Compute 6 Month Returns using Monthly Returns data

Question

I have data containing 6 months returns of a stock.

+-------------+-------------+
|    Date     |   Return    |
+-------------+-------------+
| 31/Jan/2016 | -0.06861672 |
| 29/Feb/2016 | -0.02187975 |
| 31/Mar/2016 |  0.01420873 |
| 30/Apr/2016 |  0.01858721 |
| 31/May/2016 |  0.02648122 |
| 30/Jun/2016 | -0.04780486 |
+-------------+-------------+

If I invested $100 at 31st Dec 2015 what would be my return at 30th Jun 2016?

The way I computed is I took the sum of returns which turns out to be 0.07902. Returns = 100 * (0.07902+1) = 92.09758

I am confused with the second approach where I've computed position each month using returns and than the last position that I have is my cumulative return.

If I liquidate my position using this calculation than at 30th Jun 2016 I would have 91.9867. I am confused which one is the correct approach?

+-------------+-------------+
|    Date     |   Position  |
+-------------+-------------+
| 31/Jan/2016 |   93.138328 |
| 29/Feb/2016 | 91.10048467 |
| 31/Mar/2016 | 92.39490686 |
| 30/Apr/2016 | 94.11227039 |
| 31/May/2016 | 96.60447813 |
| 30/Jun/2016 | 91.98631458 |
+-------------+-------------+

Answer 1

Calculate the product of (100+monthly_return), i.e. multiply all numbers in the second column:

|      return | return*100 |
|------------:|-----------:|
| -0.06861672 |   0.931383 |
| -0.02187975 |   0.978120 |
|  0.01420873 |   1.014209 |
|  0.01858721 |   1.018587 |
|  0.02648122 |   1.026481 |
| -0.04780486 |   0.952195 |

The cumulative return is 0.919863. Average monthly return is 0.919863^1/6-1 = -0.006937.

Answer 2

The reason for the difference is that you haven’t taken into account the impact of compounding. Summing the returns at each time period is not the correct approach here. To do this, you should multiply the returns together, not add them.