### Single Period MVO

Mean-variance optimization (MVO), also known as Modern Portfolio Theory (MPT), is a quantitative asset allocation technique that allows you to use diversification to balance the risk and return in your portfolio.

Single period MVO was developed in the pioneering work of Markowitz. The goal is to choose a portfolio for the upcoming period. The key concepts introduced by Markowitz were:

The efficient frontier is conventionally displayed on a graph with the standard deviation on the horizontal axis and the expected return on the vertical axis. As the assumed level of risk (standard deviation) increases, the obtainable expected return increases. The maximum expected return appears at the right hand end of the frontier, and always consists of a single asset (the one with the highest expected return).

Two advantages of the mean-variance formalsim are:

The efficient frontier graph may be caricatured by the icon of our single period optimizer VISUALMVO_LINK, in which we see the expected return increasing as more risk is assumed. In practice, the graph is not straight, but rather the slope decreases as the standard deviation becomes larger.

### Multi-period MVO

One problem with the conventional single period analysis is the notion of **expected return**. More correctly, this should be called the **expected single-period return**.

If we have historical data, and assume that the future will (in a statistical sense) replicate the past, then the expected return is the simple average (arithmetic mean) of the historical returns in the sample. This is different from the multi-period return (annualized return) of the historical sample, which is given by the so-called geometric mean. The geometric mean is always less than the arithmetic mean. Use of the expected return over-emphasizes the benefit of bearing increasing risk.

The difference between the expected return and the true multi-period return applies both for each individual asset and at the portfolio level. It also applies to the future as well as the past. One might expect that if:

then the calculated expected return of the portfolio would be the expected annualized return. It is not.

The distinction between the expected return and the long term multi-period return was recognized by Markowitz, but he did not develop a full treatment of multi-period MVO in which the geometric mean could be used as both an input and output of the calculation.

This is achieved in our multi-period optimizer MvoPlus. Here the concepts analogous to those of the single period analysis are:

A fundamental difference from the single period case is that the portfolio with the largest geometric mean return is often a diversified portfolio (not just a single asset), and this portfolio can have a geometric mean return that is greater than that of any of the constituent assets.

In a multi-period framework, the mean-variance formalism thus has three advantages:

The multi-period situation may be caricatured by the icon of our multi-period optimizer MvoPlus. Here we see the conventional upward sloping frontier (in red), and the Geometric Mean Frontier (in blue) which can have its maximum for an intermediate value of the standard deviation. This point is sometimes called the "critical point."

The portion of the blue curve to the right of the maximum is not actually part of the Geometric Mean Frontier, but it is displayed by MvoPlus in order to emphasize that increasing the expected return beyond the critical point is actually harmful to the long term return.

Please select the following links to learn more about single and multi-period MVO:

### References

The following are some books from which you can learn more about MVO: