# INTRODUCTION TO COMPUTATIONAL FINANCE AND FINANCIAL ECONOMETRICS PDF

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Introduction to Computational Finance and. Financial Econometrics. Chapter 1 Asset Return Calculations. Eric Zivot. Department of Economics. ECON /CFRM Introduction to Computational Finance and Financial Econometrics. Home · Syllabus raudone.info Revised. Introduction to Computational Finance and. Financial Econometrics is determined by the variable's probability distribution function (pdf).

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Menkhoff, L. Sarno, M. Schmeling, A. Example 7 In the first example. In this case. Here the annual gross return is defined as the two month return compounded for 6 months. To complicate matters. What is the annual return on this two year investment? To determine the annual return we solve the following relationship for RA: Example 8 In the second example. Example 10 Using the price and return data from example 1.

The continuously compounded monthly return. To see why rt is called the con- tinuously compounded return. Properties of logarithms and exponentials are discussed in the ap- pendix to this chapter. Notice that rt is slightly smaller than Rt.

Continuously compounded returns are very similar to simple returns as long as the return is relatively small. Example 11 In the previous example. Given a monthly continuously compounded return rt. To illustrate. For modeling and statistical purposes. Recall that with simple returns the two month return is of a multiplicative form geometric average. The second way uses the sum of the two continuously compounded one month returns.

Hence the continuously compounded two month return is just the sum of the two continuously compounded one month returns. Example 12 Using the data from example 2.

That is. What is the continuously compounded annual return on this investment? A useful summary of a broad range of return calculations is given in Watsham and Parramore Lo and MacKinlay provide a nice treatment of continuously compounded returns.

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Properties of exponentials and logarithms The computation of continuously compounded returns requires the use of natural logarithms. The exponential and natural logarithm functions have the following properties 1. Notice that ex is always positive and increasing in x. The natural logarithm function.

See the Project page on the class website for instructions on how to Figure xxx plots ex and ln x. When computing returns. You may compute annual returns using overlapping data or non-overlapping data. Make a time plot line plot in Excel of the monthly price data over the period end of December through end of December Read the data into Excel and make sure to re- order the data so that time runs forward. Do your analysis on the monthly closing price data which should be adjusted for dividends and stock splits.

I know. Then second month annual return is from the end of January. Make a time plot of the annual returns. With non-overlapping data you get a series of 5 annual returns for the 5 year period Why is a plot of the log of prices informative? Please put informative titles and labels on the graph. Make a time plot of the monthly returns. Comment on what you see eg.

Place this graph in a separate tab spreadsheet from the data. What is the annual rate of return over this five year period assuming annual compounding? Comment on what you see and compare with the plot of the raw price data. With overlapping data you get a series of annual returns for every month sounds weird. Using the data in the table. Briefly compare the continuously compounded returns to the simple returns. The second annual return is computed from the end of December through the end of December etc.

Make a time plot of the annual returns and comment. Using the continuously compounded monthly returns. Using the monthly price data over the period December through December Exercise 6. Convert this continuously compounded return to a simple return you should get the same answer as in part a.

The First Boston Corporation. Money Market Calculations: The Econometrics of Financial Markets. Princeton University Press. Convert this continuously com- pounded return to a simple return you should get the same answer as in part 5. Assuming that the continuously compounded monthly return you com- puted in part 2 is the same for 12 months.

MacKinlay Compare with your result in part 3. Com- pare with your result in part 4. Assuming that the simple monthly return you computed in part 1 is the same for 12 months. Dow Jones Irwin.

Break Evens and Arbitrage. International Thomson Business Press. Quantitative Methods in Fi- nance.

We will discuss this distribution is later chapters. It is an open question as to what is the best characterization of the probability distribution of stock prices.

The normal distribution is often a good approximation to the distribution of simple monthly returns and is a better approximation to the distribution of continuously compounded monthly returns. University of Washington January February Therefore the sample space is the set of positive real numbers bounded above by some large number.

Then the return on this investment is a random variable since we do not know its value today with certainty. As another example. In contrast to prices. Since the price of Microsoft stock next month is not known with certainty today.

For example. The price next month must be positive and realistically it can t get too large. The log-normal distribution is one possibility1. We might hypothesize that the annual return will be in! As an example. The following table describes such a probability distribution of the return. Consider two mutually exclusive events generically called success and failure.

In fact. This distribution is presented graphically in Figure 2. The probability distribution described above can be given an exact mathematical representation known as the Bernoulli distribution. Suppose that the annual return on Microsoft stock can take on any value between a and b. A That is. The p. For a continuous random variable. Further suppose that the probability that X will belong to any subinterval of SX is proportional to the length of the interval.

A typical bell-shaped pdf is displayed in Figure 3. The shape of the normal distribution is the familiar bell curve.

## Introduction to Computational Finance and Financial Econometrics

As a result. We solve Z 0. Notice that the area under the curve over the interval [a. As we shall see. This distribution is centered at zero and has in! Let X denote a standard normal random variable. The normal distribution has the annoying feature that the area under the normal curve cannot be evaluated analytically.

Given that the total area under the normal curve is one and the distribution is symmetric about zero the following results hold: The above integral must be computed by numerical approximation. Finding Areas Under the Normal Curve In the back of most introductory statistics textbooks is a table giving information about areas under the standard normal curve.

Notice that the distribution is symmetric about zero. Areas under the normal curve. The standard normal distribution is graphed in Figure 5. The cdf for the discrete distribution of Microsoft is given in Figure 6. Notice that the cdf in this case is a discontinuous step function. If FX is invertible then qa may be determined as. This inverse function must be approxi- mated numerically.

For example, we might want to know where the distribution is centered and how spread out the distribution is about the central value. We might want to know if the distribution is symmetric about the center. For stock returns we might want to know about the likelihood of observing extreme values for returns. This means that we would like to know about the amount of probability in the extreme tails of the distribution.

In this section we discuss four shape characteristics of a pdf:. Example 12 Using the discrete distribution for the return on Microsoft stock in Table 1, the expected return is. Let g X denote some function of the random variable X. Example 14 Suppose X has a uniform distribution over the interval [a. For bell-shaped or normal looking distributions the SD measures the typical size of a deviation from the mean value.

Using numerical approximations. Notice that SD X is in the same units of measurement as X whereas var X is in squared units of measurement. If X has general normal distribution. As with the standard normal distribution. The standard deviation of X. These returns are to be treated as random variables since the returns will not be realized until the end of the month.

We can invest in two non-dividend paying stocks A and B over the next month. E[Ri ]. In contrast. The high return variability of asset A re! Figure xxx shows the pdfs for the two returns. The return we expect on asset A is bigger than the return we expect on asset B but the variability of the return on asset A is also greater than the variability on asset B. Observations more than three standard deviations from the mean are very unlikely.

These cases are illustrated in Figure 6. Example 21 Using the discrete distribution for the return on Microsoft stock in Table 1. Example 23 Using the discrete distribution for the return on Microsoft stock in Table 1. Sometimes the kurtosis of a random variable is described relative to the kurtosis of a normal random variable. If excess kurtosis is greater than zero. If a distribution has a kurtosis greater than 3 then the distribution has thicker tails than the normal distribution and if a distribution has kurtosis less than 3 then the distribution has thinner tails than the normal.

Since kurtosis is based on deviations from the mean raised to the fourth power. Then the following results hold: Notice that our proof of the second result works for discrete and continuous random variables. Next consider the second result. The following proposition establishes the result. A normal random variable has the special property that a linear function of it is also a normal random variable.

These results will be used often enough that it useful to go through the derivations. In addition. We solve the problem by standardizing X as follows: This transformation is called standardizing the random variable X since. The above property is special to the normal distribution and may or may not hold for a random variable with a distribution that is not normal.

Standardizing a random variable is often done in the construction of test statistics. Let W0 denote the investment value at the beginning of the month and W1 denote the investment value at the end of the month.

Example with the log-normal distribution 1. Consider the following questions: In this example. This result is useful for modeling purposes. General functions of a random variable and the change of variables formula.

For the second method. W1 if W0 represents the initial wealth and q. The function p x. The likelihood that X and Y takes values in the joint sample space is determined by the joint probability distribution p x.

In many situations we want to be able to characterize the probabilistic behavior of two or more random variables simultaneously. Table 2 illustrates the joint distribution for X and Y. Bivariate pdf 0. Notice that the marginal probabilities sum to unity. The bivariate distribution is illustrated graphically in Figure xxx.

Notice that sum of all the entries in the table sum to unity. X depends on Y. Notice that these probabilities sum to 1. The most important shape characteristics are the conditional expectation conditional mean and the conditional variance. When such a linear relationship exists we call the regression function a linear regression.

It is important to stress that linearity of the regression function is not guaranteed. Example 28 For the data in Table 2. In this prediction context. Example 27 For the data in Table 2. Numerical approximation methods are required to evaluate the above integral. We character- ize the joint probability distribution of X and Y using the joint probability function pdf p x. Y is independent of X if knowledge about X does not in!

X is independent of Y if knowledge about Y does not in! We represent this intuition formally for discrete random variables as follows.

Proposition 34 Let X and Y be continuous random variables. Figure xxx displays several bivariate probability scatterplots where equal probabilities are given on the dots. In panel d we see a positive. In panel b we see a perfect positive linear relationship between X and Y and in panel c we see a perfect negative linear relationship. The correlation between X and Y measures the direction and strength of linear relationship between the two random variables.

The covariance between X and Y measures the direction of linear relationship between the two random variables. Some important properties of cov X. In the third quadrant. Y are 1. The plot is separated into quadrants.

Example 38 For the data in Table 2. If X and Y are jointly normally distributed and cov X. To see how covariance measures the direction of linear association.

For the example data. In the second quadrant. Covariance is then a probability weighted average all of the product terms in the four quadrants. If X and Y are independent then cov X. The last result illustrates an important property of the normal distribution: Y is not informative about the strength of the linear association between X and Y.

The second result states that variance of a linear combination of random variables is not a linear combination of the variances of the random variables.

The fourth property should be intuitive. Some important properties of corr X. By simply changing the scale of X or Y we can make cov X. Then the following results hold. This result indicates that the expectation operator is a linear operator. The third property above shows that the value of cov X.

Y depends on the scaling of the random variables X and Y. In particular. In other words. Y is informative about the direction of linear association between X and Y.

## ECON 424/CFRM 462 Introduction to Computational Finance and Financial Econometrics

Independence between the random variables X and Y means that there is no relationship. Y equal to any value that we want. Let X and Y be discrete random variables. It is worthwhile to go through the proofs of these results. Y and a and b be constants.

Distribution of Continuously Compounded Returns Let Rt denote the continuously compounded monthly return on an asset at time t. This important result states that a linear combination of two normally distributed random variables is itself a normally distributed random variable. Not all random variables have the property that their distributions are closed under addition.

The proof of the result relies on the change of variables theorem from calculus and is omitted. The details of the generalizations are not important for our purposes. The annual continuously compounded return is equal the sum of twelve monthly continuously compounded returns. Intermediate textbooks with an emphasis on econometrics include Amemiya Goldberger Port and Stone and Hoag and Craig 19xx.

For the annual standard deviation.

Ramanathan Everything you ever wanted to know about probability distributions is given Johnson and Kotz 19xx. Since each monthly return is normally distributed. The table below gives discrete probability distributions for these random variables based on the state of the economy: What is the value of y? Suppose X is a normally distributed random variable with mean 10 and variance Let X denote the monthly return on Microsoft stock and let Y denote the monthly return on Starbucks stock.

Introduction to Statistics and Econometrics. Academic Press. Harvard University Press. International Thompson Business Press.

Introduction to Probability Theory.. A Course in Econometrics. Probability Distributions. References [1] Amemiya. An Introduction to Mathematical Finance: Options and Other Topics. Harvard Uni- versity Press. San Diego. Cambridge University Press. An Introduction to the Mathematics of Financial Derivatives. Quantitative Methods in Finance. January Constant variances and covariances: No serial correlation across assets over time: Assumption 1 states that in every time period asset returns are normally dis- tributed and that the mean and the variance of each asset return is constant over time.

Normality of returns: Given assumption 1. This is the constant expected return CER model. Clearly these are very strong assumptions.

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The third assumption stipulates that all of the asset returns are uncorrelated over time1. Using the basic properties of expectation. Assumptions indicate that all asset returns at a given point in time are jointly multivariate normally distributed and that this joint distribution stays constant over time. If the news is good. Regarding the variance of returns. Given that covariances and variances of returns are constant over time gives the result that correlations between returns over time are also constant: To see this.

Rjs 0 corr Rit. In words. For covariances of returns. Since mul- tiperiod continuously compounded returns are additive we can interpret. If the news is bad. Hence these variables will be positively correlated. Consider the default case where Rit is interpreted as the continuously compounded monthly return. Then one interpretation of news in this context is general news about the computer industry and technology. Rit as the sum of 30 daily continuously compounded returns2: X X cov Rit.

Let pi0 denote the initial log price of asset i. By recursive substitu- tion. The RW model gives the following interpretation for the evolution of asset prices. In the RW model. The representation in 3 is know as the RW model for the log of asset prices. The process of creating such pseudo data is often called Monte Carlo simulation4. The sailor starts at an initial position. In order to simulate pseudo return data.

To mimic the monthly return data on Microsoft. To illustrate the use of Monte Carlo simulation. Simulated Random Walk 12 E[p t ] p t. Computer algorithms exist which can easily create such observations. Rit is a random 9. The typical size of the! If simulated data from the model does not look like the data that the model is supposed to describe then serious doubt is cast on the model.

At the beginning of every month t. Notice that the simulated return data looks remarkably like the actual return data of Microsoft.

RiT is a random variable. RiT a random sample from the CER model 1 and we call ri1. The CER model assumes that the economic environment is constant over time so that the normal distribution characterizing monthly returns is the same every month. It is assumed that the observed returns are realizations of the random variables Ri1.

RiT as a random variable. In actuality. We call Ri1. To establish some notation. Let ri1. We assume that these values are taken as given. The expected returns. The variances. We might wonder where such values come from. One possibility is that they are estimated from historical return data for the two stocks. We assume that the returns RA and RB are jointly normally distributed and that we have the following information about the means. Another possibility is that they are subjective guesses.

Assets that have returns with high variability or volatility are often thought to be risky and assets with low return volatility are often thought to be safe. We can also think of the variances as measuring the risk associated with the investments.

What does it mean for xA or xB to be negative numbers? The investor must choose the values of xA and xB. Let xA denote the share of wealth invested in stock A and xB denote the share of wealth invested in stock B. For the second result 3.

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Since RA and RB are assumed to be normally distributed. We have a given amount of wealth and it is assumed that we will exhaust all of our wealth between investments in the two stocks. The investor s problem is to decide how much wealth to put in asset A and how much to put in asset B. The portfolio problem is set-up as follows. Our investment in the two stocks forms a portfolio and the shares xA and xB are referred to as portfolio shares or weights.

Rp is also normally distributed. We summarize the expected return-risk mean-variance properties of the feasible portfolios in a plot with portfolio expected return.

If the portfolio weights are both positive then a positive covariance will tend to increase the portfolio variance. In- vestors like portfolios with high expected return but dislike portfolios with high return variance..

Notice that the variance of the portfolio is a weighted average of the variances of the individual assets plus two times the product of the portfolio weights times the covariance between the assets. Given the above assumptions we set out to characterize the set of portfolios that have the highest expected return for a given level of risk as measured by portfolio variance.

For illustrative purposes we will show calculations using the data in the table below. The portfolio standard deviation is used instead of variance because standard deviation is measured in the same units as the expected value recall.

This implies that means. First we make some assumptions: Table 1: The investment possibilities set or portfolio frontier for the data in Table 1 is illustrated in Figure 1. Here the portfolio weight on asset A, xA , is varied from We then plot these values1.

Since investors desire portfolios with the highest expected return for a given level of risk, combinations that are in the upper left corner are the best portfolios and those in the lower right corner are the worst.

Notice that the portfolio at the bottom of the parabola has the property that it has the smallest variance among all feasible portfolios. Accordingly, this portfolio is called the global minimum variance portfolio. We solve the constrained optimization problem. A negative portfolio weight indicates that the asset is sold short and the proceeds of the short sale are used to download more of the other asset.

A short sale occurs when an investor borrows an asset and sells it in the market. The short sale is closed out when the investor downloads back the asset and then returns the borrowed asset. The shape of the investment possibilities set is very sensitive to the correlation between assets A and B.

This case is illustrated in Figure 2. What this means is that if assets A and B are perfectly negatively correlated then there exists a portfolio of A and B that has positive expected return and zero variance! Very risk averse investors will choose a portfolio very close to the global minimum variance portfolio and very risk tolerant investors will choose portfolios with large amounts of asset A which may involve short-selling asset B. Now we consider what happens when we introduce a risk free asset.

In the present context, a risk free asset is equivalent to default-free pure discount bond that matures at the end of the assumed investment horizon. The risk-free rate, rf , is then the return on the bond, assuming no in! For example, if the investment horizon is one month then the risk-free asset is a day Treasury bill T-bill and the risk free rate is the nominal rate of return on the T-bill.

If our holdings of the risk free asset is positive then we are lending money at the risk-free rate and if our holdings are negative then we are borrowing at the risk-free rate.These results will be used often enough that it useful to go through the derivations.

Consider two mutually exclusive events generically called success and failure. An estima- tor of the expected return. Revised July 8, In many situations we want to be able to characterize the probabilistic behavior of two or more random variables simultaneously.

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