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 Mean > Median > Mode Mean < Median < Mode
if the distribution is symmetric if the distribution is positively skewed 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 return is influenced by the previous month’s return, then there may be a need to account for this effect in future asset 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 possible 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 Booming stock market Normal stock market Contracting stock market
Prob. 0.70 0.20 0.10
Return on X 20% 10% 5%
Compute stock X’s expected return and risk. E(Rx) = 0.70*20% + 0.20*10% + 0.10%*5% = 16.5% Varx = σ2x = [(20% - 16.5%)2 * 0.70 + (10% - 16.5%)2 * 0.20 + (5% - 16.5%)2 * 0.10 = 30.25 σx = 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 15% Compute the arithmetic and geometric means and discuss. AM = (20+25+5)/3 = 16.67% GM = [(1+.2)(1+.25)(1+.15)]1/3 – 1 = 19.93% 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 Quarter 3
(0.25/0.20)4 -1 = 1.44 (0.05/0.10)4 -1 = -0.87
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.
AMD weekly stock returns 60 40 20 0 -20 -40 -60
First, we see sharp ups and downs the continuously compounded returns. That is, we observe some leverage effects as the down spikes are larger than the up spikes. Second, we observe volatility clustering during some period such as early 2000s, 2008 and 2016. Third, we infer negative skewness and excess kurtosis in those returns. Finally, we may infer that these returns do not follow the normal distribution. 10 You are given the following data on AMD’s stock returns. Mean: 0.1106 Standard Error: 0.1835 Standard Deviation: 8.2904 Skewness: -0.2188 Kurtosis 2.8382 No. of Obs. 2040 Determine of the skewness and kurtosis coefficients are statistically significant. First, we must find the standard errors of each coefficient: Skewness standard error: √6/N = √(6/2040) = 0.0542 Test for statistical significance for Skewness: -0.2188/0.0542 =-4.034 thus statistically significant Kurtosis standard error: Test for statistical significance for Kurtosis:
√24/N = √(24/2040) = 0.1084 2.8382/0.104 = 26.168 thus statistically significant
11. Assume the following data on your investment portfolio: Current price (per unit): $100 Number of units held: 10,000 Historical volatility (1 day): 1.5% Find the VaR on your portfolio assuming both a 95% and 99% confidence intervals and a 90-day holding period. The value of the portfolio would be $100 * 10,000 = $1,000,000 VaR = (portfolio value) * (1.645 sd for one day) * (sd) * (√days) VaR(5%) = $1,000,000 * 1.645 * 0.015 * √90 = $234,087.6 VaR = (portfolio value) * (2.33 sd for one day) * (sd) * (√days) VaR(1%) = $1,000,000 * 2.33 * 0.015 * √90 = $331,564
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?
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Yes, just look at the graph above where the inverse relationship is evident. This graph is similar to the one found in AitSahalia, 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 0.325 Applying to the early 2019 (February) to April 2019, we found the following: Kurtosis
2.0014
Skewness -0.4501
Thus, yes they would both change in size and may change signs as well.
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 trends 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: Δyt = 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.