CAS Exam 9
asset allocation vs security selection
asset allocation: the allocation of a complete portfolio to the various asset categories. Within each category of assets, investors can select specific securities in order to try increase return: this is known as security selection.
Arbitrage Pricing Theory
An Arbitrage opportunity is the opportunity for an investor to earn a riskless profit without the need to make a net investment. Due to the actions of arbitrageurs, the Law of One Price exists: this states that two assets that are equivalent in all economically relevant aspects should have the same market price.
Advantage of using a broad index (like the S&P 500) for the single-index model
An advantage of using a broad index like the S&P500 is that a lot of data is available to estimate the parameters that are needed.
Main Utility function
Utility is a measurement of "benefit" U = E(r) - 0.5Aσ2 where A is the investor's risk aversion This produces the certainty equivalent rate: the rate that a risk free investment would need to offer to provide the same level of utility as the investment being analyzed.
single-index model
a version of the single-factor model, where the return on an index (eg S&P500) is used as a proxy for the common factor (m)
The Sharpe ratio of an optimally constructed risky portfolio is calculated using the equation:
(SP)^2 = (SM)^2 + [αA/σ(eA)]^2 Where SM is the Sharpe ratio of the passive market index The term αA/σ(eA) is the information ratio: it represents the contribution of the active portfolio, when held at its optimal weight, to the Sharpe ratio. To maximize the Sharpe ratio of the portfolio, it is necessary to maximize the information ratio. In order to achieve this, the investment in each security needs to be in proportion to αi/σ^2(ei), while keeping the total investment in the active portfolio equal to w*A: w*i = w*Ax[αi/σ^2(ei)]/[∑αi/σ^2(ei)]
The equation to determine the total variance (risk) of a portfolio of equally weighted assets is:
(σp)^2 = (1/n)σ¯^2 + [(n - 1)/n]Cov¯ where σ¯ if firm specific risk and Cov¯ is systematic risk
The Markowitz model (described in BKM7) has some problems:
- As the number of securities increases, the number of variables that need to be calculated/ estimated increase dramatically. - Due to the large number of required estimates, it is likely that some variables are estimated incorrectly. In this case, the model may produce nonsensical results
There are 2 ways to test models:
- Normative: tests model assumptions - Positive: examines the predictions
Market Structure assumptions are as follows:
-All assets are publicly traded & short positions are allowed. Investors can borrow/ lend at rf (the assumption that assets are tradable is necessary for investors to be able to derive identical input lists. The assumption about the interest rates is necessary for investors to derive the same tangency portfolio ) -All information is publicly available -No taxes (if investors had different tax rates, they would earn different after tax rates on the same stock, and could therefore derive different after tax optimal risky portfolios) -No transaction costs
Tests can be constructed for each version of EMH: weak, semistrong and strong. Examples of these tests are as follows: Strong-Form Tests (Inside Information)
-Markets are not expected to be strong-form efficient: people with inside information should be able to make superior profits
An alternative argument to justify the movement of prices to the equilibrium level is the "risk return dominance" argument. There are some key differences between this and the arbitrage argument:
-Risk Return Dominance argument: if prices differ from the equilibrium level, many investors will make limited changes to their portfolios, depending on their degree of risk aversion. The culmination of the relatively small transactions of many investors produces sufficient volumes to move the market price. -Arbitrage argument: the investor who discovers the arbitrage opportunity will want to maximize his position in order to maximize profits. A lower number of investors are therefore needed to move prices back to equilibrium.
Derivation of CAPM
-The contribution of an individual stock to the portfolio risk premium is: wi[E(ri) - rf] - The contribution of an individual stock to the portfolio variance is wiCov(ri, rM) The reward-to-risk ratio for the stock can therefore be calculated as: Ratio = Contribution to risk premium / Contribution to variance = [E(ri) - rf]/Cov(ri, rM) The reward-to-risk ratio for the market portfolio is known as the market price of risk: Market Price of Risk = [E(rM) - rf] / σM^2 Notice that the denominators of the reward-to-risk ratios are different: - The contribution of an individual asset to the variance of the portfolio variance is based on the asset's covariance to the market - If we look at the entire market, it is more appropriate to use the variance. Another way to look at this using the covariance of the market with itself, which is its variance.
There are 2 key implications of CAPM:
-The market portfolio is efficient -The premium on a risky asset is proportional to its beta
Tests can be constructed for each version of EMH: weak, semistrong and strong. Examples of these tests are as follows: SemiStrong-Form Tests (Market Anomalies)
-These test investigate whether publicly available information beyond the trading history can be used to generate abnormal returns. -Empirical results show that some easily available diagnostics can be used to predict abnormal returns (efficient market anomalies). -The efficient market anomalies include: Small-Firm-in-January Effect: small firms have historically generated superior returns, particularly in January. Neglected Firm Effect: Some firms are neglected by large investors, and so less information is available. A premium is required to compensate for the risk associated with less information. Note that this may explain the superior returns of small firms discussed above. Book-to-Market Ratios: High Book-to-Market firms have historically outperformed the rest of the market. Post-Earnings-Announcement Price Drift: The cumulative abnormal return of stocks has been shown to continue to increase even after the information about the event becomes public.
If can not borrow at rf, use a kinked CAL:
-the slope to the left of P will be unchanged from the regular CAL -the slope to the right of P=[E(rp)- rfB]/σP where rfB is the borrowing rate
The single index can be used to derive the return of a security using the following steps:
1. Estimate the risk premium and risk of the market index via macroeconomic analysis 2. Calculate the beta and residual variance, σ^2(ei), of each security, using statistical analysis 3. Calculate the expected return of the security (based solely on the market return) 4. Calculate alpha using security analysis
Biases that cause information processing errors:
1. Forecasting Errors: This consists of 2 parts: -too much weight is assigned to recent experience -forecasts are too extreme given the actual level of uncertainty If a firm has had recent strong performance, the forecasts of future earnings may be too high. This will lead to an overstated P/E, resulting in poorer subsequent performance once investors realize their error (P/E effect) 2. Overconfidence: Many investors overestimate their abilities. 3. Conservatism: Investors are too slow to update their beliefs in response to new evidence. 4. Sample Size Neglect & Representativeness: Investors do not account for sample size, and therefore may infer a pattern based on too small a sample.
Behavioral Biases: Behavioral biases that cause suboptimal financial decisions include:
1. Framing: Decisions are often materially impacted by how the question is framed 2. Mental Accounting: Investors may segregate decisions. For example, they may segment their investment portfolios into pieces which go towards different goals, and they will have different risk attitudes to each piece. Rationally, it would be better to optimize the risk-return profile of the entire portfolio in aggregate. 3. Regret Avoidance: Investors who make decisions that turn out badly have more regret if it were an unconventional decision. 4. Affect: The "affect" is the good or bad feeling that investors may associate with investing in a stock. For example, stocks of firms that practice socially responsible policies may generate higher affect. If investors prefer these stocks, they may drive up the prices, reducing the expected return. In fact, the stocks ranked high on Fortune's survey of most admired companies have tended to produce lower risk adjusted returns. 5. Prospect Theory: This modifies the standard financial theory's definition of risk averse investors. -under standard theory, utility depends on the level of wealth -under prospect theory, utility depends on the changes in wealth. Again, there are diminishing returns of utility as wealth increases. In addition, it shows that investors become risk seeking as they begin to lose money
There are a number of reasons why it is difficult to determine if the markets are truly efficient:
1. Magnitude Issue: Assume an investment manager of a large portfolio improves the performance of a huge portfolio by 0.1% per year. Even though this will be a big contribution in dollar terms, it is very small compared to the normal volatility of the market. It is therefore hard to assess how much the manager actually contributed. 2. Selection Bias Issue: Once an investment scheme becomes known by others, it will no longer generate abnormal returns. Therefore, only schemes that do not work will be reported. It is possible that techniques that do actually work exist, but are not being reported. 3. Lucky Event Issue: There are several cases of investors who have made huge investment returns over a period. However, this does not disprove EMH, because the number of investors is so large, just by chance, some have to make huge returns.
Tests can be constructed for each version of EMH: weak, semistrong and strong. Examples of these tests are as follows: Weak-Form Tests (Patterns in Stock Returns)
1. Returns over short horizons -Looks at whether investors can use historic trends to earn abnormal profits over the short term, by measuring the serial correlation of stock market returns -Empirical results show only small correlations over short periods (with the exception that the sectors with the best and worst returns exhibit stronger correlations, which is known as the momentum effect) 2. Returns over long horizons -Similar to the prior test, but looks at the long term returns -Empirical results have shown a negative correlation over the long term: this may be explained by the fads hypothesis -if the negative correlation is correct, a contrarian investment strategy may be profitable
Arbitrage Pricing Theory (APT) is based on the following 3 assumptions:
1. Security returns can be described by a factor model 2. There are a sufficient number of securities to diversify away idiosyncratic risk 3. Well functioning securities markets do not allow for the persistence of arbitrage opportunities
Bodie Kane and Marcus account for privately held businesses by dividing them into 2 categories
1. Those with similar characteristics to traded assets: Owners of these businesses can still achieve diversification by reducing their portfolio holdings of similar traded assets, and so they will still essentially hold the market portfolio. There is therefore little impact to CAPM. 2. Those that do not have similar characteristics to traded assets: owners of these businesses would demand a portfolio of traded assets that hedge the risk of a typical private business. The price of these hedge assets will be bid up, reducing the expected return (relative to the systematic risk), which will cause them to appear to have a negative α according to the traditional CAPM. Securities that are highly correlated with the risk will appear to have a positive α.
There are 3 versions of the EMH:
1. Weak form: stock prices reflect all information that can be derived from examining market data (past prices/volume/ etc) 2. Semistrong form: stock prices reflect all publicly available information about the firm's prospects 3. Strong form: stock prices reflect all information relevant to the firm, including that not publicly available (inside information)
Some parties believe that the results of EMH tests are signs that the markets are not efficient. Others believe that the markets are indeed efficient, but that the results of the tests arise because:
1. the properties are proxies for fundamental determinants of risk 2. the properties arise just due to data mining. One supporting factor for this argument is that many of these anomalies disappear after discovered. One way to test for this is to see if the relationship holds in a different database.
Mean-variance criterion:
A dominates B if: E(rA) ≥ E(rB) and σA ≤ σB
The APT and CAPM
APT is based on a foundation of well diversified portfolios (resulting in 0 residual risk). However, even large portfolios have non-negligible residual risk. If residual risk is small, the APT SML should still approximate the risk premium. The deviations should be unbiased, and also uncorrelated with beta or the residual SD. However, if residual risk is sufficiently high, we cannot have full confidence in the APT. Despite the above, APT is still valuable. For example, it does not rely on CAPM's assumption that investors are mean-variance optimizers. Instead, we just need a small number of arbitrageurs that look for arbitrage opportunities. A big advantage of APT is that it does not require an all inclusive portfolio. As discussed, CAPM cannot be tested as it requires this unobservable true market portfolio.
(Arbitrage Pricing Theory): Multifactor Model
Adding additional factors to the single-factor model produces a multifactor model. The advantage of this over the index model is that the varying sensitivities of securities to the different factors can be incorporated. An example of a multifactor model is a two-factor model: ri = E(ri) + βi1F1 + βi2F2 + ei Where βij is the sensitivity of the share's return to the jth factor
In order to use the single index model in practice, we need estimates of the α, β and σi of each stock, in addition to RM and σm^2. This is significantly less than the number of estimates needed in the Markowitz model
Another advantage of the single index model over the Markowitz model is that the single index model allows for specialization of effort in security analysis: for example, it is possible for one group to specialize in the computer industry, and another in the auto industry, as there is no need for anyone to estimate the covariance between stocks from the two different industries.
The 4 factor model was also used to determine if there was consistency in mutual fund performance. There only appears to be minor persistence, but a lot of this is due to expenses and transaction costs.
Another test of the persistence in performance using the 4 factor model was conducted by Bollen & Busse. Mutual fund performance was ranked to decile. Performance during the following quarter was then measured. The exhibit below demonstrates that there is only small persistence that not sufficient to justify performance chasing.
Active vs Passive Portfolio Management
As mentioned above, according to EMH, only unique insight will produce profits. It takes a lot of effort and cost to gain this unique insight, and therefore it is usually only feasible for investors with large portfolios. For smaller investors, on the other hand, the costs will exceed any potential benefits, and they therefore will need to resort to alternatives: 1. Invest in mutual funds (pooling of several investor's portfolios) 2. Passive investment Advocates of EMH would recommend the latter, as they believe that active management should not produce excess returns. Note that passive investors will trade much less often than active because according to EMH, stocks are priced fairly, which means it is pointless to buy and sell frequently.
why risk pooling can increase risk for investment portfolio
As the investor increases the risk pooling to include n assets, both the Sharpe ratio and standard deviation will increase by n^0.5. Therefore, it can be seen that risk pooling does not reduce the level of risk. In order to reduce the risk, risk pooling alone is insufficient. The investor will also need to engage in risk sharing.
Summary of the Optimization Procedure
As we saw above, the optimal risky portfolio can be created using the above steps: 1. Calculate the ratio of each security of the active portfolio: wi0 = αi / σ^2(ei) 2. Scale the above weights so total will equal 1: wi = wi0 / ∑wi0 3. Calculate the alpha, beta & residual variance of the active portfolio: αA = ∑wiαi βA = ∑ wiβi σ^2(eA) = ∑ (wi^2)x(σ^2(ei)) 4. Calculate the weight of the active portfolio: (wA)^0 = [αA/(σ^2)(eA)]/[E(RM)/(σM)^2] w*A= [(wA)^0]/[1 + (1 - βA)(wA)^0] 5. Calculate the weights of the market and each security in the optimal portfolio w*M = 1 - w*A; and w*i = w*A wi 6. Calculate the risk premium and variance of the optimal portfolio: E(Rp) = (w*M + w*A βA) * E(RM) + w*A αA σP^2 = [(w*M + w*A βA)^2] x (σM)^2 + [w*A σ(eA)]^2
Optimal Risky Portfolio of the Single Index Model
Assume that a portfolio manager has to construct an optimal risky portfolio from n actively researched firms, and a market index portfolio (S&P500). The input list includes estimates of: Risk premium of the S&P500 SD of the S&P500 Beta coefficients, stock residual values & alpha values of all n stocks The following equations can then be used: αp = ∑ wiαi βp = ∑ wiβi σ^2(ep) = ∑ (wi^2 x σ^2(ei) ) The manager will select weights that maximize the Sharpe ratio, E(Rp)/ σp, where: E(Rp) = ∑ wiαi + E(RM) * ∑ wiβi σp= [(σM)^2 (∑ wiβi)^2 + ∑ ((wi^2)x(σ^2)(ei))]^(1/2) The initial weight in the active portfolio is: (wA)^0 = [αA/(σ^2)(eA)]/[E(RM)/(σM)^2] the final weight is (i.e. not assuming beta=1): w*A= [(wA)^0]/[1 + (1 - βA)(wA)^0]
(BKM 8 Index Models) Tracking Portfolio
Assume that a portfolio manager has used the index model to derive an equation for the return of a security. She ideally would like to take advantage of her security analysis, while earning a return that is independent from the market return. To accomplish this, she can use a tracking portfolio, which will match the systematic component of P's return.
Behavioral Critique
Behavioral finance differs from conventional financial theory in that it accounts for how real people make decisions, including the irrationalities that influence decision making. There are 2 categories of irrationalities: 1. Investors do not always process information correctly, and therefore derive incorrect probability distributions 2. Even when investors do generate a correct probability distribution, they still often make inconsistent/ suboptimal decisions due to their behavioral biases.
(BKM 8 Index Models) Predicting Betas
Betas tend to change over time. It may therefore make sense to create a forecasting model for Beta. One option is to use regression: Current β = a + b * (Past β) A more sophisticated approach could be to incorporate other financial variables that may impact β. For example, Current β = a + b1 * (Past β) + b2 * (Firm Size) + b3 * (Debt Ratio) Other potential variables that can be used include: - Variance of earnings - Variance of cash flow - Growth in earnings per share - Market capitalization - Dividend yield - Debt to asset ratio
Extensions of the CAPM: CAPM with Non-Traded Assets & Labor Income
CAPM makes the assumption that all risky assets are traded. In reality, there are several assets not traded. 2 examples: - Human capital - Privately held businesses these can have a material impact on the equilibrium returns. Labor income (human capital) is a lot more difficult to hedge. One of the only options is for employees to avoid purchasing shares of their own employer, or companies from their own industry. As a result, securities offered by labor intensive firms will have a lower demand, and may appear to have a positive α according to the trad CAPM. An adjusted CAPM that accounts for labor income: E(Ri) = E(RM)x[Cov(Ri, RM) + (PH/PM)Cov(Ri, RH)]/[σM^2 + (PH/PM)Cov(RM, RH)] Where: PH = value of aggregate human capital PM = market value of traded assets the standard β is replaced by an adjusted β that also accounts for the cov with the portfolio of agg human cap. If Cov(Ri, RH) is positive, the adjusted β will be greater when the standard β is less than 1, and vice versa. Therefore, this model produces a SML that is less steep than the standard CAPM. This model may explain the negative alpha of high β securities indicated by some tests.
Even though both the SML & CML show the relationship between risk and return, there are differences between the two:
CML: graphs risk premiums of efficient portfolios (made up of the market and risk free assets) as a function of σ, because σ is the appropriate risk measure for portfolios SML: graphs risk premiums of individual assets as a function of β, because β is the appropriate risk measure for individual securities held as part of a well diversified portfolio
Indifference Curves
Curves containing portfolios with equivalent utility levels. The optimal portfolio is located at the intersection point of the CAL, and the curve tangential to the CAL.
Resource Allocation in relation to Efficient Markets
Deviations from market efficiency will generate a cost to everyone: inefficient resource allocation. This arises because securities are mispriced. One example is that firms with overpriced securities will be able to obtain capital too cheaply.
CAL Equation (Capital allocation line)
E(rc) = rf + σc([E(rp) - rf]/σp) Where c represents the complete portfolio and p represents risky assets The slope of the CAL is known as the Sharpe, or reward-to-volatility ratio [E(rp) - rf] / σp
(Markowitz Model) Once the investor has derived the weights to invest in the different assets, the expected return and variance of the optimal complete portfolio can be derived:
E(rp) = Σ wiE(ri) (σp)^2 = Σ Σ wiwjCov(ri, rj)
Advantages and disadvantages to the opt risky portfolio of the single index model
However, the advantage of using the Optimal Risky Portfolio of the Single Index Model is that by using security analysis, he can hopefully identify assets with non zero alphas, and over/underweight these relative to the market in order to increase returns. The disadvantage of this is that it will introduce firm specific risk.
Dealing with difficulty 1 of Event Studies: Leakage
If leakage of information occurs, prices may begin to move before the event. If this is the case, it may be more appropriate to measure the impact of the event by referring to the cumulative abnormal return, beginning at a time period before the actual event is made public.
Role of Portfolio Management in Efficient Market
If markets are indeed efficient, an argument can be made that portfolio management is not necessary, as stocks are fairly priced. However, portfolio management can actually still be beneficial. The text cites three uses of portfolio management: 1. Diversification: selects a diversification strategy to eliminate firm-specific risk 2. Reflects tax considerations of the individual investor 3. Adjusts portfolio to reflect the unique risk profile of the investor
Non normal returns and optimal portfolio allocation
If the returns are more heavy tailed than a normal distribution would imply, it may be more appropriate to reduce the allocation to the risky portfolio than you would under the assumption of normal returns
(Arbitrage Pricing Theory): Multifactor SML: Factor portfolios
In order to calculate the risk premiums for the risk factors, a factor portfolio can be used. This is a portfolio designed to have a β of 1 to the factor for which the risk premium is being measured, and a β of 0 on all other factors.
Individual behavior assumptions of CAPM
Individual behavior assumptions are as follows: 1. Investors are rational mean-variance optimizers (investors are only concerned about mean and variance, and are not concerned about the correlation of the asset) returns with inflation/ prices of consumption items 2. Their planning horizon is a single period: (similar to above, longer periods would result in extra-market risk factors. i.e. a change in interest rates may decrease investor's income.) 3. Investors use identical input lists (due to the market assumption that all info is public)
-Assume that a single factor market exists, where the well diversified portfolio, M, represents the market factor, F -Assume that we have identified a well diversified portfolio P, that has a positive alpha How would you construct a zero beta portfolio, Z from P and M.
Invest weight wp in P: βZ = wpβp + (1 - wp)βM = 0 -For this to be zero beta, wp = 1 / (1 - βp) wM = 1- wp = -βp / (1 - βp) Portfolio Z is riskless, as there is no residual risk (the component portfolios are well diversified), and the systematic risk is 0 (it was constructed to have zero beta). αZ = wpαp + (1 - wp)αM = wpαp Since the beta of Z is 0, E(RZ) = wpαp = αp / (1 - βp) The risk premium of Z must be 0, as Z is riskless. Otherwise it would be possible to make an arbitrage profit. For example: - If βp < 1, borrow money and invest the proceeds in Z, to make a risk free profit with zero net investment - Similarly if βp > 1, sell Z short, and invest the proceeds at the risk free rate
Disadvantage of single index model
It oversimplifies the true uncertainty: it simply divides the uncertainty into micro vs macro risk. It ignores things like the correlation between security returns. It also ignores industry events: factors that impact many firms within an industry, without materially impacting the overall economy. In the case where several stocks have correlations not accounted for by the M factor, it may be more appropriate to use a multi index model Therefore if the universe of securities from which we will construct a portfolio is small, it is possible that the two models produce different optimal portfolios: the optimal portfolio constructed by the single model will be inferior, because of the correlations that it ignores.
(Single-index model) regression equation for the premium Ri(t)
Letting the market index be M, the authors construct the regression equation for the premium, Ri(t): Ri(t) = αi + (βi)RM(t) + ei (t) Where α is the expected excess return of the security when the market excess return is 0. α will arise if the stock is underpriced. Taking the expected value of the above equation: E(Ri) = αi + (βi)E(RM) As can be seen above, the components of the risk premium include: - Systematic risk premium, (βi)xE(RM) - Nonmarket premium, αi
Equilibrium
Market prices should change to eliminate arbitrage opportunities: an investor that identifies an arbitrage opportunity will take a large position in the security that will drive the price up or down until the opportunity disappears. Prices are said to have reached an equilibrium level.
To determine to the degree to which diversification benefits a firm, risk needs to be segmented into 2 categories:
Market/ Systematic/ Nondiversifiable Risk: the risk that can not be diversified away Unique/ Firm-Specific/ Nonsystematic/ Diversifiable: the portion of the risk that can be eliminated via diversification
Extensions of the CAPM: Zero-Beta CAPM
One of the interesting properties of efficient frontier portfolios is that each portfolio on the efficient frontier has a portfolio (the zero-beta portfolio) on the inefficient side of the minimum variance frontier with which it is uncorrelated. Use if investors face borrowing restrictions of the risk free asset: E(ri) = E(rZ) + βi[E(rM) - E(rZ)] where Z is the zero-beta portfolio for M. Investors who face borrowing restrictions will invest more heavily in high beta stocks, and less in low beta. The price of high beta stocks will therefore increase. The SML in this case will be smaller than the SML of the regular CAPM. The risk premium on the market portfolio is smaller, as the expected return on the zero beta portfolio will exceed the risk free rate. As a result, there is less reward for bearing risk.
Investment for the Long Run
People believe that extending investments across time will reduce risk due to possible diversification benefits (time diversification). We can think of extending an investment horizon for an additional period to be the same as adding an additional risky asset to a pool. For example, is a 2yr investment safer than a 1yr investment? Holding the portfolio for the 2nd year simply results in more risk (risk pooling). Therefore this time diversification is not true diversification. In order to increase the Sharpe ratio, while still maintaining the same level of risk, the 2yr investor will need to halve the amount invested in the risky asset over the 2yrs.
Risk Sharing
Risk sharing involves taking a fixed amount of risk, and sharing it among several investors. This way, the investor can benefit from the higher Sharpe ratio achieved from risk pooling, without having to increase the exposure to risk.
The separation principle dictates that there are 2 steps in portfolio selection:
Selection of the optimal risky portfolio: this step is purely technical Allocation between risk free vs risky assets: this will depend on the level of risk aversion
(BKM 11) What are Event Studies and the difficulties associated with them?
Since price changes reflect new information, it is possible to determine the importance of an event by measuring the resulting price changes (this is known as an event study). There are two difficulties associated with these studies: 1. the stock price may respond to a wide range of economic news in addition to the specific event 2. information about the event may be leaked prior to the actual event.
Mutual Fund and analyst Performance: Stock Market Analysts
Stock market analysts typically work for brokers, and as a result, are usually positive in their assessments. Therefore, in order to assess their assessments, we need to look at either the relative strength of a recommendation compared to other analysts, or alternatively the change in recommendation of an analyst.
One argument supporting the lack of persistence in mutual fund returns is that funds that have good performance will attract new funds, driving the alpha down, due to the additional cost & complexity of managing the new funds.
Studies have also been conducted on bond funds. These indicate that the bond funds underperform the passive indexes by a magnitude roughly equal to expenses. Also, there is no persistence in performance. This would suggest that the bond market is efficient.
Technical Analysis is the search for predictable patterns of stock prices, which can be used to derive a profit from trading.
Technical analysts/ Chartists rely on a sluggish response of the stock price to fundamental factors, during which time the trend can be exploited. Examples of metrics include "resistance levels" (levels above prices should not rise) and "support levels" (levels below which prices should not fall). Technical analysis can only be successful if markets are not efficient.
The APT Portfolio Optimization in a Single Index Market (Treynor Black)
The APT indicates how to generate infinite profits if the risk premium of a well diversified portfolio differs from the SML. However, if the arbitrage portfolio is not perfectly well diversified, increasing the size of an investment to take advantage of an arbitrage opportunity will also increase the risk of the arbitrage position. Consider an investor who faces a single factor market, and who has identified an underpriced asset (or portfolio). The investor could adopt the Optimization Procedure discussed in BKM8 to construct an optimal risky portfolio. This procedure is said to follow a Treynor-Black (T-B) model. In that T-B model, if the residual risk of a portfolio was 0, the position in the portfolio would go to infinity. This procedure therefore has the same implication as APT: if there is no residual risk, you will take up to an infinite position. On the other hand, if residual risk does exist, the T-B model produces an optimal risky portfolio with a specific investment in the portfolio, which reflects the risk. APT however will ignore the level of residual risk. Therefore, the T-B has more flexibility than APT in reflecting the residual risk that may exist.
Capital Asset Pricing Model (CAPM)
The Capital Asset Pricing Model (CAPM) is an equation that shows the relationship between the risk and expected return of a security in equilibrium. It assumes that all investors optimized their portfolios using the Markowitz procedure, where they construct an efficient frontier based on all available risky assets, and identify an efficient risky portfolio P. Each investor faces an identical investable universe, and also uses the same input list to construct the efficient frontier. As a result, they would select the same weights for each risky asset. In addition to this, because the market portfolio is made up of the identical risky portfolios, it too will have the same weights. Thus, each investor will hold the market portfolio. This means that the CAL will also be the CML.
Security Market Line
The SML graphs the relationship between β & E(r) According to CAPM, the expected return at β=1 is equal to rM. The slope of the SML is rM - rf. If an asset is fairly priced, it will lie on the SML. An underpriced asset (positive α) will lie above the SML.
Extensions to the CAPM: CAPM with Liquidity Adjustments
The Standard CAPM ignores liquidity costs of trading like the Bid-ask spread/ Price impact. In reality, the security price should be discounted to reflect any illiquidity. The size of this discount will increase as trading costs increase. However, this increase in discount will not be proportional to the increase in the trading costs, due to the clientele effect: frequent traders tend to hold more liquid assets, and long term traders tend to hold less liquid assets. Investors are also concerned about liquidity risk (risk of unanticipated changes in liquidity). They are therefore going to demand additional return if there is a possibility that the stock will lose liquidity at a time when it is needed. The CAPM can be adjusted to include a liquidity beta: this reflects the sensitivity of the return of the security to changes in market liquidity.
Issues with CAPM assumptions that make extensions of the CAPM necessary
The assumption about no restrictions on short sales is one of the most problematic. In reality, short positions are not as easy to take as long positions: 1. There is no cap on the liability of short positions. A large short position will require significant collateral 2. There is a limited supply of stocks that can be borrowed by short sellers. 3. Many investment companies are prohibited from short sales. In addition, several countries restrict short sales. Short sales are vital to prevent prices from rising to unsustainable levels. The most unrealistic assumptions of CAPM are that: 1. All assets trade 2. No transaction costs AND single period horizon These assumptions generate the major challenges to CAPM.
Contribution of Stock to the Market Portfolio Variance
The authors derive the formula for the contribution of an individual stock to the Market Portfolio variance. Assume that the portfolio consists of n stocks. We are focusing on one specific stock, "i". The contribution of a stock to the variance of the market portfolio is: = wi [w1Cov(r1,ri) + w2Cov(r2,ri) + ... + wiCov(ri,ri) + ... + wnCov(rn,ri)] We know that ∑wyry = rM, so we can simplify the equation above: = wi [Cov(rM,ri)]
(Markowitz Model) The minimum-variance frontier is made up of the portfolios that have the lowest variance for each level of expected return.
The global minimum-variance portfolio is the single portfolio with the lowest variance.
CAPM and the Academic World
The market portfolio upon which CAPM is based is impossible to construct. To test the SML equation [E(Ri) = βiRM], we can use a regression of the excess returns of a sample of stocks against their betas, over a period t: Ri,t = λ0 + λ1βi + λ2σ^2ei + ηi,t CAPM predicts: - λ0 = 0: the average alpha = 0 - λ1 = RM: the slope of the SML equals the market risk premium - λ2 = 0: σ^2ei doesn't earn a risk premium The beta and resid variance for each stock needs to be estimated from a time series of stock returns. The problem with this is that these parameters will be estimated with large errors, which may be correlated. As a result, there may be a downward bias in λ1, and an upward bias in λ0. We may therefore reject CAPM even if it is valid. To demonstrate this, Miller & Scholes simulated rates of returns that followed the CAPM equation, but the regression test still indicated that CAPM did not hold! Another issue is that alpha, beta and the residual covariance are likely to be time varying. Again, the regression techniques, which do not recognize this, are more likely to lead to a rejection of the model.
The mutual fund theorem
The mutual fund theorem states that if all investors would hold a common risky portfolio, they would not object if all the stocks in the market were replaced with shares of a single mutual fund holding the market portfolio.
optimal risky portfolio
The optimal risky portfolio is the available portfolio that has the highest Sharpe ratio: this would be the portfolio formed where the CAL is tangential to the portfolio opportunity set (the tangency portfolio). Note that the optimal risky portfolio consists of risky assets only (no risk free assets).
Risk Tolerance & Asset Allocation
The portfolio that maximizes utility is the optimal portfolio, C. The optimal ratio, y*: y* = (E(rp) - rf)/A(σ of p)^2 (This is the amount to invest in the riskier asset)
Portfolio risk distribution vs number of stocks for the single index model
The portion of the portfolio risk from the firm-specific sources reduces as the number of stocks in the portfolio increases.
Problems with Fama French
The problem with the Fama French and other models derived from empirical evidence is that none of the factors can be identified as hedging a specific significant source of uncertainty. It is possible that the patterns observed were purely due to chance. Fama & French however argue that these variables have successfully predicted returns in alternate time periods, as well as other markets internationally.
No-Arbitrage Equation of the APT
The return on a well diversified portfolio is given by the following equation: E(RP) = αP + βPxE(RM) The action of arbitrageurs will send the α of a portfolio to 0. Therefore, E(RP) = βPxE(RM) This is known as the APT SML. It can be seen that the APT SML is the same as the CAPM SML, due to the no-arbitrage requirement of the APT Consider the following graph, which depicts the relationship between F and the return for 2 portfolios, A & B. A & B each have betas of 1, but have different expected returns: It is impossible for A & B to coexist, as this would produce an arbitrage opportunity: Portfolio A is clearly dominant to Portfolio B. An investor for example could sell short $1M of B, and buy $1M of A. This will produce a riskless payoff of: $1M * (0.1 + 1 * F) - $1M * (0.08 + 1 * F) = $1M * 0.02 = $20,000
Capital Market Line
The risky portfolio P can be determined by either an active or passive strategy: Active: determined with security analysis Passive: a benchmark portfolio is selected (therefore security analysis is unnecessary). There are certain advantages to using a passive approach: significantly cheaper than an active strategy Free rider benefit: because some investors are implementing the active strategy, mispricings should disappear. Therefore, most assets should be fairly priced at any given point in time. The Capital Market Line is simply the CAL that uses a passive portfolio as the risky portfolio.
Extensions of the CAPM: Multiperiod CAPM
The single period assumption of CAPM can also be relaxed. Instead assume that investors will optimize their consumption/ investments over their lifetime. The appropriate equation in this case is the Intertemporal CAPM (ICAPM). The authors first consider the simple case where: 1. the only type of risk is the uncertainty about portfolio returns 2. investment opportunities are constant over time They state that under this scenario, the ICAPM has the same equation as the single period CAPM. Assuming there are K sources of extramarket risk and K associated hedge portfolios, the ICAPM becomes: E(Ri) = βiME(RM) + ΣβikE(Rk) Where βik is the β on the kth hedge portfolio
(Arbitrage Pricing Theory): Single Factor Model
The single-factor model, introduced in BKM8, can be used to derive the return for a stock, based on its sensitivity to a single factor, F: ri = E(ri) + βiF + ei Where: F = deviation of a factor from its expected value βi = sensitivity of the security return to the factor
If one believed markets to be totally efficient, they also believe that research effort is not justified. This however is not correct...
There are many anomalies in prices that indicate that searching for underpriced securities can be profitable. That being said, the evidence does indicate that it is very unlikely that there is consistent superior investment strategy.
From an insurance perspective, we can reduce risk by selling shares of the insurer to investors. Assuming that the total risk per investor remains constant, the Sharpe ratio will rise as the number of policies written increases. There are some problems with this approach though:
There are some disadvantages of managing a very large firm. These disadvantages will put pressure on the profit margins The impact of any error when estimating the risk of the insured will be compounded over many policies
Efficient Market Hypothesis (EMH)
This dictates that stock prices should reflect all available information. New information relevant to the stock will cause the price to change. Since this new information is unpredictable, the stock price must move unpredictably. This process of unpredictable stock movements is called the random walk process.
Extensions of the CAPM: Consumption Based CAPM
This is based on the assumption that in a period, investors need to allocate the current wealth between consumption today; and savings/ investment to support future consumption. The optimal mix would result in the utility from an additional dollar of consumption today equal to utility associated with the future consumption generated from the investment of that dollar. Investors will value the additional income from the savings more during tough economic times (with limited consumption opportunities). Therefore assets that have a positive covariance with consumption growth (those that have a higher payoff when consumption is already high) are viewed as being riskier. The equilibrium risk premium for these assets will be higher. We can construct the CCAPM: E(Ri) = βiCRPC Where Portfolio C is a "consumption tracking portfolio", the portfolio with the highest correlation with consumption growth. Note that unlike in CAPM, the beta of the market portfolio in CCAPM is not necessarily equal to 1. Instead, it can be substantially greater than 1. This model is fairly similar to CAPM. But it has the disadvantages that the consumption growth figures are published infrequently, and are measured with significant error. Despite this, empirical evidence indicates that this model is more successful at explaining returns than the standard CAPM
Problems with Multiperiod CAPM (ICAPM)
This scenario is not very realistic. Over the long run, additional sources of risk are likely to arise, which will have an impact on the ICAPM. In particular, two types of risk considered: Changes in the parameters that describe investment opportunities: 1. some assets have higher returns during periods where economic parameters change adversely. Since the returns on these assets will counter the adverse impact of the parameters on the remainder of the portfolio, investors will bid up the price of these assets, thus reducing the expected return. 2. Changes in the prices of consumption goods: similar to the point above, investors will bid up prices of assets that hedge the increase in prices of consumption goods. Both of the assets described above that will be bid up are known as "hedge portfolios".
Fundamental Analysis
This type of analysis uses the fundamentals of a firm to determine the appropriate price. Fundamental analysts try to gain insight into the future performance of the firm that is not yet reflected by the market. Fundamentals examined include: 1. Earnings & dividend prospects 2. Future interest rate expectations 3. Risks According to EMH, fundamental analysis will usually not work. The exception is that it may be successful for investors who have a unique insight.
(Arbitrage Pricing Theory): Fama French Three Factor Model
This uses firm characteristics that seem to be a proxy for exposure to systematic risk: characteristics that have done a reasonable job of predicting average returns. Rit = αi + βiMxRMt + βiSMBxSMBt + βiHMLxHMLt + eit where: SMB = Small Minus Big; the return of a portfolio of small stocks in excess of the return of a portfolio of large stocks HML = High Minus Low; the return of a portfolio of high book to market ratio stocks in excess of the return of a portfolio of low book to market ratio stocks The market index is included as well. The purpose of its inclusion is to account for the systematic risk from macroeconomic factors. Empirical data has indicated that firm size and book to market ratio have predicted deviations from CAPM indications. They therefore may be acting as proxies for other fundamental factors. For example, high book to market ratio stocks are more likely to be in financial distress, and small stocks may be more sensitive to changes in business conditions.
Tracking portfolios need to have the same Beta as the security
To achieve this beta, the tracking portfolio would need to have a levered position in the S&P500: 1) β units in the S&P500 2) -(β - 1 )in T bills This will have an alpha of 0, since it is made of just the S&P500 and T bills The manager would purchase the security, and short sell the tracking portfolio. The manager is therefore earning the 4% alpha, without being exposed to the market risk. Note that the residual risk, eP, does still apply.
Dealing with difficulty 1 of Event Studies: market movement
To segment the movement that is due to the event, we need to calculate a proxy for what the return would have been in absence of the event (the benchmark). The movement specifically due to the event, the abnormal return, can then be calculated: Abnormal return = Actual return - Benchmark return There are several methods to calculate the benchmark return. Examples include: 1. Market return 2. Return of the stock implied by CAPM 3. Return of the stock implied by index model (et is the abnormal return)
Estimating the Single Index Model
We can use regressions to estimate the parameters of the Single Index Model. The resulting equation is called the Security Characteristic Line (SCL). Based on the regression output, we can test the hypothesis that alpha is 0. In order to reject this hypothesis, the magnitude of alpha would need to be large enough for it to be deemed economically significant. alpha would also need to be statistically significant. To reject this, we would require an absolute value of t greater than 2 Note that even if the alpha value is shown to be both economically & statistically significant, it is not appropriate to rely on it to forecast a future period, as alpha values do not persist over time. We can also test the hypothesis that beta equals 0
Impact of Diversification
We can use the single factor model to derive the return of a portfolio comprising of n stocks: rP = E(rP) + βPF + eP The authors study the impact on eP as n increases. They show that: σ^2(ep) = (1/n) σ^2(ei) As n increases, the variance of eP approaches 0. Based on this, and the knowledge that the expected value of eP is 0, we can conclude that after diversification, the realized value of eP will virtually be 0. The single factor model becomes: rP = E(rP) + βPF Because the nonfactor risk can be diversified away, the security will only receive a risk premium for the factor risk.
Mutual Fund and analyst Performance: Mutual Fund Managers
While there is no evidence that managers can consistently beat the market, there is some evidence that indicates better managers in a period tend to be better managers in following periods. The EMH tests can be modified to adjust for exposure to systematic risk factors. One option is to look at the risk adjusted returns (in excess of the CAPM prediction). However, the market index may not be an adequate benchmark: the managers may have significant holdings in small firms, whereas the market index would be dominated by large firms. A common benchmark used today is a 4 factor model, that uses the 3 Fama French factors, in addition to a momentum factor (a portfolio that is constructed dependant on the prior year's return). Again, the funds have not been shown to exceed this benchmark.
(CAPM) In equilibrium, all securities should have the same reward-to-risk ratio, and therefore the reward-to-risk ratio for an individual stock should equal the market price of risk:
[E(ri) - rf] / Cov(ri, rM) = [E(rM) - rf] / σM^2 E(ri) = rf + βi[E(rM) - rf] Where βi = [Cov(ri, rM) / σM^2] We can use the above formula to prove that βM = 1: βM = [Cov(rM, rM) / σM^2] = [σM^2 / σM^2] = 1
Single Factor Model
first consider the return of a security. This return can be divided into 2 pieces: an expected and an uncertain component: ri = E(ri) + ei where ei has a mean of 0 and a standard deviation of σi. Next, assume that the security returns are joint normally distributed (being driven by one factor, "m"). We can modify the above equation to assume that the uncertainty comes from two sources: - Uncertainty about m. This influences all securities. This factor m generates the correlation across securities. The following equations assume that the sensitivity of a stock to m is βi (this recognizes that some firms are more sensitive to changes in the economy). - Firm specific uncertainty (ei) m & ei are uncorrelated (ei is independent to shocks that impact the entire economy). This produces the single-factor model: ri = E(ri) + βim + ei where m has a mean of 0 and a standard deviation of σm. The equations for var and cov of the securities are derived from the single factor model equation: σi^2 = (βi^2)x(σm^2) + (σ^2)x(ei) Cov(ri, rj) = Cov(βim + ei, βjm + ej) = βiβj(σm)^2
Note that weak and semistrong form EMH tests require the level of risk to be reflected. As a result, if a particular portfolio is shown to generate higher returns, this could actually be caused by an inappropriate risk adjustment technique, as opposed to inefficient markets. In fact, the risk adjustment techniques are based on more questionable assumptions than the EMH.
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The process of separating the search for alpha from the choice of market exposure is known as alpha transport.
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Efficient Diversification
lowest risk level for a given return
Equation to derive the weights of the optimal risky portfolio
wD = [E(RD)σE^2 - E(RE)Cov(rD, rE)]/[E(RD)(σE)^2 + E(RE)(σD)^2 - [E(RD) + E(RE)]Cov(rD, rE)] Note that the equation refers to excess returns (excess over the risk free rate), "R", instead of the actual returns, "r".
The minimum variance portfolio is the portfolio with the lowest variance that can be constructed from assets with a certain level of correlation.
wMIN(D) = [(σE)^2 - Cov(rD, rE)]/[(σE)^2 + (σD)^2 - 2Cov(rD, rE)] As long as ρ < σD/σE, the minimum variance portfolio will consist of both bonds and stocks. If ρ > σD/σE, the minimum variance portfolio will consist exclusively of bonds (the lower variance asset).
(single index model) The equation for portfolio variance (assuming a portfolio of equally weighted securities):
σP^2 = βP^2 + σm^2 + σ^2(eP) where σ^2(eP) = (1/n)σ¯^2(e) As the equation above shows, the firm specific risk approaches 0 as the number of securities increase.
Reasons why most CAT bonds are issued offshore
• Favorable regulatory treatment • Low issuance cost and high levels of expertise in the issuance of risk-linked securities • Effective handling of issuance and settlement • Non-indemnity CAT bonds are not treated as reinsurance • Insurance regulation in the United Stated seems generally inflexible and intrusive compared to other jurisdictions
