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I want to calculate the risk of a portfolio with the following.
In order to calculate the following formula: 
However, I am not sure if I have to use log returns or simple returns to calculate the assets returns. I need the asset returns to calculate the standard deviation as well as the variance-covariance of the assets included in the portfolio.
The logarithmic method of calculating returns is frequently preferred to the obvious alternative of using the return calculated on the basis of simple interest over the period in question which, of course, is the monetary return which would actually be achieved by an investment over that period. In some cases this is due to the assumptions that prices are distributed log normally (which, in practice, may or may not be true for any given price series), then log(1 + r_i) is conveniently normally distributed.
Why Log Returns Benefits and Downsides
A Comparison between Logarithmic and Simple Returns Calculating and Comparing Security Returns
Severals thoughts: First: this topic is already covered here - mabye you have to collect parts. YOu can start here. A very good overview is given here too.
Let us define the following notions. Let $P_t^1$ and $P_t^2$ denote the prices of zwo assets at time $t$. Then $$ r_t = \frac{P_t^1-P_{t-1}^1}{P_{t-1}^1} = \frac{P_t^1}{P_{t-1}^1} -1 $$ denotes the simple return whereas $$ r_t^l = \log\left(\frac{P_t^1}{P_{t-1}^1} \right), $$ where $\log$ is the natural logarithm, denotes the log-return. Note that $r_t \in [-1,\infty)$ and $r_t^l \in (-\infty,\infty).$
Then for a portfolio where we have $q^i$ pieces of each stock we have $$ w_i = \frac{q^i P^i_t}{q^1 P^1_t+q^2 P^2_t}, $$ the fraction of asset $i$. Then the portfolio returns is given by $$ \frac{P_{t+1}}{P_t}-1 = \frac{q^1 P^1_{t+1} + q^2 P^2_{t+1}}{q^1 P^1_t+q^2 P^2_t}-1, $$ which equals $$ w_1 r^1_t + (1-w_1) r^2_t. $$ Thus the portfolio returns is the a linear combination of the asset returns. This is not true for log returns (take the log-returns and you see that the above expansion of terms does not work).
The aggregation of (portfolio) returns over time is easier: $$ \begin{eqnarray} \log(P_T/P_0) &=& \log(P_T/P_1 \cdot P_1/P_0) \\ &=& \log(P_T/P_{T-1}\cdot P_{T-1}/P_{T-2} \cdots P_1/P_0)\\ &=& \log(P_T/P_{T-1}) + \cdots + \log(P_1/P_0)\\ &=& \sum_{t=1}^T r_t^l \end{eqnarray} $$ where $r_t^l$ denotes the log-return over one period.
This is much more complicated for simple returns: For simple returns we get $$ \begin{eqnarray} \frac{P_T}{P_0}-1 &=& \frac{P_T}{P_{T-1}} \cdot \frac{P_{T-1}}{P_{T-2}} \cdots \frac{P_1}{P_{0}} - 1\\ &=& \prod_{t=1}^T (1+r_t) - 1. \end{eqnarray} $$
In Markowitz's famous paper he writes that the return on the portfolio is a weighted average of the returns on the individual assets. For the two asset case: $R_p = w r_1 + (1-w) r_2$. From this we can derive the equation you gave above, using a theorem in Statistics for the standard deviation of a weighted average. This formula for $R$ makes sense only if the returns being used are simple returns, not logarithmic returns. So that is what I would use.
I would say that one should differenciate between what the formula means, how the inputs are calculated, and how one would use it.
What is means: As noob2 points out it makes sense for simple returns' aggregation. For log returns it only holds at the limit where the time step is zero.
How to calculate input variances/covariances: I would take logs before calculating statistics, because only under this transformation my series become homogeneous in time and additive through time. Ponder on the fact that simple returns are bounded by -1 to understand how it works.
How to use: If I were to simulate, I would first draw log returns, then convert them to simple returns, and then apply the weighting as noob2 highlights.
