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Unformatted text preview: Data Analysis for Financial Engineering Professor S. Kou, Department of IEOR, Columbia University Lecture 3. Fitting Stock Return Distributions and Introduction to Dependent Structures of Stock Returns 1 Preliminary De&amp;nitions of Returns 1.1 Simple Returns Let the stock price be S ( t ) . The one-period simple return is de&amp;ned to be R t = S ( t ) &amp; S ( t &amp; 1) S ( t &amp; 1) : In other words S ( t ) = S ( t &amp; 1)(1+ R t ) . The above de&amp;nition can be easily generated to multiple periods. In fact, the k period simple return is de&amp;ned to be 1 + R t [ k ] = S ( t ) S ( t &amp; k ) = S ( t ) S ( t &amp; 1) S ( t &amp; 1) S ( t &amp; 2) S ( t &amp; k + 1) S ( t &amp; k ) = (1 + R t )(1 + R t &amp; 1 ) (1 + R t &amp; k +1 ) : In other words R t [ k ] = k &amp; 1 Y j =0 (1 + R t &amp; j ) &amp; 1 : Many &amp;nancial contracts use simple returns, e.g. credit card rates, LIBOR rates, swap rates; many treasury rates are based on simple returns. Very often we also talk about average simple return: A t [ k ] = @ k &amp; 1 Y j =0 (1 + R t &amp; j ) 1 A 1 =k &amp; 1 = exp 8 &lt; : 1 k k &amp; 1 X j =0 log(1 + R t &amp; j ) 9 = ; &amp; 1 ; which is the geometric average of the one period simple return. 1.2 Continuously Compounded Returns For mathematical convenience, we also use continuously compounded returns. Since lim n !1 (1+ x n ) n = e x , the limit of the simple return, when the compounding frequency goes to in&amp;nity, is the continuously compounded return (also called log return) r t = log S ( t ) S ( t &amp; 1) : 1 The main reason is that the continuously compounded returns are easily computed for multiple periods. In fact for k-periods, we have r t [ k ] = log S ( t ) S ( t &amp; k ) = r t + r t &amp; 1 + + r t &amp; k +1 : Mathematically, this form is simpler than that of the simple return, thus making it widely used in theoretical analysis, as well as in some practical cases. The di/erence between simple and log returns for daily data is quite small, although it could be substantial for monthly and yearly data. In fact, if R t &amp; j , then the Taylor expansions log(1 + x ) x; e x &amp; 1 x yields an approximation for the average of simple returns, A t [ k ] exp 8 &lt; : 1 k k &amp; 1 X j =0 R t &amp; j 9 = ; &amp; 1 1 k k &amp; 1 X j =0 R t &amp; j ; which is exactly the average of continuously compounded returns. 1.3 Dividend Payments After paying a dividend D ( t ) between time t &amp; 1 and t , the return becomes R t = S ( t ) + D ( t ) S ( t &amp; 1) &amp; 1 ; for the simple return, and for the continuous compounding return r t = log f S ( t ) + D ( t ) g &amp; log f S ( t &amp; 1) g : 1.4 Returns for Long and Short-Selling Positions Long means holding the asset. Short means borrowing the asset, selling it immediately, and promising to return the asset in the future. Therefore, returns for short positions are exactly opposite of otherwise identical long positions....
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