Efficient Frontier and Applications

GPS or similar navigation systems attempt to map out the most efficient travel route or the one that saves you the best combination of time and money. Similarly, mean–variance optimization is a savings philosophy between risk and reward that should lead to the most efficient portfolio, the one that maximizes return per unit of expected risk. What's another way to describe the most efficient portfolio?

Exactly! If you're establishing a given level of expected return, you want to take the minimum amount of risk to earn that return. So the efficient portfolio serves to effectively balance risk and returns. In order to do this, the mean-variance optimizer works through all possible risk coefficient values and finds the combination of assets that maximizes the investor's risk-reward ratio or utility. This leads to the efficient frontier, a graph which captures the most efficient portfolio for a given level of risk.

Not quite. If you're taking on a certain risk, you want the highest return for that risk level.

No way. You don't want maximum risk exposure per a given expected return. That's way too risky.

Take a look at the basic efficient frontier graph:  You can see that the slope of the efficient frontier shoots immediately higher off of the lowest expected return at the lowest level of risk. At this point, the lowest level of return per level of risk is essentially the broadest portfolio available. Or, put another way, it's the portfolio with the least amount of specific asset allocation. How would you describe this portfolio?

No. That's not a full portfolio; it's essentially the risk-free rate.

Indeed! It's the global minimum variance portfolio, or the portfolio with maximum diversification to reduce risk to its lowest point. In contrast, the portfolio with the highest expected portfolio return is on the farthest right-hand point in the chart. Without any constraints, that portfolio would consist of simply the highest expected returning security. In between these two points, the slope shifts upwards from left to right as the expected return increases per unit of risk. And, as you can see, the level of return per unit of risk actually decreases as you move left to right, so investors take on more risk for smaller and smaller increases in return.

Not quite. Diversification benefits must be included in the portfolio.

As the efficient frontier adds risk, investors must choose the point at which to stop adding units of risk for the expected level of return. This is where the investor's risk aversion coefficient comes in to help identify where the investor prefers to stop along the efficient frontier. But sometimes, the investor's risk aversion coefficient isn't available, which means that the investor needs to set risk parameters a different way. What's another way to establish a maximum level of risk?

Not quite. There can be various return objectives for different subportfolios along with different portfolio mixes to achieve these goals.

Yes, setting a specific return volatility level allows you to look for the best point on the efficient frontier that matches the investor's volatility level. Portfolio asset mixes can also be narrowed down through a specific return requirement, which helps you identify efficient portfolios that meet that requirement.

That's not it. A range of probabilities doesn't help you establish a basis for risk.

But one question that remains throughout the mean–variance optimization process is how to allocate cash and cash equivalents. In the simplest approach, cash is included within the optimization to map out the efficient frontier. But there are other approaches to developing the efficient frontier that exclude cash. For example, analysts can remove cash from the asset allocation process to create the efficient frontier with just the risky assets within the portfolio. Or, analysts can develop tangency portfolios that are based upon a combination of the risk-free asset and portfolios with the highest Sharpe ratio that lie upon the efficient frontier of risky assets. In practice, the tangency theory to cash asset allocation is popular because it allows you to leverage the expected return. How would you do this?

Not quite. That's going to lower both risk and returns.

Right. By borrowing funds at the risk-free rate, investors can increase their expected returns by investing the proceeds of the loan into the tangency portfolio. Or, risk-averse investors can invest more money in the risk-free asset and lower their risk and returns. For example, in breaking out tangency portfolios and cash, you may be presented with risky portfolios that combine the two closest actual portfolios along the efficient frontier. These are called __corner portfolios__, or the point at which an asset class either leaves or enters the efficient mix or a constraint becomes binding or is no longer binding.

These corner portfolios are a mix of two portfolios that form the portfolios that lie on the efficient frontier. Say you're then presented with a corner portfolio that exceeds the necessary expected return and has the highest Sharpe ratio amongst available portfolios. If the goal of utility maximization is to minimize risk per a level of return, what action would you take to achieve the client's return objective?

No way. You're already going to achieve the necessary return, so there's no need for leverage.

This will minimize risk and still achieve the client's return objective. This essentially helps the investor's overall utility. To do this, you'd set the return to the client's overall goal and solve for the proper weights between the tangency portfolio of risky assets and the risk-free rate. Other times, you might consider using a corner portfolio with the highest Sharpe ratio and leverage the expected return through risk-free asset borrowing to meet the client's return requirement.

To summarize: [[summary]]

No. That won't increase returns, it will just give you the same return you were already expecting.

Right!

Minimum expected risk per a given level of expected return

Minimum expected return per a given level of expected risk

Maximum expected risk per a given level of expected return

10-year Treasury rate

Global minimum variance portfolio

National minimum variance portfolio

Establishing a return objective

Setting a specific return volatility level

Providing a range of probabilities for specific return requirements

Borrow at the risk-free rate

Invest in just the tangency portfolio

Invest larger amounts in the risk-free rate

Borrow at the risk-free rate

Invest a portion of the portfolio in the risk-free rate

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