The model was created in 2003 and first tested in 2004.

# MPT Statistics

# (Modern Portfolio Theory)

### Beta

While standard deviation determines the volatility of a fund according to the disparity of its returns over a period of time, beta, another useful statistical measure, determines the volatility (or risk) of a fund in comparison to that of its index or benchmark. A fund with a beta very close to 1 means the fund’s performance closely matches the index or benchmark. A beta greater than 1 indicates greater volatility than the overall market, and a beta less than 1 indicates less volatility than the benchmark.

### Alpha

How do we measure the extra return rewarded to you for taking on risk posed by factors other than market volatility? Enter alpha, which measures how much if any of this extra risk helped the fund outperform its corresponding benchmark. Using beta, alpha’s computation compares the fund’s performance to that of the benchmark’s risk-adjusted returns and establishes if the fund’s returns outperformed the market’s, given the same amount of risk.

### R-Squared

The R-squared of a fund advises investors if the beta of a mutual fund is measured against an appropriate benchmark. Measuring the correlation of a fund’s movements to that of an index, R-squared describes the level of association between the fund’s volatility and market risk, or more specifically, the degree to which a fund’s volatility is a result of the day-to-day fluctuations experienced by the overall market. R-squared values range between 0 and 100, where 0 represents the least correlation and 100 represents full correlation.

# Volatility Measures

### Mean

This is the average return over the period. It can be calculated either as an arithmetic mean or as a compound mean. For financial computations the compound mean must be used.

### Standard Deviation

The standard deviation essentially reports a fund’s volatility, which indicates the tendency of the returns to rise or fall drastically in a short period of time. A security that is volatile is also considered higher risk because its performance may change quickly in either direction at any moment. The standard deviation of a fund measures this risk by measuring the degree to which the fund fluctuates in relation to its mean return, the average return of a fund over a period of time.

# Ratios

### Sharpe Ratio

The Sharpe ratio (reward-to-variability ratio) relates excess return over the risk-free rate to the additional risk taken measured by the standard deviation. The higher the Sharpe ratio, the better the performance of the portfolio under analysis. Sharpe = (Rp – Rf)/ σ

### Sortino Ratio

The Sortino ratio (reward-to-downside-variability ratio) relates excess return over the risk-free rate to the additional risk taken measured by the standard deviation of the underperforming returns. The higher the Sortino ratio, the better the performance of the portfolio under analysis. Sortino = (Rp – Rf)/ σ(underperf)

### Treynor Ratio

The Treynor ratio relates excess return over the risk-free rate to the additional risk taken measured by the beta. The higher the Treynor ratio, the better the performance of the portfolio under analysis. Treynor = (Rp – Rf)/ β