Exercise 1: Solution

1. Calculate the raw profit, raw returns, and log returns for each day relative to the previous day.

Day 2

Log Returns = ln(105)-ln(100) ≈ 0.0488 ≈ +5%
Profit = 105 - 100 = +$5
Raw Returns = 105/100 = 1.05 = 105%

Day 3

Log Returns = ln(103)-ln(105) ≈ −0.0191 ≈ -2%
Profit = 103 - 105 = -$2
Raw Returns = 103/105 ≈ 0.9810 ≈ 98%

Day 4

Log Returns = ln(107)-ln(103) ≈ 0.0381 ≈ +4%
Profit = 107 - 103 = +$4
Raw Returns = 107/103 ≈ 1.0388 ≈ 104%

Day 5

Log Returns = ln(106)-ln(107) ≈ -0.0094 ≈ -1%
Profit = 106 - 107 = -$1
Raw Returns = 106/107 ≈ 0.9907 ≈ 99%

2a. Calculate the raw profit, raw returns, and log returns for the entire period (i.e. buy at `t=0` and sell at `t=4`)

Full Period

Log Returns = ln(106) - ln(100) ≈ 0.05827 ≈ +6%
Profit = 106 - 100 = +$6
Raw Returns = 106/100 = 1.06 = 106%

2b. Demonstrate and explain how:

(i) The full-period raw profit relates to the additive and symmetrical properties of the individual raw profit from part (1).

Additive Property: adding up sequential log returns will give cumulative return for that period

Raw Profit holds: +$5-$2+$4-$1 = +$9-$3 = +$6

Symmetrical Property: the order of individual returns does not matter to the cumulative return.

Raw Profit holds: +$5-$2+$4-$1 = -$2+$4-$1+$5 =+$4-$2-$1+$5 = +$6

(ii) The full-period log return relates to the additive and symmetrical properties of the individual log returns from part (1).

Additive Property: adding up sequential log returns will give cumulative return for that period

Log Returns holds: +5%-2%+4%-1% = +9%-3% = +6%

Symmetrical Property: the order of individual returns does not matter to the cumulative return.

Log Returns holds: +5%-2%+4%-1% = -2%+4%-1%+5% =+4% -2%-1%+5% = +6%

(iii) The full-period raw return relates to the lack of additive property of the individual raw returns from part (1).

Additive Property: adding up sequential log returns will give cumulative return for that period

Raw Returns does not hold: 105%+98%+104%+99%+106% = 512 % ≠ 106%