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CHAPTER 3
THE CHARACTERISTICS OF FINANCIAL SERIES
TEST YOUR KNOWLEDGE
1 What are the differences between financial data and macroeconomic data? Financial data often differ from macroeconomic data in terms of their frequency, seasonality, revisions and other properties. Macroeconomic data such as unemployment, inflation, industrial production come in monthly frequency or lower (i.e., quarterly). GDP data come in quarterly or annually and population data only annually. By contrast, financial data such as stock prices, bond yields and interest rates are observed in daily (intra-day or even minute-by-minute (tick) basis), weekly as well as monthly frequencies. Macroeconomic series are also collected at the beginning of some period (a month, for instance) as estimates and at the end of the period the actual value of the series is recorded. This gives rise to data revisions and, potentially, measurement errors. Finally, economic data also come in as seasonality-adjusted or non-seasonally adjusted. 2 What do the relationships among the mean, mode and median of a financial series tell about the shape of the underlying probability distribution? Mean = Median = Mode if the distribution is symmetric Mean > Median > Mode if the distribution is positively skewed Mean < Median < Mode if the distribution is negatively skewed 3 Why do investors prefer that their financial investments have positive skewness than negative skewness? What are the implications of negative skewness on the asset’s risk? A negative skew of the returns’ distribution means that the asset has experienced a greater magnitude of extreme negative returns. Skewness measures the frequency of occurrence of large returns in a particular direction. In case the frequency of positive returns exceeds that of negative returns, the distribution displays a fat right tail or positive skewness. Positive skewness means that the really large returns are more likely to be positive than negative. Thus, one would much prefer a slightly reduced normal return with an occasional possibility of a pleasant high return than a slightly increased normal return with an occasional sharp decline in his asset portfolio. When the distribution is skewed to the right, the standard deviation overestimates risk, because extreme positive surprises (which do not concern investors) nevertheless increase the estimate of volatility. Conversely, and more important, when the distribution is negatively skewed, the standard deviation will underestimate risk. 4 What is autocorrelation and what are its consequences for an asset portfolio? Autocorrelation measures the similarity between measurements as a function of the time difference between them and we use it to find repeating patterns within a given time series. For example, if we are trying to predict the growth of dividends, an underestimation of growth of one year is likely to lead to an underestimation in future years. The fundamental assumption in finance is that asset returns from period to period are independent and identically distributed. However, if one month’s retu rn is influenced by the previous month’s return, then there may be a need to account for this effect in future a sset projections.
In general, serial correlation masks true asset class volatility and biases risk estimates downwards, leading to underestimation of overall portfolio risk. 5 What is volatility and are the type of volatility? Volatility is defined as the variance or standard deviation of the change in the value of a financial asset. There are several type of volatility. One is the historical volatility. There is also implied volatility, which is the market ’s estimate of the po ssible movement in a stock’s price. A third type of volatility is the volatility index, of a stock market index such as the S&P500. Finally, there is intra-day volatility. This represents the market swings during the course of a trading day and is the most noticeable and readily available definition of volatility. 6 Assume the following economic scenarios and data below: Scenario Prob. Return on X Booming stock market 0.70 20% Normal stock market 0.20 10% Contracting stock market 0.10 5% Compute stock X’s expected return and risk. E(Rx) = 0.7020% + 0.2010% + 0.10%*5% = 16.5% Varx = σ^2 x = [(20% - 16.5%)^2 * 0.70 + (10% - 16.5%)^2 * 0.20 + (5% - 16.5%)^2 * 0.10 = 30. σ x = 5.5% 7 Assume the following rates of return of data on an asset during each of the three quarters during a particular year: Quarter Return on X 1 20% 2 25 % 3 1 5% Compute the arithmetic and geometric means and discuss. AM = (20+ 25 +5)/3 = 16.67% GM = [(1+.2)(1+. 25 )(1+. 1 5)]1/3^ – 1 = 1 9 .9 3 % The GM is higher than the AM. 8 Using the data in problem 7, compute the annualized quarterly percentage change of the returns of asset X. Quarter 2 (0.25/0.20)^4 - 1 = 1. Quarter 3 (0.05/0.10)^4 - 1 = - 0. 9 Observe the graph below (pertaining to the Advanced Micro Devices, AMD, company’s weekly continuously compounded stock returns. What patterns do you see? Discuss in terms of some descriptive statistics and other stylized facts.
TEST YOUR INTUITION
1 What would happen to the distribution of continuously compounded returns of a stock if we plotted monthly or quarterly data? They would approach normality. 2 If you plotted the S&P500 closing prices and the returns, would you still see the leverage effect? Yes, just look at the graph above where the inverse relationship is evident. This graph is similar to the one found in Ait- Sahalia, Fan and Li, (2013) paper. 3 If the returns of a financial series exhibit volatility clustering, what can you say about the validity of the identically and independently distributed (iid) property? Volatility clustering implies that the variance of daily returns can be high one month (high volatility) and show low variance (low volatility) the next. This occurs to such a degree as to make an iid model of the log- prices or an asset’s log returns unconvincing. 4 If a stock’s return in the auto industry is found to have a Hurst exponent value of less than 0.5 would you expect another company’s (in the tech industry, for example) stock’s returns to also have an H value of less than 0.5? Why? Not necessarily as autocorrelation may be both negative (H < 0.5) and positive (H > 0.5). The evidence is mixed. 5 Do you think that skewness and kurtosis values of an asset’s returns would change if we computed them during contractionary periods relative to expansionary periods? Applying to S&P500 stock returns during the late 2018 (November) to early 2019 (January) subperiod, we found the following: Kurtosis 1.538 Skewness 0. Applying to the early 2019 (February) to April 2019, we found the following: Kurtosis 2.0014 Skewness - 0. Thus, yes they would both change in size and may change signs as well. 0 500 1000 1500 2000 2500 3000 3500
- 0 5 10 15 1/1/1990 1/1/1991 1/1/1992 1/1/1993 1/1/1994 1/1/1995 1/1/1996 1/1/1997 1/1/1998 1/1/1999 1/1/2000 1/1/2001 1/1/2002 1/1/2003 1/1/2004 1/1/2005 1/1/2006 1/1/2007 1/1/2008 1/1/2009 1/1/2010 1/1/2011 1/1/2012 1/1/2013 1/1/2014 1/1/2015 1/1/2016 1/1/2017 1/1/2018 1/1/ rsp S&P
CHAPTER 4
UNIVARIATE PROPERTIES OF FINANCIAL TIME SERIES
TEST YOUR KNOWLEDGE
1 What is nonstationarity and how does arise? Nonstationarity refers to the changing structure of a time series’ mean and variance over time. Examples of non - stationary processes are the random walk with or without a drift (reflecting a slow, steady change) and deterministic trends (trends that are constant, positive or negative, and independent of time). Economic relationships among variables or agents change over time. New laws or other aspects of the institutional environment can change discretely at a particular point in time, leading to changes in agents’ behavior. Examples include tre nds in knowledge accumulation and its embodiment in capital equipment, major geological or geopolitical events, and policy regime changes. Thus, nonstationarity in an economic or financial series is due to all sorts of structural changes (i.e., economic, social, political, personal etc.). 2 Prove that the random walk is difference-stationary. Proof: Δy t = yt – yt- 1 = yt- 1 + ut – yt- 1 = ut. 3 Is the random walk model stationary? Recall the random walk model: yt = yt−1 + ut In order to test this, first write yt- 1 in backshift operator notation (Byt ). Then, take this term over to the left-hand side of the model, and finally work it out: yt = Byt + ut yt – Byt = ut yt (1−B) = ut Then the characteristic equation is 1−ξ = 0, having the root ξ = 1, which lies on, not outside, the unit circle. 4 What do the autocorrelation and partial autocorrelation functions tell us? The ACF is a way to measure the linear relationship between an observation at time t and the observations at previous times. If we assume an AR(k) model, then we may wish to only measure the association between yt and yt-k and filter out the linear influence of the random variables that lie in between (i.e., yt- 1 , yt- 2 , …, yt-(k-1), which requires a transformation on the time series. Then by calculating the correlation of the transformed time series we obtain PACF. The PACF is most useful for identifying the order of an autoregressive model. Specifically, sample partial autocorrelations that are significantly different from 0 indicate lagged terms of y that are useful predictors of yt. 5 What is a moving average effect and how does it compare with the autoregressive effect? A moving average effect in a time series is best thought of as an effect that impacts the values of a series immediately and for some finite number of future periods. This contrasts with an autoregressive process where an effect impacts future values at a steadily declining rate through the correlation of values over time. The order of the moving average process is suggested by how many spikes in the ACF are sufficiently large.
The first series’ ACF and PACF exhibit a significant spike at lag 1 in both functions and thus an ARMA(1,1) model may be implied. The exchange rate series suggest a stationary process and thus resemble a random walk model. The inflation rate series displays a similar pattern as the AAA yields and thus an ARMA(1,1) may be suggested. 8 Suppose that a researcher had estimated the first five autocorrelation and partial autocorrelation coefficients for 100 observations, as follows: Lag 1 2 3 4 5 ACF 0.1 65 - 0.070 0.060 0.045 - 0. PACF 0.345 0.246 0.205 0.079 0. Test each of the individual correlation coefficients for joint significance using both the Box–Pierce and Ljung–Box tests First, we need to construct a 95% confidence interval for each coefficient using ±1.96* 1/√ T, where T = 100. The decision rule is to reject the null hypothesis that a given coefficient is zero in the cases where the coefficient lies outside the range ±0.196. From the data above, it would be concluded that only the first ACF coefficient is significantly different from zero at the 5% level and that the first 3 PACF coefficients are significant. Turning now to the joint tests, the null hypothesis is that all of the first five autocorrelation coefficients are jointly zero: H 0 : γ 1 = γ 2 = γ 3 = γ 4 = γ 5 = 0. The test statistic for the Box – Pierce (equation 50) is: ACF Q = 100 * [(0.165)^2 + (-0.07)^2 + (0.060)^2 + (0.045)^2 + (-0.01)^2 ] = 3. PACF Q = 1 00 * [(0. 3 45)^2 + (0. 2 46)^2 + (0. 20 5)^2 + (0.0 7 9)^2 + (0.0 4 9)^2 ] = 2 3. 020 The Ljung – Box test (equation 50a) is given by: ACF Q^ = 100 102* [0.165)^2 /99 + (-0.07)^2 /98 + (0.06)^2 /97+ (0.045)^2 /96+ (-0.01)^2 /95] = 3. PACF Q^ = 100 102* [0.345)^2 /99 + (0.246)^2 /98 + (0.205)^2 /97+ (0.079)^2 /96+ (0.049)^2 /95] = 23. The relevant critical values from a χ^2 distribution, with 5 degrees of freedom, are 11.1 at the 5% level, and 15.1 at the 1% level. Clearly, in both cases the joint null hypothesis that all of the five ACF coefficients are zero cannot be rejected at the conventional 5% level but not at the 1% level. As far as the PACF tests are concerned, we reject the null in both cases. 9 Why are ARMA models best suited for forecasting? ARMA models are of very useful in estimating a series structure and using it for forecasting because of its ease and flexibility. These mean that there is no need to rely on some economic theory to decide on the number of lags to use. For example, we know that both fiscal and monetary policies have lags and thus variables should be entered in the model with lags. Second, ARMA models can produce fairly accurate forecasts, static and dynamic. Third, such models can also be used for macro series where frequencies are lower than for financial data. 10 You have estimated the following ARMA(1,1) model for the random variable y: ŷ t = 0.056 + 0.8 9 yt- 1 + 0.35ut- 1 + ut Suppose that you have data for yt- 1 = 2.5 and ut- 1 = - 0. (a) Obtain forecasts for the series y for times t , t +1, and t + (b) If the actual values for the series turned out to be 1 .50 0 , 1 .775, 1 .125 for t , t +1, and t +2, calculate the mean squared error
(a) To obtain the forecasts of the time series, we define E(yt|yt- 1 ) = ft-1,1. Thus, ft-1,1 = 0.056 + 0.8 9 yt- 1 + 0.35ut- 1 + ut = 0.056 + 0.89 * 2.5 + 0.35 * (-0.05) = 2.263 for t ft-1,2 = 0.056 + 0.8 9 * 2.263 = 2.070 for t+ ft-1,3 = 0.056 + 0.89 * 2.070 = 1.898 for t+ Note that we do not know the previous u’s and construct forecasts of the series by substitutions of the previous series forec asts. (b) The MSE = ∑ T+kt=T+1 (ŷ t – yt)^2 /k Thus, 1/3 [2.263 – 1.50)^2 +(2.07 – 1.775)^2 + (1.898 – 1.125)^2 = 0. This value is also known as the out-of-sample mean squared error.
CHAPTER 5
SHORT AND LONG RUN RELATIONSHIPS AMONG TIME SERIES
TEST YOUR KNOWLEDGE
1 Consider the following price process given by the series pt. The dynamics of the process are given by pt = pt- 1 + et or, equivalently, by Δ pt = et. (a) Explain what this model implies about pt+1 and name that model. Tomorrow’s price, pt+1 , is thought of as today’s price plus some random shock that is independent of the price. Hence, this model is a random walk. (b) What could be the odds of an increase and decrease in price? An increase in price is as likely as a decrease in it. At time t, the price is considered to contain all information available. Therefore, at any point in time, next period’s price is exposed to a random shock. (c) What is the best estimate of the next period’s price? Explain why. The best estimate for the following period’s price is this period’s price. Such price processes are called efficient due to their immediate information processing. 2 Consider the following model. yt = μ + φ yt- 1 + ut Explain the values that φ might take and explain each one of them from the economics point of view. There are three possible case. First, when φ = 1, then we say that the series is non-stationary or that it contains a unit root. This case has been found to describe accurately many financial and economic time series. Second, when φ > 1, then we say that shocks to the system are not only persistent through time, but they are propagated so that a given shock will have an increasingly large influence. Put differently, the effect of a shock during time t will have a larger effect in time t +1, a larger effect still in time t +2, and so on. Such a case does not describe many data series in economics and finance. Finally, if φ > 1 then we say that the series is stationary and shocks disappear within some reasonable time frame. This is a desired case for time series. 3 Where is the variance-covariance matrix used? Provide some examples. One application of the covariance and correlation is in investments an d portfolio management. The old adage ‘don’t put all your eggs in one basket” stills rings true and implies that the investor should diversify its investments across asset classe s. In essence, this means that allocating all your money in investments whose returns are highly correlated that may all perform poorly at the same time is not a very prudent investment strategy. This is because if any one single investment performs poorly, it is very likely, due to its high correlation with the other investments, that the other investments are also going to perform poorly, leading to the poor performance of the portfolio. Another use of the variance-covariance matrix of asset returns is in utility theory. Markowitz assumed that investors order their preferences acco rding to a utility index, with utility as a convex function that takes into account investors’ risk - return
preferences. He assumed that stock returns are jointly normal and, consequently, the return of any portfolio is a normal distribution, which can be characterized by the mean and the variance (hence, his portfolio selection theory was term mean- variance analysis). 4 What do tests for unit roots and cointegration infer about the variables? Such tests infer the attributes of economic and financial variables and their relationships reflected by the characteristics of time series data. 5 Why is it necessary to test for non-stationarity in time series data before attempting to build and estimate a model? Non-stationarity can be an important determinant of the properties of a series. Also, if two series are non-stationary, we may have the problem of spurious regression, which happens when we regress one non-stationary variable on a completely unrelated non-stationary variable, despite indicating a good fit. Further, we would not able to perform any hypothesis tests in models which inappropriately use non-stationary data since the test statistics will no longer follow the distributions which we assumed they would and so any inferences are likely to be invalid. 6 Discuss the concept of cointegration for the spot and futures prices of a commodity relying on economic/finance theory. The, explain how (and why) a researcher might test for cointegration between the variables using the Engle–Granger approach. Recognizing that the spot and futures prices are essentially prices of the same asset but with different delivery and payment dates means that financial theory would suggest that they should be cointegrated. If they were not cointegrated, then this would imply that the series did not contain a common stochastic trend and that they could therefore wander apart without bound even in the long run. If the spot and futures prices for a given asset did separate from one another, market forces would work to bring them back to follow their long run relationship. The Engle-Granger approach to cointegration involves two steps. First, checking to see if the variables are individually unit root processes. Then a regression would be conducted of one of the series on the other would be conducted and the residuals from that regression collected. These residuals would then be subjected to a DF/ADF test. If the null hypothesis of a unit root in the test regression residuals is not rejected, it would be concluded that a stationary combination of the non-stationary variables has not been found and thus that there is no cointegration. On the other hand, if the null is rejected, it would be concluded that a stationary combination of the non-stationary variables has been found and thus that the variables are cointegrated. Second, if the variables are cointegrated, the second stage of the process involves forming the error correction model which, in the context of spot and futures prices, could be of the form given in equation. 7 Discuss the advantages and disadvantages between the Engle–Granger and Johansen cointegration methodologies. Which, in your view, represents the superior approach and why? The fundamental difference between the Engle-Granger and the Johansen approaches is that the former is a single-equation methodology whereas Johansen is a systems technique involving the estimation of more than one equation. The main advantage of the Engle-Granger approach is its simplicity and its intuitive interpretability. However, it has a number of disadvantages including its inability to detect more than one cointegrating relationship and the impossibility of validly testing hypotheses about the cointegrating vector. 8 When two variables cointegrate, we can define X1t = μ + β 2 X2t, and refer to X1t as the equilibrium value of X1t, and ut = X1t – X1t as the deviation from equilibrium. (a) Explain the notion of economic equilibrium and state whether it is plausible or not. (b) Define algebraically the long-run solution and the error-correction term. (a) The equilibrium value can be interpreted as the value at which there is no inherent tendency for X1t to move away, but it is important to realize that because the economy is continuously evolving and hit by shocks, the system will never settle down at the equilibrium value (X1t) and X1t will not converge to X*1t.
TEST YOUR INTUITION
1 If the correlation coefficient between tow asset portfolios is +1, would you invest in both or not? What if it was - 1? Explain using finance theory. If the correlation coefficient is +1 then there are no benefits to diversification and thus you should not hold both portfolios. If it is - 1 then you reap the full benefits of diversification. Both cases are extreme with reality being somewhere in the middle. 2 Logically, a relationship can only be interpreted as defining an economic equilibrium if the variables cointegrate, and if they don’t then there is no interpretable relationship between them. Do you agree or disagree? Agree with the above argument of cointegration or lack theorof. 3 Although there is a similarity between tests for cointegration and tests for unit roots, they are not identical. Explain why. Tests for unit roots are performed on univariate time series. In contrast, cointegration deals with the relationship among a group of variables, where (unconditionally) each has a unit root. 4 Do you suspect that globalization and financial integration would ensure cointegration among financial markets? Yes, such trends imply a closer linkage among stock markets (and bond markets) which denotes cointegration. 5 Drawing on your investments background, what do you think would happen to the benefits from diversification when assets markets cointegrate? If cointegration does not hold, markets are not linked in the long run and therefore it is possible to gain from diversification. For this reason, testing for cointegration and any changes over time in its degree is important. See, for example, Caporale Maria G., Gil-Alana, L., and Orlando, James C., (2016). Linkages between the US and European Stock markets: A fractional cointegration approach. International Journal of Finance & Economics 21, 143 – 153.
CHAPTER 6
THE EFFICIENT MARKET HYPOTHESIS AND TESTS
TEST YOUR KNOWLEDGE
1 State the argument that stock prices should follow a random walk Any information that could be used to predict stock performance should already be reflected in stock prices. As soon as there is any information indicating that a stock is underpriced and therefore offers a profit opportunity, investors rush to buy it and immediately bid up its price to a fair level, where only “normal”, ordinary rates of return can be expected. These “normal rates” are simply rates of return commensurate with the riskiness of the stock. However, if prices are bid immediately to fair levels, given all available information, it must be that they increase or decrease only in response to new information. New information, by definition, must be unpredictable since if it could be pr edicted, the prediction would be part of today’s information. Thus, stock prices that change in response to new (unpredictable) information also must move unpredictably. This is the essence of the argument that stock prices should follow a random walk, that is, that price changes should be random and unpredictable. 2 What are the sufficient and necessary conditions for an efficient market? The sufficient conditions are: (a) there are no transaction costs or market frictions in trading the asset (b) all information is available at no cost for all market participants (c) all market participants agree in the implications information has on current and future prices and dividends (d) all market agents possess homogeneous expectations and have an equilibrium model of price determination (valuation) The necessary conditions for a market inefficiency to be eliminated are: (a) that inefficiencies in a financial market should provide the basis for an investment strategy to beat the market and earn abnormal returns as long as the cost of transactions are smaller than the expected profits from the strategy (b) that there should be rational, profit-maximizing investors who can replicate the market-beating strategy and trade until the inefficiency vanishes 3 What value for b do we expect for this regression, Rt+1 = a + bXt + ut+1, in the classic “efficient markets” view? Xt can be any variable. Interpret a test on b. If stock prices are not predictable, or loosely-speaking , the “random walk” view , we should see b = 0, for any variable Xt. A test of b = 0, would give evidence on the ‘informational efficiency’ part of the EMH. 4 If you run a regression of returns on lagged (past) returns, explain the possible values of the slope coefficient.
8 Develop an argument about the predictability of stock returns from dividend or earnings yields in an efficient and irrational market The predictability of stock returns from dividend yields or earnings yields is not in itself evidence for or against market efficiency. In an efficient market, the forecast power of dividend yields says that prices are high relative to dividends when discount rates and expected returns are low, and vice versa. On the other hand, in an irrational world, low dividend yields signal irrationally high stock prices that will move predictably back toward fundamental values. Therefore, to judge whether the forecast power of dividend yields is the result of rational variation in expected returns or irrational bubbles, other information must be used. 9 What is the behavioral critique? The foundation of behavioral finance is that conventional financial theory ignores how people make decisions and that people make a difference (Barberis and Thaler, 2003). Economists have begun to recognize that there exist irrational investors who either do not always process information correctly, thus making erroneous inferences (of the probability distributions) about future rates of return, or that even given a probability distribution of returns, they often make inconsistent or systematically suboptimal decisions. These arguments form the crux of the behavioral critique. 10 Discuss why some studies have found that value stocks tended to have higher returns than growth stocks, based on price-earnings and price-to-book-value ratios Stocks with low price-earnings multiples (or value stocks) appear to provide higher rates of return than stocks with high price-to-earnings ratios (or growth stocks) as documented by Ball (1978) and Basu (1983). This finding is consistent with the views of behavioralists that investors tend to be overconfident of their ability to project high earnings growth and thus overpay for growth stocks (Kahneman and Riepe, 1998). The ratio of stock price to book value (or the value of a firm’s assets minus its liabilities divided by the number of shares outstanding) has also been found to be a useful predictor of future returns. Low price-to- book is considered to be another mark of value stocks and is also consistent with the bahavioralists’ view that investors tend to overpay for growth stocks that subsequently fail to live up to expectations (see Fama and French, 1993).
11 Inspect the graphs below and explain what you see in terms of market efficiency (and its 3
forms).
(a) This graph shows SP500’s current and one-day lagged returns from 8/4/2019 to 8/2/
(b) This graph shows the hypothetical path of a stock’s cumulative abnormal return (CAR) ten
days before a public announcement and five days after the announcement
(c) same as (b)
(a) (b) (c)
-. -. -. -. -. . . . . . -.05 -.04 -.03 -.02 -.01 .00 .01 .02 .03. return at day t return at day t-
(a) We do not observe some significant relationship between the return on successive days, and thus the evidence is supportive of the weak form of market efficiency. (b) Prior to the announcement, the actual stock return is equal to the expected (thus near zero abnormal return), whereas at day 0 when the new information is released the abnormal return is around 3%. The adjustment in the stock price is immediate. In the days following the release of information, there is no further drift in the stock price, either upward or downward. The semi-strong form of market efficiency predicts that stocks prices should react quickly to the release of new information, and thus one should expect the abnormal stock return to occur around the news release. This is confirmed above. (c) The strong form of market efficiency predicts that the release of private information should not move stock prices. If insiders trade on private information, we should see a pattern close to the one illustrated in (c). Prior to the announcement, CAR run-up occurs since insiders have an incentive to take advantage of the private information.
CHAPTER 7
THE CAPITAL ASSET PRICING MODEL
TEST YOUR KNOWLEDGE
1 The CAPM asks what would happen if all investors shared an identical investable universe and
used the same input list to draw their efficient frontiers. What is the key insight of CAPM? Please
answer the question using the capital allocation line and the efficient frontier concepts.
Investors facing the same risk-free rate they would then draw an identical tangent CAL and all would arrive at the same risky portfolio, P. Thus, all investors would choose the same set of weights for each risky asset. Because the market portfolio is the aggregation of all identical risky portfolios, since all investors choose the same risky portfolio, it must be the market portfolio, that is, the value-weighted portfolio of all assets in the investable universe. Therefore, the capital allocation line based on each investor’s optimal risky portfolio will in fact also be the capital market line. 2 Discuss the components of an asset’s total risk. Give some examples of each type of risk.
The total risk of a financial asset, therefore, can be expressed as the sum of the systematic and idiosyncratic risk:
Total Risk = Systematic (or non-diversifiable or market) risk+ Idiosyncratic (or firm-specific) risk.
Systematic risk : Systematic risk is the risk that arises from the market structure and general economic conditions and
more importantly, affects all market players. Examples of such risk include business cycles, inflation, budget deficits, and interest rates. Sometimes, however, whether risk is systematic depends on the broad context.
Idiosyncratic (or firm-specific) risk : Idiosyncratic risk is the risk that is only exposed by a specific firm or industry.
For example, the success or failure in research and development, personnel (management) changes in a company affect the company only and not significantly other firms in the economy. 3 Assume you have the following data on some portfolios, their risk premia, expected returns and risk (as measured by their standard deviation), all expressed in decimals: Portfolio Risk Premium Expected Return Risk (St. Dev) L (low risk) 0.03 0.08 0.0 5 M (medium risk) 0.06 0.10 0. 10 H (high risk) 0.10 0. 15 0. 20 Using the utility function (text (equation 7.6), U = E(r) – 0.5 A σ^2 , evaluate each portfolio (investment) using utility scores produced by the utility function. Assume the investors have values of risk aversion, A, of 2 and 5. The risk-free alternative is assumed to be 5%. Answer A Utility Score of L Utility Score of M Utility Score of H
2 E(r) =.0 8 – 0. 5 2(.05)^2 = .0775 E(r)=. 1 – 0.52(.1)^2 =.09 E(r)=.15-0.52(.2)^2 =. 11
5 E(r) =.08 – 0.55(.05)^2 = .07 37 E(r)=.1 – 0.55(.1)^2 = .0 75 E(r)=.15-0.5* 5 *(.2)^2 =. 05
The portfolio with the highest utility score for each investor appears in bold. Notice that the high-risk portfolio, H, would be chosen only by the investor with the lowest degree of risk aversion, A =2, while the medium-risk portfolio, M, would be passed over even by the most risk-averse of the two investors. All three portfolios beat the risk-free alternative for the investors with
levels of risk aversion given in the table. More risk-averse investors (who have the larger values of A) penalize risky investments more severely. Investors choosing among competing investment portfolios will select the one providing the highest utility level. 4 What does the slope of the capital allocation line represent and how can you use it to allocate your wealth? How else can you call this characteristic? The slope of the CAL shows the excess return per unit or risk (or the reward-to variability ratio) and is known as the Sharpe ratio. All combinations (allocations) of the risk asset P with risk-free borrowing or lending have the same Sharpe ratio. The Sharpe ratio is maximized when the steepest CAL is just tangent to EF and above its min variance point. Then, according to your risk tolerance, you allocate your wealth between this highest Sharpe ratio portfolio and risk-free lending or borrowing. This feature of EF is referred to as fund separation, as we mentioned above, according to which investors with the same beliefs about expected returns, risks and correlations all will invest in the portfolio or “fund” of risky assets that h as the highest Sharpe ratio. Investors differ only in their allocations between this fund and risk-free lending or borrowing based on their risk tolerance. Notice in that in this case, the composition of the optimal portfolio of risky assets does not depend on the investor’s tolerance for risk. 5 What is the Treynor ratio, what is its relation to the Sharpe ratio and how can you use it? If CAPM holds, then another measure of the (ex-ante) excess return per unit of risk, but this time the risk is measured by the incremental portfolio risk given by the portfolio’s beta, is the so - called Treynor ratio: TRx = {E(rx) – rf )} / β x = E(rm) – rf and the value of TRx should be the same for all portfolios of securities. As with the Sharpe ratio, the Treynor ratio is used to compare the historic performance of alternative portfolios (investment strategies), and th e ‘best’ portfolio is the one with the highest Treynor ratio. The Treynor ratio can also be used to rank
alternative risky portfolios and although there are difficulties in interpreting it when βx < 0.
6 What is Black’s zero-beta model and how does it resemble with the CAPM? Every portfolio on the efficient frontier, except for the global minimum- variance portfolio, has another ‘mirror’ portfolio on the bottom (or the inefficient) part of it with which it is uncorrelated. That mirror portfolio is referred to as the zero-beta portfolio of the efficient portfolio. If we choose the optimal portfolio P and its zero-beta portfolio z, then we obtain the following equation: E(rx ) - E(rz ) = [E(rp ) - E(rz)] {cov(rx,rp )/σ^2 p } = β x E(rp) - E(rz ) which resembles the CAPM. In this case, the risk-free rate is replaced with the expected return on the zero-beta portfolio of the optimal risky portfolio. The beta of the portfolios that have returns uncorrelated with the efficient, mean-variance (including the market) portfolio returns will be zero. 7 What are the assumptions of the simple linear regression model and what their implications for the OLS estimators? The five assumptions are: (1) E(ut ) = 0 The errors have zero mean (2) Var(ut ) = σ^2 <∞ The variance of the errors is constant and finite over all values of xt (3) Cov(ui,uj ) = 0 The errors are linearly independent of one another (4) E(ut,xt ) = 0 There is no relationship between the error and corresponding x variate
(5) ut ∼ N(0 , σ^2 ) The error term is normally distributed with 0 mean and constant variance
If the above assumptions hold, then the estimators â and bˆ determined by OLS will possess the following desirable properties, known as best linear unbiased estimators (BLUE):
(a) Best : means that the OLS estimator bˆ has minimum variance among the class of linear unbiased estimators (this is
famous Gauss – Markov theorem which states that the OLS estimator is best by examining an arbitrary alternative linear unbiased estimator and showing in all cases that it must have a variance no smaller than the OLS estimator)
