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For any fund that evolves randomly with time, volatility is defined as the standard deviation of a sequence of random variables, each of which is the return of the fund over some corresponding sequence of (equally sized) times. Thus, "annualized" volatility σ annually is the standard deviation of an instrument's yearly logarithmic returns. [2]
Under the assumption of normality of returns, an active risk of x per cent would mean that approximately 2/3 of the portfolio's active returns (one standard deviation from the mean) can be expected to fall between +x and -x per cent of the mean excess return and about 95% of the portfolio's active returns (two standard deviations from the mean) can be expected to fall between +2x and -2x per ...
Stock A over the past 20 years had an average return of 10 percent, with a standard deviation of 20 percentage points (pp) and Stock B, over the same period, had average returns of 12 percent but a higher standard deviation of 30 pp. On the basis of risk and return, an investor may decide that Stock A is the safer choice, because Stock B's ...
are the residual (random) returns, which are assumed independent normally distributed with mean zero and standard deviation These equations show that the stock return is influenced by the market (beta), has a firm specific expected value (alpha) and firm-specific unexpected component (residual).
is the standard deviation of the stock's returns. This is the square root of the quadratic variation of the stock's log price process, a measure of its volatility. Option related: (,) is the price of the option as a function of the underlying asset S at time t, in particular:
The MPT is a mean-variance theory, and it compares the expected (mean) return of a portfolio with the standard deviation of the same portfolio. The image shows expected return on the vertical axis, and the standard deviation on the horizontal axis (volatility). Volatility is described by standard deviation and it serves as a measure of risk. [7]
It is defined as the difference between the returns of the investment and the risk-free return, divided by the standard deviation of the investment returns. It represents the additional amount of return that an investor receives per unit of increase in risk. It was named after William F. Sharpe, [1] who developed it in 1966.
σ M = standard deviation of the market portfolio σ P = standard deviation of portfolio (R M – I RF)/σ M is the slope of CML. (R M – I RF) is a measure of the risk premium, or the reward for holding risky portfolio instead of risk-free portfolio. σ M is the risk of the market portfolio. Therefore, the slope measures the reward per unit ...