returnCalculations - Logarithms and Exponential Functions x...

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Unformatted text preview: Logarithms and Exponential Functions x -0.5 -0.4 -0.3 -0.2 -0.1 0 0.01 0.05 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 ln(x) exp(x) ln(exp(x)) 0.61 -0.5 0.67 -0.4 0.74 -0.3 0.82 -0.2 0.9 -0.1 1 0 -4.61 1.01 0.01 -3 1.05 0.05 -2.3 1.11 0.1 -1.61 1.22 0.2 -1.2 1.35 0.3 -0.92 1.49 0.4 -0.69 1.65 0.5 -0.51 1.82 0.6 -0.36 2.01 0.7 -0.22 2.23 0.8 -0.11 2.46 0.9 0 2.72 1 0.1 3 1.1 0.18 3.32 1.2 0.26 3.67 1.3 0.34 4.06 1.4 0.41 4.48 1.5 0.47 4.95 1.6 0.53 5.47 1.7 0.59 6.05 1.8 0.64 6.69 1.9 0.69 7.39 2 6 4 2 y 0 -2 -4 -6 -1 -0.5 0 Note: graph is created using chart wizard. Chart type is xy scatter. y = ln(x), y=exp(x) 6 4 2 y 0 -2 -4 -6 -1 -0.5 0 0.5 x 1 1.5 2 2.5 Column D Column E Note: graph is created using chart wizard. Chart type is xy scatter. Time Value of Money Compounding Frequency annual quarterly weekly daily continuously R FVnm = $V ⋅ 1 + m mn V $1,000.00 R 0.1 m Periodic Rate 1 10.00% 4 2.50% 52 0.19% 365 0.03% infinity 0.00% FVnc = $V ⋅ e R⋅n R $V ⋅ 1 + m mn Value of V Effective at end of Annual year Rate $1,100.00 10.00% $1,103.81 10.38% $1,105.06 10.51% $1,105.16 10.52% $1,105.17 10.52% R RA = 1 + −1 m m e R −1 FVnc = $V ⋅ e R⋅n Asset Return Calculations I. Simple Returns month t 1 2 3 Rt = Price Pt 80 85 90 Pt − Pt −1 Pt −1 One Period Return Rt 0.0625 0.0588 Two Period Return Rt(2) 0.1250 Two Period Return Rt(2) 0.1250 Rt (2) = Pt − Pt −2 Pt −2 Rt (2) = (1 + Rt −1 )(1 + Rt ) −1 II. Continuously Compounded Returns month t 1 2 3 Price Pt 80 85 90 ln(Price) ln(Pt) 4.38 4.44 4.5 One Period Return rt One Period Two Period Return Return rt rt(2) 0.0606 0.0572 rt = ln(1 + Rt ) 0.0606 0.0572 0.1178 rt (2) = ln(1 + Rt (2)) rt (2) = ln( Pt ) − ln( Pt −2 ) rt = ln( Pt ) − ln( Pt −1 ) Note: this formula is not very practical for computations. )(1 + Rt ) −1 Two Period Return rt(2) 0.1178 Two Period Return rt(2) 0.1178 rt (2) = rt −1 + rt rt (2) = ln(1 + Rt (2)) ) = ln( Pt ) − ln( Pt −2 ) ...
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