## Mean-Variance Optimization

Given the inputs (returns, standard deviations, and correlations), AllocationADVISOR determines the applicable range of possible returns by finding the return of the minimum variance portfolio (MinVarReturn) and the return of the asset with the greatest expected return (MaxReturn). AllocationADVISOR’s optimizer finds the combination of assets that minimizes the risk for each of 100 return points within this range. Thus, the efficient frontier is made up of 100 efficient portfolios.

## Historical Correlation

Correlation measures how closely related the variances of two series are.

where:

correlation of asset i with asset j
covariance of asset i with asset j

## Historical Forecast Risk (Standard Deviation)

The historical Forecast Risk is the annualized standard deviation of the historical Forecast Return.

*

where:

= Forecast risk for asset i
rt  = tth arithmetic return
T = number of arithmetic returns

## Historical Forecast Return

The historical Forecast Return is the annualized arithmetic mean return. (This number is different from the annualized return in StyleADVISOR, which uses the geometric mean return.)

Number of periods per year

where:

E[Ri] = forecast return for asset i
rt = tth arithmetic return
T = number of arithmetic returns

## Zephyr Defined Universes

Zephyr creates domestic equity and investment-grade fixed income universes based on investment style / fund behavior using the Morningstar, Mobius, Nelson’s, PSN, and eVestment Alliance databases.

The universes are standardized in the sense that they are created the same way each time. A five year scan is performed for each database using a single computation. The coefficients from the Style Benchmark are then used to create the universe.

Zephyr Domestic Equity Universes

## Volatility of Rank

To understand this calculation, it is necessary to recall that there are two ways of measuring the "instability" of a sequence of numbers:

## Variance Explained

The variance explained is also referred to as Standard R2 in StyleADVISOR. This is usually very close to the correlation squared. To understand what variance explained means, think of a manager and a Style Benchmark. Any variance in the difference between manager and Style Benchmark, i.e., any variance in the excess return of manager over benchmark, represents a failure of the Style Benchmark variance to explain the manager variance. Hence, the quotient of variance of excess return over variance of manager represents the unexplained variance.

## Variance

Variance can be calculated in two ways:

## Up and Down Capture

The up and down capture is a measure of how well a manager was able to replicate or improve on phases of positive benchmark returns, and how badly the manager was affected by phases of negative benchmark returns.

## T-Statistic

The T-Statistic of a manager series vs. a benchmark series is the information ratio multiplied by the square root of the number of years.

T-Statistic = Information Ratio *

The T-Statistic is used to calculate the significance level.

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