The efficient frontier is a portfolio analysis concept designed to assess risk vs. return for an investment portfolio.

**James Letzelter** is a partner at Webb, Scott & Quinn Inc. Contact him at Jim_Letzelter@wsqi.com.

The efficient frontier is a portfolio analysis concept designed to assess risk vs. return for an investment portfolio. While the financial projections of individual assets are key to the analysis, the end result is critical to successful structuring of the portfolio as a whole. Key to the efficient frontier is that it represents the highest level of a portfolio's return for any given level of risk. It can be applied to physical assets, as well as financial instruments—simultaneously. Figure 1 displays a typical efficient frontier chart. The horizontal axis represents risk (typically in the form of standard deviation of return in dollars), while the vertical axis represents return (typically in the form of the mean return expected of the portfolio). The line, which is the efficient frontier itself, represents the highest level of expected return for any given level of risk for any possible combination of assets for the portfolio. The area under the curve is populated by every other "non-efficient" portfolio that can exist.

Plotting a power generation portfolio is an extremely useful approach to understanding risk and return—and the power industry is well suited for it. For one, power markets are generally volatile—even in the long term. Recent poor financial performance by individual assets-and portfolios as a whole-has highlighted the need for more insightful approaches to asset acquisition and portfolio structuring.

Fortunately, some key elements of portfolio assessment exist for the power industry: access to powerful stochastic power market models, and both historical and projected data. These are critical in performing efficient frontier analysis, which relies heavily on stochastic inputs for individual returns. Moreover, since many energy portfolios consist of both physical assets and financial instruments such as contracts, the ability of the efficient frontier to consider both simultaneously makes it an excellent portfolio assessment choice.

By identifying the efficient frontier, energy investors have the ability to quickly assess their existing energy portfolios. Additionally, the outputs also are readily usable by other risk analysis and risk management tools, enabling more powerful decision making on asset utilization—such as how to market a power plant's output in the trading arm of the organization. Most of all, the efficient frontier provides the information upon which portfolio changes should be made. The information enables an energy portfolio owner to restructure its portfolio to be efficient-producing the highest expected return for any level of risk—by buying or selling the power generation assets that comprise an ideal portfolio.

As previously mentioned, data is critical in efficient frontier analysis. First and foremost, a stochastic projection of the cash flow from individual assets is required. This includes all of the existing assets in the current portfolio, plus projections for the potential new entrants to the portfolio. WS&Q develops this data with a generation dispatch model also known as a production cost model. There are many models currently available, but the model must be one that has probabilistic forecasting capabilities. In short, these models simulate the production of energy by individual plants and the power markets as a whole. They predict power market prices (based on supply, demand, and transmission constraints) and power plant production, revenue, and costs. These parameters, when modeled stochastically, provide the critical risk and return parameters that drive the efficient frontier. The key probabilistic variables that drive the market models are generally fuel prices, demand growth, hydroelectric capability (in some regions), and nuclear capability (in some regions). Other variables also can be modeled probabilistically for particular regions and study emphasis.

It is worth noting that in addition to individual risk and return metrics, these models will enable the calculation of asset correlations. Correlation among various asset classes is absolutely critical in the development of meaningful inputs. Indeed, the performance of a portfolio relies on how various assets perform relative to one another under varying market conditions.

### Planning the Model

In addition to access to good data, another key to successful efficient frontier analysis is the planning of the model. Once data has been mined and refined, the actual development of an efficient frontier model is relatively straightforward. It is largely driven by a single data table that defines the key parameters of all of the investments (power-plant assets) in the current and future potential portfolio. This can be comprised of specific individual investment options, or asset classes. When the latter is chosen, it is because specific power plants are not known, but an organization may generically consider, for example, a coal-fired power plant, a gas-fired combined-cycle power plant, or any number of types of assets to get a feel for the role of that asset type in the portfolio.

The first step is to identify all of the power-plant assets, or asset classes available. These generally are broken down into the primary fuel consumed by the generation unit (*e.g.*, coal, gas, oil, nuclear, hydro, wind). These are further differentiated by parameters such as the region in which the plant is located, constraints such as an organization's preference for clean or renewable assets, and the financial and operating profile that makes an asset or asset class unique.

The second step is to define the return metric to be used as the analytical focal point. The return metric is the specific profit or return measure as defined by the user. It may be based on net income, free cash flow, or other financial statistics—based on the preference of the user. This return can be conveniently expressed in percent, so that actual return, in dollars, is the simple product of the investment size and percent return.

Third, the return of each individual asset must be expressed stochastically—again, most conveniently based on the outputs from a production-cost model. The table must be populated with (assuming a normal distribution of return) both a mean and standard deviation of the percent return for each asset. Practically any distribution function can be modeled in Crystal Ball, but we will assume for convenience that a normal distribution applies to all assets.

The final input table parameters are related to investment magnitude. The user must provide constraints on the amount of dollar investment in the entire portfolio. Also, the user may choose to limit the amount of investment in any particular type of power plant.

### Setting Up Crystal Ball

In order to set up Crystal Ball for the efficient frontier, the first step is to define assumptions, decisions, and forecasts. The individual power-plant asset returns, modeled stochastically as earlier described, will be the assumptions. The variables that define the amount of investment in any single asset will be the decisions. The total portfolio investment (the sum of the individual asset investments) will serve as the forecast. In this manner, the model is able to adjust the amount of investment in each asset type—which has probabilistic return projections—while capturing the forecasted risk and return for every possible portfolio combination.

Once these parameters have been defined, Crystal Ball's OptQuest module is launched. The OptQuest wizard systematically prompts for the required setup information. First, the decision variables, previously defined in standard Crystal Ball mode, will be displayed. OptQuest prompts for the range of investment (dollars) possible for each asset from the data input table, and also for an initial value, or guess. Next, a constraint equation is requested. This is where the total portfolio investment is capped. Simply put, the equation states that the sum of all decision variables is equal to the desired portfolio investment value.

When these steps are complete, OptQuest provides immediate graphical feedback on solution status. During the simulation, OptQuest honors all constraints while testing various combinations of asset investments. For each level of risk (as defined by the standard deviation of return on the horizontal axis), a maximum expected (mean) return value is captured. As such, OptQuest collects the points on the efficient frontier—and stores the asset allocation (investment choices) for each portfolio point on the efficient frontier.

### Interpreting Results

When the simulation is complete, the efficient frontier curve is displayed and the individual asset investment levels that comprise the portfolio at each point on the curve are provided. While in reality there are infinite points (and allocations) on the curve, OptQuest provides this information for discrete points along the curve. Referring back to the efficient frontier curve in Figure 1, the curve represents efficient portfolios at increasing levels of both risk and return (from left to right).

It is important to note that every point on the curve is in fact efficient, but that the best portfolio is a matter of an organization's overall risk profile and strategy. What also is clear is that a portfolio under the curve (not on it) is inefficient—that is, the portfolio does not expect maximum returns for the level of risk to which it is exposed. Said another way, the portfolio is too risky for the level of return.

So which point on the curve is the best place to be? Again, this is largely subject to the strategy at the highest level of the organization: conservative investors to the left—risk-takers to the right. However, there is a general school of thought regarding the default "best' portfolio—based on the "Sharpe Ratio" *(see Figure 2)*. Every point on the efficient frontier has a Sharpe Ratio, which is calculated as the slope of the line that connects an efficient frontier point to the value of risk-free return. The rule of thumb is that the point with the highest Sharpe Ratio represents the best portfolio allocation. In its most basic interpretation, this point provides the highest return-to-risk ratio-the most bang for the buck.

### Implementing Results

Now that an efficient portfolio point has been selected, how is it actually achieved? In short, by divesting of, and investing in, specific power plant assets. Based on the current energy portfolio—and the desired efficient portfolio—the organization must add assets and remove them as defined by the selected efficient point.