# Asset Allocation

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Asset allocation in finance is the task of dividing an investment portfolio among various asset classes, and accounts for over 90% of the variation in different investors' returns. Since returns from various assets tend to move somewhat independently, the risk from a diversified portfolio can be lower than for any of the individual components. Give your own chosen weights to ten classes of assets, visualize the location of the resulting blend in risk-reward space, and measure its past and recent returns.

Contributed by: Jason Cawley (March 2011)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

The ten asset classes are cash equivalents (T-bills), bonds, large cap value stocks, large cap blend (e.g. the S&P 500), large cap growth, small cap value, small cap growth, commercial real estate, commodities, and international stocks.

The cash asset does not have its own control, but is a residual. If the other assets sum to less than 1, the balance is considered to be in cash. If the other assets sum to more than 1, they are normalized to 1 by dividing each by the sum of all the control weights. The weight of cash is 0 whenever the total of the other controls is 1 or more.

Past returns are taken from the BlackRock survey of 20 years of asset class returns, supplemented by recent 2007 total return information, and from Nareit's REIT and the CRB commodity total return indices. The international stock asset class is based on the capitalization-weighted EAFE sector (also known as "Europe, Americas, and the Far East"). Standard deviations and the covariance matrix used are based on annual figures; monthly numbers would be more exact for more frequent portfolio rebalancing.

Two views are provided: the efficient frontier and past performance. The efficient frontier chart plots the 20-year average annual return on the axis against the annual standard deviation of those returns on the axis. Strictly speaking, the axis in such a chart should be the expected return, however arrived at. Here the expectation is taken from the past 20 years of historical returns. The past performance chart shows the total return of the chosen asset weights given the actual past returns of each of the components. An equally weighted portfolio of 10% in each of the 10 asset classes is provided for reference in both views.

You can see the performance of a single asset class by setting all other weights to 0. On the efficient frontier view, notice that varying the weight in a single asset between 0 and 1 moves you along the line between cash and that asset's position in risk-return space. Note that cash does have a nonzero standard deviation in its return (about 2%). While the principle in T-bills is risk-free in credit terms, short-term interest rates do fluctuate and the return of cash is not a constant.

The efficient frontier is constructed from the set of all portfolios that maximize expected return for a given level of risk. Associated to each portfolio is a point in risk-return space. The collection of points that maximize return for a given level of risk or, equivalently, minimize risk for a given level of expected return, is the efficient frontier. Positions up (higher expected return) and to the left (lower risk) are preferable to positions lower and to the right. The purple line shows the efficient frontier.

The slope of the line drawn from cash to a given asset gives the Sharp ratio of that asset, a measure of the increment to expected return achieved by acceptance of an increase in the level of risk, using that asset. Theoretically, assets with an inferior Sharp ratio are dominated by those with a better Sharp ratio, if one can either reduce cash positions to hold more of the asset, or borrow at the cash expected return. One then holds a higher Sharp ratio asset with leverage, instead of using a dominated asset, if a higher expected return is desired. In practice, the position of an asset in risk-return space is uncertain, and not all investors can borrow against risky assets at the cash rate of return.

Modeling expectations using past average returns can be risky, since future returns need not look like past returns. To explore this sensitivity, the actual returns for each of the assets in 2007 are at first hidden in this Demonstration. You can choose to include them in either view or not. It is a good test to try selecting an asset allocation based on the results through the end of 2006, and only afterward look at the 2007 results of that asset mix.

The snapshots show, first, a portfolio chosen to overweight the asset classes showing the best risk to reward characteristics in the past 20 years, while aiming for a target return of 11%, above the return of the equally weighted benchmark. Once an 11% return is achieved, the standard deviation is minimized. But past performance is no guarantee of future results—the value stocks and real estate this process favors had a poor year in 2007.

Another snapshot shows the risk involved in choosing the sole asset class with the best return over the 20-year baseline period. This was the small cap value asset class. The standard deviation of returns that would result from a 100% allocation to that asset is twice that of the more balanced portfolio, and small cap value stocks lost nearly 10% in 2007.

Another snapshot shows a contrarian portfolio that favors assets that look poor in risk-reward terms over the 20-year baseline period, while also keeping a significant bond weighting to limit overall risk. This contrarian portfolio did quite well in 2007. But it might do poorly again in 2008.

The last snapshot shows a simple old-fashioned approach to asset allocation: a 60-40% mix of large cap blend stocks like the S&P 500 (60), and the whole bond market (40). This allocation results in an overall risk-reward position close to the equally weighted portfolio, using just two asset types. It did suffer a considerable decline in the 1999 to 2002 period, however—given a much rougher ride in that stretch than the more diversified portfolio shown in the first snapshot. On the other hand, this simple 60-40 allocation did better in 2007.

## Permanent Citation

"Asset Allocation"

http://demonstrations.wolfram.com/AssetAllocation/

Wolfram Demonstrations Project

Published: March 7 2011