day1 1000 day2 10 51 day3 1083 day4 1102 day5 1100a Calculate the series of daily log returns in b What is the accumulated return over the five days Compare this value with what you would have obtai

To calculate the daily log returns in %, we can use the formula:

Daily Log Return = ln(Pt/Pt-1) * 100

where Pt is the price of the stock on day t, and Pt-1 is the price of the stock on the previous day.

a) Let's calculate the series of daily log returns in %:

Day 1: log return = ln(10.00/10.00) * 100 = 0.00% Day 2: log return = ln(10.51/10.00) * 100 = 5.11% Day 3: log return = ln(10.83/10.51) * 100 = 3.04% Day 4: log return = ln(11.02/10.83) * 100 = 1.75% Day 5: log return = ln(11.00/11.02) * 100 = -0.18%

b) To calculate the accumulated return over the five days, we can use the formula:

Accumulated Return = (1 + log return 1/100) * (1 + log return 2/100) * ... * (1 + log return n/100) - 1

Accumulated Return = (1 + 0.00/100) * (1 + 5.11/100) * (1 + 3.04/100) * (1 + 1.75/100) * (1 - 0.18/100) - 1

Accumulated Return = 0.0000 * 1.0511 * 1.0304 * 1.0175 * 0.9982 - 1

Accumulated Return = 0.0013 or 0.13%

If we had used arithmetic returns instead, the accumulated return would be calculated as:

Arithmetic Return = ((Pt - Pt-1)/Pt-1) * 100

Arithmetic Return = ((10.51 - 10.00)/10.00) * 100 + ((10.83 - 10.51)/10.51) * 100 + ((11.02 - 10.83)/10.83) * 100 + ((11.00 - 11.02)/11.02) * 100

Arithmetic Return = 5.1% + 3.0% + 1.8% - 0.2% = 9.7%

The accumulated return using arithmetic returns would be 9.7%, which is significantly different from the accumulated return using log returns.

c) Assuming the weekly returns do not change over time, we can calculate the price of a share by the end of the year using the arithmetic returns.

Arithmetic Return for a year = Average Arithmetic Return * 52 weeks

Average Arithmetic Return = (5.1% + 3.0% + 1.8% - 0.2%)/4 = 2.425%

Arithmetic Return for a year = 2.425% * 52 = 126.1%

Price at the end of the year = Initial price * (1 + Arithmetic Return for a year/100)

Price at the end of the year = 10.00 * (1 + 126.1/100) = 22.61

The price of a share by the end of the year, assuming the weekly returns do not change over time, would be $22.61.

d) To calculate the average arithmetic return over the week, we sum up the daily arithmetic returns and divide by the number of days in the week.

Average Arithmetic Return = (5.1% + 3.0% + 1.8% - 0.2%)/4 = 2.425