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Unformatted text preview: 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 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 scalefree , meaning that they do not depend on monetary units (dollars, cents, etc.) not unitless unit is time; they depend on the units of t (hour, day, etc.) 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 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 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 Returns 100.20.1 0.1 0.20.20.1 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. 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 Returns 17 Asset pricing models (e.g. CAPM) use the joint distribution of crosssection { 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....
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 Spring '11
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