Can you explain why sometimes the returns are calculated in the formula as x_n/x_(n-1) - but other times it is calculated as ln(x_n/x_(n-1))? which one should i use in any specific occasion? thanks. This is for the covariance and volatility update formulas.
The difference is the assumed compounding. \( x_n/x_{n-1}-1 \) assumes compounding at the base frequency of your series. i.e. daily, monthly etc. \( \ln(x_n/x_{n-1}) \) assumes continuous compounding. Bearing this in mind, it's a case of how you anticipate the numbers will be used.
In your case, volatility and covariance would not normally be interpreted in terms of continuously compounded returns for direct use. i.e. reporting to a client. They would be expecting numbers with daily compounding for example.
Where log returns have an advantage is that the cumulative return between two time points is simply the sum of the log returns whereas for non-continuously compounded returns you would need to compound them together in the usual way. Obviously the latter is more computationally intensive so log returns are better suited for simulation or modelling activities where they are internal to the process rather than a direct output. They also play nicely with a lot of the mathematics around stochastic processes.
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