; there is no other possible portfolio in quadrant 1 or north-west of any of these portfolios. For all these portfolios the criteria form dominance referred to earlier is met. In the sense that there is no portfolio that dominates the portfolios on the frontier beyond portfolio P3, this part of the frontier can be referred to as an efficient frontier. We now want to add a risk free asset to the portfolio. The risk free rate is indicated by rf . The risk free portfolio is indicted on the y-axis. Why do you think this so? Because by definition the risk free asset has zero standard deviation. An investor can decide to allocate his investment between risk free assets and risk assets, remember risk assets is represented by efficient frontier. Lets say that he chooses to allocate his funds between the risk free asset and portfolio P3 of the risky assets. This would imply that his portfolio will be allocated anywhere between rf and portfolio P3. the more risk averse the investor the more of the risk free asset he will include in his portfolio. His portfolio will lie closer to the y-axis indicating a greater proportion of risk free assets in it. Conversely, a less risk averse investor will include more of the risky asset in his portfolio; his portfolio would lie closer to portfolio P3. You should be familiar with this from one of our earlier lectures. But this is not the most optimal decision by the investor. Lets assume he decides to invest in the risk free asset and portfolio P4. Any portfolio on the line between rf and portfolio P4 will domi- 102 nate portfolios represented by portfolio P3 and the risk free asset. This should be clear from our earlier discussion. One could follow the same argument for at portfolio that consist of the risk free asset and portfolio P5 or the risky assets, but not for portfolio P7 of the risky assets…Why? Well that is quite simple. Any portfolio that is on the line between portfolio P5 and P7 of the risky portfolios is dominated by a portfolio on the efficient frontier. The lines rf – P3, rf – P4, and rf – P4 are all Capital Allocation lines. We can continue to move our CAL to the left until we reach our optimal risky portfolio (P6) as defined by the tangent between the CAL and the efficient frontier. There is no risky portfolio that offers a better reward-volatility (risk) ratio than this. Recall that the reward to volatility ratio is the Sharpe Ratio. As we move our CAL to the left our sharp ratio increases (slope of the CAL). Finally, we can introduce the level of risk aversion of the investor. In a previous lecture we were introduced to the calculation of the proportion of assets that should be allocated to risky assets give the level of risk aversion “A”. This will give us the complete optimal portfolio which is diagrammatically this is represented by the tangent between the CAL and the investors highest indifference curve. Note: The objective of an investor interested in the minimum variance portfolio is to minimize the risk of the portfolio. Whereas the objective of an investor interested in the optimal risky portfolio is to obtain the best risk and return trade-off available. Efficient(Fron,er(and(3(Possible(“op,mal”( Por8olios( Capital Allocation Line Expected Return P8 Indifference Curve P7 Optimal Risky Portfolio P6 Optimal Complete Portfolio P5 P4 Global Min-Variance Portfolio rf Standard Deviation Based on the mean-variance criterion and the concept of portfolio dominance we can see that portfolio P1 is not dominant, portfolios P2 to P 6 can offer higher returns at lower risk. As a result, portfolio P does not form part of the efficient frontier. We can extend this argument to portfolios P2 and P3 as well (portfolio P3 just about dominated by portfolio P4). Portfolios P2 and P3 therefore do not form part of the efficient frontier. 103 However, portfolios P4 to P8 meet all the criteria of dominant portfolios; there is no other possible portfolio in quadrant 1 or north-west of any of these portfolios. For all these portfolios the criteria form dominance referred to earlier is met. In the sense that there is no portfolio that dominates the portfolios on the frontier beyond portfolio P3, this part of the frontier can be referred to as an efficient frontier. We now want to add a risk free asset to the portfolio. The risk free rate is indicated by rf . The risk free portfolio is indicted on the y-axis. Why do you think this so? Because by definition the risk free asset has zero standard deviation. An investor can decide to allocate his investment between risk free assets and risk assets, remember risk assets is represented by efficient frontier. Lets say that he chooses to allocate his funds between the risk free asset and portfolio P3 of the risky assets. This would imply that his portfolio will be allocated anywhere between rf and portfolio P3. the more risk averse the investor the more of the risk free asset he will include in his portfolio. His portfolio will lie closer to the y-axis indicating a greater proportion of risk free assets in it. Conversely, a less risk averse investor will include more of the risky asset in his portfolio; his portfolio would lie closer to portfolio P3. You should be familiar with this from one of our earlier lectures. But this is not the most optimal decision by the investor. Lets assume he decides to invest in the risk free asset and portfolio P4. Any portfolio on the line between rf and portfolio P4 will dominate portfolios represented by portfolio P3 and the risk free asset. This should be clear from our earlier discussion. One could follow the same argument for at portfolio that consist of the risk free asset and portfolio P5 or the risky assets, but not for portfolio P7 of the risky assets…Why? Well that is quite simple. Any portfolio that is on the line between portfolio P5 and P7 of the risky portfolios is dominated by a portfolio on the efficient frontier. The lines rf – P3, rf – P4, and rf – P4 are all Capital Allocation lines. We can continue to move our CAL to the left until we reach our optimal risky portfolio (P6) as defined by the tangent between the CAL and the efficient frontier. There is no risky portfolio that offers a better reward-volatility (risk) ratio than this. Recall that the reward to volatility ratio is the Sharpe Ratio. As we move our CAL to the left our sharp ratio increases (slope of the CAL). Finally, we can introduce the level of risk aversion of the investor. In a previous lecture we were introduced to the calculation of the proportion of assets that should be allocated to risky assets 104 give the level of risk aversion “A”. This will give us the complete optimal portfolio which is diagrammatically this is represented by the tangent between the CAL and the investors highest indifference curve. Note: The objective of an investor interested in the minimum variance portfolio is to minimize the risk of the portfolio. Whereas the objective of an investor interested in the optimal risky portfolio is to obtain the best risk and return trade-off available. Objec-ve:# Maximise:'' 'Sp'='E(rp);'rf'/'δp' Op#mal'Risky'Por0olio' Step 1: Wmin(D) = E(RD)δ2E – E(RE)Cov(rD,rE) [R#=#Risk#Premium]# E(RE)δ2D + E(RD)δ2E – [E(RD) + E(RE)]Cov(rD, rE) Step 2: Wmin(E) = 1- Wmin(D) CAL E(R) IC Step 3: ORP OCP E''(r'p)'='wD#E#(rD#)#+'wE#E#(rE#)# Objec-ve:# Minimise&Risk& Global&Min*Variance&Por0olio& G M-V P r f Step 4: δ2P = W2Dδ2D + W2Eδ2E + 2WDWECov(rD,rE) SD Step 1: δ2E – Cov(rD,rE) Wmin(D) = CAL E(R) δ2D + δ2E – 2Cov(rD, rE) IC ORP Step 2: Wmin(E) = 1- Wmin(D) OCP G M-V P r f Step 3: SD E&&(r&p)&=&wD#E#(rD#)#+&wE#E#(rE#)# Step 4: δ2P = W2Dδ2D + W2Eδ2E + 2WDWECov(rD,rE) Cov(rD,rE) = !DE!D!E 105 Determine the allocation of investment funds in the complete portfolio for an investor with a coefficient of risk aversion of 5, given an opportunity to invest in risk free treasury bills with an interest rate of 6% Op#mal'Complete'Por.olio'' Step 1: CAL E(R) Percentage of Capital in Risky Portfolio E(rp)'–'rf' Y= ' ''''A'δ2p'' IC ORP OCP Repeat the above for an investor who has a coefficient of risk aversion of 3 G M-V P r f Step 2: Percentage of Capital in Risky Portfolio Risk Free Portfolio = 1-Y Step 3: Split of Securities SD 25% 45% Split Risky Portfolio into its percentage Debt and Equity given Capital Allocation between Risky and Risk free Portfolio Solu'on! 30% Risk free portfolio Debt (Bonds) Equity Risky Portfolio Assume the following characteristics of an optimal risky portfolio: E (rp) = 15.12% σp = 15.37% Objec've:! Allocate'investments' consistent'with'level'of' risk'aversion' Op#mal'Complete'Por.olio'! Step 1: Percentage of Capital in Risky Portfolio Step 2: E(rp)'–'rf' Y= ' ''''A'δ2p'' Percentage of Capital in Risky Portfolio Risk Free Portfolio = 1-Y = -28.81% Answer'='Y'='77.29%' Step 3: Split Risky Portfolio Bonds: 48.19% Equity: 29.10% The split between Bonds and equity in this portfolio is: Bonds - 62.35% Equity – 37.65% Required: 106 portfolio would be made up of all the risky assets in proportion to their market values, but it would not contain any risk-free asset. Indifference Curve (A=3) Expected Return Optimal Complete Portfolio CAL/ CML (if all risky assets i.e. the market) Indifference Curve (A=5) Optimal Complete Portfolio Optimal Risky Portfolio Global Min-Variance Portfolio rf Standard Deviation Lending Portfolio Borrowing Portfolio Based on our earlier discussion, the capital market line (CML) provides the best risk and return tradeoff for an investor. Along the CML, a potential portfolio for the investor will be the market portfolio. However, it is not necessary for the investor to pick that portfolio because he/she can pick any portfolio along the CML. This investment decision will be heavily influenced by the investor’s risk preference. The following graph illustrates how the risk preference of an investor affects his/her investment decisions: 2. If the investor’s risk preference is represented by indifference curve 2, he/she will pick portfolio L, which has a lower level of risk than the market portfolio. The only way to reduce the risk of the portfolio is to include the risk-free security in it. In essence, the investor is “lending” part of his/her money at the riskfree rate. As a result, portfolio L is also known as a lending portfolio. From the graph above, we know that any portfolio along the CML that is below the market portfolio can be termed a lending portfolio. 3. If the investor’s risk preference is represented by indifference curve 3, he/she will pick portfolio B, which has a higher level of risk than the market portfolio. The only way to increase the risk of the portfolio is to borrow at the risk-free rate to increase the investment base (i.e. margin trading). As a result, portfolio B is also known as a borrowing portfolio. From the graph, we know that any portfolio along the CML that is above the market portfolio can be termed a borrowing portfolio. 1. If the investor’s risk preference is represented by indifference curve 1, he/she will pick portfolio M, which is the market portfolio. In other words, the investor’s Business 442: Investments Chapter 5-21 Dr. Siaw-Peng Wan 107 Identify the risk-return combinations available from the set of risky assets. CAL/ CML (if all assets i.e. the market) Expected Return Indifference Curve Optimal Risky Portfolio Identify the Optimal portfolio of risky assets by finding the portfolio weights that resulted in the steepest CAL. Individual Assets Optimal Complete Portfolio free asset against a single optimal risky portfolio (i.e. the market portfolio). From the graph, we know that in the situation where a risk-free asset is not available, the investor can only reduce the risk of the portfolio by shifting his/her funds towards “safer” risky assets, which will move the investor downward along the efficient frontier. Global Min-Variance Portfolio rf Chose an appropriate complete portfolio by mixing the risk-free asset with the optimal risky portfolio Standard Deviation • Efficient Frontier of risky assets • Common risky portfolio for all clients • Risk aversion factored in when a decision is made between proportion of risk free and risky portfolio (separation property) • The best risk Portfolio is the same for all clients regardless of risk aversion • Different managers may have different “optimal portfolios • Optimal Risky portfolios may be affected by constraints e.g. client preferences Optimal Risk Portfolio does not take into account risk aversion whereas your optimal complete portfolio does According to James Tobin’s Separation Theorem, an investor’s decision making process is actually made up of two separate decisions: i. To be on the CML, the investor initially decides to invest in the market portfolio (i.e. portfolio M). This is the investment decision. ii. Based on the investor’s risk preference, he/she makes a separate financing decision on whether to lend or borrow at the riskfree rate to get to the desired point. In the above three scenarios, we have seen how an investor can adjust his/her return or risk by borrowing or lending with a risk- However, when the investor has access to the risk-free asset, he/ she can reduce the risk of the portfolio by investing or lending at the risk-free rate, which will move him/ the portfolio at the expense of a lower return. If you look at the graph carefully, you will realize that the portfolios on the CML provide a higher return than the portfolios on the efficient frontier for the same level of risk. In other words, the availability of the risk-free asset helps an investor reduce the risk of his/her portfolio but also helps preserve most of the return of the portfolio. Similarly, we know that in the absence of the risk-free asset, the investor can improve the return of the portfolio by shifting his/ her funds towards “riskier” risky assets, which will move the investor upward along the efficient frontier. However, with the presence of the risk-free asset, the investor can improve the return of the portfolio by borrowing at the riskfree rate, which will move the investor along the CML. We know both strategies will improve the return of the portfolio at the expense of a higher level of risk. 108 If you look at the graph carefully, you will realize that the portfolios on the CML experienced a smaller increase in risk level than the portfolios on the efficient frontier for the same level of increase in return. In other words, the availability of the risk-free asset helps an investor improve the return of his/her portfolio without taking on as much risk. her downward along the CML. We know both strategies will reduce the risk of Allocate funds between the risky portfolio and the risk-free asset • Calculate the fraction of the complete portfolio allocated to the risky portfolio and the risk-free portfolio. • Calculate the share of the portfolio in each risky asset and the T.bill (risk-free) Conclusion Summary: Specify the return characteristics of all securities: • expected returns, • variances, and co-variances • correlation coefficients Establish the risky portfolio, P. • Calculate the optimal risky portfolio, P • Calculate the properties of portfolio P using the weights determined in the pervious step. In this chapter, we have looked at three different portfolios and we learned three different lessons from these portfolios: 1. With a portfolio that contains a risky asset and a risk-free asset, we derived the basic relationship between risk and return through the capital allocation line. We found that the return of a portfolio can be broken down into two components: a guaranteed return and a compensation for taking risk. 2. With a portfolio that contains multiple risky assets, we derived the efficient frontier that helped explain the concept of diversification. In other words, it is possible for an investor to put together a portfolio that has a risk level lower than the individual risk levels of the assets it contained. 3. With a portfolio that contains multiple risky assets and a riskfree asset, we showed that it is possible for an investor to improve his/her risk or return situation by lending or borrowing at the risk-free rate. 109 Chapter 14 Index Models Section 1 Index Models n = 50 estimates of variances; (n2- n)/2 = 1,225 estimates of covariances; 1,325 estimates. ! Markowitz!Por,olio!Selec1on! ! CAL/ CML (if all risky assets i.e. the market) Expected Return Indifference Curve Optimal Risky Portfolio Individual Assets Optimal Complete Portfolio Global Min-Variance Portfolio N=50 1325 estimates rf Standard Deviation Examples)of)single)factors:) • Business)cycles) • Interest)rates) ) The problem with this approach This approach calls for a high level of data inputs. Consider the following: • For a 50 security portfolio, the Markowitz model requires the following parameter estimates: n = 50 estimates of expected returns; The process is time consuming and error prone. An investment analyst will find it very difficult to manage such a huge number of inputs. Index Models are designed to overcome this problem, but at a cost of accuracy. This will become evident later. • For a 50 security portfolio, the single-index model requires the following parameter estimates: n = 50 estimates of expected excess returns, E(R); n = 50 estimates of sensitivity coefficients, i; n = 50 estimates of the firm-specific variances, 2(ei); 1 estimate for the variance of the common macroeconomic factor, 2M; or (3n + 1 = 151) estimates. In addition, the single-index model provides further insight by recognizing that different firms have different sensitivities to macroeconomic events. The model also summarizes the distinction between macroeconomic and firm-specific risk factors. 111 ri = E (ri) + m + ei Building(a(Single(Factor(Model( Return Firm-specific (unique) ri(=(E((ri)((+( ei( “e” has a: Mean = 0 SD = δ Expected return Unexpected return ri(=(E((ri)((+(((((((+(ei( m Uncorrelated “m” has a: Mean = 0 SD = δ Expected return Unexpected return Return emanating from the inclusion of a common macro-economic factor Whilst(“m”(is( common(to(all( securi=es,(ei(will(be( unique(to(firm(“i”,( therefore,(by( defini=on(m(and(ei( are(not(correlated( with(each(other;( any(shock(resul=ng( from(the( macroeconomic( factor(will(not( impact(the( unexpected(frimI specific(return.( Whilst “m” is common to all securities, ei will be unique to firm “i”, therefore, by definition m and ei are not correlated with each other, therefore any shock resulting from the macroeconomic factor will not impact the unexpected frim-specific return. Actual return on security I = Expected return of security i + Surprise return on the security Macro economic surprise return tends to zero as time passes. ei = Firm specific surprise return. Return: Let us consider the rate of return on a single security.. ri = E(ri) + ei The return for this security can be broken down into two portions, an expected return [E (ri )] and an unexpected return [ei ]. The unexpected return is assume to have a zero mean. We can include a common macroeconomic factor into the model. We call this factor “m”. Note that this factor will have an impact on all securities. Note further that the return associated with this macroeconomic factors will be unexpected as well. We can therefore re-write the return on the portfolio as; 112 Building(a(Single(Factor(Model( Risk Because returns m and ei are uncorrelated we can write the variance as… Variance Because “m” is a common market factor, which by definition affects all securities, it will be correlated with all securities. This is not the case with “e”; “e” is frim specific and will by definition be uncorrelated across firms (or securities). δi2$=$$δ2m$$+$δ2ei$ Systematic factors (market factors) Unsystematic factors (firm specific factors) Covariance Cov(ri,$rj)$=$Covm$+$ei,$m+ej$ Assume(1(macro(factor.(No( correla9on(between(firms.(Simply( variance(of(security(to(the(market,( rather(than(to(the(stocks(and(the( market(as(with(morkovits.( e is uncorrelated across firms Cov(ri,$rj)$=$Covm,m$=$δ2m$ Significantly reduced number of estimates because we are now relating the variance of each security to the market and not to each other as in the Markowitz model Risk: Variance: We measure risk by looking at the variance of the return of the stock. Within the portfolio the variance will be driven by the market δ2m and the firm δ2i. This defines the systematic factors (affects all securities) that impact stock returns and unsystematic (or firm specific) factors that impact the stock returns , respectively. As we have seen under the section on returns m and ei are uncorrelated. The variance is written as: Covariance: Let’s assume we have 2 securities, “i” and “j”. The covariance will come from the uncertain parts of the returns. We can therefore concentrate on m and e for security i and j. we can therefore write the covariance as: Cov(ri, rj) = Covm + ei, m+ej We said that e was uncorrelated across firms, we can therefore drop e from the expression which would then make the covariance equal to… Cov(ri, rj) = Covm,m = δ2m This reduces variance of securities i and j to the variance of the common macro-economic factor (m). δ2 = δ2m + δ2ei 113 m = a common macroeconomic factor that affects all security returns. The S&P 500 is often used as a proxy for m. Building(a(Single(Factor(Model( Beta = sensivity of security to the market Systema8c#Risk#<#β2iδ2m# ## Firm#Specific#Risk##<#δ2ei### Return ri#=#E#(ri)#+#βm#+#ei# Variance We can ignore ei and ej we assume them to be uncorrelated. δ2i#=#β2iδ2m#+δ2ei### Covariance Cov(ri,#rj)#=#Cov#(βi#m+ei,#βj#m+ej)# ei = firm-specific surprises Variance: We should factor the sensitivity coefficient (β) into the equation for the variance as well. The variance will be as follows: δ2i = β2iδ2m δ2ei Cov(ri,#rj)#=#βi#βj#δ2m########################### [δax,by = ab δxy] 8-8 Return β2δ2m will define the systematic (market) risk to which security returns is exposed. δ2ei will define the unsystematic (firm specific) risk. We can now re-write the single factor model by including beta. The will be as follows: ri = E (ri) + βm + ei Covariance: We should factor the sensitivity coefficient (β) into the equation for the covariance as well. The covariance will be as follows: This expression defines the single factor model where… βi = response of an individual security’s return to the common factor, m. Beta measures systematic risk. Cov(ri, rj) = Cov (βi m+ei, βj m+ej) We can ignore ei and ej because, as before, we assume them to be uncorrelated. Remembering that [δax,by = ab δxy], we can rewrite the covariance as follows: Cov(ri, rj) = βi βj δ2m 114 From Single Factor Model to the Single Index Model: • To make the single factor model operative we relate a broad market index such a the S&P500 or the JSE all share index as a proxy for the macroeconomic factor. • We will end up with an equation for the index model that is similar to the single factor model…the reason we refer to as the single index model that it uses the market index as a proxy for the common macroeconomic factor. • We have an abundance of historical information from which to estimate market risk and market risk premium for the single index model. • The market risk premium will be denoted as: RM = rM – rf and the associated risk is denoted as σM • We can now regress the excess returns of the security (Ri = ri – rf) on the excess returns of the market (RM). This will tell us how sensitive the return of the security in relative to that of the market. To get the single index model we regress excess market returns on excess index returns. Now we want to estimate the single index model at time period “t”. This will be stated as: Ri (t) = αi +βiRM (t) +ei (t) α i represents the intercept or the security return when the market return is zero. If market return is zero then βiRM (t) equal to zero and the return on the security will be equal to α. βi gives us the slope of the security, it is a reflection of the sensitivity of the security to movements in the market (sensitivity coefficient); is a measure of systematic risk. The Single Index Model: The error term (ei ) averages out to zero over the long term. This is also referred to as the firm specific unexpected (surprise) return. Because ei is zero we can re-write the equation as: E(ri) – rf = Ri and the E(rM) – rf = RM [note: we are using capital “M” to denote the market and not small “m” which denoted a single economic factor; R= excess return] Ri (t) = αi +βiRM (t) or 115 E(Ri) = αi +βi E (RM) βi E (RM) tells us that security’s risk premium is due to the risk premium of the index. The extent to which the security “mimics” the index risk premium is given by the sensitivity coefficient (βi ). This is referred to as the systematic risk premium. The second part of the risk premium is given by α. This is a nonmarket risk premium. This is what investment managers will look for…companies whose stock exhibit positive α. Generally is a security is under-priced you will see a positive α for it. Note that if one company gains +ve α another loses. The sum total of α in the market is zero i.e it is competed away. Single"index"model" Return! 1" The"stock’s"expected"return"if"the"market"is"neutral"ie."When" excess"return"(Rm;Rf)"="zero" α" 2" The"component"of"return"due"to"movements"in"the"overall" market"(β)"i.e."the"securiCes"responsiveness"to"market" movements"" βi(rM;rf)" 3" The"unexpected"component"of"return"due"to"unexpected"events" that"are"relevant"only"to"this"security"(firm"specific)" ei" !(#$)="&$+"'$!(#()! Summary"of"result"of"the"single"index"model" 1" The"stock’s"expected"return"if"the"market"is"neutral"ie."When" excess"return"(Rm;Rf)"="zero" " α" 2" The"component"of"return"due"to"movements"in"the"overall" market"(β)"i.e."the"securiCes"responsiveness"to"market" movements"" " βi(rM;rf)" 3" The"unexpected"component"of"return"due"to"unexpected"events" that"are"relevant"only"to"this"security"(firm"specific)" " ei" 4" The"variance"aIributable"to"the"uncertainty"of"the"common" macroeconomic"factor"" " β2iσ2M" 5" The"variance"aIributable"to"firm;specific"uncertainty" σ2(ei)" " Risk Premium 116 For'a'single'security:'δ2i'='δ2'(ei)' Index Model and Diversification: Advantage&of&the&index&model&–& diversifica1on&gets&rid&of&the& unsystema1c&risk& δ2p = β2pδ2M δ2 (ep) Index&Model&and&diversifica1on& The variance determined by the market is represented by β2pδ2M and the variance represented by company specific components is represented by δ2 (ep). Por$olio'Variance' δ2p'='β2pδ2M'+δ2'(ep)' It does not matter how much effort we put into diversification, the variance that is determined by the market will persist, simple because those factors that determine the variance will impact on all securities within the portfolio. However, the factors that impact on the variance attributable to that which is company specific as represented by δ2 (ep), can be diversified because these factors do not affect all securities; the firm specific components (ei) are independent. The more stocks we add to the portfolio the greater the change that the firm specific components will cancel each other out. Because of this property of independence we can state that the variance of an equally weighted portfolio is equivalent to the average of the variance of the portfolio x number of securities in the portfolio…. Which would imply that as the number of securities in the portfolio increase the variance of the portfolio would decrease. If “n” increases the average variance decreases to a negligible amount Risk (σ ) δ2'(ep)'='δ2'(e)'/'n' δ2'(ep)' β2pδ2M' Market risk Number of companies (n) 8-17 What we would like to do is break this excess expected rate of return down into its components to see which parts would be subject to variance (risk). If we assume that each security in our portfolio is equally represented then the weight of each security (Wi) must be equal to 1/n and the sum of 1/n securities must be equal to the number of securities in the portfolio. We can therefore write the excess return as… Rp= Σ WiRi or Rp= 1/n Σ Ri (since wi = 1/n) Note that we have changed the subscript in reference to the excess return from “p” to “i” and we are stating that the portfolio excess returns equate to the sum of all equally weighted excess 117 returns of each security within the portfolio. Because we made reference to the excess returns of security “i” in the equation above we can now substitute Ri with αi + βi RM + ei in the above equation since we have stated above that Ri=αi + βi RM + ei. We therefore say that… Rp= 1/n Σ Ri = 1/n Σ (αi + βi RM + ei) Alpha*and*Security*Analysis* Macro8Economic'Analysis:'' Es9mate*of…* • risk*premium**E(RM)'and** • risk*of*the*market*index*–*β2iδ2M' ** We can break this down further as follows: Rp = 1/nΣi α + (1/nΣβi)RM + 1/nΣei Now we can clearly see that the excess return consists of a company specific return [1/nΣi α ] a market return [(1/nΣβi)RM ] and a company specific “surprise” return [1/nΣei], this last element of return is driven by events that are relevant to a specific security (i). The variance of returns therefore come from [(1/nΣβi)RM ] and [1/nΣei], (market and the company). For a single security we represented the variance by…. δ2 = δ2m δ2(ei) For the portfolio, by…. δ2p = β2pδ2M δ2 (ep) The variance determined by the market is represented by β2pδ2M and the variance represented by company specific components is represented by δ2 (ep). Sta@s@cal'analysis:'Passive' por$olio'analysis' Es9mate*of*…* • Beta*coefficients*(β)'and* • Residual*variances*(δ2(ei)'of*all* securi9es* iste Cons ncy' 2 sources of premium Por$olio'Manager:'' Establishes*a*market*driven* expected*return*which*is*use*as*a* benchmark*for*performance* measurement*8*βiE(RM)' Security'analysis:'Ac@ve'por$olio' analysis' Es9mate*of*…* • Security*α*–*This*represents* incremental'risk'premium'over8 and8above*the*market*risk* premium*which*is*βiE(RM)' This separation of macroeconomic analysis of market risk premium from statistical analysis of the alpha has some significant advantages: It allows us to separate the risk premium attributable to the market from that attributable to the security. The risk premium attributable to the market can now be set as a benchmark against which performance could be measured. If alpha is positive it provides a risk premium over and above that of the market…the security will be a good buy and should in theory be “over-weighted” in the portfolio. This security is referred to as under-priced. 118 The opposite is also true, a security with an alpha that is negative is generally a bad buy and therefore its portfolio weight should be reduced. This security is referred to as over-priced. • When%introducing%the%index%por0olio%we%assume%that%this%also% contributes%as%risk%premium%in%propor8on%to%its%risk%(E(RM)&/& σ2M),&leaving%the%por0olio%variance%undisturbed.% Optimal Risky Portfolio: • As%a%result,%the%assump8on%of%a%beta%of%1%for%the%overall% por0olio%s8ll%holds.% The investor has a choice to invest in the market only, in which case he invest in the index portfolio, or he can invest in the security that gives him additional alpha as well as the market (i.e. index portfolio). In exchange for additional alpha, the investor sacrifices some of the benefit of diversification which is assumed in the index (market) portfolio. Adding an additional asset to the portfolio introduces company specific risk. We start off with an assumption that the active portfolio has a beta of 1. When introducing an additional asset to the portfolio we assume that the asset contributes a return (alpha) in proportion to its risk (αA / σ2A), leaving the portfolio variance undisturbed. • The%weight%in%the%ac8ve%por0olio%given%these%assump8ons%is% given%by….% !!"=#$"/&$" /(()*)/&$* & Or better stated…we can view this as the ratio of investment in the active portfolio relative to the investment in the passive (market or index) portfolio. • For$any$level$of$σ2A$the$correla2on$between$the$ac2ve$and$ passive$por9olio$is$greater$when$beta$of$the$ac2ve$por9olio$is$ higher.$If$so,$the$diversifica2on$benefit$in$the$passive$por9olio$ is$diminished$and$as$a$result$its$weight$in$the$ac2ve$por9olio$is$ reduced.$This$can$be$easily$seen$from$….$ $!∗"=#$$"/%+(%−'")!$" $ ! 119 Information ratio: !∗"=#$$"/%+(%−'")!$" ! Having decided on the weight of the active portfolio within the overall risky portfolio, how do we ascertain whether we have made the right decision, otherwise stated…how do we know what value our security analysis has added to our portfolio? • ….when!the!βA#is#>1,!the!weight!of!the!ac*ve#por0olio#increases#/! the!weight!in!the!passive#por0olio#therefore#decreases# • ….when!the!βA#is#<1,!the!weight!of!the!ac*ve#por0olio#decreases#/!! the!weight!in!the!passive#por0olio#therefore#decreases# • Note!that!(∗)#is!invested!in!the!ac5ve!por8olio!and!18#(∗)##is! invested!in!the!passive!por8olio! # To answer that we look at the following equation….. The information ratio is ((alpha)(A)/(sigma)(eA))2 portion of the equation. It tells us how much extra return we can obtain from security analysis relative to the risk incurred when we over or under-weight securities relative to the passive market index. To#re&iterate….# • !!"=#$"/&$" /(()*)/&$* # Regardless#of#beta,#when#w#0# decreases,#so#does#w*.# 2 1 Other#things#held#equal#,w#0#is#smaller#the# greater#the#residual#variance#of#a# candidate#asset#for#inclusion#in#the# por=olio.# • !∗"=#,!"/(+((−.")!!" ## 3 4 We can also refer to this as the information ratio of the active portfolio; this is what we need to maximise in order to maximise the overall Sharpe ratio. Increased#firm&specific#risk#reduces#the# extent#to#which#an#acAve#investor#will# be#willing#to#depart#from#an#indexed# por=olio.# Therefore,#other#things#held#equal,# the#greater#the#residual#variance#of# an#asset#(σ2),#the#smaller#its# posiAon#in#the#opAmal#risky# por=olio.# Why?# The Sharpe ratio indicates that: a higher alpha makes a security more desirable alpha, the numerator of the Sharpe ratio, is a fixed number that is not affected by the standard deviation of returns, the denomi- 120 nator of the Sharpe ratio. Hence, an increase in alpha increases the Sharpe ratio. Since the portfolio alpha is the portfolio-weighted average of the securities’ alphas, then, holding all other parameters fixed, an increase in a security’s alpha results in an increase in the portfolio Sharpe ratio. The index portfolio is only efficient if all alpha values are zero. When adding a security to the overall risky portfolio that security brings with it systematic risk, which is compensated for by the market risk premium as well as firm specific risk which is compensated for by the alpha. Therefore, if the alpha is zero, adding the security to the overall portfolio makes it less efficient…i.e. there is no risk premium to compensate for the addition risk associated with the new security. When market analysis uncovers positive alphas, the index (market) portfolio is no longer efficient. Why? Because there are more positive gains out there that the market index does not capture Optimal Risky Portfolio: • Weight'in'Por,olio:' • W*M$=$1'$W*A$ $ • Return' • E(Rp)$=$+$W*A$αA$+$(W*M$+$W*A$βA)$E(RM)$ $ • Risk' • δ2p$=$(W*M$+$W*A$βA)2$δ2p$$+$(W*A$δA)2$ If all securities have an alpha of zero the optimal weight of the active portfolio in the overall portfolio is zero and the weight in the index (market) portfolio will be 1. 121 Optimal risky portfolio of the single index model: Combination of: • Market-index portfolio, the passive portfolio denoted by M • If we are only interested in diversification we will hold the index portfolio. In theory, all diversifiable risk will have been reduced to zero in this portfolio. We call this the Passive Portfolio / Passive Investment • Active portfolio denoted by A • However, we may feel that specific securities have the potential to offer extra firm specific returns (i.e. positive alpha). If so, we would want to include this in our portfolio. We call this the Active Portfolio. It is made possible by security analysis. of your active portfolio is now 50% higher than originally anticipated. Explain the significance of this change. 3. Explain why adding a security with a zero alpha to an efficient index portfolio will make the portfolio less attractive. Solution: 1. The initial position of the active portfolio in the overall risky portfolio is given by: W0A = 46% 2. The adjusted position of the active portfolio in the overall risky portfolio is given by: Example: W*A = 57.7% Assume that you have a market (index) portfolio with an expected return to variance ratio (E(RM) / δ2M ) of 26%. Assume further that you have an active portfolio with a portfolio alpha to portfolio variance ratio (αA / δ2A ) of 12%. The weight of the active portfolio in the optimal risky portfolio would increase from 46% to 57.7% and the weight in the passive portfolio would decrease to from 54% to 42.3%. 1. What would the initial position of your active portfolio in your optimal risky portfolio be? The correlation between the active and the passive portfolio is greater when the beta of the active portfolio is higher. This implies less diversification benefit from the passive portfolio, hence a lower position in it. 2. How would the initial position of your active portfolio in your optimal risky portfolio change if you discovered that the beta 122 3. Adding a security to the portfolio adds systematic risk to the portfolio, but this risk is compensated for by the market premium (through beta). The security also adds company specific risk to the portfolio variance, if the security’s beta is zero, there is no compensation for this risk. This makes the portfolio less attractive. • Satrix Divi - The Satrix DIVI is a convenient way to invest in a tracker that tracks the FTSE/JSE Dividend Plus Index • Proptrax - Property Index Tracker or PropTrax is the a property ETF in South Africa. It gives the investor access and good exposure to the high performance real estate sector of the JSE Funds that track the JSE Top 40 • Satrix 40 - The Satrix40 ETF is a convenient way to invest in a composite of the top 40 large cap shares on the JSE Securities Exchange • NewFunds Shari’ah Top 40 - This FTSE/JSE Shari'ah Top 40 Index is designed to reflect the Shari'ah compliant companies identified from the FTSE/JSE Africa Top 40 index • StanLib Top 40 - This ETF consists of the shares that constitute the FTSE/JSE Top40 index of the Johannesburg Stock Exchange Funds that track commodities, dividends and property • NewGold - NewGold gives investors the opportunity to benefit from the performance of the value of Gold Bullion Debentures 123 Chapter 15 The Capital Asset Pricing Model Section 1 The Capital Asset Pricing Model Capital market theory (CMT): by each investor is small relative to the aggregate wealth of all investors. • CMT builds on Portfolio Theory leading to the Markowitz Efficient Frontier. 2. Single-period investment horizon – This implies that investors are not concerned with anything that happens after the single period; it would not affect his current investment decision (not very realistic) • Recall that in Portfolio Theory we assumed that we can derive a set of risky assets and an aggregate efficient frontier. • We also assumed that all investors will want to maximise their utility in terms of risk and return. • To do this they will select a risky portfolio on the efficient frontier which is tangent to their utility map. • CMT takes this one step further and develops a model for pricing all risky assets • This leads to the development of the CAPM which we could use to determine the required rate of return for any risky asset. Assumptions: 1. Individual investors are price takers – this ensures that no single investor can influence the market price of securities through his actions. This assumption pre-supposes that the wealth held 3. Investments are limited to traded financial assets – Investment is therefore confined to stock, bonds and other tradable securities. Lending and borrowing at the risk free rate is considered part of these investments. 4. No taxes and transaction costs – In reality these are relevant factors that investors would consider in their investment decision. The tax bracket in which investors fall may dictate their investment behavior and transaction costs have a direct bearing on the final performance measure of an investment. 5. Information is costless and available to all investors – this assumption is far from true; there is a high cost attached to macroeconomic and security analysis. 6. Investors are rational mean-variance optimizers – If all investors are mean-variance optimizers it would imply that all of them use the Markowitz portfolio selection model. Their preferences 125 would be described by the utility function. People will take on risk relative to the potential reward. 7. There are homogeneous expectations – This would imply that all investors would use the same expected returns and covariance matrix to generate their efficient frontier and the optimal risky portfolio. A further implication is that all inputs used in estimating these returns and risk measures use the same set of inputs. 8. For every Borrower in the market, there is a lender – This would imply that collectively, all investors hold the market portfolio./ Characteristics of equilibrium under CAPM: E (r) M rf QM 2 The Market portfolio will be on the CML, tangent to the Efficient Frontier. All investors hold “M”, the market portfolio in their optimal risky portfolio or some combination of M and the risk free asset. 3 The equilibrium risk premium on the market portfolio is determined by the average level of risk aversion of investors' and the market portfolio’s variance (risk). E(rM)-rf = Āδ2M 4 CML CAL E (rM) 1 All investors will hold the proportion of risky assets in their portfolio in proportion to that assets weight in the market portfolio. Q The risk premium on the an asset within the portfolio will be determined by the Beta Coefficient of that asset. (remember: the beta coefficient is the asset’s covariance with the market relative to market risk) E(ri)-rf = βi [E(rM)-rf ] 1. Because all investors are assumed to use the same assumptions to build their portfolios they will end up having the same risky portfolio; this portfolio will be a mirror image of the market portfolio as far as the percentage assets allocation within the portfolio is concerned. As an example, if the risky portfolio of investor A had 3% of security XYZ in its portfolio, investor B will have 126 the same percentage of security XYZ in its portfolio and the market portfolio will contain the magnitude of XYZ as a percentage of its portfolio. Of course the rand or dollar value of the portfolios may differ. 2. If all investors hold the market portfolio, there is no need for complicated security analysis; the market portfolio, by virtue of its tangency with the CML, is the efficient market index portfolio. We introduced the “Separation Property” of portfolio selection when we looked at Optimal Risky Portfolios earlier; the same principal can be applied here. The investor s’ portfolio selection decision is segregated into… a decision to adopt the risky portfolio (in this case the market portfolio) and a decision between the percentage of the risk free asset to include in the complete portfolio i.e. what percentage of the risky portfolio would be substituted with the risk free asset. This will be determined by the investors level of risk aversion as we saw in the discussion under the construction of a risky portfolio. different investment managers may well come up with different “efficient risky portfolios” because in reality the input variables will differ. 3. The equilibrium risk premium in the market portfolio is determined by the average risk aversion of investors in the market portfolio and the market risk (variance). One of the assumptions of CAPM is that risk-free investments involve borrowing and lending among investors and that for every borrower there was a lenders. Therefore in aggregate borrowers and lenders cancel out i.e. they net to zero. If the investor opts to invest “y” in the optimal portfolio “M” it would be denoted by.. y- E(rM)-rf/ Aδ2M If borrowers and lenders “cancel each other out” the average position in the risky portfolio will be equal to 1; the average risk aversion will be Ā; the above equation then becomes… E(rM) – rf = Ā δ2M i.e The equilibrium risk premium in the market portfolio is determined by the average risk aversion of investors in the market portfolio and the market risk (variance). A passive investor will view the market index portfolio as a reasonable approximation of an efficient risky portfolio. Remember 127 Understanding)the)Market)Por=olio) E (r) Includes)all)risky)assets)–)stocks,)binds,) op5ons,)real)estate)etc.)therefore))a) Completely)Diversified)Por2olio)i.e.)no) unique)risk) Propor5on)of)individual)stock)in)the) Market)Por=olio)is)determined)by)its) value)rela5ve)to)that)of)the)market) i.e.)the)propor5on)is)value9 weighted.))This)must)be)the)case)if) the)market)is)in)equilibrium.) CML E (rM) M CML)=)The)(new))Relevant) efficient)fronGer)viz.)only) risky)assets) rf QM Lending Portfolio = Borrowing Portfolio The)price)at)which)a)stock)is)included) in)the)op5mal)por=olio)is)the)only) point)of)concern)for)the)investor) Q All)investors)must)hold)the)same) market)por2olio))(on)average)) because)they)use)the)same)input) data)to)determine)the)op5mum) por=olio.) The)equilibrium)risk)premium)on)the) market)por=olio)is)determined)by) the)average)level)of)risk)aversion)of) investors')and)the)market)por=olio’s) variance)(risk).) E(rM)-rf = Āδ2M Expected return-beta relationship under CAPM: A stock’s contribution to the variance of the market portfolio is what matters ! Portfolio risk is what matters to investors and what governs risk premium they demand. ! Therefore, what matters is the contribution of the risk to the overall portfolio from holding a particular asset, in this cases assume that assets is a stock (share) in GE ! We can measure this contribution by looking at the covariance of the stock with the market. We get… WGE [ w1Cov(r1,rGE) = +w2Cov(r2, rGE) + (wGECov(rGE,rGE) + (wnCov(rn,rGE)] ! Notice that the covariance of GE with all other assets in the market dominates the variance of the GE stock. GE’s contribution to the variance of the market portfolio is determined by its covariance with the market. GE’s contribution to the variance of the market portfolio 9-11 = WGE Cov(rGE,rM) Reward–to-risk ratio’s of the stock and portfolio ! We know from previous discussions that the excess return of an asset (or the risk premium of the asset) is driven by the weight of the asset within the portfolio. Therefore the contribution of holding a percentage of GE in the portfolio to the risk premium of the market portfolio is.. WGE [ E(rGE) – rf] ! The reward –to-risk ratio for investment in GE is therefore.. GE’s contribution of risk premium GE’s contribution to portfolio variance [ E(rGE) – rf] Cov(rGE,rM) = ! The reward –to-risk ratio for the portfolio is… Market risk premium Market variance = [ E(rM) – rf] δ 2M 9-12 ! CAPM states that in equilibrium all investments will offer the same reward–to– risk ratio. Therefore… [E(rGE) – rf] Cov(rGE,rM) = [ E(rM) – rf] δ 2M ! We can determine the risk premium of an individual stock (let’s say GE) from this equation… E(rGE) – rf = Cov(rGE,rM) [ E(r ) – r ] M f δ 2M ! The contribution of an asset’s stock to the variance of the market portfolio is also know as the stock’s beta…we can therefore E(rGE) = rf + βGE [ E(rM) – rf] E(rM) = rf + βM CAPM!! [ E(rM) – rf] 128 The Security Market Line: The SML and Forecast Performance: E (r) SML E (rM) M [ E(rM) – rf] = Slope The SML is a graphical depiction of the individual asset risk premium as a function of asset risk. How does this differ from the CML? 18% Risk Premium – βGE [ E(rM) – rf] 16% Beta (market) – Slope - rf βM=1 β Analyst(Forecast( return( βM SML α=#+2# Fair(Return( s1pulated(by(SML( (CAPM( M E (rM) = 14% rf = 10% [ E(rM) – rf] SML = Fairly priced assets i.e. return is commensurate with risk 1 E(rGE) = rf + E (r) Assets in Market Equilibrium βGE [ E(rM) – rf] 9-14 SML – interested in the properties of the security more than we are interested in the properties of the portfolio. Have a different measure of risk to the CML - Beta CML – measure of risk was standard deviation Measure of stock variance to MP & Risk premium f(beta) – Relationship as confirmed by CAPM. βM=1 1.5 β • This(stock(is(under+priced.#CAPM(says(it(should(return(16%,(but(the(analyst( forecast(18%.( • Based(on(the(analyst(forecast,(this(stock(is(a(good#buy## • UnderGpriced(stock(plot(above#the#SML# • Difference(between(fair(return(and(analyst(forecast(return(is(the(alpha((here(+2%)( E (r) Fair(Return( s1pulated(by(SML( (CAPM( SML M E (rM) = 17% 15% α=#$8%# rf = 10% 7% Analyst(Forecast( return( 0.71 βM=1 β • This(stock(is(over$priced.#CAPM(says(it(should(return(15%,(but(the(analyst(forecast( 7%.( • Based(on(the(analyst(forecast,(this(stock(is(a(bad#buy## • OverHpriced(stock(plot(below#the((SML( • Difference(between(fair(return(and(analyst(forecast(return(is(the(alpha#(here((H8%)( 129 The SML and Actual Performance: • The$SML$defines$all$“Fairly$priced$assets”$ E (r) Actual'Asset' Return' 18% 16% SML α=#+2# • Nega9ve$Alpha$(over>priced$stock)$ • This$stock$is$over&priced.+CAPM$says$it$should$return$15%,$but$the$analyst$ forecast$7%.$ • Based$on$the$analyst$forecast,$this$stock$is$a$bad+buy++ • Over>priced$stock$plot$below+the$$SML$ Fair'Return' s/pulated'by'SML' (CAPM' M E (rM) = 14% rf = 10% βM=1 1.5 • Posi9ve$Alpha$(under>priced$stock)$ • This$stock$is$under&priced.+CAPM$says$it$should$return$16%,$but$the$ analyst$forecast$18%.$ • Based$on$the$analyst$forecast,$this$stock$is$a$good+buy++ • Under>priced$stock$plot$above+the+SML+ β • Actual$Performance$ • Posi;ve+Alpha++=$over$–$performance$ • Nega;ve+Alpha+=$under>performance$ $ Over*Performed# E (r) Fair'Return' s/pulated'by'SML' (CAPM' SML • You can treat the SML as a passive investment. / Use SML as a Benchmark M E (rM) = 17% 15% • Based on the analyst forecast you can adjust your portfolio (active Investment). α=#$8%# rf = 10% 7% Actual'Asset' return' 0.71 βM=1 Under$Performed# Potential investment strategy: • Increase assets with a positive alpha. β • Reduce assets with a negative alpha. • When all non-positive alphas have been included / excluded the average portfolio held will resemble the market portfolio again. 130 Estimation of alpha: A security has an forecast rate of return of 0.13 and a beta of 2.1. The market expected rate of return is 0.09 and the risk-free rate is 0.045. The alpha of the stock is ? • Es#mate(of(Expected(return(based(on(SML:( E(ri)(=(rf(+(βi((rM=rf)(((((((((((( E(ri)(=(4.5%(+(2.1((9%=4.5%)(((((((((((( E(ri)(=(13.95%( • Es#mate(of(alpha( α(=(Forecast(Return(–(CAPM(Expected(Return( α(=13%–(13.95%(=(=0.95%( E(r)&=&D1&+&P1&–&P0&/&P0& && 0.18&=&P1&-75&+&6&/&75&& & P1&=&82.50& & E(ri)&=&rf&+&βi[E(rM)-rf]& & E(ri)&=&6%&+&1.2(10%)& & E(ri)&=&18%& & Can these two assets exist in equilibrium? Por$olio' Expected'return' Beta' A" 20" 1.4" B" 25" 1.2" Estimation of share price: E (r) SML A share sells for $75 today. It will pay a dividend of $6 per share at the end of the year. Its beta is 1.2. What do investors expect the stock to sell for at the end of the year? Assume a risk free rate of 6% and E(rM) = 16% A??? 20% 16% E (rM) B ??? M Por-olio"A"has"a" higher"beta"than" Por-olio"B,"but"the" expected"return"for" Por-olio"A"is"lower" than"the"expected" return"for"Por-olio"B."" Thus,"these"two" por-olios"cannot" exist"in"equilibrium." rf βM=1 1.2 1.4 β 131 Under CAPM is it possible to have the following example? Por$olio' Expected'return' Standard'Dev.' A" 30" 35" B" 40" 25" Is the following consistent with CAPM? Por$olio' Risk%Free% Market% A% Expected'return' Beta' 10% 0.0% 18% 1.0% 16% 0.9% E (r) If"the"CAPM"is"valid,"the"expected'rate'of'return' compensates'only'for'systema:c'(market)'risk,"represented" by"beta,"rather"than"for"the"standard"devia>on,"which" includes"nonsystema>c"risk.""" " Thus,"PorFolio"A’s"lower"rate"of"return"can"be"paired"with"a" higher"standard"devia>on,"as'long'as'A’s'beta'is'less'than'B’s.' E(r)%=%10%+%β%×%(18%–%10)% E(r)%=%10%+%[1.5%×%(18%–%10)]% E(r)%=%22%% % SML 22% A = 16% M E (rM) rf βM=1 1.5 The%expected%return%for% PorKolio%A%is%16%;%that%is,% PorKolio%A%plots%below% the%SML%(!%A%=%–6%),%and% hence,%is%an%overpriced% porKolio.%%% % This%is%inconsistent%with% the%CAPM.% % β Impact of selected relaxed assumptions: Assump5on:%Investors%can%borrow%and%lend%at% the%risk%free%rate%% AssumpAon:& Borrow%at%the%Risk.Free%Rate%and%invest%in%the%%Risky%asset,%essen5ally,%Lending at 7% and Borrowing at 9% E&(r&)& (CAL)& P& E&(rp&)&=&15%& rB&=&9%& rf&=&7%& S&(y≤&1)&=&0.36& S&(y&<&&1)&=&0.27& Sharpe'ra(o' Slope'=&[E(rp')'–'rf]/'σ'p' ' F& Slope'=&(0.15'–'0.09)/'0.22' ' Slope'=&0.27' σp&=&22%& σ 132 Assump&on:*Investors*invest*in*a*risk*free* asset*and*the*risky*por7olio* E (r) E(ri))=))E(r2))+)Betai(E(rm))–)E(R2)))SML M E (rM) Remember…. • CAPM%is%based%on…% – …%the%Markowitz%full%covariance% model%in%which%the%theore;cal% market%por=olio%consists%of%every% available%asset.% – …an%assump;on%of%mean@ variance%efficiency%i.e.%The% op;mal%por=olio%mean@variant% efficient,%implying%a%loca;on%on%a% the%efficient%fron;er.% E (rM) - E (rZ) E (rZ) βM=1 β Zero%Beta)Asset) Assump&on:*There*are*no*transac&on*costs*and* investors*have*heterogeneous*expecta&on* – …an%assump;on%about%expected% return@beta%rela;onship%which% cannot%be%observed.%% Given%that%%the%theory%is% based&on&expected& returns,%which%are%not% observable…% % 1. How%do%you%%test%the% validity%of%CAPM?% 2. %How%do%you%test%for% mean%variance% efficiency?% With an Index Model…. E (r) SML E (rM) • We# look# at# actual# returns# based# on# historical# data…this# is# observable.# • The#rela6onship#between#excess#return#and#market#variance#(i.e.#the# Cov#(Ri,#RM)#is#the#same#as#that#of#CAPM# M • The#only#difference#in#the#above#rela6onship#between#the#2#models# is# that# the# market# under# CAPM# is# the# (unobservable)# market# of# every#single#risky#porHolio#whereas#that#under#the#index#model#the# market#is##the#market#index#(a#proxy#for#the#market#porHolio).# E (rf) or E (rZ) βM=1 β 133 With an Index Model…. • The$excess$return$rela.onship$under$the$2$models$is$also$almost$iden.cal.$$ – CPM$:$E(ri)$–$rf$=$σi[E(rM)$–$rf]$ – Index:$$E(ri)$–$rf$=$α$+$σi[E(rM)$–$rf]$ • The$only$difference$is$the$alpha….$ – under$CAPM$alpha$is$assume$to$be$zero.$$ – Under$the$Index$model$alpha$is$assume$to$average$out$to$zero.$$ • Actual$data$shows$that$alpha$is$cantered$around$zero$and$slightly$posi.vely$ skewed$as$a$result$of$pressure$on$porPolio$managers$to$deliver$posi.ve$ alpha.$$ • However$this$skew$is$rela.vely$small,$which$explains$how$difficult$it$is$to$ outperform$the$market…giving$support$to$CAPM$theory.$On$a$sample$basis$ alpha$tends$to$zero,$even$under$the$Index$model.$ Example(of(alpha(distribu3on( 134 Chapter 16 Arbitrage Pricing Theory and Multifactor Models of Risk and Return Section 1 Arbitrage Pricing Theory and Multifactor Models of Risk Review&'&Building&a&Single&Factor&Model& Return Review&'&Building&a&Single&Factor&Model& Firm-specific (unique) ri&=&E&(ri)&&+& ei& “e” has a: Mean = 0 SD = δ Expected return Unexpected return m i& ri&=&E&(ri)&&+&&&&&&&+&e Uncorrelated “m” has a: Mean = 0 SD = δ Expected return Unexpected return Return emanating from the inclusion of a common macro-economic factor Whilst&“m”&is& common&to&all& securiAes,&ei&will&be& unique&to&firm&“i”,& therefore,&by& definiAon&m&and&ei& are&not&correlated& with&each&other;& any&shock&resulAng& from&the& macroeconomic& factor&will&not& impact&the& unexpected&frim' specific&return.& Risk Because returns m and ei are uncorrelated we can write the variance as… Variance δi2$=$$δ2m$$+$δ2ei$ Systematic factors (market factors) Unsystematic factors (firm specific factors) Covariance Cov(ri,$rj)$=$Covm$+$ei,$m+ej$ e is uncorrelated across firms Cov(ri,$rj)$=$Covm,m$=$δ2m$ Significantly reduced number of estimates because we are now relating the variance of each security to the market and not to each other as in the Markowitz model 136 What would the impact/change on expected return be based on the following assumptions ? Review&'&Building&a&Single&Factor&Model& Beta Systema8c#Risk#<#β2iδ2m# ## Firm#Specific#Risk##<#δ2ei### Return ri#=#E#(ri)#+#βm#+#ei# Variance We can ignore ei and ej we assume them to be uncorrelated. δ2i#=#β2iδ2m+#δ2ei### Assume: • the Macro- economic factor is GDP • we expect GDP of 5% • βi = 1.5 Covariance • Actual GDP = 3% Cov(ri,#rj)#=#Cov#(βi#m+ei,#βj#m+ej)# Cov(ri,#rj)#=#βi#βj#δ2m########################### [δax,by = ab δxy] 8-4 Single Factor Model equation: ri = E(ri) + βiF + ei Result: Given a beta of 1.5, Ri will be 1.5( 5%-3%) or 3% lower than anticipated. The$security’s$surprise$ return$is$/3%$from$the$ macro$factor$PLUS$any$ firm$specific$return.$ ri!=!Return!on!security!! βi=!Factor!sensi1vity!or!factor!loading!or!factor!beta! F!=!Surprise!in!macro;economic!factor! !(F!could!be!posi1ve!or!nega1ve!but!has!expected!value!of! zero)! ei%=!Firm!specific!events!(zero!expected!value)! 137 Multifactor Models: Is the single factor a good representation of systematic risk to which expected returns are exposed? Even if an index is used as a proxy for the market? Multifactor models can provide a better description of security returns. This means that the risk measures that you put in place could be better and more effective. They allow us to better explain the individual stock returns, build models of equilibrium security pricing and can expose various macroeconomic risks and manage these risks. (Hedging strategies become more effective) – creating an opposite factor exposure Mul$factor+Model+Equa$on+ ri = E(ri) + βiGDPGDP + βiIRIR + ei + + + ri++=+Return+for+security+I$ Surprise+Return+ $ βGDP$=+Factor+sensi$vity+,+loading+or+beta+for+GDP++ + βIR++=+Factor+sensi$vity,+loading+or+beta+for+Interest+ Rate+ $ $ei$=+Firm+specific+events+ Compare+SAA+to+Eskom+–+will+a+single+index+model+do?+ + + 10-9 Surprise returns (Expected return = 0) – anything that differs from what we assumed when we calculated our securities Negative Beta – e.g. IR Return on security = Expected return – Any surprise return on that seurity 138 Required: Mul$factor+SML+Models+ • Calculate the risk premium that is attributable to each factor. E(ri ) = rf + βiGDPRPGDP + βiIRRPIR ! ! !! ! • Calculate the total factor risk premium. • Assume a T Bill rate of 4%.what would the Expected return be for Company A βiGDP+=+Factor+sensi$vity+for+GDP++ + RPGDP=+Risk+premium+for+GDP+ + βiIR!=+Factor+sensi$vity+for+Interest+Rate+ RPIR+=+Risk+premium+for+Interest+Rate+ 10-11 Example: Assumptions: • Company A has a GDP beta of 1.2 and an interest rate beta of -0.3. Risk%Premium%for%GDP:% βGDPRPGDP=%1.2%x6%% βGDPRPGDP=%7.2%% % Risk%Premium%for%Interest:% βIRRPIR=%<0.3%x(<7%)% βIRRPIR=%2.1%% % Total%Factor%Premium:% βGDPRPGDP%+%βIRRPIR%=%9.3%% Expected(Return:( E((r)(=(rf+(βGDPRPGDP(+(βIRRPIR(( E(r)(=(4%+1.2(6%)(+(>0.3)(>0.7)( E(r)(=(13.3%( % • The risk premium for one unit of exposure to GDP is 6% • The risk premium for one unit of exposure to interest rate risk is -7% 139 Arbitrage Pricing Theory: • Arbitrage occurs if there is a zero investment portfolio with a sure profit. • Since no investment is required, investors can create large positions to obtain large profits. • Regardless of wealth or risk aversion, investors will want an infinite position in the risk-free arbitrage portfolio. • In efficient markets, profitable arbitrage opportunities will quickly disappear. • Occasionally mispricing can occur/arise. Well-diversified portfolios: rP#=#E#(rP)#+# PF#+#eP# ! F#=#some#factor# # ! For#a#well4diversified#por:olio,#eP## ◦ approaches#zero#as#the#number#of#securiAes#in#the# por:olio#increases## ◦ and#their#associated#weights#decrease,#therefore#we# can#write…# rP#=#E#(rP)#+# PF# • Two assets must have the same beta before we can create the arbitrage portfolio. • Efficient market hypothesis says that this position is unstable and so will be quickly eliminated. Expected return from the portfolio + Surprise return from a macro-economic factor +Surprise return from the market 140 Returns(as(a(Func,on(of(the(Systema,c( Factor( Return (%) A Return (%) Returns(as(a(Func,on(of(the(Systema,c(Factor:( An(Arbitrage(Opportunity( B A Return (%) A"="1" A"="1" (0.10 + 1 x F) x R1M Long A C -(0.08 + 1 x F) x R1M Short C C="1" 10 10 10 F 0 +0.20 x R1M = R20K 8 F 0 • Risk Free Profit because factor F risk cancels out over the long and short positions 0 rA"="E"(rA)"+" AF( • Net Investment is zero rA"="10%+1F( Well5diversified( Single5Stock( • As long as well diversified portfolios with equal betas have different expected returns an arbitrage 10-18 opportunity exists 10-17 Well diversified portfolio – Non-factor risk (systematic risk) is diversified out of the portfolio. Return determined by systematic risk only and the beta is 1. Expected return of the portfolio is 10%. At 10% the systematic factor is zero. An#Arbitrage#Opportunity# • Construct portfolio D with 50% A and 50% F Expected Return (%) • Beta of D = (0.5X0)+(0.5X1) = 0.5\ A 10 D 7 6 r f= 4 C • Return of D = (0.5*4) + (0.5*10%) = 7% Risk Premium • Beta is the same as Portfolio C but return is higher – an arbitrage opportunity exists F 0.5 1 Beta with respect to the Macro-factor Strategy:) • Long#D# • Short#C# 10-19 Therefore we can say that to rule out the arbitrage opportunities, well diversified portfolos must lie on the line F-A 141 Therefore we can say that to rule out the arbitrage opportunities, well diversified portfolios must lie on the line F-A APT$and$CAPM$ APT$ Security)Market)Line)(Mkt.)Index)) E(rp) = rf + [E(rM) – rf]βp Expected Return (%) E(rM) [E(rM - rf)] rf • There are a sufficient number of assets to construct a welldiversified portfolio– all asset are on the SML • No arbitrage opportunity exists 1 Beta with respect to Index • Equilibrium$means$no$arbitrage$ opportuni8es.$ CAPM$ • Model$is$based$on$an$inherently$ unobservable$“market”$ porDolio.$ • APT$equilibrium$is$quickly$ restored$even$if$only$a$few$ investors$recognize$an$arbitrage$ • Rests$on$meanHvariance$ efficiency.$The$ac8ons$of$many$ opportunity.$ small$investors$restore$CAPM$ equilibrium.$ • The$expected$return–beta$ rela8onship$can$be$derived$ without$using$the$true$market$ • CAPM$describes$equilibrium$for$ porDolio.$ all$assets.$ 10-23 10-20 APT Model: APT applies to well diversified portfolios and not necessarily to individual stocks. With APT it is possible for some individual stocks to be mispriced - not lie on the SML. APT can be extended to multifactor models. 142 Example( Example( Table&1& Table&1& Factor& Ifla+on( Industrial(Produc+on( Oil(Prices( Factor&Beta& Factor&Risk&Premium& 1.2( 6%( 0.5( 8%( 0.3( 3%( Table&2& Factor& Ifla+on( Industrial(Produc+on( Oil(Prices( QuesAon:& Expected&Rate&of&Change&Actual&Rate&of&Change& 5%( 4%( 3%( 6%( 2%( 0%( Suppose(that(the(market(expected(the(values(for(the(3(macro(factors(given( in(column(1(of(table(2,(but(the(actual(values(turn(out(as(given(in(column(2(of( the(same(table.(Calculate(the(revised(expected(returns(on(the(stock(once(the( surprises(become(known( Factor& Ifla+on( Industrial(Produc+on( Oil(Prices( Factor&Beta& Factor&Risk&Premium& 1.2( 0.5( 0.3( 6%( 8%( 3%( Actual&Rate&of&Change& 5%( 3%( 2%( 4%( 6%( 0%( Table&2& Factor& Ifla+on( Industrial(Produc+on( Oil(Prices( Expected&Rate&of&Change& SoluAon& E(r)(=(6%(+((1.2(×(6%)(+((0.5(×(8%)(+((0.3(×(3%)(=(18.1%( ( Surprises&in&the&macroeconomic&factors&will&result&in&surprises&in&the&return& of&the&stock:& Unexpected(return(from(macro(factors(=(( [1.2(×((4%(–(5%)](+([0.5(×((6%(–(3%)](+([0.3(×((0%(–(2%)](=(–0.3%( ( E((r)(=18.1%(−(0.3%(=(17.8%( ( Beta * risk premium: Explains Return Beta * (Actual – Expected): Explains Risk 143 Where%Should%We%Look%for%Factors?% • Need%important%systema:c%risk%factors% – Chen,%Roll,%and%Ross%used%industrial%produc:on,%expected% infla:on,%unan:cipated%infla:on,%excess%return%on% corporate%bonds,%and%excess%return%on%government%bonds.% – Fama%and%French%used%firm%characteris:cs%that%proxy%for% systema:c%risk%factors.% Fama$French*Three$Factor*Model* • SMB*=*Small*Minus*Big*(firm*size)* • HML*=*High*Minus*Low*(book$to$market*raAo)* • Are*these*firm*characterisAcs*correlated*with*actual* (but*currently*unknown)*systemaAc*risk*factors?* rit = α i + βiM RMt + βiSMB SMBt + βiHML HMLt + eit 144 Chapter 17 Financial Risk Management Risk v.s. Return Higher risk <=> higher returns Section 1 Introduction Financial Risk Management: Market risk: The practice of creating economic value in a firm by using financial instruments to manage exposure to risk. The risk of losing value in our investment due to changes in market factors: • Firm only adds value when performing activities shareholders couldn't undertake themselves • So risk management only add value when mitigating risks shareholders couldn't mitigate themselves • In perfect markets, hedging of risks has no value as price of bearing risk within firm is same as price of bearing risk outside it • Risks unique to firm are the best candidates for FRM - diversification • Changes in asset prices, interest rates (yield curves, credit spreads), currencies, inflation and commodity prices Measurement: Value at Risk (VaR) Credit risk: Risk arising from failure of counterparty to meet its obligation • Pre-settlement risk: default risk that arises prior to settlement • Settlement risk: default risk that arises at settlement 1. Exposure: in event of default how large will outstanding obligation be? 2. Default probability 3. Recovery rate: if counterparty defaults, how much of obligation can be recovered e.g. Through bankruptcy proceedings 146 Liquify risk: Basel Committee on Banking supervision: Arises from situations in which party wanting to trade in an asset unable to do so because nobody in the market wants to trade that asset • Set international standards for banks to guard against losses as a result of financial and operational risk Asset liquidity risk: risk arising from the inability to liquidate assets at a reasonable price • Value at Risk: measures how much is an entity at risk of losing with a given probability level ad over a specific time horizon Funding liquidity risk: risk arising from the inability to raise cash to meet obligations at a reasonable cost • Expected shortfall: alternative method that Basel is currently trying to implement - given that an entity’s loss exceeds a certain threshold, what amount is it most likely to lose? Operational risk: Normal distribution: Risks arising from current or future losses due to failed or inadequate people, processes or systems or external events Problems: Failed/ inadequate people: employee fraud or collusion, mistakes, strikes, lack of knowledge or expertise, death and disability Failed/ inadequate processes: accounting error, project risk, transaction error • Allows any value, even negative, but share prices cannot be negative • Does not account for compounding So use Lognormal Failed/ inadequate systems: data quality, security breaches, system failure, system capacity constraints External events: government action, natural disaster 147 Lognormal distribution: Example: Describes random variable that grows every instant by a rate that is itself a normal random variable. Hypothetical returns on Telkom SA Limited given in table below, together with the simple returns and continuously compounded returns: Progression of lognormal random variable reflects continuous compounding. Day Price 100.0 102.5 107.0 103.0 105.0 1 2 3 4 5 Simple Return Log Return 2.50% 4.39% -3.74% 1.94% 2.47% 4.30% -3.81% 1.92% Whole Week 5.00% 4.88% Calculate the simple return over the last day as: Sum of Returns 5.09% 4.88% 102.50-100 = 2.5% 100 Notice from table, sum of log-returns is same as log returns over whole week (calculated using the prices from day 1 and day 5). Example: Suppose stock price of Telkom SA Limited yesterday was R100 per share, and share price today is R102.50. ( ( ( 100e0.0247+0.043-0.0381+0.0192 = 105 Log-return is rate r that satisfies: Same does not apply to simple returns. er = 1.025 i.e. r = ln(1.025) = 2.47% This property is convenient because we assume that share returns are normally distributed. Sum of normal is still normal Difference is usually small, but lognormal return has important properties. 148 Suppose future daily log-returns on ABC Co.’s shares are independent of each other, and normally distributed with mean of 1% and a standard deviation of 2% Lets say tomorrow’s return is r1 and the next day’s return is r2, because of property of log-return, we know that two-day return r is just r1 + r2. We know that r is also normally distributed with a mean of 2(0.01) and a variance of 0.022 + 0.022 = 2(0.02)2. The standard deviation of the two day return r is √2 x (0.02). General rule: If log-returns over next t days are all independent and identically normally distributed with mean of μ and standard deviation σ, then t-day log-return is normally distributed with mean tμ and deviation √t.σ 149 Section 2 Value at Risk Value at risk (VaR) • Measure of how the market value of an asset or of a portfolio of assets likely to decrease over a certain time period under usual conditions. The pre-specified probability: Level of certainty that we attach to expected loss figure. Very important to realize with VaR there is always (small) chance real loss could exceed VaR. Pre-specified probability can also be stated in terms of a confidence level, which is just one minus probability. Or more formally: • The maximum loss over a target horizon, such that there is a low, pre-specified probability that the actual loss will be larger. Example: Investment bank holding portfolio reports that it has 1-day VaR of R2 million at a 95% confidence level. VaR therefore asks: “What is the worst case scenario?” The maximum loss figure: This is the VaR figure (and is reported as positive number). VaR is described as measure of downside risk. Provided that normal conditions prevail, this implies bank can expect that, with a probability of 95%, change in value of its portfolio would not result in a decrease of more than R2 million over 1 day ( ( ( ( ( OR with probability of 5%, value of its portfolio will decrease by (lose) R2 million or more over 1 day. The target horizon: This is time period over which loss is being measured. VaR is most often calculated over short time periods (1- week or 1-day). 150 Common VaR calculation models: Delta-Normal method: • Assumes that underlying risk factors have normal distribution. • Variance-covariance also known as Delta-Normal: assumes risk factor returns always (jointly) normally distributed and that change in portfolio value are linearly dependent on all risk factor returns • Historical simulation: assumes asset returns in future will have same distribution as they had in past (historical market data) • Monte Carlo simulation: future asset returns more or less randomly simulated to derive approximation of distribution • Uses these assumptions to determine distribution of underlying portfolio profits and losses, which is also normal. • Once distribution obtained, we use standard mathematical properties of normal distribution to determine loss that will be equalled or exceed x percent of the time, i.e. Value at Risk. • Method typically easy to calculate due to simplifying assumptions. Drawbacks: All models calculate VaR using same formula – difference lies in how distribution of investment returns derived. Assumption that portfolio value linearly related to risk factors • Can be dangerous assumption when portfolio contains instruments with non-linear exposure to risk factors. E.g. Options. Assumption of a normal distribution of asset returns • Unfortunately, there is large body of empirical evidence that suggests that this assumption is not realistic. 151 Historical Simulation method: Monte Carlo Simulation method: • Calculated by taking the current portfolio and subjecting it to actual changes in market over past period. • Rather than simulating observed risk factor changes like Historical simulation, we choose distribution believed to adequately approximate possible changes in risk factors. • Results indicates what our losses and gains would have been if our current portfolio was held in past periods instead. • We make the assumption that this distribution will repeat itself in future. • From this distribution’s mean and standard deviation we can then compute VaR. • Then use random number generator to generate thousands of hypothetical changes in risk factors. • Use these to construct thousands of hypothetical profits and losses on current portfolio and, from this, distribution of profits and losses (law of large numbers). • Use this distribution to calculate VaR. Benefits: • Simple to implement (preferred method by majority of banks) • Does not assume normal distribution of returns (more realistic) • Able to approximate options and option-like instruments without difficulty 1. Choose risk factors that will affect instrument/situation 2. Assign distributions/values to each factor 3. Create/Determine the ‘function’ that relates the risk factors to the instrument/situation 4. Generate random numbers from 0 to 1 for each of the factors Drawbacks: 5. Use RN’s and the CDF’s of each factor, obtain a value for each factors. • Requires large data set to compute 6. Calculate function value for that iteration • Can be computationally intensive 7. Go back to step 4 and repeat process n times • Relies on assumption that past distributions will be repeated 8. Average all the simulated path values 152 Benefits: • Does not assume normal distribution of returns • Able to approximate options and option-like instruments without difficulty Drawbacks: • Complex method requiring substantial computational power. • Hence, not preferred by most financial institutes. • Selecting appropriate distribution and relevant parameters requires high degree of expertise and judgement. ! Simulating Quarterly Profits (look @ Excel sheet) ! Assign Distributions to Profit Factors: " " " ! Quantity (Q) Variable Costs per unit (VC) Price per unit (P) ~ Uniform(8000, 12000) ~ Normal (7,2) ~ Normal(10,3) Fixed Costs are R5000 Profit Function: 153 Section 3 The Delta-Normal Method Step 1: Mapping the Exposures First step involves mapping exposures to risk factors using sensitivities to obtain approximate linear relationship between risk factors and changes in value of portfolio. It is the combination of these three components that determine how much we can lose on our portfolio. The Risk factor: this is primary source of uncertainty. Examples of risk factors are stock prices, credit spreads, exchange rates, etc. The Exposure: This is how much exposure we have to particular risk factors. For a well-diversified portfolio of stocks, use beta of portfolio to obtain relationship between changes in value of portfolio P and changes in level of stock market M : ∆P = β ∆M We are treating each rand invested in portfolio as β Rands invested directly in market. E.g. If we had a portfolio worth R1 million with a beta of 1.25 we would treat it for purposes of calculating VaR as R1.25 million exposed to the market. The Sensitivity: This is ‘link’ between risk factor and exposure, and tells us the extent to which any unit change in risk factor will affect our portfolio (stronger if more sensitive). 154 Step 2: Obtain distribution of risk factors For purposes of simplicity, we will always assume that changes in risk factors (stock prices, indices, bond yields etc.) are normally distributed. Will therefore need to know the mean and standard deviations of the risk factors, as well as correlations between them when more than one risk factor present. Sometimes, you may be given distribution of daily returns and have to calculate distribution of weekly or monthly returns using square-root of time rule. Step 3: Obtain distribution of changes in value of position Derive using variance and covariance rules. Step 4: Get the VaR statement Once we have calculated mean µ∆V and standard deviation σ∆V, (1-x)% VaR is given by: VaR = µ∆V + σ∆V αx% Always check that distribution of risk factors corresponds with horizon over which VaR is being calculated. Example: Need to calculate VaR for 1-day period but mean and standard deviation of the distribution given in months. Need to convert relevant variables to 1-day period so horizons match. If log-returns over next t days are all independent, and identically normally distributed with mean of µ and standard deviation σ, then the t -day log-return is normally distributed with mean tµ and standard deviation √t. Σ 155 Basic example: Suppose portfolio manager manages a portfolio which consists of a single asset. The return of the asset is normally distributed with annual mean return 10% and annual standard deviation 30%. The value of the portfolio today is R100 million. With 1% probability what is the maximum loss at the end of the year? VaR = µ∆V + σ∆V α1% ( = (100 x 0.1) + (100 x 0.3) x – 2.326 ( = -59.78 Thus, annual 99% confidence VaR for our portfolio is R59.78 million. Note: have defined αx% to be negative, so VaR (being a loss figure) will typically be negative. We often report the VaR number as a positive number, so don’t get confused. 156 Section 4 Multiple Exposures The Square root of time rule: Q: Assume that the returns on portfolio are normally distributed with a mean monthly return of 3% and standard deviation of 12%. You wish to calculate the 5% annual VaR for this portfolio. What is the adjustment that needs to be made to the variables in order to match time periods? More frequently, however, we need to convert from annual or monthly values to daily values to compute VaR. Q: Assume we have a normally distributed asset with an annual mean of 22% and standard deviation of 32%. We want to calculate the 1-week VaR so we need to convert as follows: If we assume A:( 1 1-week mean = 22 x 52 Stationarity: The distribution of the portfolio changes will remain the same for the next 12-months. We can then get ( 1-week standard deviation = = 0.423% 1 × 32 52 = 4.44% ( = 12 x 3% = 36% Independence: The changes in the value of the portfolio over each month are independent of the other months. = 12 ×12 = 41.6% 157 Multiple Exposures: What is the 5% one-month VaR of your combined position? Assume we have $110 million invested in a well-diversified portfolio of equities. Returns on our equities are perfectly correlated with the S&P-500 index, and the portfolio has beta of 1. In addition, the portfolio pays a fixed dividend of 1.4% per year. We also have a short position of 200 (hypothetical) S&P-500 Index futures contracts which mature in 6 months. We have computed the value of the position to be $55.643 million. Our first exposure is clearly a $110m well-diversified portfolio. Lastly, we have a long position of 500 FTSE-100 Index futures contracts invested in the UK market with a current value of ₤29.696 million. The expected monthly return on the S&P-500 is 1% and that on the FTSE-100 is 1.25%. The monthly standard deviation on the S&P-500 is 6.1% and that of FTSE-100 is 6.5%. The correlation between the two indices is 0.55 and the current exchange rate is 1.6271 $/₤. The portfolio has a beta of 1 and is perfectly correlated with the S&P 500 index so we treat the investment exposure as $110m in the S&P 500. Keep in mind at this stage that the portfolio pays dividends. Our second position is a short position in the S&P-500, worth $55.643 million. Given that both of these initial investments are in the same underlying asset (S&P-500) we can net them off against each other: $110m – $55.643m = $54.357m net exposure in the S&P500 Lastly, we have a long position in FTSE-100 futures.. We will need to bring these two exposures together to compute a VaR for our combined position, however, so we need to convert the FTSE-100 exposure to dollars before proceeding: Our FTSE-100 position is worth ₤29.696m which converts at exchange rate to $48.319m. (exch rate = 1.62712) 158 We have been told that the returns are normally distributed and the mean and standard deviations have been given. The only factor left to consider is the dividends payable on the portfolio. As the dividends are certain this will influence the final expected portfolio value. µΔV( = 54.357 x (0.01) + 48.319 x (0.0125) + 0.128 ( = $1.2759m σ2ΔV= (54.3572 x 0.0612) + (48.3192 x 0.0652) + (2 x 54.357 x 48.319 x 0.061 x 0.065 x 0.55) ( = 32.3138 Our monthly dividend is therefore: 0.014 * 110 = 0.128m 12 σΔV= $5.6845m Our VaR is therefore: The change in our value is therefore based on changes in each of the underlying positions plus the dividend which is received with certainty: where X1 is the exposure to the S&P 500 and X2 is the exposure to the FTSE-100. From this, our mean and deviation can be expressed in dollar terms (millions) as follows: VaR = (1.2759 - 1.645*5.6845) = -8.0751 So, the 5% 1-month VaR of the portfolio is $8.0751m Concept Check: If we were to calculate the VaR for our exposures separately, they would not add up to the answer above. Why? Correlation between assets needs to be considered. 159 VaR in terms of relative losses: Up until now looked at VaR as absolute loss number expressed in monetary terms. Therefore instead of ΔV we use ΔV:( ( ( ( V ( ( ( ( X ΔV X 1 0.128 = (Δ ALSI ) + 2 (Δ S&P ) + V V V V When comparing investments of different sizes, however, how do we determine whether the difference is significant or not? Example: A VaR of $8.0751m on a $50m portfolio is a lot riskier than the same VaR on a $100m portfolio. Computing Relative VaR Instead of calculating the absolute loss, we can use the relative loss to ensure a more accurate comparison. For our previous example, we could have found the relative VaR by: VaR = (0.01243 - 1.645*0.05536) = -7.864% 8.0751/102.68 = 7.864% of portfolio value. Alternatively, we can work with relative losses from our initial calculations so that the final answer is expressed as a relative figure. 160 Risk Decomposition: When holding multiple positions we might be interested in decomposing our combined VaR to find the contributions from each individual position. E.g. For our earlier example, how much VaR stems from portfolio, how much from the short position in S&P futures and how much from the long position in FTSE futures? The Final VaR of the whole portfolio was calculated as $8.0751m. The FTSE position has therefore increased the risk from $4.7825m to $8.0751m. Initial VaR -$9.8096 Final VaR -$ 8.0751 Intermediate VaR -$ 4.7825 Impact of S&P Futures Impact of FTSE Futures Calculating VaR for just the portfolio we would find a value of $9.8096m. (Remember Dividends!) We could then calculate the VaR of the combined position of the portfolio and short position in the S&P 500 futures. (Remember Dividends!) While intuitive, the biggest problem with this approach is that the risk contributions of different parts of the complete portfolio will depend on the order in which the positions are listed. Concept check: Why does the above statement hold? Self Assessment: Re-order exposures & re-calculate This would yield a VaR of $4.7825m. The addition of the short position has therefore reduced VaR by $5.0271m A short $55.643m position index futures on S&P 500 A well diversified $110m portfolio of American shares A long ₤29.696 position index futures on FTSE-100 161 Summary: ! Calculating VaR for share portfolio only: " " ! Exposure of $110m to S&P 500 (Remember Dividends!) Calculate $ Mean and Std Dev from S&P distribution Calculate VaR of the combined position of the share portfolio and short pos. in S&P 500 futures. " " Used combined exposure: $110m – $55.643m = $54.357m Again, only looking at S&P distribution (Remember div’s!) Marginal risk decomposition: Found that incremental risk decomposition suffers from problem related to order in which investments added to overall portfolio. Can avoid problems with incremental risk decomposition by looking at how much risk changes if we increase size of exposure by one unit. E.g. What would impact be on VaR if we increased holdings in FTSE by $1? This will give us marginal risk decomposition. Calculating VaR for just the portfolio we would find a value of $9.8096m. (Remember Dividends!) Calculating the VaR of the combined position of the portfolio and short position in the S&P 500 futures would yield a VaR of $4.7825m. (Remember Dividends!) The addition of the short position has therefore reduced VaR by $5.027m ! First, we write VaR in terms of portfolio elements. ! Let " " XC1 = S&P equity portfolio XF1 = Short futures position in the S&P 500 Since we aggregated them in our previous discussion, we’ll also indicate the net S&P position as X1 ( = XC1 + XF1). " X2 = Long FTSE futures position 162 The 95% VaR can be written as follows: Thus to find the risk contribution of each investment we find the partial derivatives with respect to each element: A convenient property of VaR is that it can be written as: Notice that the risk contributions sum to the total VaR. If we increase the amount of the FTSE portfolio from X2 to X*2, the amount of risk in the portfolio changes by: If we multiply top and bottom by X2 we find: So: = (0.01117 ) − 1.645 × 54.357 × 0.0612 + 48.319 × 0.55 × 0.061 × 0.065 54.357 2 × 0.0612 + 2 × 54.357 × 48.319 × 0.55 × 0.061 × 0.065 + 48.319 2 × 0.065 2 = 0.01117 − 1.645 × 0.202 + 0.1054 10.994 + 11.455 + 9.864 = -0.0778 = the increase in portfolio risk for each percentage increase in the amount of X2 held in the portfolio. 163 Therefore the marginal risk contribution from the original portfolio is: $110 x -0.0778 = -$8.564m (with some rounding error) = (0.0125 ) − 1.645 × 48.319 × 0.065 2 + 54.357 × 0.55 × 0.065 × 0.061 54.357 2 × 0.0612 + 2 × 54.357 × 48.319 × 0.55 × 0.061 × 0.065 + 48.319 2 × 0.065 2 = 0.0125 − 1.645 × And: 0.204 + 0.1185 10.994 + 11.455 + 9.864 = -0.081 Therefore the marginal risk contribution from the original portfolio is: = (0.01) − 1.645 × = 0.01 − 1.645 × 54.357 × 0.0612 + 48.319 × 0.55 × 0.061 × 0.065 54.357 2 × 0.0612 + 2 × 54.357 × 48.319 × 0.55 × 0.061 × 0.065 + 48.319 2 × 0.065 2 $48.319 x -0.081 = -$3.908m (with some rounding error) 0.202 + 0.1054 10.994 + 11.455 + 9.864 = -0.079 Therefore the marginal risk contribution from the original portfolio is: -$55.643 x -0.079 = +$4.397m (with some rounding error) Baring’s Bank: ! Long $7.7b NIKKEI futures ! Short $16b JGB Futures ! σNK = 5.83%, σJGB = 1.18%, ρ = 0.114 ! Var95% = 1.645·σp = $835m ! Var99% = 2.326·σp = $1.18b ! Actual loss was $1.3b – exceeding both the 95% and 99% VaR 164 Example: You are responsible for a portfolio comprising three investments, two of which are held in derivative instruments. Your current equity portfolio investment of R45m comprises a position in the Satrix Top 40 ETF. In addition, you hold a long position in 150 index futures contracts on the Top 40 Index with each futures contract having an exposure of R4371. You have also attempted to diversify your risk and to this end have sold 200 put options on the S&P 500 Index. Each option contract has an exposure of $2062 to the S&P 500 Index. The contract multiplier for both the options and futures contracts is 10. The expected monthly rate of return on the Top 40 is 1.1% and that of the S&P 500 is 1.25%. The standard deviation of the monthly returns of the Top 40 index is 4.45%, and that of the S&P 500 is 5.85%. The correlation between the monthly rates of return is estimated to be r = 0.55. Ignore currency risk. x% αx% 0.01% -3.719 1% -2.326 2.50% -1.96 5% -1.645 10% -1.282 25% -0.674 50% 0 Solution: First determine the exposures: R45m invested in Satrix Top 40 ETF. Since the ETF overall return is the same as the Top 40’s, can treat as a direct position in the Top 40 index Long position in 150 Top 40 index futures contracts: Given that exposure per contract is R4371 and contract multiplier is 10 This means that Top 40 futures exposure is: a) Assuming the exchange rate is 7.15 R/$, what is your 1-month 95% VaR? b) Calculate the marginal risk contribution of each of your exposures. 150 x 10 x 4371 = R6 556 500 Net Exposure to Top 40 is thus: R45m + R6 556 500 = R51 556 500 165 Short position on 200 S&P Put options: VaR = R0.936m – 1.645 x R3.549m = -R4.902m Given that exposure per option is $2062 and contract multiplier is 10 Therefore the 1-month 95% VaR is R4.902m. This means that S&P 500 exposure is: 200 x 10 x 2062 = $4 124 000 Marginal Risk Decomposition for 3 Factors: Satrix Top 40 ETF Long Top 40 Index futures Convert to Rand terms : Short S&P 500 Index futures $4 124 000 x 7.15 = R29 486 600 Now determine Portfolio Mean and Std Deviation: µ∆V( = (51.557 x 0.011) + (29.487 x 0.0125) All Subscripts “1” represents Top 40 ( = 567 121.5 + 368 582.5 All Subscripts “2” represents S&P 500 ( = R935 704 σ2∆V (= (51.557 x 0.0445)2 + (29.487 x 0.0585)2 + (2 x 51.557 x 29.487 x 0.0445 x 0.0585 x 0.55) ( If we consider the Satrix Fund exposure and the Top 40 Futures exposure i.t.o the previous derivative formula, we can see that we will get the same marginal risk sensitivity rate for both exposures respectively. = R12.592m σ∆V = R3.549m Why? The input values used in the equation are identical! 166 Thus, only decompose risk to two factors, namely: ! • Top 40 Net exposure = 0.0125 – 1.645 x (29.487 x 0.05852) + (51.557 x 0.0445 x 0.0585 x 0.55) 3.549 = -0.0685 • S&P 500 exposure ! ! For the Top 40, the marginal risk contribution is: = 0.011 – 1.645 x (51.557 x 0.04452) + (29.487 x 0.0445 x 0.0585 x 0.55) 3.549 = -0.0559 ! For the S&P 500, the marginal risk contribution is: Therefore the rand risk contribution is: 0.0685 x R29.487 = R2.020m ! Final Check: Sum of Marginal Values = Total VaR R2.882m + R2.020m = R4.902 ! Therefore the rand risk contribution is: 0.0559 x R51.557 = R2.882m 167