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# returns_overhead - 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 12 ancient Greeks: would have thought of returns as determined by Gods or Fates (three Goddesses of destiny) did not realize random phenomena exhibit regularities (law of large numbers and the central limit theorem) did not have probability theory despite their impressive math
Returns 13 Peter Bernstein — Against the Gods: The Remarkable Story of Risk development of probability theory and understanding of risk these took a surprisely long time

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Returns 14 Probability arose out of gambling during the Renaissance. University of Chicago economist Frank Knight (1916 Cornell PhD) distinguished between measurable uncertainty or “risk proper” (e.g., games of chance) — probabilities known unmeasurable uncertainty (e.g., finance) — probabilities unknown
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

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Returns 16 Leap of Faith: Future returns similar to past returns So distribution of P t can estimated from past data
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 .

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

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