Measurement of Return and Risk¶
Risk is quantified by the variability of returns, through assigning probabilities and calculating the expected values of returns. The standard deviation, which represents the spread of a set of numbers, is calculated as the square root of the average of the squared deviations from the mean. These deviations are the differences between each outcome and the mean value of all values.
While standard deviation is useful when means are identical, the coefficient of variance provides a relative measure of risk, especially useful for comparing datasets or projects that vary in size and mean. The coefficient of variance represents the degree of variation relative to the mean, offering a more accurate gauge of risk in comparisons. Generally, a higher coefficient of variation signals greater risk.
Rate of Return¶
The rate of return is determined using the formula:
- Rate of Return:
- \(R_i = \frac{(P_1 - P_0) + D_1}{P_0}\)
Where: - \(R_i\) = Rate of return, - \(P_1\) = Ending price, - \(P_0\) = Beginning price, - \(D_1\) = Dividend.
Expected Rate of Return¶
The expected rate of return is calculated by:
- Expected Rate of Return:
- \(\bar{X} = \sum_{i=1}^{n} X_i p(X_i)\)
Where: - \(\bar{X}\) = Expected value of return, - \(X_i\) = Return in state \(i\), - \(p(X_i)\) = Probability of state \(i\).
Risk¶
Variance¶
Variance, which measures risk, is calculated as follows:
- Variance:
- \(\sigma^2 = \sum_{i=1}^{n} [(X_i - \bar{X})^2 p(X_i)]\)
Where: - \(\sigma^2\) = Variance, - \(X_i\) = Return in state \(i\), - \(\bar{X}\) = Expected value of return, - \(p(X_i)\) = Probability of state \(i\).
Standard Deviation¶
The standard deviation, or the square root of the variance, is represented by:
- Standard Deviation:
- \(\sigma = \sqrt{\sigma^2}\)
Where: - \(\sigma\) = Standard deviation.
This formulaic approach allows for a systematic evaluation of the potential return and associated risks of investment opportunities, providing investors with the tools needed to make informed decisions.
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