CAIA Topic 2
LO 2.6.6.6: Demonstrate knowledge of option exposures.
A covered call strategy combines being long the underlying security and being short a call option on the security. It has limited upside and limited downside. A protective put strategy combines being long the underlying security and being long a put option on the security, and has unlimited upside and limited downside. An option spread involves (1) either calls or puts and (2) both long and short positions on the underlying security. A bull spread combines a long position in a lower strike price option and a short position in a higher strike price option. A bear spread combines a long position in in a higher strike price option and a short position in a lower strike price option. Option combinations involve both calls and puts on the underlying security. An option straddle is a position in a call and a put (either both long or both short) on the same underlying security, same expiration date, and same strike price. An option strangle is a position in a call and a put (either both long or both short) on the same underlying security and expiration date, but with different strike prices. A risk reversal consists of a long out-of-the-money call combined with a short out-of-the-money put on the same asset and with the same expiration. A collar includes a long position in the underlying security, which is combined with a long put option and a short call option. Put-call parity is a no-arbitrage relationship between two sets of positions with identical payoffs: (1) a long underlying asset and (2) a long call, short put, and long risk-free bond. call + risk-free bond - put = underlying asset
LO 2.3 .3.2: Demonstrate knowledge of returns based on notional principal.
A forward contract is a bilateral contract that obligates one party to buy and one party to sell a specific quantity of an asset, at a set price, on a specific date in the future. The initial value of a forward contract is zero to both parties at contract initiation. Notional principal is the face amount on the underlying asset, upon which cash flows on a derivative instrument are based. The return on notional principal equals the gain or loss on the derivative instrument divided by the notional principal. The return on notional principal is misleading because the initial cash outflow does not equal the notional principal. In many alternative investments (e.g., forwards), there is no cash outflow at the initiation of the investment. Using a fully collateralized basis, the return is calculated by matching a collateral amount equal to the notional principal. Assuming continuous compounding, the return on the fully collateralized position equals ln(1 + R) + Rf, where Rf is the risk-free rate and R is the return on the derivative instrument. Alternatively, the return can be expressed on a partially collateralized basis, in which the forward contract is matched with capital equal to a percentage, p, of the forward contract's notional principal. The return on a partially collateralized position equals [L × ln(1 + R)] + Rf, where L equals the leverage factor 1 / p.
LO 2.8.8.2: Demonstrate knowledge of the concepts of ex ante and ex post alpha.
Alpha can be categorized into two types: Ex ante alpha is a forecast of incremental return after adjusting for time value of money and systematic risk effects. Ex ante alpha is the return attributable to the skill of the manager. Ex post alpha measures the realized incremental return after adjusting for the time value of money and systematic risk effects. Ex post alpha may be attributable to skill, luck, or a combination of the two.
LO 2.4.4.1: Demonstrate knowledge of the characteristics of return distributions.
An ex post distribution refers to frequencies associated with historical data. An ex ante distribution refers to frequencies associated with anticipated future data. The normal distribution is the familiar bell-shaped function that is used to calculate probabilities and assess risks. The normal distribution is symmetric, has no skewness, and its kurtosis equals 3 (i.e., zero excess kurtosis). Characteristics of the lognormal distribution include: Log returns, ln(1 + R), are normally distributed, or, equivalently, (1 + R) are lognormally distributed. The distribution is continuous. The distribution is skewed to the right. The distribution is bounded from below by zero so it is useful for modeling asset prices which never take negative values.
LO 2.7.7.2: Demonstrate knowledge of various types of asset pricing models.
Asset pricing models describe relationships between risk and expected return, produce expected returns that are used to determine asset intrinsic values, describe the variability of returns, and separate risks into diversifiable and non-diversifiable sources. A normative model is one that attempts to explain how investors should behave. In contrast, a positive model is one that attempts to explain how investors actually do behave. Theoretical models are based on assumptions and logic that presumably captures underlying behavior. In contrast, empirical models are based on historically observed behavior. Applied models are pragmatic in nature and are designed to address real-world problems, such as how to achieve efficient diversification. In contrast, abstract models are theoretical models designed to describe behavior under hypothetical, often unrealistic, circumstances. Alternative investments are best addressed with applied models, because they are designed to address real-world problems. Cross-sectional asset pricing models are used to identify key sources of return differentials across assets and aid in the identification of peer groups. In contrast, time-series asset pricing models are used to identify key sources of return differentials over time for an individual asset or portfolio and aid in the measurement of performance relative to risk-adjusted benchmarks. Panel data sets refer to data spanning multiple time periods and multiple assets (a combination of cross-sectional and time-series data).
LO 2.4.4.4: Demonstrate knowledge of standard deviation (volatility) and variance.
Assuming portfolio returns are normally distributed, the standard deviation can be viewed approximately as the typical deviation of the portfolio return from the mean. The variance of an n-asset portfolio equals: where wi and wj are the allocation weights for assets i and j, respectively, and Cov(Ri, Rj) is the covariance of returns between assets i and j. If returns are uncorrelated across assets, then If returns are perfectly correlated across assets, then the portfolio standard deviation is simply the weighted average of the standard deviations of the individual asset returns: The standard deviation of the portfolio return for a two-asset portfolio is: If the second asset is risk-free, then σ2 = 0, the portfolio variance equals w12 σ12 , and the standard deviation equals w1σ1. If returns are uncorrelated over time, then the variance of a multiperiod log return equals the sum of the inter-period log-return variances. In addition, if the variances of daily log returns are identical (homoskedastic), then the T-period log-return variance equals T multiplied by the variance of the daily log return. This also implies that the T-period log-return standard deviation equals the square root of T multiplied by the standard deviation of the daily log return.
LO 2.4.4.5: Demonstrate knowledge of methods used to test for normality of distributions.
Autocorrelation, illiquidity, and nonlinearity cause data to be non-normal. Autocorrelation refers to the correlation among lagged values of a random variable, which might cause outcomes to be more extreme than predicted by the normal distribution. Prices of illiquid assets are often estimated using appraisals, which frequently exhibit positive autocorrelation. Non-linearities in returns often introduce skewness into the distribution. The Jarque-Bera statistic is used to test data for departures from the normal distribution. The Jarque-Bera statistic tests the null hypothesis that the skewness and excess kurtosis for the data distribution jointly equal zero.
LO 2.3 .3.1: Demonstrate knowledge of return and rate mathematics.
Continuous compounding refers to the continuous reinvestment of interest, in which case the simple return will equal R = e^R(m)→∞ - 1, where Rm→∞ is the continuously compounded return (i.e., the return upon which continuous compounding is applied). The continuously compounded return (i.e., log return) equals ln(1 + R). Compounding in which interest is not continuously reinvested is known as discrete compounding. The geometric mean return can be calculated using log returns: e^(M) - 1, where M is the arithmetic average log return.
LO 2.4.4.3: Demonstrate knowledge of various measures of correlation of returns.
Correlation measures the strength of the linear relationship between the returns of two assets. The correlation equals the covariance of returns for the two assets divided by the product of the standard deviations of returns for the two assets. The correlation ranges from -1 to +1. Portfolio diversification improves as the correlation moves toward -1. The Spearman rank correlation is the correlation of the rankings of the asset returns and is the preferred correlation measure when the data series contains outliers. In the context of the CAPM, the beta measures the sensitivity of an asset's returns to changes in the broad market return. The formula for the beta is the covariance of returns between the asset and the market portfolio divided by the variance of market portfolio returns. The correlation over time for an asset is called autocorrelation. The k-order autocorrelation is defined as: First-order autocorrelation (or serial correlation) is tested using the Durbin-Watson statistic, which can be approximated by: DW ≈ 2(1 - correlation). The Durbin-Watson statistic will be close to 4 in the presence of strong negative serial correlation, close to 0 in the presence of strong positive serial correlation, and close to 2 in the presence of no serial correlation.
LO 2.9.9.1: Demonstrate knowledge of single-factor regression models.
Describe the problem autocorrelation poses to regression analysis = observations are not independent Describe the problem heteroskedasticity poses to regression analysis = OLS assumes variances are constant...means they aren't...can mean high residual variance and invalidate standard errors and t-stats. Interpret a regression's goodness of fit = Goodness of fit or R-squared ○ 0 to +1 § e.g., if 0.80 means 80% of variation in the asset's excess returns is explained by market's excess return □ TO MEASURE IDIOSYNCRATIC RISK.... 1- R SQUARED Calculate the t-statistic = estimate - hypothesized/standard error....if > 1.96 = statistically significant A simple linear regression is a statistical method that estimates a linear relationship between a dependent variable and a single independent variable. In a simple linear CAPM-based regression, the asset's excess returns are regressed against the market's excess returns. The slope coefficient measures the asset's beta, which is the sensitivity of the asset's returns to changes in the market portfolio returns. The intercept of the CAPM-based regression equals the incremental performance of the asset relative to the CAPM benchmark return and is called the asset's alpha. Ordinary least squares (OLS) is an estimation method that minimizes the sum of squared regression residuals. Outliers have disproportionately large effects on regressions due to the residual squaring process of OLS regressions. The OLS method assumes that regression residuals are not correlated with their lagged values. Violation of the assumption is called serial correlation, which causes standard errors and t-statistics to be incorrectly calculated. The OLS method assumes that the variance of the residuals is constant. Heteroskedasticity refers to a violation of the constant error variance assumption. Conditional heteroskedasticity is related to the level of the independent variables, and causes standard errors and t-statistics to be incorrectly calculated.
LO 2.9.9.5: Demonstrate knowledge of approaches to analyzing hedge fund returns using multifactor models.
For example: Describe how style analysis and asset class groupings can be used to analyze fund performance. Describe how performance of a fund can be analyzed using returns of funds with similar strategies. Describe how marketwide factors can be used to analyze performance of a fund. Describe how specialized market factors can be used in hedge fund replication. Multifactor models explain fund returns relative to: returns of asset classes held by the fund, returns of funds with similar strategies, market factors that drive asset returns, and specialized market factors. In regression-based style analysis, a multiple linear regression is estimated in which portfolio returns are regressed against asset class index returns. Each slope coefficient measures the extent to which the portfolio is exposed to each asset class included in the regression. Principal components analysis is a multivariate statistical method that groups funds that correlate highly with each other. Studies show that funds can be classified into one of five trading style groups: distressed, global macro, value, opportunistic, and trend-following. Hedge fund replication identifies investment strategies mimicking a particular fund's returns. In the fund replication method, a fund's return is explained by a specialized set of market-based (as opposed to marketwide) factors.
LO 2.9.9.6: Demonstrate knowledge of estimating hedge fund performance persistence.
For example: Discuss approaches to estimating hedge fund performance persistence. Performance persistence can be examined with regression tests and measures of skill tests. Results of empirical testing regarding the performance persistence of hedge funds are mixed largely due to inadequate measures.
LO 2.4.4.6: Demonstrate knowledge of time-series return volatility models
For example: Identify various measures used in time-series models (e.g., price levels, price variation, risk). Define the concepts of heteroskedasticity and homoskedasticity. Recognize the key components of the generalized autoregressive conditional heteroskedasticity (GARCH) method. Describe how the GARCH method is used to model risk evolution through time. Contrast the GARCH method with the autoregressive conditional heteroskedasticity (ARCH) method. ARCH models are used to forecast variances based on historical unexpected outcomes. In contrast, GARCH models are used to forecast variances based on historical unexpected outcomes and historical variances.
LO 2.9.9.4: Demonstrate knowledge of methods for modeling changing correlation.
For example: Recognize and describe the concept of conditional correlation. Describe the rolling window approach to modeling changing correlation. Alternative investment returns are nonstationary, implying that means, variances, and/or correlations are not constant over time. Conditional correlation is the correlation between two variables relative to a specific set of circumstances. A positive conditional correlation exists when the correlation between a fund's returns and the market index returns is higher in up-markets versus down-markets. A negative conditional correlation exists when the correlation is lower during up-markets than during down-markets. A positive conditional correlation is indicative of a good market timer. Time-varying correlation and regression estimates can be derived using a rolling window analysis, in which a moving window of time is used to derive periodic correlation and regression slope estimates.
LO 2.6.6.4: Demonstrate knowledge of arbitrage-free financial models.
In the simplest forward pricing model: F(T) = S for all maturities, T Cost of carry is a measure of the financial difference between holding a position in the spot market and holding a position in the forward market. Any difference between the spot and forward price is due to the cost of carry, which causes the term structure of forward prices to have a slope or curve. The cost of carry relationship is as follows: F(T) = S + carrying costs Financial forwards include futures contracts on stock indices, U.S. Treasury bond futures, and Eurodollar CD futures. The pricing relationship for financial forwards is as follows: F(T) = S × e(r-d) × T A binomial tree model shows the possible values an option can take at each given time period, and reflects the uncertainty in outcome by modeling an upward and downward movement at each state.
LO 2.5.5.1: Demonstrate knowledge of measures of financial risk.
Key risk measures that are used in addition to standard deviation include the following: Semistandard deviation equals the volatility of returns falling below the mean. Target semistandard deviation measures the volatility of returns falling below a prespecified target. Shortfall risk is the probability that the investment return will fall below the target. Tracking error measures the extent to which investment returns deviate from the benchmark returns over time. Drawdown equals the percentage decline in asset value from its previous high. Value at risk (VaR) is a measure of potential loss relative to a prespecified confidence level. Conditional VaR is the expected loss given that the portfolio return already lies below the pre-specified "worst case" quantile return (i.e., below the 5th percentile return).
LO 2.9.9.2: Demonstrate knowledge of multifactor regression models.
Multicollinearity refers to the condition in which two or more of the independent variables are highly correlated. When independent variables are correlated, the intercept and slope standard errors are biased upward, which, in turn, biases the t-statistics downward. The stepwise regression method chooses independent variables based on each variable's explanatory power. The first independent variable chosen is the one with the highest t-statistic for its slope. Then, additional variables are added sequentially depending on the magnitude of their t-statistics. Limited economic theory is applied when using stepwise regression, which can lead to overfitting.
LO 2.6.6.3: Demonstrate knowledge of multifactor and empirical asset pricing models.
Multifactor asset pricing models describe the relationship between expected returns for an asset and its exposures to multiple risk factors: E(Ri) - Rf = βi,1[E(R1) - Rf ] + βi,2[E(R2) - Rf] + ...+ βi,K[E(RK) - Rf]. Theoretical models are based on assumptions and logic that presumably captures underlying behavior. In contrast, empirical models are based on historically observed behavior. The main weakness with empirically derived multifactor models is that the factors may have been identified from spurious correlations (correlations between factors that are purely coincidental). The steps to derive an empirical multifactor model are: Derive excess returns for the security (the dependent variable of the ex post regression). Identify a set of potential factors. Perform tests of significance to identify the important "priced" factors. Assuming the factors are tradable assets (i.e., rates of return can be observed), the intercept of the ex post multifactor model reflects the abnormal performance of the security. The Fama-French three-factor model is an empirical multifactor model which states that E(Ri) - Rf = β1(Rm - Rf) + β2 E(SMB) + β3 E(HML), where Rm - Rf is the excess return, SMB is the size factor equal to the difference in returns between small and big firms (Rs - Rb), and HML is the book-to-market factor equal to the difference in returns between high and low book-to-market firms (Rh - Rl). The Fama-French-Carhart four-factor model states that E(Ri) - Rf = β1(Rm - Rf) + β2 E(SMB) + β3E(HML) + β4 E(UMD), where UMD (or the momentum factor) equals the difference in returns between stocks that were up the most and those that were down the most during the prior year. Factors should be selected based on solid theoretical rationale or based on rigorous statistical testing. Three caveats are in order when deriving factors. First, we should not indiscriminately test a multitude of factors. Second, we should avoid identifying factors based on spurious correlation. Third, the CAPM does not work well for alternative assets, which could have large idiosyncratic risks that are not easily diversified away.
LO 2.5.5.2: Demonstrate knowledge of methods for estimating value at risk (VaR).
Parametric VaR refers to a VaR calculation derived from the normal distribution. When calculating VaR, volatility can be estimated based on historical data or implied option volatility. Leptokurtic distributions require that VaR is modified to either utilize a distribution that can account for larger tails or a higher standard deviation value for a given confidence level. Historical VaR uses past return data and ranks the returns to determine which fall below a given confidence level. Monte Carlo analysis for VaR simulates values for risk factors and estimates how changes in risk factors might affect the fund's returns. A model is used to randomly generate possible future outcomes for the fund, and those simulated outcomes indicate what types of losses are possible for the fund. Monte Carlo VaR is similar to historical VaR in that both determine the potential losses from a set of returns. The difference is that historical VaR uses realized returns from the fund's past, while Monte Carlo VaR uses hypothetical returns generated from the simulation. The VaR of a two-asset portfolio where VaRs are the same equals the sum of the individual asset VaRs if the returns for the individual assets are perfectly positively correlated. The portfolio VaR equals zero if the individual asset returns are perfectly negatively correlated. If the assets are uncorrelated, the portfolio VaR is calculated as the square root of the sum of the squared VaRs.
LO 2.5.5.3: Demonstrate knowledge of ratio-based performance measures used in alternative investment analysis.
Ratio-based performance measures include the Sharpe ratio, Treynor ratio, Sortino ratio, information ratio, and return on VaR. The Sharpe ratio equals the portfolio's expected return in excess of the risk-free rate, divided by the portfolio's total risk. The Sharpe ratio is appropriate if the portfolio is the investor's total "standalone" portfolio. The Treynor ratio equals the portfolio's expected return in excess of the risk-free rate, divided by the portfolio's systematic risk. The Treynor ratio is appropriate when comparing components of a well-diversified portfolio. The Sortino ratio equals the portfolio's expected return in excess of the target return, divided by the portfolio's target semistandard deviation. The information ratio equals the portfolio's expected return in excess of the benchmark return, divided by the portfolio's tracking error. The return on VaR is the expected return on the portfolio divided by its VaR.
LO 2.7.7.3: Demonstrate knowledge of various approaches to performance attribution.
Return attribution is the process of ascribing returns to different components of the asset's performance. The active return equals the difference between the managed fund's return and its benchmark. Using the ex post CAPM equation (a single-factor model), the active return equals the deviation of the realized return from the ex post CAPM expected return, which equals the intercept of the regression of excess returns for the investment against the excess returns for the market. The Fama-French and Fama-French-Carhart multifactor models can be used to benchmark fund performance. The fund's incremental performance equals the difference between the historical excess return earned by the fund versus the return generated by the ex post model.
LO 2.8.8.6: Demonstrate knowledge of return drivers.
Return drivers of an investment are investments, products, and strategies that generate the investment's risk and return. Return drivers are divided into two classes: beta drivers and alpha drivers. Beta drivers are exposures to market risk factors that compensate investors for bearing nondiversifiable market risk. The excess return that a stock provides above the risk-free rate is referred to as the equity risk premium (ERP). The tendency of the ERP to exceed its expected value based solely on risk aversion is known as the equity premium puzzle. Passive beta drivers are return drivers from a pure play on beta, which refers to passive investing such as a buy-and-hold strategy to replicate a benchmark index. Alpha drivers are exposures to active return factors and determined by investment strategy. Most alternative assets are alpha drivers. Product innovators are alpha drivers that create new investment opportunities, while process drivers are beta drivers that deliver beta as cheaply and efficiently as possible.
LO 2.5.5.4: Demonstrate knowledge of risk-adjusted performance measures used in alternative investment analysis.
Risk-adjusted performance measures include Jensen's alpha, M2, and average tracking error. Jensen's alpha equals the difference between the portfolio mean return and CAPM ex-post mean return. The M2 equals the expected return on a leveraged portfolio that has the same standard deviation as the market index. The average tracking error equals the average return difference between the portfolio and the portfolio's benchmark.
LO 2.8.8.8: Demonstrate knowledge of sampling and testing problems.
Selection bias refers to the exclusion of certain observations from the sample, causing distortions in the relevant characteristics of the population. Survivorship bias, a type of selection bias, occurs when funds or companies no longer in existence are excluded from the sample. Another related bias is self-selection bias, in which fund managers make the decision to report or not report performance. As a result of selection bias, most hedge fund and private equity databases underrepresent poorer-performing funds. Affected databases will likely exhibit an upward bias. Data mining refers to the practice of vigorously testing data until valid relationships are found. Data dredging refers to the practice of overusing statistical tests (e.g., running hundreds of tests) to identify significant relationships with little regard for underlying economic rationale. The main problem with data dredging relates to the failure to take the number of tests into account when examining the results (i.e., placing too much confidence on the results). Backtesting is the process of applying models on historical data to determine how well the models would have explained the actual results. Backtesting, when combined with data dredging, implies that too many hypothetical strategies are tested, which can lead to false predictions. Overfitting occurs when many parameters are used to fit a model to historical data. Backfilling refers to updating the database by inserting returns that pre-date the date of entry in the database. The danger with backfilling is backfill bias, also known as instant history bias, which occurs when funds and strategies added to the database are not representative of the population. In this case, backfilling creates an upward return bias because it is more likely that successful funds will backfill. Cherry-picking is the process of selectively reporting results, biasing the reporting toward results that support a particular view. Chumming refers to scattering disparate investment predictions in the hopes that some of the predictions are correct and then luring unsuspecting investors with marketing material focusing on the winning predictions while concealing the losing predictions.
LO 2.3 .3.5: Demonstrate knowledge of the distribution of cash waterfall.
The cash distribution waterfall is the provision describing how capital is distributed to the providers of capital (i.e., investors or limited partners) and decision makers (i.e., the managers or general partners). The distribution waterfall sets the rules and procedures for the distribution of profits. The hurdle rate, or preferred return, is the rate of return that must be distributed to the limited partners before general partners can earn any incentive fees. A soft hurdle rate allows the general partner to share in all profits if the performance of the fund is above the hurdle rate. A hard hurdle rate allows the general partner to share only in profits in excess of the hurdle rate. Carried interest is an incentive fee equal to the percentage split of profits that general partners earn after meeting the minimum hurdle rate and is paid on top of management fees. Catch-up provisions give the general partner a larger distribution of the profits upon passing the hurdle rate. The catch-up rate is the percentage of profits that will be distributed to the general partner to catch up to the incentive fee once the hurdle rate is surpassed. The fund-as-a-whole carried interest arrangement calculates the carried interest on the performance of the entire fund. In contrast, a deal-by-deal carried interest arrangement pays the general partner profits on each investment, independent of the performance of other investments. A deal-by-deal carried interest arrangement is advantageous for the general partners but not for the limited partners. A fund-as-a-whole carried interest arrangement is more protective of the limited partners but might dilute the ability of the firm to attract talented general partners. Incentive fees are similar to the payoffs of a long call option. The general partner can earn extremely high returns, similar to the payoffs of a deep in-the-money option. The hurdle rate is analogous to the "strike price." The higher the hurdle rate, the lower the value of the call option. Incentive fees, like call options, become more valuable as greater risk is experienced by the fund. Therefore, incentive fees can create perverse incentives for the management of the fund. Vesting denotes the process and timetable by which the general partners are legally transferred their incentive payments. A clawback clause is a provision whereby limited partners have the right to reclaim incentive fees from the general partner. A clawback is the opposite of vesting. Management fees are fees paid to the general partner to cover basic fund operating costs such as salaries, research, travel, rent, and utilities and are paid regardless of the performance of the fund.
LO 2.6.6.2: Demonstrate knowledge of single-factor asset pricing models and ex ante pricing.
The ex ante form of the capital asset pricing model (CAPM) is an equilibrium model that derives the expected return on a stock given the expected return on the market portfolio, the stock's beta coefficient, and the risk-free rate: E(Ri) = Rf + βi[E(Rm) - Rf]. The ex post CAPM focuses on realized returns: Ri,t - Rf = βi(Rm,t - Rf ) + εi,t. The CAPM implies that the expected return on any asset is solely determined by its systematic risk (beta) and that no additional expected return will be earned by bearing non-systematic or idiosyncratic risk. The CAPM also implies that all investors should use a two-fund strategy of investing a portion in the risk-free fund and the remainder in a well-diversified risky fund. Asset pricing models can be used to separate risks and returns into diversifiable (idiosyncratic) and non-diversifiable (systematic) components and to quantify the compensation expected to be received for risk.
LO 2.8.8.7: Demonstrate knowledge of statistical methods for locating alpha.
The four steps in hypothesis testing are: State the hypothesis. Form an analysis plan. Analyze sampled data. Interpret the results. The null hypothesis is the statement that the analyst attempts to reject. The alternative hypothesis is the opposite claim of the null hypothesis, and it represents the behavior that exists if the null hypothesis is false. The null hypothesis is examined using a test statistic, which is a function of the observed values of the random variables of interest. Large test statistics indicate sampled data are far from expected, in which case the null hypothesis is rejected. If the test statistic is not large, then we fail to reject the null hypothesis. The p-value equals the probability of observing a sample estimate as extreme as the one observed, assuming the null hypothesis is true. The significance level denotes the probability that a significant result may be due to random chance. The confidence level equals 100% minus the significance level. Statistical tests do not offer evidence to prove a null hypothesis. Statistical tests are designed to disprove rather than to prove the null hypothesis statement. Outcomes with lower p-values often are mistakenly interpreted as having a stronger relationship. Statistical significance is often mistaken for economic significance. The test statistic might exceed its critical value because the standard error is small, not because the estimate is large. Economic significance describes the extent to which a variable in an economic model has a meaningful impact. The confidence level is misinterpreted to equal the probability that a relationship exists. In reality, the probability that a true relationship exists, even if the statistical test indicates one exists, generally is unknown. A type I error occurs when wrongly rejecting a true null hypothesis. A type II error occurs when failing to reject an untrue null hypothesis.
LO 2.8.8.4: Demonstrate knowledge of return attribution.
The primary goal of return attribution analysis is to attribute returns to systematic risk (beta), skill (ex ante alpha), and idiosyncratic risk (luck). Model misspecification refers to the use of models that do not properly capture systematic and idiosyncratic risks. Alphas will be misdiagnosed in the presence of model misspecification. Types of model misspecification include misestimated betas, nonlinear relationships, and omitted or misidentified factors. Beta nonstationarity refers to the tendency for beta to shift over time. Nonstationarity of beta raises a key question regarding whether a fund's superior return is attributable to alpha or beta. Examples of beta nonstationarity include the following: Beta creep refers to gradual increase in beta over time. Beta expansion refers to increases in beta as market conditions change. Market timing refers to attempts of the fund manager to alter the fund beta in anticipation of changes in market conditions. Alpha and beta effects are difficult to disentangle. In most cases, performance is attributable to a commingling of alpha and beta.
LO 2.4.4.2: Demonstrate knowledge of moments of return distributions (i.e., mean, variance, skewness, and kurtosis).
The shape of a probability distribution is characterized by its raw moments and central moments. The first raw moment is the mean of the distribution. The second central moment is the variance. The third central moment divided by the cube of the standard deviation measures the skewness of the distribution, and the fourth central moment divided by the fourth power of the standard deviation measures the kurtosis of the distribution. Skewness for the normal distribution equals zero. If skewness is positive, then the distribution is elongated to the right. If skewness is negative, then the distribution is elongated to the left. Kurtosis of the normal distribution equals 3. Positive values of excess kurtosis indicate a distribution that is leptokurtic (i.e., more peaked with fat tails), whereas negative values indicate a platykurtic distribution (i.e., less peaked with thin tails).
LO 2.6.6.1: Demonstrate knowledge of the concept of informational market efficiency.
The three forms of market efficiency include weak form (security prices fully reflect all available security data on past prices and volumes), semistrong form (security prices fully reflect all publicly available information) and strong form (security prices fully reflect all publicly and privately available information). Six factors that result in improved informational market efficiency are: (1) larger asset values, (2) higher frequency of trades, (3) little or no trading frictions, (4) low levels of regulatory constraints, (5) easier access to quality information, and (6) lower levels of uncertainty about asset valuation.
LO 2.6.6.5: Demonstrate knowledge of the term structure of forward contracts.
There are four cost-of-carry model scenarios for financial securities. In a simple scenario with no interest costs and no dividends, there are no differences in forward prices and all forward prices equal the spot price. This results in a flat term structure of forwards. The interest rate equals the dividends rate, but both are positive. This also results in a flat term structure of forwards. The interest rate exceeds the dividend rate. When that happens, the no-arbitrage forward price must exceed the spot price and the term structure of forward prices is upward sloping. The dividend rate exceeds the interest rate, and the no-arbitrage forward price is lower than the spot price. The term structure of forward prices is downward sloping.
LO 2.8.8.3: Demonstrate knowledge of empirical approaches to inferring ex ante alpha from ex post alpha.
There are two steps when analyzing the ex post alpha to estimate ex ante alpha: Identify the appropriate ex post asset pricing model or benchmark. Statistically examine the ex post alpha to determine the extent to which the alpha is attributable to luck or skill. Potentially confounding challenges to the empirical method for determining ex ante alphas are invalid inferences, non-normality of returns, and sample selection biases.
LO 2.8.8.9: Demonstrate knowledge of statistical issues in analyzing alpha and beta.
If returns are normally distributed and the null hypothesis of zero alphas is correct, then the percentage of abnormal performers should equal the significance level. However, if returns are non-normal, then the percentage of funds with abnormally high alphas may be higher or lower than the percentage predicted by the normal distribution. Spurious correlation is correlation that does not result from a true or direct relationship. A spurious correlation is coincidental or idiosyncratic and is limited to the set of observations being examined, causing the correlation to vary over time. Causality refers to a reliable and direct cause-and-effect relationship among variables. Causality exists when one variable at least partly determines the value of another variable. Models that do not properly account for nonlinearities are misspecified and can lead to erroneous conclusions regarding correlation and causality. Correlation is a measure of the strength of linear relationship between variables. If a nonlinear relationship exists, the correlation between the untransformed variables will be low. A potentially powerful relationship indeed exists, which is misdiagnosed by the misspecified model. Beta estimation is affected by the choice of factors used in the model. If thousands of tests are performed, hundreds of factors might seem statistically significant merely by chance. Therefore, significant factors might be found where no true relationship exists. Lessons regarding alpha estimation include the following: Returns should be compared against a proper risk-adjusted benchmark rather than compared against each other. Alpha calculations are only as reliable as the asset pricing model used to estimate performance. The probability that a fund alpha is nonzero, given that the test indicated a statistically significant alpha, is generally unknown. Lessons regarding beta estimation include the following: The probability that a fund beta is nonzero, given that the test indicated a statistically significant beta, is generally unknown. Using a linear regression model, a zero beta does not necessarily imply a lack of relationship; a zero beta only implies a lack of linear relationship. A statistically significant beta does not necessarily imply causality; the prices of two variables might be linked through inflation or some other common factor, while no causality exists in the relationship between the two price variables.
LO 2.9.9.3: Demonstrate knowledge of dynamic risk exposure models.
Nonlinear models examine nonlinear relationships between dependent and independent variables. Dynamic risk exposure models examine nonlinear relationships caused by factor risk exposures that change over time. Three dynamic risk exposure models are: dummy variable regression model (using up market and down market beta), separate regressions model (overlapping regressions to estimate beta for each regression period), and the quadratic curve regression model (or dynamic risk exposure model; excess returns for the asset are regressed against the square of the market excess returns resulting in a quadratic curve.
LO 2.7.7.4: Demonstrate knowledge of the limitations of the CAPM approach for analysis of alternative investments.
The CAPM fails to account for common alternative investment characteristics such as multiperiod non-stationarity, non-normality of returns distributions, and investment illiquidity.
LO 2.6.6.7: Demonstrate knowledge of option pricing models.
The basic option pricing model values an option on a portfolio that contains both long and short positions. The option gives the option holder the right to either purchase the entire portfolio or walk away and let the option expire. Po = PlN(d) − PsN(d − v) The Black-Scholes call option formula values a call option as a function of the price of the underlying security, the option strike price, the volatility of security returns, the option's time to expiration, and the risk-free rate: c = SN(d1) − e−rTKN(d2) A variation of this formula is the well-known Black-Scholes put option formula. The Black forward option pricing model replaces the underlying security in the Black-Scholes call option formula with the forward contract. c = e−rT[FN(d1) − KN(d2)] In the currency option pricing model, there are two risk-free interest rates that correspond to the two currencies that are exchanged. In other words, the model prices an option that gives the right to exchange S* units of one currency with its associated r* risk-free interest rate, for S units of another currency at r risk-free interest rate: option price = e−r TS∗N(d1) - e−rTSN(d2)
LO 2.3 .3.3: Demonstrate knowledge of the internal rate of return (IRR) approach to alternative investment analysis.
The internal rate of return (IRR) is the discount rate that equates the present value of an investment's cash inflows to the present value of the investment's cash outflows. There are four types of IRRs. The lifetime IRR is the IRR if all of the cash flows are available from start to finish during the investment. The interim IRR is the IRR that assumes an appraised terminal value for investments. The point-to-point IRR is the IRR if the time 0 and time T cash flows are appraised values or other cash flows during the investment's lifetime. The since-inception IRR is used to determine the performance of funds rather than investments.
LO 2.8.8.5: Demonstrate knowledge of ex ante alpha estimation and return persistence.
Abnormal return persistence refers to the tendency of idiosyncratic performance to be positively correlated over time. A positive correlation suggests that most superior returns are attributable to skill, rather than to luck.
LO 2.6.6.8: Demonstrate knowledge of option sensitivities.
"Option Greeks" measure option sensitivities to four underlying factors: the underlying security, the return volatility of the asset, time to expiration, and the risk-free interest rate. The most widely used sensitivities are delta, vega, theta, rho, and gamma. Delta measures the sensitivity of the option price to changes in the price of the underlying security. Vega measures the sensitivity of the option price to changes in the price volatility of the underlying security. Theta measures the sensitivity of the option price to changes in time to expiration. Rho measures the sensitivity of the option price to changes in the risk-free rate. Gamma measures the rate of change in delta relative to changes in the price of the underlying security.
LO 2.7.7.1: Demonstrate knowledge of benchmarking and its role in the analysis of risk and return of investments.
Benchmarking is the process of identifying the appropriate index against which a portfolio's performance is evaluated. For any fund, the appropriate benchmark is one that matches the fund's objectives and constraints. A benchmark is often selected by an external investor or analyst. Benchmarks might be formed based on peer groups or on indices. A peer group is a group of funds with objectives and constraints that are similar to the fund under examination. Many indices exist, but most are value-weighted averages of the index components. The MSCI World Index and Russell 2000 are two popular equity benchmarks.
LO 2.3 .3.4: Demonstrate knowledge of problems with the use of IRR in alternative investment analysis.
The IRR may not be correct when examining complex cash flow patterns that involve borrowing type cash flow patterns or multiple sign change cash flow patterns. Difficulties also arise when comparing IRRs across investments with scale differences, which are differences in timing of cash flows, differences in investment size, or both. IRRs also are problematic when aggregating the results of several funds because the portfolio IRR is not simply a weighted average of the component IRRs. The IRR calculation assumes all cash flows are reinvested in the original investment and earn a return equal to the original investment's IRR. In contrast, in the modified IRR calculation, the investment's cash inflows are compounded at an assumed reinvestment rate, and the investment's cash outflows are discounted at an assumed financing rate, which offers a more realistic measure of the performance of the investment. The dollar-weighted return is the investment's IRR, taking into account all cash inflows and outflows. In contrast, the time-weighted return is an averaged return that ignores the effects of the timing of cash distributions or withdrawals.
Normal Distribution
○ Normal curve is symmetrical ○ 2 halves are identical ○ Doesn't hit zero but gets close ○ Mean = mu ○ Width = standard deviation = sigma ○ Values often thought as distance from the mean in standard deviations § Z value...how many standard deviations away from the mean
