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Covariance is a statistical measure of the extent that 2 variables move in tandem relative to their respective mean (or average) values.In the investment world, it is important to be able to measure how different financial variables interact together.Covariance can provide clues to the following two questions:Are there common factors affecting the returns of your investments?How can you measure the magnitude of this relationship?It is calculated by taking the product of two variables’ deviations from their average values.The practical applications of covariance are quite significant in statistics, economics, finance, and portfolio management.Investment decision-making based on covariance analysis can have serious financial implications, and as such, it is important to be well-grounded in its understanding.Formula of CovarianceQuite often, covariance analysis aims to assess historical relationships among variables of interest.If we obtain a sample of monthly returns for two stocks, X and Y, covariance can be calculated as:     Where,Xi = return (%) for stock X for period iYi = return (%) for stock Y for period iX = sample mean or average value of X for sample nY = sample mean or average value of Y for sample bn = sample size (12 observation implies n=12)Note: the formula above is the covariance computation for a sample of data. When working with a population (the entire data), the denominator changes to (n) rather than (n-1).We calculate covariance using the formula above:     Therefore, there is a positive relationship between the returns of Stock X and Stock Y. In other word, the returns of both stocks tend to move in the same direction for the sample of interestLimitations of CovarianceThe major limitation of the covariance measure is in its interpretation. While we can gauge the directional relationship between the variables, the magnitude in itself, is not very informative. From the above example, a covariance of 0.91 does not tell us how strong the relationship between the returns of Stock X and Y is, and as such, our conclusions are limited. One way we can work around this shortcoming is to determine the correlation coefficient.Another short-coming in covariance is that the result is highly sensitive to the volatility of the variables’ variances. For example, the presence of just a few outliers in a data set can significantly skew its result, rendering it as a potentially misleading statistic, in terms if interpretation.Covariance in Portfolio Management TheoryPortfolio management theory (or Modern Portfolio Theory (MPT)), as developed by Harry Markowitz in the 1950s, makes extensive use of the covariance measure. The theory posits that an efficient frontier exists, which is derived from the expected returns and variances (or standard deviation) of sets of investment portfolios. Given varying weights in two asset classes (e.g. stocks and bonds), we can determine risk-efficient points on a graph plotting the expected returns and standard deviations of the portfolio. The line of the graph is the efficient frontier. In other words, the EF plots the maximum return possible given a level of risk (variance).The role of covariance in MPT lies in its impact on the diversification effects of adding an individual investment, portfolio, or different asset class to an existing portfolio. Thus, given an existing portfolio, it is possible to reduce its inherent risk (for a given expected return) by adding an investment whose returns exhibits a low covariance with those of the existing portfolio.ConclusionCovariance is a statistical measure of the extent and direction of co-movement between two variables deviations from their respective means. It can be used to assess the association between important economic and financial data such as stock returns, equity indexes, bond returns, inflation, interest rates, and a multitude of relationships of interest. Covariance analysis can also be used to assess the diversification benefits of adding different asset classes into our portfolio. Its limitation is the difficultly in its interpretation, since the strength of the relationship cannot be strongly gauged from its result.

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The Moving Average Convergence-Divergence (MACD) indicator is one of the easiest and most efficient momentum indicators you can get. It was developed by Gerald Appel in the late seventies. The MACD moves two trend following indicators and moving averages into a momentum oscillator by subtracting the longer moving average from the shorter moving average. The result is that the MACD gives the best of both worlds: trend following and momentum. The MACD is continually changing above and below the zero line while the moving averages come together, cross and diverge. Traders can search for signal line crossovers, centerline crossovers as well as divergences to generate signals. For that reason the MACD is unbounded, it is not necessarily useful for identifying overbought and oversold levels. Note: MACD is pronounced as either “MAC-DEE” or “M-A-C-D”. Take a look at the sample chart with the MACD indicator in the lower panel:MACDCalculationMACD Line12-day EMA - 26-day EMA)Signal Line:9-day EMA of MACD LineMACD Histogram:MACD Line - Signal LineThe MACD Line is the 12-day Expotential Moving Average(EMA) minus the 26-day EMA. Closing prices are used for these moving averages. A 9-day EMA of the MACD Line is plotted with an indicator to function as a signal line and identify turns. The MACD Histogram denotes the difference linking MACD and its 9-day EMA, the Signal line. The histogram stays positive when the MACD Line is above its Signal line and negative when the MACD Line is below its Signal line. The values of 12, 26 and 9 are the typical setting used with the MACD, but other values can be exchanged depending on your trading style and goals.InterpretationThe MACD is about the convergence and divergence of the faster and slower moving averages. Convergence occurs when the averages move towards each other. Divergence occurs when the averages move away from each other. The shorter moving average is faster and more responsive. The longer moving average is slower and less reactive to price changes. The MACD Line moves above and below the zero line – also known as the centerline. The direction, of course, depends on the direction of the moving average cross. A positive MACD is when the shorter moving average crosses above the longer moving average. As the shorter moving average moves further above the longer moving average (diverges) this means the stock price upside momentum is increasing. When the short moving average drops below the long moving average, it demonstrates that the stock shows a downward momentum.The yellow area shows the MACD Line in negative territory as the short line is below the long line. In this chart, the crossing occurred at the end of September (see the black arrow) and the MACD moved diverged further into negative territory as the short moving average moves further away from the long moving average. The orange area highlights the period of positive MACD values, which is when the short moving average moves above the long moving average. Notice that the MACD Line stayed below during this period (red dotted line). The red line means that the distance between the slow EMA and long EMA was less than 1 point, which is not a much of a difference.DivergencesDivergence forms when the MACD line moves away from the price line of the stock. Bullish divergence are formed when a stock’s price records a lower low and the MACD hits a higher low. The lower low for the stock confirms the downtrend, but the higher low for the MACD line shows less downward momentum. Downside momentum still outpaces the upward momentum as long as the MACD remains negative. When the downward momentum slows, it can foreshadows a trend change or a upside rally. The next chart uses a Google (GOOG) chart with a bullish divergence for Oct-Nov 2008. Notice that there were clear lower troughs as both Google’s price line and its MACD line bounced in October and late November. Notice that the MACD line formed a higher low as Google’s price line formed a lower low in November. MACD is signalling a bullish divergence as the signal line crosses over in early December. Google’s price line confirmed the reversal with a breakout.A bearish divergence forms when a stock price records a higher high and the MACD line forms a lower high as the faster MA crosses the slower MA. The higher high for the stock price is quite normal for uptrends but when the MACD shows a lower high, this illustrates less upside momentum. Even though upside momentum may have declined, upward momentum is still out performing downside momentum as long as MACD is positive. Declining MACD upward trends can foreshadow a trend reversal or forecast a large price decline. Below we see a chart for Gamestop (GME) with a large MACD bearish divergence from Aug to Oct. The stock chart demonstrates a higher high above 28, but the MACD line falls short of the previous high and shows a lower high. The following MACD crossover is bearish. On the GME price chart, notice how the support is broken and turned into resistance on the following bounce in Nov as we see with the red dotted line. This momentary price bump provided another chance to sell or sell short.We should be careful interpreting a MACD divergences. Bearish divergences are quite common for strong uptrends as do bullish divergences during a strong downtrend. Price uptrends quite often begin with a strong advance which will produce strong upside momentum for MACD. Even though we can see that the uptrend continues, it continues at a slower pace than started the uptrend which causes the MACD to decline. Even when upside momentum is not as strong, upside momentum still outpacing the downside momentum as long as the MACD line is above zero. We can see the opposite occuring when a strong downtrend begins. The next chart shows SPY which is the S&P 500 ETF. This chart shows four bearish divergences from Aug to Nov 2009. Despite the slower upside momentum, SPY’s price line continued higher because the uptrend was strong. Notice how SPY’s price continues a series of higher highs as well as higher lows. Remember, as long as MACD is positive the upside momentum is stronger than downside momentum.ConclusionsMACD is a special indicator as it brings together both momentum and trend in one technical indicator. This unique combination of trend and momentum can be used with daily, weekly and monthly charts. The standard moving average lines for MACD use the difference between the 12 and 26-period EMAs. Chartists that are looking for a more responsive indicator can use a shorter short-term moving average and a longer long-term moving average. A MACD(5,35,5) is far more responsive than the more standard MACD(12,26,9) and can be a better indicator for weekly charts. Chartists looking for a less sensitivity indicator can use lengthening the moving averages. A less responsive MACD will still oscillate above/below zero but the frequency of the crossovers centerline and signal line crossovers will decline. Finally, remember that MACD is calculated using the difference between two moving averages. This means that the MACD line is dependent on the price of the stock. For example, the MACD line for a \$20 stock may move from -1.5 to 1.5 while the MACD line for a more expensive \$100 stock can move from -10 to +10. You cannot compare the MACD charts for several stocks with far different prices. If you want to compare the momentum of various stocks you should probably use the Percentage Price Oscillator (PPO) rather than MACD.

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DefinitionVolume-Weighted Average Price (VWAP) is often used as a trading benchmark by traders, pension funds, mutual funds and market makers. It can allow traders to get a sense of how successful they were in obtaining a good price. A buy order filled below the VWAP would be considered a good trade.Using VWAPA lot of people will wonder why not just use the average price, it’s a lot easier to calculate and isn’t it essentially the same thing anyway? As we can see, the difference is that it tracks volume as well. By tracking volume you also get information about liquidity as well as the amount of money that traded, not just the price.CalculationVWAP calculation can vary greatly depending on the time frame and time horizon you choose, as well as the price calculation. However, VWAP is typically used within one trading day and uses a one minute time frame.The formula for VWAP is:VWAPformulaThe price can be used as the last price in a time frame or the price calculated using the high, low and close in a given time frame. Here is an example of the calculations:Volume Price Price * Vol Time VWAP 1 min20 10.15 10.15*20 = 203 9:30 203/20=10.15030 10.21 10.21*30 = 306.3 9:31 (203+306.3)/(20+30)=10.18675 10.22 75*10.22 = 766.5 9:32 (203+306.3+766.5)/(20+30+75)=10.206We can see from the calculations that the VWAP is cumulative and thus a price at the beginning with high volume will have more effect than a price at the end of the day, since it is now just a drop in all the trades placed over the day. It would take a very large volume and/or price change to change the VWAP at the end of the day (depending on the time frame you use). Another important thing to notice is the time frame, a smaller time frame (such as every trade or tick) will be more accurate, but for a stock that trades a lot this could be data for 50,000 trades. If you are doing multiple stocks, this could easily slow down or even crash your computer if you started storing and calculating enough days.GraphThe greatest change in the results and calculation of VWAP is the time frame. If we look at the difference below between a 1 minute time frame and a two minute minute time frame we will see there is a discrepancy.   As we can see the two minute does not follow as closely as the 1 minute since it is only “checking” the price every two minutes.Another ratio we can use similar to VWAP is the moving VWAP (MVWAP) that is similar to a simple moving average. This will use a different period and will sometimes be carried over from day to day depending on the period used. Essentially, instead of starting at one day, we will calculate our MVWAP using the data over the period we wish to study. For example, we can say we wish to take a period of 10 minutes and thus we will essentially have a VWAP that started it’s “day” ten minutes before.

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