Stock Price Analysis: Daily Log Returns, Accumulated Return, and Year-End Price Prediction

To\u0020calculate\u0020the\u0020daily\u0020log\u0020returns\u0020in\u0020%,\u0020we\u0020can\u0020use\u0020the\u0020formula:\u0020\n\nDaily\u0020Log\u0020Return\u0020=\u0020ln(Pt/Pt-1)\u0020*\u0020100\n\nwhere\u0020Pt\u0020is\u0020the\u0020price\u0020of\u0020the\u0020stock\u0020on\u0020day\u0020t,\u0020and\u0020Pt-1\u0020is\u0020the\u0020price\u0020of\u0020the\u0020stock\u0020on\u0020the\u0020previous\u0020day.\n\na)\u0020Let's\u0020calculate\u0020the\u0020series\u0020of\u0020daily\u0020log\u0020returns\u0020in\u0020%:\n\nDay\u00201:\u0020log\u0020return\u0020=\u0020ln(10.00/10.00)\u0020*\u0020100\u0020=\u00200.00%\nDay\u00202:\u0020log\u0020return\u0020=\u0020ln(10.51/10.00)\u0020*\u0020100\u0020=\u00205.11%\nDay\u00203:\u0020log\u0020return\u0020=\u0020ln(10.83/10.51)\u0020*\u0020100\u0020=\u00203.04%\nDay\u00204:\u0020log\u0020return\u0020=\u0020ln(11.02/10.83)\u0020*\u0020100\u0020=\u00201.75%\nDay\u00205:\u0020log\u0020return\u0020=\u0020ln(11.00/11.02)\u0020*\u0020100\u0020=\u0020-0.18%\n\nb)\u0020To\u0020calculate\u0020the\u0020accumulated\u0020return\u0020over\u0020the\u0020five\u0020days,\u0020we\u0020can\u0020use\u0020the\u0020formula:\n\nAccumulated\u0020Return\u0020=\u0020(1\u0020+\u0020log\u0020return\u00201/100)\u0020*\u0020(1\u0020+\u0020log\u0020return\u00202/100)\u0020*\u0020...\u0020*\u0020(1\u0020+\u0020log\u0020return\u0020n/100)\u0020-\u00201\n\nAccumulated\u0020Return\u0020=\u0020(1\u0020+\u00200.00/100)\u0020*\u0020(1\u0020+\u00205.11/100)\u0020*\u0020(1\u0020+\u00203.04/100)\u0020*\u0020(1\u0020+\u00201.75/100)\u0020*\u0020(1\u0020-\u00200.18/100)\u0020-\u00201\n\nAccumulated\u0020Return\u0020=\u00200.0000\u0020*\u00201.0511\u0020*\u00201.0304\u0020*\u00201.0175\u0020*\u00200.9982\u0020-\u00201\n\nAccumulated\u0020Return\u0020=\u00200.0013\u0020or\u00200.13%\n\nIf\u0020we\u0020had\u0020used\u0020arithmetic\u0020returns\u0020instead,\u0020the\u0020accumulated\u0020return\u0020would\u0020be\u0020calculated\u0020as:\n\nArithmetic\u0020Return\u0020=\u0020((Pt\u0020-\u0020Pt-1)/Pt-1)\u0020*\u0020100\n\nArithmetic\u0020Return\u0020=\u0020((10.51\u0020-\u002010.00)/10.00)\u0020*\u0020100\u0020+\u0020((10.83\u0020-\u002010.51)/10.51)\u0020*\u0020100\u0020+\u0020((11.02\u0020-\u002010.83)/10.83)\u0020*\u0020100\u0020+\u0020((11.00\u0020-\u002011.02)/11.02)\u0020*\u0020100\n\nArithmetic\u0020Return\u0020=\u00205.1%\u0020+\u00203.0%\u0020+\u00201.8%\u0020-\u00200.2%\u0020=\u00209.7%\n\nThe\u0020accumulated\u0020return\u0020using\u0020arithmetic\u0020returns\u0020would\u0020be\u00209.7%,\u0020which\u0020is\u0020significantly\u0020different\u0020from\u0020the\u0020accumulated\u0020return\u0020using\u0020log\u0020returns.\n\nc)\u0020Assuming\u0020the\u0020weekly\u0020returns\u0020do\u0020not\u0020change\u0020over\u0020time,\u0020we\u0020can\u0020calculate\u0020the\u0020price\u0020of\u0020a\u0020share\u0020by\u0020the\u0020end\u0020of\u0020the\u0020year\u0020using\u0020the\u0020arithmetic\u0020returns.\n\nArithmetic\u0020Return\u0020for\u0020a\u0020year\u0020=\u0020Average\u0020Arithmetic\u0020Return\u0020*\u002052\u0020weeks\n\nAverage\u0020Arithmetic\u0020Return\u0020=\u0020(5.1%\u0020+\u00203.0%\u0020+\u00201.8%\u0020-\u00200.2%)/4\u0020=\u00202.425%\n\nArithmetic\u0020Return\u0020for\u0020a\u0020year\u0020=\u00202.425%\u0020*\u002052\u0020=\u0020126.1%\n\nPrice\u0020at\u0020the\u0020end\u0020of\u0020the\u0020year\u0020=\u0020Initial\u0020price\u0020*\u0020(1\u0020+\u0020Arithmetic\u0020Return\u0020for\u0020a\u0020year/100)\n\nPrice\u0020at\u0020the\u0020end\u0020of\u0020the\u0020year\u0020=\u002010.00\u0020*\u0020(1\u0020+\u0020126.1/100)\u0020=\u002022.61\n\nThe\u0020price\u0020of\u0020a\u0020share\u0020by\u0020the\u0020end\u0020of\u0020the\u0020year,\u0020assuming\u0020the\u0020weekly\u0020returns\u0020do\u0020not\u0020change\u0020over\u0020time,\u0020would\u0020be\u0020$22.61.\n\nd)\u0020To\u0020calculate\u0020the\u0020average\u0020arithmetic\u0020return\u0020over\u0020the\u0020week,\u0020we\u0020sum\u0020up\u0020the\u0020daily\u0020arithmetic\u0020returns\u0020and\u0020divide\u0020by\u0020the\u0020number\u0020of\u0020days\u0020in\u0020the\u0020week.\n\nAverage\u0020Arithmetic\u0020Return\u0020=\u0020(5.1%\u0020+\u00203.0%\u0020+\u00201.8%\u0020-\u00200.2%)/4\u0020=\u00202.425%