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SlidesChapter3

# SlidesChapter3 - Returns 1 RETURNS Prices and returns Let...

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Returns 1 RETURNS Prices and returns Let P t be the price of an asset at time t . Assuming no dividends the net return is R t = P t P t - 1 - 1 = P t - P t - 1 P t - 1 The simple gross return is P t P t - 1 = 1 + R t

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Returns 2 Example: If P t - 1 = 2 and P t = 2 . 1 then 1 + R t = P t P t - 1 = 2 . 1 2 = 1 . 05 and R t = 0 . 05
Returns 3 The gross return over k periods ( t - k to t ) is 1 + R t ( k ) := P t P t - k = P t P t - 1 P t - 1 P t - 2 · · · P t - k +1 P t - k = (1 + R t ) · · · (1 + R t - k +1 ) Returns are scale-free , meaning that they do not depend on monetary units (dollars, cents, etc.) not unit-less — unit is time; they depend on the units of t (hour, day, etc.)

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Returns 4 Example: Time t - 2 t - 1 t t + 1 P 200 210 206 212 1 + R 1.05 .981 1.03 1 + R (2) 1.03 1.01 1 + R (3) 1.06 1+R 1.05 = 210/200 .981 = 206/210 1.03 = 212/206
Returns 5 Example: Time t - 2 t - 1 t t + 1 P 200 210 206 212 1 + R 1.05 .981 1.03 1 + R (2) 1.03 1.01 1 + R (3) 1.06 1+R(2) 1.03 = 206/200 1.01 = 212/210 1+R(3) 1.06 = 212/200

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Returns 6 Log returns log prices : p t := log( P t ) log( x ) = the natural logarithm of x Continuously compounded or log returns are logarithms of gross returns: r t := log(1 + R t ) = log P t P t - 1 = p t - p t - 1 where p t := log( P t )
Returns 7 Example: Suppose P t - 1 = 2 . 0 and P t = 2 . 06. Then 1 + R t = 1 . 03, R t = . 03, and r t = log(1 . 03) = . 0296 . 03

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Returns 8 Advantage — simplicity of multiperiod returns r t ( k ) := log { 1 + R t ( k ) } = log { (1 + R t ) · · · (1 + R t - k +1 ) } = log(1 + R t ) + · · · + log(1 + R t - k +1 ) = r t + r t - 1 + · · · + r t - k +1
Returns 9 Log returns are approximately equal to net returns: x small log(1 + x ) x therefore, r t = log(1 + R t ) R t Examples: * log(1 + . 05) = . 0488 * log(1 - . 05) = - . 0513 see Figure

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Returns 10 -0.2 -0.1 0 0.1 0.2 -0.2 -0.1 0 0.1 0.2 x log(1+x) x Comparison of functions log(1 + x ) and x .
Returns 11 Behavior of returns What can we say about returns? cannot be perfectly predicted — are random.

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Returns 15 Uncertainty in returns At time t - 1, P t and R t are not only unknown, but we do not know their probability distributions. Can estimate these distributions: with an assumption
Returns 16 Leap of Faith: Future returns similar to past returns So distribution of P t can estimated from past data

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Returns 17 Asset pricing models (e.g. CAPM) use the joint distribution of cross-section { R 1 t , . . . , R Nt } of returns on N assets at a single time t . R it is the return on the i th asset at time t . Other models use the time series { R 1 , R 2 , . . . , R t } of returns on a single asset at a sequence of times 1 , 2 , . . . , t . We will start with a single asset.
Returns 18 Common Model — IID Normal Returns R 1 , R 2 , . . . = returns from single asset. 1. mutually independent 2. identically distributed 3. normally distributed IID = i ndependent and i dentically d istributed

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Returns 19 Two problems : The model implies the possibility of unlimited losses , but liability is usually limited R t ≥ - 1 since you can lose no more than your investment 1 + R t ( k ) = (1 + R t )(1 + R t - 1 ) · · · (1 + R t - k +1 ) is not normal sums of normals are normal but not products but it would be nice to have normality, so math is simple
Returns 20 The Lognormal Model Assumes: r t = log(1 + R

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SlidesChapter3 - Returns 1 RETURNS Prices and returns Let...

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