A vital part of asset management business is portfolio optimization. The more efficient a portfolio is optimized for a given level of risk, the better your chances of realizing expected returns. Robo advisors also use optimization process to provide you the maximum return you can get for a given level of risk, while at the same time minimizing investment costs for you. Let’s review some of the popular models used in investment finance (geek alert!) and how robo advisors apply them in their own investment process.

**Mean Variance Optimization**

Harry Markowitz introduced a mathematical formulation for investment diversification in his paper “Portfolio Selection” that was published in 1952. He showed how diversification in a portfolio could lead to better returns for the same amount of risk. In effect, the model solves for an “efficient frontier” which is a set of portfolios, each one offering maximum return for a given level of risk.

Efficient Frontier (Source: ToolsForMoney.com)

Below the curved efficient frontier, portfolios offer a sub-optimal risk adjusted return. That is to say, diversification has not been done efficiently enough to reap the rewards of maximum risk adjusted returns. This trade-off between risk and return is at the heart of the MVO theory, which is designed optimize a portfolio for a single period.

In order to find a portfolio on the efficient frontier that maximizes return for a given level of risk, following inputs are needed.

- Forecasts for asset returns for a single period
- Forecasts for standard deviations for a single period
- Forecasts for correlations between the assets

These forecasts are based on historical data. Once you have these forecasts, next step is to calculate the minimum variance portfolio and maximum return portfolio. Finally between these two portfolios, portfolio that has a minimum risk for each of the 98 portfolios between minimum risk and maximum return portfolios is calculated, which in turn provides the efficient frontier.

__MVO Limitations __

Despite the significant utility of the MVO theory, there are some major limitations in this model.

- It is difficult to forecast asset returns with accuracy using historical data, which tends to be a poor forecasting source. As return estimations have a much larger impact on MVO asset allocations, small changes in return assumptions can lead to inefficient portfolios. Therefore, MVO tends to lead to highly concentrated portfolios that do not offer as much diversification benefits in practice as they seem to provide in theory.
- The model assumes that asset correlations are static or linear. In reality, asset correlations move dynamically, changing with the market cycles. During the global financial crisis, asset correlations approached almost to 1, so if anything, diversification seemed to have insignificant impacts on the portfolios.
- Last but not the least, MVO assumes normality in return distributions. Therefore, it does not factor in extreme market moves which tend to make returns distributions either skewed, fat tailed or both. Without optimizing the portfolio for asset that may actually have skewed distributions or fat tailed, MVO could lead to a riskier allocation that is intended.

As a result of these limitations, portfolios optimized using the MVO approach are rather concentrated and fail in achieving maximum diversification.

**Black Litterman Model**

To counter the limitations of the Mean Variance framework which often results in concentrated portfolios, economists Fisher Black and Robert Litterman at Goldman Sachs developed the Black Litterman model. It’s a step ahead of the MVO framework in that it allows an investor to incorporate his views on expected returns into the market implied asset returns. After all, if an investor seems to specialize in a certain asset class and possesses unique insight into how that particular asset class will perform over a future period, there needs to be a way to forecast returns considering the investor’s own return expectations. Thus, Black Litterman Model tries to overcome high-concentration (normal distribution assumption), input sensitivity, and estimation error maximization problems that are inherent in the MVO theory.

Black Litterman builds on the MVO framework. It starts with neutral, equilibrium asset weights that are implied from the market portfolio assumed to be on the efficient frontier. Investment professionals often tend to have expert views on the performance of certain asset classes in a portfolio, which may deviate from the market portfolio implied weights. An investor can then enter his views into the model, which uses *reverse optimization *process to calculate CAPM (Capital Asset Pricing Model) equilibrium returns for each of the assets in a market portfolio. An investor has the option to express views regarding one or all of the assets in a portfolio, though because investor views on all the assets in a portfolio are not mandatory, estimation error is minimized to a certain extent.

Given the level of confidence as well as the magnitude of the views an investor inputs into the model, a portfolio is created that may have asset weights different from the market portfolio. The degree of investor confidence in the original market portfolio weights also has an impact on the portfolio optimization process using this model, which means if you have a high confidence in the market portfolio implied weights, your optimized portfolio will reflect that. The resulting optimized portfolio is a weighted average of the market equilibrium portfolio and “investor views portfolio” – the stronger these views are, the greater the divergence of the optimal portfolio will be from the market equilibrium portfolio. However, in the absence of any specific views, an optimal portfolio will be the *market portfolio*.

__Black Litterman Limitations__

Black Litterman model brought the MVO model one step further and has usage in the practical world. However, it still faces a few limitations, which are discussed next

- It is difficult to define the market portfolio. Risky assets trading on public markets are far from a complete representation of the overall market. For instance, REITs do not encompass the private real estate deals, which are a much larger chunk of the overall REITs market. Therefore, asset allocation based on publicly available securities may render a portfolio with less than optimal allocation to certain asset classes.
- Normality assumption in the model again leads to the same problems suffered under the MVO model. Asset returns do not always behave normally and the need for a model that uses other than normal return distributions, and factors in asset return skewness and fat tails, still remains.
**Gordon Growth Model**

Thus far we discussed two models that are central in portfolio optimization process. Gordon growth model, though not a tool for portfolio optimization per say, can help an investor boost individual asset returns by hand picking asset classes, indices, or securities that pay a high fixed payment growing at a constant rate in perpetuity. This could either be dividend paying stocks (typical application of the model), asset classes with a high payout ratio to investors, or indices such as ETFa paying out dividends. The model was first published in 1956 by a professor at the University of Toronto, Myron J. Gordon along with Eli Shapiro.

Figure 3: Gordon Growth Model Equation

The model posits that the value of a stock/security is the net present value of all of its future payouts. For a dividend paying stock, given the value of a company’s dividend in one year, it is possible to find out the intrinsic value (true value based on fundamentals) of a stock today. The model makes the following two assumptions

- Stock is dividend paying
- And dividends grow at a constant rate in perpetuity

It does not require inputs on stock return expectations nor requires information on the market conditions. Thus due to its simplistic approach, it’s fairly easy to value a company, provided it pays out a dividend. Though the model finds its applications in an index or an asset class valuation as well, it is mostly used to value dividend paying stocks.

__Gordon Growth Limitations__

As it turns out, the model’s simple approach to valuation is one of its main drawbacks. The model suffers from the following disadvantages.

- It can only be used for stable companies that pay a dividend growing at a constant rate, which is rarely the case as companies go through different business cycles that make growing dividend at a constant rate difficult. Extending it to indices or asset classes, the problem is multiplied as the whole indices or asset classes can suffer from different economic stages.
- The growth rate in the formula can’t be bigger than the rate of return used to discount the future payments. Otherwise, it ends up with a negative number in the denominator which will not mean anything.
- Despite the simplicity, Gordon Growth model still requires forecasting of inputs. Specifically, growth rate and required rate of return both need to be forecasted as precisely as possible because small changes in these two inputs can have major effects on the value of a company’s equity.
**Robo Advisors Optimization process**

Finally, it’s time to turn to the primary question – what is the process used by robo advisors to optimize their portfolios? Most robo advisors use one form or the other of an optimization process and the details vary as to the extent of the optimization. In the end, it comes down to each robo advisors individual investment philosophy and views on the market.

Figure 4: Robo Advisors’ Investment Philosophy (Source: ETF.com)

In order to explain how robo advisors go about portfolio optimization, consider Wealthfront and Betterment.

**Wealthfront** uses Black Litterman model to get implied market weights and combines them with investment views of Burton Malkiel, Wealthfront’s Chief Investment Officer, to create asset allocation for portfolios. In order to optimize the portfolio, the company also puts constraints such as minimum and maximum asset weights. Thus, Black Litterman model serves their optimization well as it takes the mean-variance approach and allows the company to input its own views as reflected by Malkiel to arrive at an optimal portfolio.
**Betterment **also uses the Black Litterman model. However, they do not insert investor views into the optimization process and rather try to stick to the market equilibrium portfolio weights, specifically on the equities side of the portfolio. It also does not constraint portfolio weights. The resulting optimal portfolio resembles the global market portfolio to a certain extent. The one thing that Betterment does unique, however, is its Fama French style tilt towards small cap and value asset classes.

Although robo advisors are known for their minimal human intervention in the investing process, their product offerings are in fact affected by the views and market insights of their respective investment committees or experts. In the end of the day, it is the mind behind the robo that is making investing calls, and you need to familiarize with him sooner rather later.