The Scalar Kalman Filter
In this section, we will discuss the estimation of the value of a constant. Let us first examine how to do it in normal course. The typical method would be to take multiple measurements of the value and use the average of the measured values as an estimate of the constant. The reasoning behind the approach is that the measurements could have errors associated with them that is, some measured values could be greater than the true value of the constant, and others could be lower. By taking an...
Geometric Interpretation
Key to doing the geometric interpretation is the idea of eigenvalue decomposition of the covariance matrix F. A brief discussion on eigenvalue decomposition is provided in the appendix. Let the eigenvalue decomposition of F be given as UDUT. If xA and xB are the factor exposure vectors of the two stocks, let us consider a transformation of the two vectors as shown in Equations 6.18. This is the transformation from the factor exposure space to the factor return space. Now, using simple matrix...
Lag 1 Correlation in a AR1 Series
The variance at each time instant is given as var yt E yt2 a2 a4 a var et If a gt l, the series explodes and the variance becomes infinity. However, when a gt l, the variance can be calculated as the sum of an infinite geometric series and written as cov gt yt-i _ E yjt-1 _ E t-1 t -1 _ avar yt
Appendix Eigenvalue Decomposition
Consider a scalar l and a corresponding vector v. They are an eigenvalue, eigenvector pair of a square matrix A if they satisfy the equation The equation means that the vector v is special with respect to the matrix A. Multiplying the vector with the matrix does not change the direction or orientation of the vector. Its magnitude, however, is multiplied by the scalar l. A square n x n matrix may have n eigenvalues l1r12, ,ln and n corresponding eigenvectors vi,v2, ,vn Therefore Let us write...
The Random Walk Process
An important and special ARMA series that merits discussion is the random walk. The random walk has been studied extensively by scientists from various disciplines. Phenomena ranging from the movement of molecules to fluctuations of stock prices have been modeled as random walks. Let us therefore discuss this in some detail. A random walk is an AR 1 series with a 1. From the definition of an AR series given, the value of the time series at time t is therefore yt et e-i e-2 e yt-i 2.10 In words,...
Arbitrage Pricing Theory
Arbitrage pricing theory was originally proposed by Stephen A. Ross in 1976. Unlike the preceding introduction, in which APT was presented as an extension of CAPM, the original proposal by Ross is actually embedded in an arbitrage argument and is appropriately reflected in the name of the theory. In this chapter, however, we will avoid an elaborate discussion on the foundations of APT. For that, we direct the reader to the material listed in the references. Instead, we will provide simple...
Market Neutral Strategy
Having discussed CAPM, we now have the required machinery to define market neutral strategies They are strategies that are neutral to market returns, that is, the return from the strategy is uncorrelated with the market return. Regardless of whether the market goes up or down, in good times and bad the market neutral strategy performs in a steady manner, and results are typically achieved with a lower volatility. This desired outcome is achieved by trading market neutral portfolios. Let us...
Application Calculation Of Portfolio Beta
The topic of this section is more of a misnomer, and it likely to strike the reader as an anomaly. We had earlier stated that the APT is a more advanced version of CAPM. Then, if we have access to the APT model, why would we want to calculate the parameters of a simpler CAPM model Is that not going back full circle That would indeed be true. However, if we are looking to hedge our portfolio with the market portfolio, then the hedge ratio that provides the best possible hedge is given by the...
The Capm Model
CAPM is an acronym for the Capital Asset Pricing Model. It was originally proposed by William T. Sharpe. The impact that the model has made in the area of finance is readily evident in the prevalent use of the word beta. In contemporary finance vernacular, beta is not just a nondescript Greek letter, but its use carries with it all the import and implications of its CAPM definition. Along with the idea of beta, CAPM also served to formalize the notion of a market portfolio. A market portfolio...
Application Tracking Basket Design
A tracking basket is a basket of stocks that tracks an index. If the basket is the same as the index, then the prices of both will be the same at all times. However, if the tracking basket is composed of fewer stocks than the index, then there is likely to be tracking error that is, the returns for the tracking basket are not exactly the same as the returns for the index. The discrepancy in the returns is expressed in terms of tracking error. Tracking error may be defined as the standard...