By: Dr K C Gupta, YBB Personal Finance

MPT is the use of statistics in portfolio analysis (1950s- ). Several terms were customized for portfolio applications. Only basic math is assumed as various software tools make it all easy to use.

A LINEAR REGRESSION is applied to 2 data series, one with fund returns (daily, weekly or monthly) & another with benchmark returns (e.g. SP500 for stock & hybrid funds). In a graph of values showing benchmark returns on horizontal axis & fund returns on the vertical axis, the slope of the best-fit line is BETA (short-term volatility) & its intercept on the vertical axis is ALPHA.

Alpha > 0 means that fund outperforms its beta-bogey (beta x benchmark), alpha = 0 means that fund just keeps up, & alpha < 0 means that fund underperforms. Alpha is manager’s skill or magic.

Beta > 1 means that the fund’s short-term moves (e.g. % changes daily) are greater than those for the benchmark, beta = 1 means the moves are comparable, & beta < 1 means the moves are smaller. So, if a fund’s beta is 0.60, & the SP500 on a day fell by -2%, then the fund likely fell by around -1.2% (= -2 x 0.6).

CORRELATION r has the meaning of angle or direction when two data series have only 2 or 3 components but becomes abstract with larger data series. Two funds with a high correlation tend to move in the same direction. If r = 0, then the two funds are uncorrelated or move independently. If r = -1, then two funds are negatively or inversely correlated. The correlations may vary with time. The MPT statistics are most meaningful when the correlation r is near +1 or -1.

STANDARD DEVIATION (SD) is a measure of dispersion of the data points in a series about the average. There are probabilities associated with the number of data points contained within the intervals of average +/- 1*SD, +/- 2*SD, +/- 3*SD. The SD may be thought of as longer-term volatility (e.g. for 1 year or more). A ratio of SDs,
the Relative SD = SD/SDbenchmark = beta/r,
indicates the relative longer-term volatility of the fund with respect to the benchmark. This ratio may be considered as Effective Equity & used for asset allocation purposes. So, if the relative SD is 0.7, you can expect that you have about 70% exposure to the market (benchmark).

Note that correlation r & standard deviation SD can be calculated independently of the MPT or regression analysis. They are also much more stable than beta or alpha.

The monthly returns & SDs can be ANNUALIZED as,
   Annualized Return = 12 x Monthly Return,
   Annualized SD = sqrt(12) x Monthly SD = 3.46 x Monthly SD.

There are similar relations for weekly data with factors 52 & sqrt(52) = 7.21, & daily data with factors 250 & sqrt(250) = 15.81, assuming 250 trading days in a year. The annualized SD based on daily returns would be higher than the SD based on weekly or monthly returns. 

SHARPE RATIO (SR) is defined as the ratio of the excess fund return (vs risk-free 3m T-Bills) & the fund SD, or, the excess return per unit of volatility (risk). It depends highly on the time period & fund category. It’s hard to find funds with positive SR in aggressive categories because it’s hard to beat indexes over time.

There are several PORTFOLIO ANALYTICS SOFTWARE that provide MPT statistics: Portfolio Visualizer (PV), Stock Rover (SR) Premium, TestFol, MFO Premium (MFOP), Morningstar Portfolio. Be careful to use data from a single source only because some assumptions made may be different. You can also develop your own Excel or Google Spreadsheets, if so inclined.

As an EXAMPLE of MPT statistics from PV, consider the 3-yr period from 01/2022 to 12/2024.
   LC-growth VIGAX: alpha = -0.70, beta = 1.20, SD = 22.08, r = 0.95, SR = 0.33
   LC-blend VFIAX (SP500): alpha = +0.10, beta = 1.00, SD = 17.40, r = 1.00, SR = 0.35
   Allocation 60-40 VBINX: alpha = -2.88, beta = 0.74, SD = 13.16, r = 0.98, SR = 0.04

As SP500 is the benchmark used, note that VIFAX has beta = 1.00 & r = 1.00; aggressive LCG VIGAX has higher beta & SD; allocation/hybrid VBINX has lower beta & SD.
Relative SD (or, Effective Equity) shows that VGIAX has 1.2690x or 126.90% long-term volatility of SP500, & VBINX has 0.7563x or 75.63%. Relative SD can be used for portfolio design &/or risk assessment.

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