What Statistical Concepts are Important in Portfolio Management?
All answers and calculations are from ZOONOVA
Here are some of the important ones with their definitions.
β
Beta (Β) is a measure of the volatility, or systematic risk, of a security or a portfolio in comparison to the market as a whole. Beta is used in the capital asset pricing model (CAPM), which calculates the expected return of an asset based on its beta and expected market returns.
BMrk α
Benchmark Alpha (α), often considered the active return on an investment, gauges the performance of an investment against a market index used as a benchmark, since they are often considered to represent the market’s movement as a whole. The excess returns of a fund relative to the return of a benchmark index is the fund's alpha.
CAPM α
CAPM Alpha (α) is determined by the difference between how much a security, or portfolio, should be returning according to the Capital Asset Pricing Model (CAPM) vs the actual security or portfolio. This is sometimes referred to as Jensen's Alpha.
CAPM Price
The expected price of the underlying stock using the Capital Asset Pricing Model. It uses the S&P 500 benchmark from the start of the year to the current day, and the CAPM calculated expected return (CAPM RoR).
CAPM RoR
The expected return of the stock using the CAPM, or Capital Asset Pricing Model, for the calculation. The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The risk-free rate is customarily the yield on government bonds like U.S. Treasuries.
CVaR
Conditional Value At Risk (CVaR) is also known as mean excess loss, mean shortfall, tail Var, average value at risk or expected shortfall. CVaR was created to serve as an extension of value at risk (VaR). The VaR model allows managers to limit the likelihood of incurring losses caused by certain types of risk, but not all risks. The problem with relying solely on the VaR model is that the scope of risk assessed is limited, since the tail end of the distribution of loss is not typically assessed. Therefore, if losses are incurred, the amount of the losses will be substantial in value. Zoonova calculates CVAR for both 95% and 99% confidence levels.
Drawdown
Take the minimum stock price over the 2 year period and look at the most recent price. If Max is the maximum and Min is the minimum price over the past umpteen years, and P is the current price, we look at: LOSS = 1 – P / Max and GAIN = P / Min – 1 The more positive the ratio the better. A maximum drawdown (MDD) is the maximum loss from a peak to a trough of a portfolio before a new peak is attained. (Trough Value – Peak Value) ÷ Peak Value.
ExpReturn
Expected Return is the annual expected return of a stock using 2 years of closing prices.
Info Ratio
The information ratio (IR) is a ratio of portfolio returns above the returns of a benchmark – usually an index – to the volatility of those returns. The information ratio (IR) measures a portfolio manager's ability to generate excess returns relative to a benchmark but also attempts to identify the consistency of the investor. The higher Info ratio the better.
M - Squared
(a.k.a. M2, M-Squared) In simple words, it measures the returns of an investment portfolio for the amount of risk taken relative to some benchmark portfolio. Popularly known as Modigliani Risk Adjustment Performance Measure or M2, it was developed by Nobel prize winner Franco Modigliani and his grandaughter Leah Modigliani in the year 1997.
Pds Down
How many periods the stock was down.
Pds Up
How many periods the stock was up.
R - Squared
(a.k.a. R2, R-Squared) R-squared is a statistical measure that represents the percentage of a fund or security's movements that can be explained by movements in a benchmark index. For example, an R-squared for a fixed-income security versus the Barclays Aggregate Index identifies the security's proportion of variance that is predictable from the variance of the Barclays Aggregate Index. The same can be applied to an equity security versus the Standard and Poor's 500 or any other relevant index. R squared is measured from 0–1 or 0–100%. A level of 1, or 100%, means a perfect correlation to the benchmark.
Sharpe
The Sharpe Ratio is a measure that indicates the average return minus the risk-free return divided by the standard deviation of return on an investment. The Sharpe Ratio is a measure for calculating risk-adjusted return, and this ratio has become the industry standard for such calculations. The higher the Sharpe Ratio the better.
C‑Sharpe
The Conditional Sharpe Ratio is defined as the ratio of expected excess return to the expected shortfall. CVaR is used as the denominator in the C-Sharpe calcuation whereas the standard Sharpe ratio uses Standard Deviation as the denominator.
Sortino
The Sortino ratio is the excess return over the risk-free rate divided by the downside semi-variance, and so it measures the return to "bad" volatility. (Volatility caused by negative returns is considered bad or undesirable by an investor, while volatility caused by positive returns is good or acceptable.) A higher Sortino Ratio is better.
Treynor
The Treynor ratio, also known as the reward-to-volatility ratio, is a metric for returns that exceed those that might have been gained on a risk-less investment, per each unit of market risk. The Treynor ratio, developed by Jack Treynor, is calculated as follows: (Average Return of a Portfolio – Average Return of the Risk-Free Rate)/Beta of the Portfolio. The higher the Treynor Ratio the better. P&L. This is the return of the stock from the beginning of the year to the present.
VaR
Value At Risk (VaR) is a statistical technique used to measure and quantify the level of financial risk within a firm or investment portfolio over a specific time frame. This metric is most commonly used by investment and commercial banks to determine the extent and occurrence ratio of potential losses in their institutional portfolios. VaR calculations can be applied to specific positions or portfolios as a whole or to measure firm-wide risk exposure. Zoonova calculates historical VAR using 2 years of daily prices and returns both VAR at 95% and 99% level of confidence.
Variance
Variance is used in statistics for probability distribution. Since variance measures the variability (volatility) from an average or mean and volatility is a measure of risk, the variance statistic can help determine the risk an investor might take on when purchasing a specific security. A variance value of zero indicates that all values within a set of numbers are identical; all variances that are non-zero will be positive numbers. A large variance indicates that numbers in the set are far from the mean and each other, while a small variance indicates the opposite.
Variance Neg
The calculated negative variance of the stock over the specified period, 2 years.
Volatility
Standard deviation is a statistical term that measures the amount of variability or dispersion around an average. Standard deviation is also a measure of volatility. Generally speaking, dispersion is the difference between the actual value and the average value. The square root of the variance equals the standard deviation.
Volatility Neg
The negative volatility, or standard deviation, of the sock over the specified period of time.
YTD%
Profit or loss of the stock during the current year.
Another important calculation is Portfolio Optimization, Modern Portfolio Theory.
The Capital Asset Pricing Model (CAPM) describes the relationship between risk and expected return and that is used in the pricing of risky securities.
r̅a = rf + βa (r̅m − rf)
Where
rf = Risk-free rate
βa = Risk of the security
r̅m = Expected market return
The general idea behind CAPM is that investors need to be compensated in two ways: Time value of money and risk. The time value of money is represented by the risk-free (rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk measure (βa) that compares the returns of the asset to the market over a period of time and to the market premium (r̅m − rf).
CAPM states that the expected return of a security or a portfolio equals the rate on a risk-free security plus a risk premium. If this expected return does not meet or beat the required return, then the investment should not be undertaken. Using the CAPM model and the following assumptions, one can compute the expected return of a stock:
rf = 3%, βa = 2, r̅m = 10%, the stock is expected to return 17% (3% + 2 (10% − 3%)).
The Modern Portfolio Theory (MPT), or mean-variance optimization (a.k.a. "Portfolio Optimization"), is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk, defined as variance. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. A mean-variance analysis is the process of weighing risk (variance) against expected return. By looking at the expected return and variance of an asset, investors attempt to make more efficient investment choices- seeking the lowest variance for a given expected return, or seeking the highest expected return for a given variance level.
Portfolio Optimization determines the optimal percentage of contribution, given a user-defined minimum return on investment (ROI), for each asset ("Optimal"). In the case of stocks, the difference between the optimal and current position is also calculated ("Δ Shares", NOTE: multiple positions of a single stock are aggregated into a single value). Users then have the option to display output ("As Output") based on optimized positions.
There are more calculations but this is a good start.
Cheers.
