7. Log Returns in Stata

1. Log Returns in Stata: How to calculate and interpret them In investing, the log return of a security is the natural logarithm of the ratio of the price of the security at the beginning of the period to the price at the end of the period. Log returns can be used to calculate the return of a security over arbitrarily small time periods, as opposed to simple returns, which can only be used to calculate returns over complete periods. To calculate the log return of a security in Stata, the user must first load the data into memory. For this example, we will use data from the Federal Reserve Bank of St. Louis. The user can then type in the following commands: log return = ln(price at t/price at t-1) tsset date log using returns.csv, replace The first line calculates the log return for each period. The second line tells Stata that the data is time-series data, meaning that the observations are time-ordered. The third line saves the log return data in a CSV file called “returns.csv”. Log returns can be interpreted just like regular returns. A positive log return indicates that the security’s price increased from the 1. To calculate log returns in Stata, you can use the "logreturn" function. Log returns are a type of relative return that are often used when analyzing financial data. They are calculated by taking the natural logarithm of the difference between the current price and the previous price. Log returns can be used to measure the performance of a security or portfolio, and they are often used in statistical analyses. To calculate log returns in Stata, you can use the "logreturn" function. This function takes two arguments: the name of the variable to be used in the calculation and the lag. The lag is the number of periods back that you want to use in the calculation. For example, if you want to calculate the log return for the last 5 days, you would use a lag of 5. The "logreturn" function will return the log return for the specified period. You can then use this value to calculate statistics or to make comparisons between different securities or portfolios. 2. Log returns are simply the natural logarithm of the ratio of the current price to the previous price.

2. To calculate log returns in Stata, you simply take the natural logarithm of the ratio of the current price to the previous price. For example, if the current price of a stock is $100 and the previous price was $90, the log return would be ln(100/90) = 0.0953. interpreting log returns can be a little bit trickier. Generally speaking, a positive log return indicates that the price of the security has increased, while a negative log return indicates that the price has decreased. However, it's important to remember that the log return is a relative measure, so a positive log return doesn't necessarily mean that the security is doing well - it could just be outperforming the market. There are a few other things to keep in mind when interpreting log returns. First, the log return is not additive - that is, you can't simply add up the log returns over time to get the total return. Second, the log return is not symmetric - a two percent increase followed by a two percent decrease is not the same as a four percent decrease. Finally, the log return is not necessarily linear - a doubling in price is not the same as two times the log return. Despite these caveats, log returns are still a useful way to measure price changes, and can be helpful in identifying trends and spotting outliers. 3. Log returns are often used because they are approximately additive. Log returns are often used because they are approximately additive. This means that if you have a series of log returns, you can simply add them up to get the overall return. This is convenient when you are trying to compare different investment strategies, or when you want to see the performance of an investment over time. There are a few things to keep in mind when using log returns. First, they are only approximate. This means that if you are trying to compare two investments, the one with the higher log return may not actually be the better investment. Second, log returns can be negative. This doesn't mean that your investment has lost money, it just means that it didn't make as much money as it could have. Overall, log returns are a convenient way to compare investment strategies and to track the performance of an investment over time. 4. That is, if you have two assets with log returns x and y, then the log return of the portfolio is approximately x+y. Log returns allow us to easily calculate the return of a portfolio. That is, if we have two assets with log returns x and y, then the log return of the portfolio is approximately x+y. This is because the log function is additive. This is a useful property because it allows us

3. to calculate the return of a portfolio without having to weight the individual assets. For example, consider a portfolio with 40% in asset A and 60% in asset B. The return of the portfolio can be approximated as: 0.4 * x + 0.6 * y = (0.4 * x) + (0.6 * y) = 0.4x + 0.6y So, we can see that the return of the portfolio is simply the weighted sum of the log returns of the individual assets. There are a few things to keep in mind when using log returns. First, they are only an approximation. In reality, the return of a portfolio will be slightly different. Second, log returns can be negative. This doesn't mean that the value of the portfolio has decreased, only that the percentage change is negative. 5. Log returns are also often used because they are approximately normal. When we say that something is "approximately normal", we mean that it follows a normal distribution fairly closely. Why is this important? Well, one of the main advantages of using a normal distribution is that we can use some well-known statistical properties to help us make predictions. For example, if we know that a certain quantity is approximately normally distributed, then we can use the 68-95-99.7 rule to estimate the percentage of values that fall within a certain range. This can be extremely useful when trying to predict things like stock prices or interest rates. Another advantage of using the normal distribution is that we can use it to calculate "z-scores". Z-scores tell us how many standard deviations a given value is from the mean. This can be useful for identifying outliers or unusual values. So, how does this all relate to log returns? Well, it turns out that log returns are approximately normally distributed. This means that we can use all of the properties of the normal distribution when working with log returns. One final point to note is that the normal distribution is just one of many possible distributions that data can follow. In some cases, data may not follow a normal distribution at all. In these cases, we would need to use a different distribution to make predictions. 6. That is, the distribution of log returns is approximately normal, with a mean of zero and a standard deviation of one. In order to calculate log returns in Stata, you first need to calculate the natural logarithm of the closing price of the asset for each period. This can be done using the command "ln(price)". Next, you need to calculate the difference between the natural logarithms of the closing prices for each period. This can be done using the command "diff(ln(price))".

4. The distribution of log returns is approximately normal, with a mean of zero and a standard deviation of one. This means that if you take a large enough sample of log returns, approximately 68% of the values will fall within one standard deviation of the mean, 95% of the values will fall within two standard deviations of the mean, and 99.7% of the values will fall within three standard deviations of the mean. The fact that the distribution of log returns is approximately normal is extremely important for financial analysts, because it means that many statistical techniques can be used to analyze data sets consisting of log returns. For example, analysts can use regression analysis to study the relationships between different asset prices, or they can use time-series analysis to study how asset prices change over time. 7. Finally, log returns are used because they are often easier to interpret than raw returns. Log returns are a measure of how much a security has increased or decreased over a period of time. They are calculated by taking the logarithm of thesecurity's price at the end of the period and subtracting the logarithm of the security's price at the beginning of the period. Log returns are often used instead of raw returns because they are easier to interpret. A security with a log return of 0.5 has doubled in value, while a security with a log return of -0.5 has halved in value. This is not the case with raw returns, where a security with a return of 100% is not necessarily twice as valuable as a security with a return of 50%. Log returns are also useful for comparing returns across securities of different prices. A security with a price of $10 and a return of 10% has the same log return as a security with a price of $100 and a return of 1%. However, log returns are not without their drawbacks. One is that they can be difficult to work with when dealing with securities with large price swings. For example, a security with a price of $10 that doubles to $20 has a log return of 0.693, while a security with a price of $1,000 that doubles to $2,000 has a log return of 0.301. Another drawback of log returns is that they can mask the true underlying volatility of a security. For example, a security with a log return of 0.5 over a period of one year has an underlying annualized volatility of approximately 69%. Despite these drawbacks, log returns are still a widely used measure of security performance. There are many ways to calculate log returns in Stata, but the most popular method is the "log" function. This function calculates the natural logarithm of the change in a value. For example, if the stock market index value goes from 10,000 to 11,000, the natural logarithm of the change is ln(11,000/10,000) = ln(1.1) = 0.095. Log returns are useful because they are often used to calculate other important measures, such as volatility. In addition, log returns are easy to interpret - a return of 0.095 means that the

5. stock market index increased by 10% over the period. Overall, log returns are a helpful way to measure changes in a value over time, and are particularly useful for comparing different investments.