CFA Level 1 - Book 1
*Effective annual rate* - actually being earned after adjustments made for diff compounding period. Formula? Compute the EAR if the stated rate is 12% compounded quarterly.
(1+ rate / m) ^m - 1 (1 + .12 / 4) ^ 4 - 1 = 12.55%
Code of Ethics - 6 pillars
*A PUPPI* ■ *Act* with integrity, competence, diligence, respect, and in an ethical manner with the public, clients, prospective clients, employers, employees, colleagues in the investment profession, and other participants in the global capital markets. ■ *Place* the integrity of the investment profession and the interests of clients above their own personal interests. ■ *Use* reasonable care and exercise independent professional judgment when conducting investment analysis, making investment recommendations, taking investment actions, and engaging in other professional activities. ■ *Practice* and encourage others to practice in a professional and ethical manner that will reflect credit on themselves and the profession. ■ *Promote* the integrity of and uphold the rules governing capital markets. ■ Maintain and *improve* their professional competence and strive to maintain and improve the competence of other investment professionals.
*P*rofessionalism.
*KIMM*el a. Knowledge of the Law / Code. Members understand and comply with all laws - KVIOA i. Know laws in *all countries* in which they trade securities. ii. Responsible for violations in which they *knowingly* participate. iii. If *illegal* actions of client, customer or personal Code Violation - Must dissociate. iv. If violation of Code by fellow member - strongly encourage to report. No requirement to report unless law. v. When going with the 'applicable law' - Code vs country of residence vs. country of business... go with the strictest! But if 'law of locality' is stated, then go with that. (Ie. Member resides in MS (more strict than Code) country, does business in LS (less strict than Code) country; MS law applies, but can state that law of locality where business is conducted governs.) Member must adhere to the Code and Standards. Not the MS laws. b. Independence and Objectivity. (GSBPT) i. Modest gifts and entertainment are acceptable. Must be disclosed. If from client, must be disclosed to employer (if contingent of future perform.. see ACA need consent too) ii. Sell side analysts. Need effective firewalls bw investment banking. No buy side influence allowed (if they have position in the security and try to convince you to issue favorable report). iii. Fund Manager Relationships (Secondary Fund Managers giving gifts for hiring, etc) iv. Public Companies - can't engage in retaliatory practices by restricting access to managers / conference calls, etc... v. Travel Funding - use commercial travel options when talking to executives, unless have to use charter c. Misrepresentation. i. Investing through outside managers - must disclose their intended use of external managers and must not represent those managers' investment practices as their own. ii. Guarantees are misleading bc there is some risk. (unless in the structure and losses covered - ie CDs). iii. No plagiarism - ****acknowledge the author of a model, but members are not required to acknowledge information from a recognized statistical and reporting service. iv. Completed for Employer - using the work of others within the same firm is permitted, even if the analyst has left the firm - property of the firm. The analyst cannot re-release it as his own. d. Misconduct. i. disclose bankruptcy that involves fraudulent or deceitful business conduct
Integrity of Capital Markets
*M&M* a. Material Nonpublic Information - illegal - all rest is good. Have to wait a little bit until trading. **** ONLY APPLIES TO NON-PUBLIC INFO ABOUT THE ISSUER (ie analyst that is influential can trade on own name). **** failing to prevent transfer of info is a violation. i. Mosiac Theory - analysts get their information from a wide variety of sources - public / non-public. b. Market Manipulation i. No Spreading false rumors. No non-fact based report during quiet period. ii. Transaction-Based Manipulation. Trading a position between intercompany positions to create the illusion of a price rise.
Standards of Professional Conduct
*PC DESIRE* - *P*rofessionalism. - Integrity of *C*apital Markets. - *D*uties to Clients. - Duties to *E*mployers. - Investment Analysi*S*, Recommendations, and Action. - Conflicts of *I*nterest. - *Re*sponsibilities as a CFA Institute Member or CFA Candidate.
Duties to Clients
*PRoSPECt* a. Loyalty, *P*rudence, and Care. i. Follow IMAs / Govering Docs. COST BENEFIT FOR VOTING PROXIES. ALSO VOTE IN BEST INTEREST FOR BENEFICIARIES NOT SPONSOR ii. Recommend submitting at least quarterly statement. iii. Disclose conflicts of interest - ie you use a broker that pays your utility bill in exchange for using them. b. Fai*R* Dealing i. Not necessary equal treatment of clients but should not advantage / disadvantage any clients against another client. Make premium levels of service available. MUST DISTRIBUTE 'HOT' SHARES ON A WEIGHTED / PRO RATA BASIS. ii. Investment Recommendations - Advise clients of changes in recommendations. Shorten timeframe from decision to dissemination. IF ORDER COMES IN INFORM THAT CHANGE MADE EVEN IF IT IS BEFORE OTHERS.. IE MAILED OUT. iii. No participation in OVERSUBSCRIBED equity IPO c. *S*uitability i. Reasonable inquiry into client's investment experience. ii. Determine that an investment is suitable to the client's financial situation / context of the client's total portfolio. UPDATE INFO REGULARLY. IF EXECUTING UNSUITABLE INV. FOR CLIENT, GET AFFIRMATIVE STATEMENT iii. Management capacity - make sure investments made in line with objectives / constraints. d. *Pe*rformance Presentation - *Past misleading gross terminator eliminated* i. No misleading *performance* indicators. Should not indicate the ability to achieve a ROR similar to that achieved in the *past*. Brief presentations - indicate that the presentations have offered *limited* information. ii. Include all *terminated* accounts in performance. Include disclosures (*gross of fee*s). e. *C*onfidentiality: Unless i. Illegal ii. Disclosure required iii. Disclosure permitted
GIPS? Requirement? What is a composite? VERIFICATION (of GIPS compliance)? Key Points: - Min years of initial compliant performance presentation? After this? - Country Version of GIPS? - Nine major sections of GIPS? FICC DP RPW Other?
-Global Investment Performance Standards -No, just voluntary for investment firms to follow. -Grouping of individual portfolios representing a similar investment strategy. IE "Large cap growth stocks".. includes ALL portfolios (current and past) that followed that strategy. Verification - OPTIONAL, but if they pursue it...: - Must be third party that attests: -- 1) firm has complied with all GIPS requirements for composite construction for ENTIRE firm --2) Procedures established to present performance that is methodology, data, and format-compliant with GIPS - 5 years or since inception; include an additional year for every year until reach 10 years. - CVG can be used and say "GIPS" compliant before 2006. If there was a conflict, must disclose. 0. Fundamental of compliance (statement used, definition of firm, etc..). 1. Input data 2. Calc Methodology 3. Composite construction - use correct classification and asset weighted averages. 4. Disclosures 5. Presentation 6. Real estate 7. Private equity 8. Wrap fee/SMA (Separately Managed Accounts) - No partial compliance, - No statements mentioning "in compliance with GIPS." Only statement used is "_____ has prepared and presented this report in compliance with GIPS" - All prospects must receive - Composite list must be presented on request -- including discontinued ones for 5 years. - Total firm assets must include all account including non-discretionary and non-fee paying. - Only firms, not individuals. CAN SEPARATE FIRM THAT HOLDS ITSELF TO BE A DISTICT BUSINESS ENTITY FOR COMPLIANCE!
Different charts for technical analysis
1. Line charts of closing price 2. Bar charts (High / Low --- Closing price, dash on right, opening price, dash on left) 3. Candlestick charts (same as bar charts but have boxes with high low, if closing price higher than opening price, then box clear, if closing price lower than opening, then filled). 4. Point and Figure Chart. (Drawn on graph paper, horizontal axis is not time, but number of changes in direction. Analyst must decide "reversal size" for the chart -- usually 3 times the box size. X's represent increase of 1 box size. O's decrease of 1 box size.)
Hypothesis testing two means: Frequently want to know if there is a difference bw two population means. There are two t-tests for this: -Application of both of these tests requires 3 facts? -Appropriate statistic for both? -Describe the two types. Frequently want to know if there is a difference bw two population means (as above) BUT DEPENDENT ON EACH OTHER (other 2 facts same)? Example? Uses same methodology as prior techniques, just take the difference of each observation to calc mean and std deviation.
1. Samples are independent 2. 2 populations are normally distributed 3. Assumes population variances are unknown Always t-statistic. 1. Assumes unknown variances are equal. (Pooled variance) 2. Assumes unknown variances assumed not equal. Called "paired comparisons" test. Just a test of whether the average difference between returns is significantly different from zero, based on the standard error. For example, comparison of returns of two steel firms equal over a a 5 year period? Not independent samples, returns dependent on market / industry risk.
Desirable properties of an estimator (3 + explain)
1. unbiased estimator - one where expected value of the estimator is equal to the parameter (E(sample mean) = µ) 2. efficient estimator - one where variance of the sampling distribution is smaller than other unbiased estimators of parameter. 3. consistent estimator- where accuracy increases w sample size
State probability of event in terms of "odds" o If probability of event is 1/8? Unconditional probability vs Conditional probability? Independent vs. dependent events?
1/8 = .125. then the odds that the event WILL occur is 1/8 / 7/8 = "the odds for the event occurring are one-to-seven" Unconditional probability (marginal probability) - probability of an event regardless of past of future occurrence of other events. Probability of an economic recession, regardless of the changes in interest rates or inflation P(A) Conditional probability - occurrence of one event affects the probability of the occurrence of another event. The probability of an economic recession, given changes in interest rates. P(A|B) - notation used, probability of A, given B. Events A and B are independent only if P(A|B) = P(A) and P(B|A) = P(B) (Dice / coin flips have 'no memory' and are thus independent events.)
Larger is not always better: biases, costs, etc. 5 main biases (describe)
ASSauLT Dat*A* mining bias - using the same DB to search for patterns until one that 'works' is discovered. Ex: Value stocks outperform growth stocks. Data on historical stock returns in limited and difficult to parse this out. *S*ample selection bias -- some data excluded from analysis, usually bc of lack of availability. *S*urvivorship bias - most common --- study of mutual fund performance, most databases only include funds in existences and thus exclude closed funds and merged funds. *L*ook ahead bias -- study tests a relationship using sample data that wasn't available at the test date. Ex: trading strategy based on EY book value -- not available until the audit. *T*ime period bias -- too long or short. Don't make a 2 quarter return a trend, also don't include stock returns from 1920.
Absolute vs. Relative vs. Cumulative frequencies? Histogram vs. Frequency polygon?
Absolute -- Total number of observations Relative -- % of total absolute frequency Cumulative. --- sum of total relative frequency thus far. -Histogram - graphical representation in bar graph of absolute frequency distribution (vertical) and intervals (horizontal) -Frequency polygon - same as histogram but: ---the INTERVAL MIDPOINT is plotted on the horizontal axis ---not a bar graph but a line graph.
Money-weighted returns Define? Ex: Buy a purchase of stock for $100 today, buy an additional share in a year for $120. End of each year, Dividend of $2 for each share. Sells for $130 / share at the end of year 2. What is the money-weighted rate of return? Time-weighted rate of return (or _________) Define? Three steps? Ex: Same as above but for TWR.
Applies concept of IRR to investment portfolios. BV is considered a cash inflow, All withdrawals / EV is considered a cash outflow. o 0----------1-----------2---------3 o 100 120 o - -2 -4 o - -260 o ________________________ o 100 118 -264 o o NOW SOLVE FOR THE IRR --- Same as before o IRR = 13.86% The geometric mean return. This is preferred method in the investment management industry of performance measurement because it is not affected by the timing of cash inflows and outflows. Step 1: Value portfolio immediately preceding significant additions / withdrawals. Form subperiods that correspond to these dates Step 2: Compute the holding period return (HPR) for each subperiod Step 3: Compute the product of (1+HPR) for each subperiod to obtain a total return for the TOTAL measurement period. Step 1 Period 1: BV = 100, Dividends paid = 2, EV = 120 Period 2: BV = 240, Dividends paid = 4, EV = 260. Step 2 HPR for 1: (120+2) / 100 - 1 = 22% HPR for 2: (260+4) / 240 - 1 = 10% Step 3 (1 + time-weighted rate of return)^2 = (1.22)(1.10) Time-weighted RoR = (((1.22)(1.10))^1/2) - 1 15.84%
Measures of central tendency. Define? Mode - value ______. Data set may have more than one mode. If one mode, the set is ______. If two / three modes then ______, ________. Geometric mean - often used when calculating investment returns over multiple periods. Formula? Geo mean is always ________ and geo mean and arithmetic mean ___________________. Harmonic mean (or _____________)- certain situations like ____________. Formula? Ex: Investor purchases $1,000 of stock each month, and over last three months prices paid per share were $8, $9, and $10. What is the average cost per share for the shares?
Average of a data set. Can be Mean, sample mean, median, etc... that occurs most frequently; unimodal; bimodal, trimodal G = (X1 x X2 x X3...)^(1/n) (If geo mean of returns then necessary to add 1 to the returns.) less than or equal to the arithmetic mean; always different unless all observations equal. dollar cost averaging avg cost shares over time N / (∑ 1/ Xi) • 3 / (1/8 + 1/9 + 1/10) = $8.926 / share • CAN CHECK THIS MATH: Arithmetic mean on total shares purchased. o 1000 / 8 + 1000 / 9 + 1000 / 10 = 336.11 shares o 3000 / 336.11 shares = 8.926 per share
Calculating NPV in the Calculator. Calculating IRR in the Calculator.
CF, 2nd, CLR WORK --- Clears memory XX, Enter - Initial cash outlay. *down*, XX, Enter -- Period 1 CF *down*, *down*, XX, Enter - Period 2 CF. Continue as needed until done [NPY], YY, Enter --- Discount rate *down*, CPT --- Calculate NPV. Same as before except second to last step IRR, CPT - calculates the IRR.
Conflicts of Interest
CONTRE a. Disclosure of Conflicts i. Must ensure that such disclosures are prominent, are delivered in plain language and communicate the relevant information. ii. Typically Broker/Dealer, board service, disclosure of ownership of actual stock. Also special compensation arrangements. INCLUDES FRIENDS / ANY PRINCIPAL in FIRM. iii. Disclose all matters that COULD potentially impair Independence. b. Priority of Transactions -- Clients over personal. Must have adequate opportunity. i. Includes situations where the member is a 'beneficial owner' ii. Firms should have blackout periods - to avoid 'frontrunning' iii. RECOMMEND, not require getting supervisor approval before participating in IPO. c. Referral Fees --- must disclose to the customer. i. Members should update clients at least quarterly on nature and value.
Sentiment Indicators? Opinion polls and.... -Put/Call Ratio? -Volatility Index? -Margin debt? (required to be reported by brokers) -Short Interest Ratio? (required to be reported by brokers) Flow of funds and... - Arms index (or ST trading index)? - Mutual Fund Cash Position? - Offerings?
Can be used to discern the views of potential buyers and sellers. -Put/Call ratio is put volume divided by call volume. Increases in this ratio indicate a more negative sentiment. -VIX: Chicago Board of Options measures this -- measures volatility of options on S&P. High levels suggest fear of declines. -Tends to follow trends -- more debt from broker means that more bullish bets / increasing prices. less debt means having to pay margin call and thus lowering prices further. - Increases indicate a strong negative sentiment. Short interest is the number of shares borrowed and sold short, while ratio is this number divided by average daily trading volume. HOWEVER, is bearish but also indicates a future demand for purchases. - # of advancing issues / # of declining issue / Volume of adv. issues / Volume of decl. issues 1 mean even trading, > 1 means majority of volume in declining, < 1 -- majority of volume in increasing. - Ratio of MF cash to total assets. During uptrends, managers want have less cash, etc.. (TAs think contrarian, more cash on hand -- more money to increase stock value in future). - More IPOS / secondary when the stock prices are high so could indicate a peak?
Uptrend? Downtrend? Trendline? If the price crosses trendline in a significant way, if from uptrend, then called a ______ if from downtrend, then called a _______. Trendline is though to represent a level of _____ (buying expected that prevents further price decreases) or _______ (selling expected to prevent further price increases). Explain change in polarity
Consistently reaching higher highs and higher lows. Consistently reaching lower highs and lower lows. Determines whether a trend is continuing or reversing. Ie Uptrend connects increasing lows, downtrend connects the decreasing highs. breakdown; breakout. support; resistance; In some markets there will be a resistance point at a certain round number / historical high, etc. And then once that number is breached significantly, it becomes a support level from the new high.
Investment Analysis, Recommendations and Actions
DCR a. Dilligence and Reasonable Basis i. Independence for making decision / must be thorough and documented. ii. Must be able to explain the basic nature of it, consider alternate negative scenarios. iii. Make sure external advisors have adequate compliance and IC / don't deviate from strategies iv. Group Decisions - can disagree but not remove your name bc there is some basis in the decision. b. Communication i. Disclose the basic format used to analyze ii. Use reasonable judgment - identify the important factors for clients. (RECOMMENDATIONS DON'T NEED ALL RESEARCH, JUST MAINTAIN) iii. Distinguish between fact & opinion iv. REMEMBER: The argument that clients 'won't care' could also be interpreted as 'theres no reason NOT to disclose' c. Record Retention i. Requires GENERALLY that members retain records. Recommends a 7 year holding period ii. A member who changes firms must recreate the analysis documentation supporting the recommendation using publicly available information. Can't rely on memory or prior firm's materials. iii. Generally is the firms responsibility.
Ethics: The ______________ of the CFA BoD has overall the Professional Conduct Program and enforcement of the Code and Standards Several circumstances can prompt an inquiry from the ________________ (through the Professional Conduct staff), such as: (4 things). The Officer may decide one of 3 decides upon investigation.
Disciplinary Review Committee CFA Institute Designated Officer *SWPP* -*Self* Disclosure by members / candidates on their professional Conduct Statements of involvement in a civil / criminal complaint. -*Written* Complaints about a member or candidate's professional conduct received by the staff -Evidence of misconduct received through *public* sources (media) -CFA exam *proctor* report of a violation i) that no disciplinary sanctions are appropriate, ii) to issue a cautionary letter, or iii) to discipline the member or candidate
Null hypothesis / test statistic example. EX: Researcher gathered data on daily returns of call options over 250 days. Mean daily return = 0.1% and the sample st. deviation has been 0.25%. Researcher believes that mean daily portfolio > 0. Construct a hypothesis test at 5% level of sign.
Example: 1. H₀: µ₀ <= 0, Ha: µ₀ > 0 2. test statistic = mean / z-statistic 3. 5% level of sign. = One tailed test would be 1.645 critical z-value. 4. Decision rule is to reject if: test statistic > 1.645 5. Calc 6&7. Test statistic = (0.001 - 0) / ( 0.0025 / √ 250 ) = 6.33 Decision rules: 6.33 > 1.645. REJECT NULL
Hypothesis testing: Differences between sample variances of populations. What new statistic? Application requires 2 facts? Hypothesis? Calc of F-statistic? 2 NOTES? Ex: Researcher believes textile companies' EPS are more divergent than paper companies. Took a sample of 31 textile companies and 41 paper companies. Sample standard deviation for textile was $4.30 and paper $3.80. Earning for textile have greater st. deviation than paper industry?
F-distributed test statistic (F-test) 1. Samples are independent. 2. Normally distributed populations ie (two tailed). H₀: σ²₁ = σ²₂ Ha: σ²₁ ≠ σ²₂ F = s²₁ / s²₂ s²₁ = variance of the sample of n₁ observations drawn from population 1 s²₂ = variance of the sample of n₂ observations drawn from population 2 NOTE: n₁ -1 & n₂ -1 are the DoF used to identify the critical value from F-Table. Shape of F-distribution is dependent on 2 DoFs. NOTE: Always put the larger variance in the numerator above. Only have to consider the critical value for the right hand tail as rejection region 1. H₀: σ²₁ <= σ²₂ Ha: σ²₁ > σ²₂. 1 = textile, 2 = paper. 2. F-test: F = s²₁ / s²₂ 3. LoS: Use 5% as a default 4. Decision rule: Reject if F-test > 1.74 5. F = 4.30² / 3.80² = 1.2805 6. Fail to reject null 7. Earnings variance of the industries is not statistically significantly different from one another.
PV / FV of a single sum Formula? Interest rate per compounding period? Future Value Factor? Present Value Factor? Annuity - stream of equal cash flows that occur over equal intervals. Different bw ordinary and annuity due? Future value of an annuity due - IMPORTANT TO REMEMBER?
FV = PV (1 + I/Y) ^ N (I/Y) ------ interest rate per compounding period (1 + I/Y) ^ N ------- known as the future value factor 1 / (1 + I/Y) ^ N -------- known as the present value factor Ordinary annuities (CF at end of each period) vs. Annuity due (CF at the beginning of each period - including t=0). Convention is at the END of the N years, despite receiving cash flows at the beginning.
Central limit theorem. Important because? n is sufficiently large at ____ Important properties: (3)
For simple random samples of size n from a population with mean µ and variance σ², the sampling distribution of the sample mean approaches µ and a variance equal to σ²/n as the sample size becomes large. Specific inferences can be made about a population mean, regardless of the populations distribution, as long as sufficiently large. n>= 30. 1. n is large, then the sampling distribution of the sample means will be approx. normal 2. µ and mean of of all possible sample means are equal. 3. variance of sample means is pop variance / sample size.
Consider the use of arithmetic vs geometric mean Best times to use each?
Geometric --- appropriate measure for PAST performance. - Compounded returns Arithmetic - best estimator for the NEXT YEARS returns (geometric for MULTI year returns --- forecast over the next 3 years).
Reversal Patterns. Head-and-shoulders pattern? Double top and triple top pattern? (can also be the be the reverse of these for downturns - inverse H&S, double/triple bottom) Continuation patterns? Triangles? Rectangles?
H&S - upturn happens but the demand is fading. Reach a 'neckline' as a support level before it reaches multiple times and then back down to where it was. Double top / triple top -- same concept. just reaching the resistance level multiple times. Pause in trend rather than a reversal. Forms triangles (symmetrical: HL and LHs, ascending: HL and Resistance, or descending: LH and Support). Triangles do not imply a change in direction of the trend. Rectangles indicate a range bw a support level and a resistance level.
Holding period yield (HPY). Formula? Ex: What is the HPY for a T-Bill priced at $98,500 with a FV of $100,000, 120 days from maturity? (D=0 bc T-Bills are pure discount) Effective annual yield (EAY). Formula? And main differences? Ex: Compute the EAY using the 120-day HPY of 1.5228%. Money market yield (or _____________). Formula? Useful for? Ex: What is the MMY for a 120 day T-bill that has a bank discount yield of 4.5%? Bond Equivalent yield. Refers to? Ex: A 3 month yield has a holding period yield of 2%. What is the yield on a bond-equivalent basis?
HPY = ((P1 + D1) / P0) - 1 P0 = Initial price of the instrument P1 = Price received for the instrument at maturity D1 = Interest payment (Distribution) o 100,000 / 98,500 - 1 = 1.5228% Accounts for compound interest and based on 365 day year. EAY = ((1+HPY)^(365/t)) - 1 o EAY = ((1+0.015228)^(365/120)) - 1 o EAY = 4.7042% CD equivalent yield Useful for making the quoted yield on a T-Bill comparable to yield quote for interest bearing, 360-day instruments. rMM = HPY x (360 / t) • rBD = X / Y x 3 • 4.5% / 3 = X / Y = 1.5% • (pick an even number for the HPY) • HPY = 100,000 / 98,500 = 1.5228% • rMM = 1.5228% x 3 = 4.569% Refers to 2 x the semiannual discount rate (coupon interest paid twice a year). First, convert to semi-annual effective rate. 1.02^2 - 1 = 4.04% Second, double the rate 4.04% x 2 = 8.08% = bond-equivalent basis.
Hypothesis testing - 7 steps? What is a null hypothesis? What is an alternate hypothesis? One-tailed vs Two tailed test examples? Two tailed test notation? What is a test statistic? Formula? Decision rule for two tailed test? One tailed?
HSS DSHR • State the *H*ypothesis • Select the appropriate test *S*tatistic • Specify the level of *S*ignificance • State the *D*ecision rules regarding the hypothesis • Collect the *S*ample and calculate the sample statistics • Make a decision regarding the *H*ypothesis • Make a decision based on the *R*esults of the test. The hypothesis that the researcher wants to reject. H₀. Typical statement is H₀: µ ≥ µ₀. µ is the population mean, µ₀ is the hypothesized value of the population mean (always includes 'equal to' notation). What is concluded if the null hypothesis is rejected. Ha One tailed test --- test whether return on stocks is greater than zero. Two tailed test --- test whether return is different from 0 (hypothesized value). H₀: µ = µ₀. or Ha: µ ≠ µ₀ Point estimate of the population parameter with the hypothesized value of the parameter. test statistic = (sample statistic - hypothesized value) / standard error of the sample statistic. Reject the null hypothesis (otherwise 'fail to reject') if the: test statistic > upper critical value or test statistic < lower critical value. One tailed -- same as above but different critical values and only upper or lower test.
Holding period return ? T-Bill purchased for $980 and sold 3 months later for $992. Holding period? Total return ? Stock purchased for $30, sold for $33 6 months later, $.50 dividends. TR?
Holding period return HPR = (Ending Value / Beginning Value) - 1. HPR = 992/980 - 1 = 1.22% Total return - percentage change similar to above --- but if the asset has cash flows associated with it. TR = ((EV + CF received) / Beg Value) - 1 TR = (33+ 0.50) / 30 - 1 = 11.67%
Oscillators: ROC oscillator RSI Index Moving average convergence / divergence Stochastic oscillator
Indicators that tell technical analysts whether market is oversold / overbought. ROC: rate of change: MOMENTUM -- calculated as 100 times the difference between the latest closing price and the closing price n periods earlier. Thus oscillates around 0. Relative Strength Index: based on the ratio of total price increases to total price decreases. Scaled to 0-100, and highest meaning overbought, lowest oversold. MACD: exponentially smoothed moving averages. Greater weight to recent price flux. Stochastic: Calculated from latest closing price and recent high and lows for a period.
Duties to Employers
LiAR a. Loyalty i. From Employers - give all employees handbook and no unethical incentives ii. From Employees - must act in best interest until resignation. Cant keep any firm materials / client lists on home computer etc... However, general names of clients worked with isn't confidential. iii. Independent practice - allowed if employee discloses compensation, duration and nature AND employer agrees before it begins. b. Additional Compensation Arrangements i. Must obtain written consent from employer (email OK). c. Responsibility of Supervisors i. Must take actions to prevent violations from employees violating. Not enough to speak and receive assurances ii. Must make reasonable effort to detect violations iii. If compliance system inadequate, must bring to the attention of management. MUST DECLINE supervisory responsibilities until addressed.
Percentiles: Formula? Ex: What is the third quartile for the following distribution of returns • 8%, 10%, 12%, 13%, 15%, 17%, 17%, 18%, 19%, 23%
Ly = (n+1) y /100 Y = given percentile N = data points Ly = (10+1) 75 / 100 Ly = 8.25 • This means that, when sorted in ascending order, the 3rd quartile is in between the 8th and 9th item in the set and it is weighted 25% toward the 8th item. Therefore, 8th item = 18%, 9th item = 19%. The 3rd quartile is 18.25%.
Why a large difference between two previous TWR and MWR returns? TW RoR = 15.84%, while MW RoR = 13.86%. Preference generally for each method?
MW RoR gave greater weight to year 2 because there was money invested but the return was less (10% vs 22% for yr 1). MW method distorts the portfolio measurement returns. If a lot of money is contributed right before a period of high performance can distort the MW RoR upwards. The time-weighted method removes this distortion. However, if the manager has complete control of the money flows into and out of an account, then MW method likely better.
Confidence interval: 90% confidence? 95% confidence? 99% confidence? Ex: Avg return for mutual fund is 10.5% per year and st deviation is 18%. If returns are approx normal. what is the 95% confidence interval? Z-value -- simlar to above. To standardize an observation from a given normal distribution, the z-value of the observation must be calculated. The z-value represents ___________. Formula? Calculating probabilities using z-Values Refer to ________________. This calculates the ___________. If you have a negative z-value, then just calc using ___________ Ex: EPS w mean of $6, σ = $2. Probability that EPS will be $9.70 or greater?
Mean +/- 1.68s Mean +/- 1.96s Mean +/- 2.58s 10.5% +/- (1.96)(18) -24.78% to 45.78% the number of standard deviations from the pop mean. z = (x - population mean) / standard deviation standardized table. cumulative probability; F(-X) = 1 - F(X) (9.70 - 6) / 2 = 1.85 Look up in table = .9678 1 - .9678 = .0322 3.2%
Moving averages? Usually? In uptrend / downtrend? What are MAs usually viewed as? What are Bollinger bands? Volatility? Price is below lower Bollinger band? above upper?
Mean of last n days. 20 days bc # of trading days in a month. In uptrend -- MA below price, vice versa. Support / resistance levels. Based on st. deviation of closing prices over the last n periods. Usually high / low -- chosen # of deviations over /under the moving average. In periods of high volatility -- wider bands; low, narrower. Below lower: oversold, likely to increase in the near term. Above high: overbought, likely to decrease in near term.
Combinatorics. You can order A OR B. # of combinations? You can order A AND B. # of combinations? SIMPLE: # of ways of arranging n objects, if no restrictions is n! How many ways to arrange 4 people in 4 chairs in a row? REPETITION: see above, but if m members of a group identical then divide total number of arrangements by m!. 7 ppl in race, 1st gets plat, 2nd gold, next 2 silver, next 3 bronze. # of ways to award medal? REPETITION PRT 2. If ANY IDENTICAL MEMBERS HAVE TO COUNT FOR THEM TOO (A, in addition to NOT A). Ex: 8 member Card Club sending 3 people to nationals? How many possibilities? Multiple Groups / Decisions. • Think about the ________? • Ex1: Frat has to send 3 seniors and 2 juniors to conference. Have 6 seniors and 5 juniors. • Ex2: 7 possible colors for yearbook. Can choose at most 2 colors.
OR means ADD -- # of A+ # of B combos AND means MULTIPLY -- # of A x # of B combos. SIMPLE: 4! = 4x3x2x1=24 REPETITION: 7! / 3! * 2! 5040 / 12 = 420 REPETITION PT 2: 8! / 3! * 5! = 8 * 7 = 56. multiple possibilities (AND vs OR). • Ex1: 6! / 3! 3! x 5! / 3! 2! • Ex2: 7! / 6! 1! + 7! / 5! 2!
Loan amortization and payments. -Company wants to borrow 10000 for five years. Interest rate 10%. Draw an amortization table for the 5 years -At an expected RoR of 7%, how much must be deposited at the end of each year for the next 15 years to accumulate $3000? Compound annual growth rate (CAGR). Forumla? Sales at A, Inc have been 4.5, 5.7, 5.3, 6.9, 7.1 million for the last 5 years - What is the compound annual growth rate? Funding a Future Obligation Suppose you must make 5 $1,000 payments, the first one starting at the beginning of Year 4. To accumulate the money to make these payments you want to make three equal payments into an investment account, the first to be made one year from today. Assuming a 10% ROR, what is the amount of these payments? REMEMBER!?
PV = 10,000, I/V = 10%, N = 5. CPT = PMT Beginning Principal balance - annual Interest - Payment - Reduced Principal --- Ending Principal FV = 3000, I/V = 7%, N = 15, CPT = PMT CAGR = ((EV / BV) ^ (1 / # of years)) - 1 (7.1 / 4.5 ) ^ (1/4) - 1 Be careful with the # of years!!!!! Only 4 years, because the first year of sales was at the end of the year. Also --- FV = -7.1, PV = 4.5, N = 4, CPT --- I/Y 0 ----- 1 ------ 2 ------ 3 ----- 4 -----5 ----6 -----7 _____X ____x ____x _______________1K ___1K __1K __1K _1K PMT = 1000, I/V = 10%, N = 5, CPT - PV2 0 ----- 1 ------ 2 ------ 3 ----- 4 -----5 ----6 -----7 _________PV2 PV2 * 1.1 = PV3 0 ----- 1 ------ 2 ------ 3 ----- 4 -----5 ----6 -----7 _______________PV3 FV = PV3, I/V = 10%, N = 3, CPT - PMT T DOES NOT EQUAL YEAR! - ie beginning of year 1 = t=0.
Perpetuity - pays a fixed amount at set intervals over an infinite period of time. Forumula? RoR that an investor would realize if she paid $75 for preferred share and received a $4.50 annual dividend beginning next year and forever?
PV = PMT / I/Y (note - this is its value one period before its payment). 4.5 / 75 = 6%
Define a parameter, a sample statistic, and a frequency distribution. How to construct a frequency distribution? The interval w the greatest frequency is called the _____ interval.
Parameter - measure used to describe a population Sample statistic - measure used to describe a sample. Frequency distribution - tabular presentation of statistical data by assigning it to intervals. • Step 1. Define the intervals (Must be mutually exclusive, each observation only placed in one.) • Step 2. Tally the observations. After intervals have been defined, observations must be tallied or assigned to their appropriate interval. • Step 3. Count the observations. Indicate the absolute frequency in each interval. modal.
P-value? Consider a 2-tailed hypoth test at 95% significance, where test statistic is 2.3 (greater than 1.96 z critical value). P-value? What does this mean?
Probability of obtaining a test statistic that would lead to a rejection of the null hypothesis, assuming the null is true. Look up the Z table at 2.3 critical value. 1.07. Therefore p-value is 2.14% This means that we would reject this test statistic at 2%, 1% significance level, but not 3%, 4%, 5%, etc.. Sometimes easier to report this that arbitrarily choose a significance level.
Probabilitty definitions DICE EXAMPLE Random variable? Outcome? Event? Mutually exclusive events? Exhaustive events? Two defining properties of probability? Empirical probability? A priori probability? Subjective probability?
Random variable - uncertain quantity Number that comes up Outcome - Observed value of Random Variable Rolling a 4 - that is outcome Event - single outcome or a set of outcomes Rolling a 4 or rolling an even number - Event Mutually exclusive events - events that can't happen at same time Rolling a 4 and a 6 is mutually exclusive Exhaustive events - those that include all possible outcomes Rolling an even and rolling an odd number are both mutually exclusive and exhaustive events. Probability of any event (Ei) is bw 0 and 1. (0 <= P(Ei) <= 1) If a set of Events, E1, E2... is mutually exclusive and exhaustive, the probabilities of those events sum to 1. o Empirical probability - established by analyzing past data (objective) o A priori probability - determined using a formal reasoning and inspection process. (objective) o Subjective probability - least formal / involves personal judgment.
Parametric tests? Non-parametric tests? Situations for non-parametric tests? Spearman rank correlation test?
Rely on assumptions regarding the distribution of the population. Example: Z test require population large or population normally distributed. Also relies upon a mean and st. deviation to define normal. Do not consider a particular population parameter or have few assumptions about population sample. Used when can't support parametric tests. Also used for data not suited for parametric tests (ranked observations). 1. Cannot support a parametric test. Ie. hypothesis test for a non-normal distribution and a small sample sample size (neither z-test or t-test appropriate). 2. Data is ranked, rather than a specific value. 3. Hypothesis doesnt involve parameters of distribution. Ie. testing to estimate if data / series of changes are random. Can be used when data is not normally distributed. Ie. performance rank of 20 mutual funds for 2 years. Large positive value (0.85) means that a high (low) rank one year is associated with a high (low) rank in the next. And vice versa.
Cycle Theory? Elliot wave theory?
Some TAs think that can apply naturally occurring cycles to financial markets (ie 4 year presidential cycles or 54 year Kondratieff cycles). Wave = chart movement (ie reversal: generally up or generally down). If uptrend, then upward moves consist of 5 waves and downward waves consist of 3 waves. If downtrend, then downward moves consist of 5 waves and upward waves consist of 3 waves. Uses Fibonnaci ratios (0,1, then adding together, 1,2,3,5,8,13,21).
Chebyshev's inequality. Define? Ex: What is the minimum percentage of any distribution that will lie within +/- 2 standard deviations of the mean?
States that for any set of observations, whether sample or population, and regardless of the shape of the distribution, the percentage of the observations that lie within k standard deviations of the mean is at least 1- (1/k^2 ) for all k > 1. 1 - 1/(2^2) = 1 - ¼ = ¾ = 75%
Technical Analysis? Implies? Based on? As opposed to? Advantages?
Study of collective market sentiment (expressed by buying / selling assets). Efficient Market Hypothesis. Based on idea that market prices reflect both by the interaction of supply and demand. As opposed to fundamental analysis which attempts to determine the intrinsic value of an asset. (FA -- use company's financial data to determine data. TA -- uses firm share price and trading volume data to project a target price). Adv: -Actual Price and Volume data is observable. (FA -- subject to assumptions) -Can be used for Commodities / other assets that don't produce a Cash Flow in the future. - Useful when fraud occurs bc takes a while to restate, TA -- may reflect the true value of company. Disadvantage: - Limited usefulness in markets where price / volume data might not truly reflect supply and demand (illiquid markets, currency markets subject to manipulation by central banks)
Bayes formula Rule for updating prior probability of an event. Example: • Elec Inc, is mulling overseas expansion (O) and price increases (I) • Probabilities: o P(I) = .3 o P(not I) = .7 o P(O | I) = .6 o P(O | not I) = .4 • What is new probability of price increase if overseas expansion announced (P(I|O))
Updated prob of an event given new info= • Prob of new info for a given event / Unconditional prob of new info x Prior probability of event • P(A|B) =( P(B|A) / P(B)) * P(A) In this case the old probability is P(I), want to know P(I|O) o P(I|O) = (P(O|I) / P(O)) * P(I) Alternate way P(I), P(not I) = PRIORS P(IO) = P(O|I) * P(I) - multiplication rule • P(IO) = P(I|O) * P(O) P(IO) = .6 * .3 P(IO) = .18 .18 = P(I|O) * P(O) • P(O) = P(O|I) P(I) + P(O|not I) P(not I) • P(O) = .6 * .3 + .4 * .7 • P(O) = .18+.28 = .46 .18 = P(I|O) * .46 P(I|O) = .3913
Hypothesis testing: Chi-square test? Hypothesis? Chi-square test statistic? Characteristics? Formula? Ex: Co advertises monthly returns w st. dev. of 4% from '90-'98. Want to verify for 24 months b/w '98-'00, st. dev is 3.8%. Is the current st. dev. diff from old?
Used for hypothesis test of a variance of a normally distributed population. ie (two tailed). H₀: σ² = σ²₀ σ² = True population variance σ²₀ = Hypothesized variance X² = chi-square test statistic. - Asymmetrical - Approaches normal as the degrees of freedom increase - Bounded by 0. X² (ⁿ⁻¹) - with n-1 degrees of freedom = (n-1)*s² / σ²₀ n = sample size s² = sample variance Ex: 1. H₀: σ² = σ²₀ Ha: σ² ≠ σ²₀ 2. Test statistic: chi-square. (n-1)*s² / σ²₀ 3. LoS? Use 5% as a default 4. Decision rule. Reject if X² < 11.689 or X² > 38.076. (23 = DoF -- look up in table) 5. Calc. 23*(.038²) / .04² = 20.7575 6. Fail to reject. 7. Recently measured st. dev is close enough to advertised 4%. w 5% LoS.
Hypothesis testing: Type I vs. Type II errors? Probability of rejecting null when true? Probability of failing to reject null when false? Describe this more.
When drawing inferences from a hypothesis test -- 2 types of errors: Type I: Rejection of a null hypothesis when it is actually true. Type 2: The failure to reject a false hypothesis when it is actually false. Level of significance (Type I) Power of a test (Type II) Power of a test is actually: 1 - P(Type II Error). Two main factors into P(Type II Error) = Sample size & Level of significance. LoS actually has a negative effect on power of a test: If you reduce LoS from 5% to 1%, then less likely to make a Type I Error, but also reduce the power of the test (increase the probability of failing to reject a false null).
z-statistic vs. t-statistic? Population mean with a: Normal distribution with known population variance? Normal distribution with unknown population variance? Nonnormal distribution with known population variance? Nonnormal distribution with unknown population variance?
_______Small (N<30)__ Large (N>=30) N, KV _____z ___________z N, UKV ____t___________ t (conservative, but can use z (technically the same)) NN, KV ____n/a_________ z NN, UKV___ n/a_________ t (conservative, but can use z (technically the same))
Responsibilities as a CFA Institute Member or CFA Candidate
a. Conduct as Members / Candidates in the CFA Program i. Cheating on Exam ii. Revealing ANYTHING about Exam / breaking any rules. iii. Improperly using the designation to further goals. (Volunteer on committee and use it to your biz advantage) b. Reference to CFA Institute, CFA designation, CFA Program. i. Must not make promotional promises bc you are CFA member ii. Must sign PCS & Pay dues! iii. Reference 'candidacy' in the program. Can state that he/she finished the program in xx years, but not claim that superior ability. (say candidate as long as the person is registered for the next CFA exam) iv. Chartered Financial Analyst and CFA marks must always be used either after a charterholder's name or as adjectives, but not as nouns in written / oral communications. The CFA designation should always be capitalized and shown without periods v. Cannot put the CFA logo on company letterhead.
Confidence intervals t distribution -- most appropriate to use when? t distribution properties (4)? When looking up t-distribution crictical value --- make sure _________ Degree of confidence vs level of significance? Confidence Interval formula for normal distribution w known pop variance? Confidence Interval formula for normal distribution w unknown pop variance? Ex: 99% CI, N = 36, X = 80, σ = 15. Construct a CI for the mean value Confidence intervals can be interpreted from a _______ or _______ perspective.
constructing confidence intervals based on 1. small samples 2. with populations with unknown variances 3. a normal / approx normal distribution. - It is symmetrical - Defined by a single parameter: degrees of freedom (equal to number of observations minus 1) - "Fatter tails" than normal distribution - As DoF (sample size) gets larger, t-distribution becomes z-distribution. pay attention to one-tailed vs two-tailed probabilities. Confidence interval estimates range of values that actual value lies with a probability of 1- α Degree of confidence --- 1 - α Level of significance --- α x +/- zα/2 * σ/√n x +/- tα/2 * s/√n α/2 = prob of right hand side of the tail / 2. 80 +/- 2.58 * 15/√36 = 73.55 to 86.45 probabilistic = 99% of the intervals after re-sampling will include the pop mean. practical = 99% confident this interval includes the pop mean.
o Discrete distribution ? o Continuous distribution ? o Cumulative probability function? Ex: X={1,2,3,4}. P(x) = x / 10. F(3)? Discrete uniform random variable? • Ex: DUR distr. X={1,2,3,4,5}, what is prob of x = 4? F(4)? Binomial random variable -- # of successes in a number of trials. Formula? Example • Compute the probability of drawing 3 black beans out of 5 selections (only white and black beans). Probability for selecting a black bean is 0.6. EV and variance of a binomial random variable? Ex: 2/3 chance Dow going up. Over 5 days EV and Variance?
countable, probability random variable X is equal to x. Ex: Prob of raining 33 days in June = 0, but some value for 25 days. infinite, therefore measured as P(x1 < X < x2), because if you split it enough x1=x2=0 F(x) = P(X<=x) Therefore cumulative distribution is F(3) = .1 + .2 + .3 = .6 One for which all of the probabilities for all possible outcomes are equal. any p(x) = .2. cumulative probability for the nth outcome F(x) = n*p(x) p(4) = .2 F(4) = .8 P(x) = (n! / (n-x)!x!) (p^x) (1-p)^(n-x) n = total trials x = successes p = probability of success o P(x) = (n! / (n-x)!x!) (p^x) (1-p)^(n-x) o P(x) = (5! / (2! 3!) (0.6^3) (0.4^2) = 0.3456. EV of X = np Variance of X = np(1-p) (5)(2/3) = 3.35 (5)(2/3)(1/3) = 1.12
Lognormal distribution. Formula? Characteristics vs. Normal? Lognormal distributions uses "_______" "_______" of 0 is the same as holding period return of _____. Continuously compounding. Effective rate? Ex: Based on contin. compounded annual rate. of 10%, HPR? Based on stated cc rate of 10.571%, stated annual rate? Ex: Stock purchased for 100, sold year later for 120. RoR on continuously compounded basis? Continuously compounded rates of return is ______ for multiple periods. Formula? Ex: Invest. appreciated over 2 years from 1000 to 1221.40 over 2 years. Annual continuously compounded rate?
e^x, where x is normally distributed. (ln e^x = x) - Lognormal skewed to the right - Bounded from below by zero so that its useful for modelling asset prices that will never fall below zero. "Price relative" - end of period price of asset / beginning. "Price relative"; -100% effective annual rate (HPR) = (e^Rcc) - 1 Rcc = contin. compounded annual rate. Ex: HPR = e^0.10 - 1 = 10.571% Rcc = ln (1 + 0.10571) = 10% ln (120/100) = ln (1.2) = 18.232% additive; HPTt = (e^((Rcc)(t))) - 1 Ex: HPR = 1.2214 1.2214 = e^(2x) - 1 ln (1.2214) = 2x 0.2 = 2x x = 0.1
Calc EV, variance and std dev. of PORTFOLIO RETURNS If dealing w a portfolio: first need to ___________. EV of portfolio? Portfolio variance? Best explained via examples: • 2 asset portfolio A&B • 3 asset portfolio A,B,&C
figure out weights: wi = MV of investment in i / MV of the full portfolio E(Rp) = ∑(w1 * E(R1) + w2 * E(R2)...) Var(Rp) = (wa)(wa)(Cov(Ra,Ra))+ (wa)(wb)(Cov(Ra,Rb))+ (wb)(wa)(Cov(Rb,Ra)) + (wb)(wb)(Cov(Rb,Rb)) o Cov(Ra, Rb) = Cov(Rb, Ra) o Cov(Ra,Ra) = o^2(Ra) Var(Rp) = wa^2*o^2(Ra)+wb^2*o^2(Rb)+2(wa)(wb)(Cov(Ra,Rb)) o Cov(Ra,Rb) = o(Ra)*o(Rb)*Corr(Ra,Rb) Var(Rp) = wa^2*o^2(Ra)+wb^2*o^2(Rb)+2(wa)(wb)(o(Ra) o(Rb)Corr(Ra,Rb)) • Same as before but 9 clusters o DO EXAMPLES ON NON-FLASHCARDS
Covariance - measure of ________? Formula? Similar to concept of variance, in fact ______ Example: o 3 possible states (S) next year: Boom, Normal, Slow o P(B) = .3, P(N) = .5, P(Slow) = .2 o Returns of stock A and B as follows o Event ----------P(S) ---------Ra ---------Rb o Boom ----------.3------------.2------------.3 o Normal---------.5-----------.12-----------.1 o Slow------------.2------------.05----------.00 o Covariance? Problem with covariance is _______? Correlation Formula? Measures? Correlation range and interpretations? Example: Same facts as before. Correlation?
how two assets move together - extent to which two random variables tend to be above and below their respective means Cov(A, B) = ∑(Pi (( Ai- mean A)(Bi - mean B))) Cov(A, A) = variance(A) EV of Ra = (.3 .2) (.5 .12) (.2 .05) = EV of Rb = (.3 .3) (.5 .1) (.2 .0) .3(.07*.16) + .5(-.01*-.04) + .2(-.08*-.14) Cov(Ra, Rb) = .00580 Similar to variance - how to interpret? (+/- indicate positive/neg relationships, but no scale) Correlation - much easier Corr(A, B) = Cov(A,B) / (o(A) x o(B)) • Measures the strength of the linear relationship bw two random variables • Correlation of -1, perfect negative correlation, means that movement of random variable results in proportionally opposite movement relative to its mean. • Correlation of 0, no linear relationship • Correlation of 1, perfect positive correlation means that movement of random variable results in proportionally positive movement relative to its mean. • Cov(Ra, Rb) = .00580 • o^2(Ra) = .0028 • o^2(Rb) = .0124 • .00580 / sqrt(.0028) x sqrt(.0124) • Corr(Ra, Rb) = .984
Skewness Distribution is symmetrical if it is shaped identically on both sides of its mean. Skewness or skew - refers to the extent to which a distribution ____________. Positively (____) skewed distribution --- _____________ Negatively (_____) skewed distribution --- _____________ Relative locations (mean, median, mode) Symmetrical? Positively skewed distribution? Negatively skewed distribution?
is not symmetrical. right ; more outliers on the right tail left ; more outliers on the left tail Symmetrical- Mean, median and mode are equal Positively skewed distribution- Mode is less than median. Median less than the mean. Negatively skewed distribution- Mode is more than median. Median is more than the mean.
Explain the key properties of the normal distribution o Normal distribution properties Completely defined by ___________ Skewness? Kurtosis? Tails? Univariate vs. Multivariate distribution. o Univariate distribution - distribution of a __________ o Multivariate distribution - specifies the probabilities associated with a _____________________________. Between 2 discrete random variables - use ____________. Between continuous random variables, a multivariate ______________________________. Role of Correlation in Multivariate: Can be useful to see the correlations of n assets. For each number of n assets, there are 0.5n(n-1) correlations. Ie. 6 assets, then ___ correlations. When building portfolio of assets -- desirable to have assets with a ____ correlation to ensure ____ variance of portfolio
mean and variance Skewness = 0, ie Normal distribution is symmetric around mean. Median = Mean = Mode Kurtosis = 3 Tails get very thin but extend indefinitely -single random variable. -group of random variables. Is only meaningful when each random variable is someway dependent upon the behavior of the others - joint probability tables. . - normal distribution may be used to describe them if all of the variables are normally distributed. 15; lower; low
Kurtosis. Define? Three types of kurtosis? Two additional facts? Sample skewness (sK) for large populations? Formula? Evaluations? Measure of sample kurtosis? Formula? Evaluations?
measure of the degree to which a distribution is more or less "peaked" than a normal distribution Lepokurtic - More peaked than normal distribution • More returns clustered around the mean and more returns with large deviations from the mean. Platykurtic -Less peaked than normal distribution Mesokurtic- Same kurtosis as normal distribution Investment returns rarely normal distributions Positive kurtosis and negative skew indicates greater risk. sK = (1/n) (∑( (Xi - sample mean)^3) / s^3) s = sample standard deviation • Denominator always positive (std deviation). But Numerator either positive of negative. o Therefore, positively skewed - tend to be positive (large positive outliers). Negatively skewed - tend to be negative (large negative outliers). Sample kurtosis = (1/n) (∑(Xi - sample mean) ^4) / s^4) 'Excess kurtosis' - if it has either more or less kurtosis than normal • Computed kurtosis for all normal distributions is 3. • Excess kurtosis is kurtosis less 3. (lepokurtic is positive, platykurtic is negative).
Joint Probability of two events (___________ of probability) Example: o P(I) = 40% chance on authority increases rate o P(R|I) = 70% chance of a recession given an increase in rates o What is P(RI)? (the joint prob of a recession AND an increase in interest rates) Probability that at least 1 of 2 events will occur (_______ of probability) Example: o Using prior info, also given P(R) = 34% o Determine the probability that either interest rates will increase or a recession will occur. Probability of any number of independent events: IMPORTANT? • Example: o What is the probability of rolling three 4's in one simultaneous toss of three die? Total probability rule - can calculate the unconditional probability
multiplication rule P(AB) = P(A|B) x P(B) • Joint probability of A & B is equal to the conditional prob of A given B times the unconditional probability of B P(RI) = P(R|I) x P(I) .28 = .7 x .4 addition rule P(A or B) = P(A) + P(B) - P(AB) Doesn't matter if mutually exclusive (then P(AB) is 0) P(R or I) = .34 + .4 - .28 P(R or I) = .46 When dealing with independent events: "and" indicates multiplication while "or" addition 1/6 x 1/6 x 1/6 = 1/216 P(R) = P(R|S1)* P(S1) + P(R|S2)*P(S2)... • Holds when S1.. Sn is mutually exclusive and exhaustive • Similar to Expected Value
Descriptive statistics vs Inferential statistics? Different statistical methods use different levels of measurement or measurement scales. 4 scales and an example of each?
o Descriptive statistics - used to summarize the important characteristics of large data sets. (ie know population) o Inferential statistics - pertain to procedures used to make forecasts, estimates and judgments. (ie don't know population) NOIR (french for black) Nominal scales - least information. Observations classified with no particular order. Example: Assign a 1 to a muni bond fund, 2 to a corporate bond fund. Ordinal scales - Every observation is assigned to one of several categories. And then then categories are ordered by characteristics Example: Of 1000 small cap stocks, assign a 1 to the 100 best performing stocks, number 2 to the next 100 performing stocks. Can understand that a 3 is better than a 4, but no sense of scale. Interval scale - provide relative ranking, plus assurance that differences in scale are equal. Example: Temperature. Difference between 50 F and 65 F is the same as 50F and 35 F. However, measurement of 0 does not measure the absence of what we are measuring, therefore ratios are irrelevant. 30 F is not necessarily three times hotter than 10 F. Ratio scale - represent the most refined level of measurement. Provide ranking and equal differences between scale values, and they also have a true zero point as the origin. Example: Measurement of money. If you have $4, twice as much as $2.
LOS --- Real risk free rate --- DEFINE? T-Bill / Treasuries are risk free rates but NOT _________ Nominal risk-free rate = ?
o Real risk free rate - investor's increase in purchasing power NO EXPECTATION OF INFLATION. o T-Bill / Treasuries are risk free rates but NOT real rates of return. Nominal risk-free rate = real risk free rate + expected inflation rate.
T-Bills = _____________ (unlike other US govt bonds) T-Bills quoted on a ____________, which is based on the ________ of the instrument instead of the __________. Bank discount yield (BDY). Formula? 3 reasons bank discount yield is NOT REPRESENTATIVE of return earned by investor? Ex: Calculate the bank discount yield for a T-Bill priced at $98,500, with a FV of $100,000 and 120 days until maturity
pure discount instrument bank discount basis; face value; purchase price rBD = (D / F) x (360 / t) where rBD = annualized yield on a bank discount basis D = the dollar discount (difference bw face value and purchase price) F = the face value (par value) of the bill t = number of days until maturity 360 = bank convention with # of days. • Assumes no compounding, uses simple interest • Expresses the dollar discount from par as a fraction of FV, NOT the market value • 360 day convention instead of 365 day year o rBD = 1,500 / 100,000 x 360 / 120 = 4.5%
Roy's safety first criteria -- -- States that the optimial portfolio minimizes that the ___________ falls below __________. Formula? Only difference between the Sharpe ratio? Short fall risk? Example: 120 million endowment, min threshold after 1 year is 123.6 million. 3 portfolios being considered: • Port A B C • E(rp) 9% 11% 6.6% • Op 12% 20% 8.2% Which portfolio most desirable? For that portfolio -- what is % chance less than min threshold?
return on a portfolio; minimum threshold level. SFRatio = (E(rp) - rl) / σp Where rp = portfolio return rl = threshold level return σp = st dev of portfolio Sharpe uses the risk-free rate instead of threshold level (if these two were equal than Sharpe and RSF would be the same) One reason to use RSF is to calculate the downside risk. Short fall risk is the one-sided tail probability of a normal distribution. (123.6 - 120) / 120 = 3% = min threshold A B C SF Ratio 0.5 0.4 ~0.44 CHOOSE A % chance less than min threshold = F(-0.5) = 1 - F(0.5) = 1 - .6915 = 0.3085
Simple random sampling -- everyone same likelihood. Systematic sampling -- ___________________ Sampling error formula? Stratified vs simple random sampling? Time-series data? Cross sectional data? Longitudinal data? Panel data?
selecting every nth member of a population sampling error = sample mean - pop mean Stratified classifies the population into categories then samples to get to a final sample, simple random does not. - observations over period time. ie. monthly returns of stock from 1994 -2004 - sample of observations at a period of time. ie. EPS of all NASDAQ companies as of Dec 31st, 2004. - observations over time of multiple characteristics of the same entity. ie. US unemployment and inflation #s over 10 years. - observations over time of the same characteristic for multiple entities .ie. D/E ratios for 20 companies over 24 quarters.
Monte Carlo: Best with Example: Valuation of stock options: _________ values for __________. Chose stock price and relevant interest rate as two main factors in option valuation. Would do the following: 1. ________________ 2. _______________(100,1000, or 10,000) 3. _______________ 4. _______________ Major applications? (5) Limitations (2)? Historical simulation --- difference with MC?
simulate; main risk factors 1. Specify the probabilities of stock price, interest rate, as well as as parameters of distributions (skewness, mean, variance). 2. Randomly generate values for stock price and interest rate. 3. Value the options for each of the risk factor values 4. Average the option value 1. Value complex securities 2. Simulate the profits / losses from a trading strategy 3. Calculate estimates of VAR (value at risk) to determine riskiness of portfolio 4. Simulate pension assets / libailities to see the variation between the two 5. Value portfolio that is not a normal distribution 1. Will provide answers no better than the assumptions used. 2. Simulation is a statistic measure, not analytical one. Based on actual changes in the past -- but not necessarily always indicative of future.
Standard error of the sample mean Formula -- SE of SM (known pop st deviation)? SE of SM (unknown pop st deviation)? Ex: mean wage = $13.50, w population standard dev of $2.90, calc. the standard error of sample mean for sample size of 30. Ex: mean return= 2%, w sample standard dev of 20%, calc. the standard error of sample mean for sample size of 30.
standard deviation of the distribution of sample means. Known pop st deviation σ₉ = σ / √n σ₉ = Standard error of the sample mean σ = pop st deviation Unknown pop st deviation s₉ = s / √n s₉ = Standard error of the sample mean s = sample st deviation Ex1: $0.53 Ex2: 3.6%
Dispersion - define? Range? Mean absolute deviation (MAD) ? Population Variance (θ^2) ? Population standard deviation (θ)? Sample Variance (s^2) ? Sample standard deviation? Variance of an individual stock with exhaustive probability weights?
the variability around the central tendency. (relates more to risk) Max - Min ∑ ( |Xi - mean| ) / n - Interpret as, on average, an individual return will deviate +/- from the mean return by the MAD ∑( (Xi - mean)^2 ) / n - nonsense metric but gets us to standard deviation (If you use decimals vs. whole numbers, get different answers! Standard dev better) Sqrt (pop variance) ∑ ( (Xi - sample mean)^2 ) / n - 1 - Called an 'unbiased estimator' bc using only n significantly underestimates population variance Sqrt (sample variance) ∑ ( P(Xi)(Xi - mean)^2 )
Difficult to compare 2 different dispersions (ie 8% mean for retail stocks and 16% for real estate portfolio). Relative dispersion is needed. Commonly measured with: 1. Coefficient of variation (CV). Measures? Formula? Ex: T-bills have average return of 0.25% and standard deviation of 0.36%. S&P 500 is 1.09% with a standard deviation of 7.30%. Compute the CV for these two. 2. Sharpe ratio (or the reward-to-variability ratio) Measures? Formula? - Portfolios with ____ Sharpe ratios are preferred bc ___ return with __ variability. Two Sharpe ratio limitations?
variation per unit return. CV = Sx / X o Sx = standard deviation of x o X = Average value of X T-bills = 1.44 S&P = 6.7 Means that there is less dispersion per unit of monthly return for T-Bills than for S&P 500. excess return per unit of risk. (rp - rf ) / op • rp = portfolio return • rf = risk-free return • op = standard deviation of portfolio returns o Excess return known as (rp - rf) large; high; low 1. If comparing 2 negative Sharpe ratios, not necessarily true that higher Sharpe ratio implies superior risk-adjusted performance. 2. The ratio is useful when standard deviation is an appropriate measure of risk. However, investment strategies with option characteristics have asymmetric return distributions. (ie large probability of small gain, small probability of large gain). Standard deviation may underestimate risk / produce a bad Sharpe ratio
Need to construct a binomial tree to describe stock price movement o Binomials applied to stock price movement Example: • Stock with price S, will over the next period increase in value 10% or decrease in value 10%. • Therefore u = 1.1, d = 1/1.1 • Probability (U) = .7, therefore probability (D) = .3. S = 50 Tracking error? Ex: If mutual fund of US stocks return was 4% and the index was 7%, . Continuous uniform distribution Spans from lower limit, a, to some upper limit, b. Outcome P(x1 <=X<=x2) = (x2 - x1)/(b-a) Example: o X is uniformly distributed between 2 and 12. Calculate the probability that X will be between 4 & 8. ? o Cumulative Distribution Frequency - Linear?
• ----- uuS • -----/ • ---uS • --/--\ • S----udS • --\--/---- • ----dS • -----\--- • -------ddS • - • EV uuS = 50 (1.1)(1.1) = $60.50. Probability = (0.7)^2 = 49% • EV udS = 50 (1.1)(1/1.1) = $50. Probability = (0.7)(0.3) x 2 = 42% • EV ddS = 50 (1/1.1)(1/1.1) = $41.32. Probability = (0.3)(0.3) = 9% Difference between the total return on a portfolio and the total return on the benchmark Ex: tracking error for that period is -3% o (8-4) / (12-2) = 4/10 Prob o | / o | /| o | / | o | / | o |/ | | o 2 4 8 12 o x
TVM problems Calculating Annuity Due in the Calculator. If you are calculating Annuity Due on the Calc. the FV is the value at the END OF THE PERIOD! Ex. The FV at the beginning of period 9 is $19,603. Want to know how much in payments that it would take to get to this amount over beginning period (10) payments starting today. I/Y = 8%.
• 0---1---2---3---4---5---6---7---8---9—10 • X---x---x---x---x----x---x---x---x---x---FV o Have to take 19,603 and make it the end of t=9 instead of beginning: 21,171. o Then change calc to annuity due (instead of ordinary) o Then FV = 21,171; I/Y = 8%, N=10, CPT PMT = 1,353