Based on the excel, use DMA, SES, Holts and regression to forcast the performance.
Select up to 5 explanatory variables (but not necessarily 5) and estimate the corresponding regression model to forecast volatility in June 2018. Briefly explain why these variables are selected. You need to ensure that the 3 regression model satisfies the underlying assumptions. You can use the full sample or a sub-sample to estimate your model. Please justify your sample selection and report the in-sample estimated coefficients in Table 1 below. Fill in the sample period and replace X’s with variable names. Compare the forecasting performance of the best of DMA and Holt’s models against the linear regression in the hold-out period. Use the best model from above analyses to forecast volatility in June 2018. Based on daily returns in June, the team with the lowest forecast error gets 2 extra marks. Investment Research Predicting Volatility Stephen Marra, CFA, Senior Vice President, Portfolio Manager/Analyst Uncertainty is inherent in every financial model. It is driven by changing fundamentals, human psychology, and the manner in which the markets discount potential future states of the macroeconomic environment. While defining uncer- tainty in financial markets can quickly escalate into philosophical discussions, volatility is widely accepted as a practical measure of risk. Most market variables remain largely unpredictable, but volatility has certain characteristics that can increase the accuracy of its forecasted values. The statistical nature of volatility is one of the main catalysts behind the emergence of volatility targeting and risk parity strategies. Volatility forecasting has important implications for all investors focused on risk-adjusted returns, especially those that employ asset allocation, risk parity, and volatility targeting strategies. An understanding of the different approaches used to forecast volatility and the implications of their assumptions and dependencies provides a robust framework for the process of risk budgeting. In this paper, we will examine the art and science of volatility prediction, the characteristics which make it a fruitful endeavor, and the effectiveness as well as the pros and cons of different methods of predicting volatility. 2 Introduction: Statistical Properties of Volatility “The starting point for every financial model is the uncertainty facing investors, and the substance of every financial model involves the impact of uncertainty on the behavior of investors and, ultimately, on market prices. The very existence of financial economics as a discipline is predicated on uncertainty.”1 A big part of risk management, asset allocation, and trading in finan- cial markets is quantifying the potential loss of assets. In order to measure these potential losses and make sound investment decisions, investors must estimate risks. Volatility is the purest measure of risk in financial markets and consequently has become the expected price of uncertainty. The trade- off between return and risk is critical for all investment decisions. Inaccurate volatility estimates can leave financial institutions bereft of capital for operations and investment. In addition, market volatility and its impact on public confidence can have a significant effect on the broader global economy. Volatility targeting and risk parity are asset allocation methodolo- gies that are directly impacted by volatility forecasting. Funds that are managed for the insurance industry with a volatility band of 8%–12%—for example—use asset allocation aiming to control the overall fund returns remaining within that range of volatility, as inves- tors with different levels of risk tolerance and time horizons demand differentiated levels of volatility. Maintaining this range of volatility requires that a view be taken on the expected future volatility for the asset classes in the fund. In addition, funds which use risk parity are focused on the allocation of risk rather than the allocation of capital, assuming that each asset class contributes the same degree of volatility to the overall fund. As the volatility of each of these asset classes is not constant, a forecast for the expected volatility for each is required to maintain this type of investment approach. It is well established that volatility is easier to predict than returns. Volatility possesses a number of stylized facts which make it inherently more forecastable. As such, volatility prediction is one of the most important and, at the same time, more achievable goals for anyone allocating risk and participating in financial markets. The volatility of asset returns is a measure of how much the return fluctuates around its mean. It can be measured in numerous ways but the most straightforward is historical, observed volatility, which is measured as the standard deviation of asset returns over a particular period of time. When volatility is calculated by reverse-engineering options market prices, it essentially becomes both a market price for and an expectation of uncertainty. The stochastic or random nature of asset prices and returns necessitates the use of statistics and statistical theory to help describe and predict these market fluctuations. The entire field of financial econometrics is predicated on the integration of the theoretical foundations of economic theory with finance, statistics, probability, and applied mathematics to make inferences about the financial and market relationships critical in the disciplines of asset allocation, risk management, securities regula- tion, hedging strategies, and derivatives pricing. Volatility is forecastable because of a number of persistent statistical properties. Volatility Clustering There is a delay for large or small changes in the absolute value of financial returns to revert back to mean levels. In other words, the magnitude of financial returns have latency—large changes in financial returns tend to be immediately followed by large changes and small changes tend to be immediately followed by small changes. This can lead to volatility clusters over time. Many studies have found that volatility clustering is likely due to investor inertia, caused by investors’ threshold to the incorporation of new information. Except during times of extreme market turmoil, only a fraction of market participants are actually trading in markets at any given point in time. As such, it takes a period of time for these investors to engage in the market and implement their changing views as new information is revealed. The evidence for volatility clustering is shown by the positive serial correlation (correlation of a return series with itself lagged) in the absolute value of returns which eventually decays over a period of observations (Exhibit 1). Volatility cluster- ing can enhance the ability to forecast volatility. This clustering can be shown by plotting a scatter chart of current month versus next month’s volatility (Exhibit 2). Exhibit 1 Autocorrelation of Global Equity Returns Autocorrelation of Absolute Value of Returns -0.1 0.0 0.1 0.2 0.3 0.4 MSCI EM MSCI ACWI S&P 500 Index 35302520151050 Lag (weeks) For the period January 1999 to October 2015, weekly returns The performance quoted represents past performance. Past performance is not a guarantee of future results. This is not intended to represent any product or strategy managed by Lazard. It is not possible to invest directly in an index. Source: Bloomberg Exhibit 2 Past Volatility May Be Indicative of Future Volatility… 0 40 80 120 0 40 80 120 Following Monthly Volatility (%) Monthly Volatility (%) R squared: 0.35 For the period February 1984 to September 2015, monthly returns Data are based on the S&P 500 Index. Source: Bloomberg 3 In contrast, if one plots the current month’s return versus the next month there is no linear relationship and the serial correlation of actual returns—not the absolute value of returns—remains insignificant (Exhibit 3). Leverage Effect The hypothesized leverage effect along with the volatility feedback effect describes the negative and asymmetric relationship between volatility and returns. The mathematical calculation of volatility is indifferent to the direction of the market. However, volatility is negatively correlated to returns. At the same time, negative returns result in larger changes in volatility than positive returns. The beta of the CBOE Volatility Index (VIX) to the S&P 500 Index on negative return days is -3.9 with an r-squared of 0.36 whereas the beta of VIX to the S&P 500 Index on positive return days is -2.8 with an r-squared of 0.23 (Exhibit 4). The volatility feedback effect suggests that as volatility rises and is priced into the market, there is a commensurate rise in the required return on equity as investors place a higher hurdle rate on returns to achieve their desired risk-adjusted upsides. This leads to an instant decline in stock prices as the volatility immediately reduces the risk- adjusted attractiveness of equities. As stock prices fall, companies become more leveraged as the value of their debt rises relative to the value of their equity. As a result, the stock price becomes more vola- tile. This effect is more pronounced in well-developed markets that have more analyst coverage. Mean Reversion Another stylized property of volatility is that it reverts to the mean over time. The half-life of volatility is measured as the time it takes volatility to move halfway towards its long-term average. Volatility has a half-life of about 15–16 weeks—based on autoregressive models which we will discuss later. With regards to implied volatility, the degree of mean reversion is both asymmetric and accelerated (Exhibit 5). The half-life of VIX mean reversion is about 11 weeks and is consider- ably less than the half-life for equity returns, which is roughly 15 to 16 weeks (shown by the autocorrelation in Exhibit 1). In addition, VIX mean reversion is far more pronounced when the VIX reaches higher levels than when it dips below its long-term average. So, historically the VIX has dropped with greater frequency and magnitude when ele- vated than it has increased when at depressed levels. This suggests that Exhibit 3 ...But Not Indicative of Future Returns Following Monthly Return (%) Monthly Return (%) -30 -20 -10 0 10 20 -30 -20 -10 0 10 20 30 R squared: 0.00 Autocorrelation of Returns -0.2 0.0 0.2 0.4 S&P 500 Index MSCI EM MSCI ACWI 35302520151050 Lag (weeks) Top chart: for the period February 1984 to September 2015, monthly returns and data are based on the S&P 500 Index. Bottom chart: for the period January 1999 to October 2015, weekly returns The performance quoted represents past performance. Past performance is not a guarantee of future results. This is not intended to represent any product or strategy managed by Lazard. It is not possible to invest directly in an index. Source: Bloomberg Exhibit 4 There Is a Negative Relationship between Returns and Volatility Change in VIX on Negative S&P 500 Index Days Change in VIX on Positive S&P 500 Index Days Change in VIX (%) Change in S&P 500 Index (%) -40 -20 0 20 40 60 -12 -10 -8 -6 -4 -2 0 Change in VIX (%)