# Variance drain

Variance drain, also called volatility drag,[note 1] operates under the theory that between two portfolios with the same beginning and same average return, the one with the greater variance will have a lower compound return and less-ending wealth.[1]

In statistics, a fund's risk is characterized by the variation (volatility) of its return. If we measured the deviations compared to the average return over time, the positive and negative deviations will tend to cancel each other out. By squaring the deviations, the numbers always remain positive when they are summed. This averaging of the sum of the squared deviations is known as the variance.[2]

## Arithmetic versus geometric returns

Returns over time are calculated as follows:

First, by an arithmetic average of the returns. This is the simple sum of returns divided by the number of periods.

Second, by a geometric return. The returns are compounded over a period of time, which is measured by the starting and ending values of the investment.

For example, a stock is purchased for $100. At the end of the first year, its price is$200 (100% gain)[3] In the second year, the stock price drops to $100 (50% loss). • The Arithmetic return over 2 years is 25% = (100% - 50%) / 2 • The Geometric return over 2 years is is 0% =$100 (final value) - \$100 (starting value)

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, the arithmetic mean is always greater than the geometric mean unless the numbers are identical.[4] In terms of investing, this inequality can be restated as: A return which has some risk (volatility) will generate lower returns than the arithmetic return.

## Variance drain

The arithmetic mean - geometric mean inequality was utilized in a 1995 paper titled "Variance Drain - Is your return leaking down the variance drain?"[5] "Drain" refers to active managers who generate a variance in excess of a passive alternative. Active managers needed to cover not only their fees and transaction costs, but the cost of the "drain" due to the variance.[2]

Some authors have observed that the more variable a given asset's return is, the greater the difference between the arithmetic and geometric averages. One popular approximation:

${\displaystyle{\displaystyle r_{g}\approx r_{a}-{\frac{1}{2}}\sigma^{2}}}$
where rg = geometric return, ra = arithmetic return, ${\displaystyle{\displaystyle\sigma^{2}}}$ = its variance
${\displaystyle{\displaystyle{\mbox{Variance Drain}}=r_{a}-r_{g}\approx{\frac{% \sigma^{2}}{2}}}}$

These expressions suggest that the variance of returns drains the arithmetic average returns to produce the smaller, realized, compound returns over the holding period.[6]

In the above example, the Variance Drain is 25 % = 25% (arithmetic average) - 0% (geometric average).