Introduction to Efficient Frontiers

Section One: All Eggs in One Basket.

Firstly, let's begin by understanding what is meant by portfolio optimisation. Optimising a portfolio essentially refers to creating a set of assets, which minimises your risk while also maximising your returns on investment. This may be done through multiple different ways, such as Portfolio Diversification or even complex mathematical models. Generally, diversification is considered the most common method, as it simply involves having assets of different categories. To illustrate this, a well-diversified portfolio will contain real estate, shares, bonds, artwork and watches, all of which commit to a varied set of possessions.

It is important to understand why portfolio diversification has such an effect on lowering risk. The simplest reason behind it is that failure of one market will not affect the entire portfolio, as it will only represent a small fraction of it. If let's say, there is a real estate bubble and housing prices take a large hit, an investor who solely has buildings as their investments will face large losses, as there is no other type of assets to counteract the adverse effect of the bubble. On the other hand, an investor with lots of different categories of assets will suffer much less as real estate will make up a certain percentage of the portfolio, but there will be many other assets that will keep the value up.

Section Two: The Efficient Frontier

The figure above shows an Efficient Frontier for a portfolio of Three assets.

Every point on an Efficient Frontier represents a different combination of weights of assets in the portfolio. For example, 80% of Bonds, 10% of Shares and 10% of Real Estate. The X-axis shows the standard deviation, which quantifies the risk of the combination of assets. Every single asset has a certain degree of volatility or in other words the fluctuation of the market price for it. The volatility is measured in standard deviation, which measures how much on average a point on the graph will vary from the trend line. Volatility comes with risk, as it is unknown whether the share will fluctuate downwards or upwards, which creates uncertainty leading to risks. Thus, the standard deviation of a portfolio means the relative risk associated with it. The Y-axis shows the expected return of the combination, meaning the profit or potential loss that will result from the combination used.

First, with efficient frontiers, investors only focus on the outside points of the curve, as ones on the inside carry a higher risk while maintaining the same expected return. Thus, the points inside the frontier aren't fully optimal, so an investor would stay on the outside. To illustrate the aforementioned case, the expected return of 4.1 there are points of 1.7 and 2 standard deviations. A rational investor would choose the point with 1.7 standard deviations, rather than 2, as the risks are much less, whereas the return is the same.

Secondly, it is important to note, that for the same value of risk, there may be two possible points that one could utilise. For example, at a standard deviation of 2, there are potential returns of 3.7 and 4.5. In this case, it wouldn't be optimal to be at the point with 3.7, as with the same risk the investor will get a higher return, so the portfolio should be created with an expected return of 4.5.

Section Three: Further Efficient Frontiers

Next, it is important to consider the implications of shorted assets on portfolio management. Shorting refers to borrowing an asset, causing it to be labelled as a liability. This concept is used by Hedge Funds, which use shorting stocks to generate profits. The risk of shorting is higher than of simply purchasing, as the losses are potentially unlimited, as well as interest may have to be paid on it.

On an efficient frontier, shorting assets extend the tails of the curve further, as the risk increases with more and more liabilities. The frontier shown above shows a simplified portfolio of two assets which also have been shorted. For example, at the point with a standard deviation of 0.45 and an expected return of 52, asset 1 has been shorted, whereas asset 2 was purchased in full. Not only is this point not optimal as it doesn't maximise the return, as it is on the lower tail of the curve, but it also carries immense risk related to the shorting.

Section Four: Construction of Frontiers

Constructing an efficient frontier is very simple. An efficient frontier is defined by two parametric equations, one for each axis. Using predetermined constants of the standard deviation of each asset used, as well as the expected returns and correlation coefficient between the assets we can calculate the portfolio standard deviation and portfolio expected return for as many different weights as desired. Then, using the values obtained the points are plotted, and a frontier is drawn. The formulae are quoted below: