# Global Minimum Variance Portfolio

Table Of Contents

## What Is Global Minimum Variance Portfolio?

A Global Minimum Variance Portfolio (GMVP) refers to an investment portfolio with the least possible spread compared to other potential portfolios of assets with high-risk profiles. Hence, it determines the risk threshold, marking the point where an investor cannot build a more ideal or suitable portfolio below the applicable risk level.

Different securities or assets may have distinct or similar risk profiles individually. However, when these investment products are combined in a portfolio, their overall risk can be mitigated by evaluating the mean-variance efficient portfolios lying on the efficient frontier. The global minimum variance portfolio is the point on the left of the boundary of this efficient frontier.

##### Table of contents

- A Global Minimum Variance Portfolio or GMV portfolio is a portfolio of assets with a high-risk profile that has minimal volatility or variance than all other optimal portfolios.
- Such portfolios are built to achieve the minimum possible portfolio variance for a given set of risky assets by selecting asset weights that minimize the variance.
- Harry Markowitz proposed the concept of portfolio selection in 1952.
- The GMV portfolio is represented as the left-most point on an efficient frontier consisting of the minimum variance portfolios presenting all the feasible investment opportunities.

### Global Minimum Variance Portfolio Explained

The Global Minimum Variance Portfolio (GMVP) is a version of the mean-variance portfolio theory proposed for portfolio selection by economist Harry Markowitz in 1952. Since then, the field of finance has witnessed rapid expansion.

Markowitz's novel concept of minimizing portfolio variance while adhering to budget constraints has given rise to the widely acknowledged and commonly employed GMV portfolio for determining risk-adjusted returns. Budget constraints comprise factors like capital available for investment, transaction costs, liquidity expectations, etc.

It is a popular portfolio optimization strategy among risk-averse investors. It outlines the lowest achievable portfolio variance or risk given the assets chosen by an investor. Since the focus is on minimizing the risk, investors may, at times, accept lower returns on their portfolio than the returns achievable through other points or portfolio choices.

To understand the GMV portfolio, one must first understand the efficient frontier. It is the curve that is formed of all the optimal or efficient portfolios that represent various return possibilities based on investors' risk-taking capacity.

Out of all these potential investment opportunities, the most efficient portfolio is the global minimum variance portfolio that has the lowest return volatility, i.e., minimal risk and fair return potential. Before making a decision, an investor must check their risk tolerance level and study various points (portfolio choices) on the efficient frontier. Let us go through the following graph for a clearer picture.

While the GMV portfolio looks promising on theoretical grounds, its practical use or implementation poses various complications. The foremost concern is ascertaining the asset return distribution criteria.

Moreover, it is challenging to determine the correlation coefficient, as analysts cannot determine the population covariance matrix of the asset returns with any level of accuracy or reliability. Taking only historical information into account may lead to the wrong conclusions. It must be noted that despite these hurdles, risk-averse investors prefer applying the GMVP strategy.

### Formula

In this section, let us study this modern portfolio theory concept in detail. The GMV portfolio can be derived using a global minimum variance portfolio Excel template.

Consider that a portfolio has only two assets; the formula for determining the global minimum variance portfolio is as follows:

Here,

**σ**is the global minimum variance portfolio;_{P}^{2}**w**is the weight of Asset 1;_{1}**w**is the weight of Asset 2;_{2}**σ**is the variance of returns of Asset 1;_{1}^{2}**σ**is the variance of returns of Asset 2;_{2}^{2}**P**is the covariance of returns of Assets 1 and 2, where the coefficient correlation of the assets is the standard deviation of the returns of Assets 1 and 2, respectively._{1,2}σ_{1}σ_{2}

Similarly, one can compute the GMV portfolio for more than two assets by extending the above equation. Also, it is notable that the global minimum variance portfolio weights formula signifies that the sum of the weights of all the assets in a portfolio is equal to 1, i.e., 100%.

Let us check this formula below:

### Examples

Let us study some examples that emphasize the practical implications of the GMV portfolio optimization method.

#### Example #1

Suppose a small-cap fund comprises 100 high-risk stocks. The portfolio manager, Jenny, often uses a global minimum variance portfolio calculator to determine the change in the risk-return profile of the underlying assets.

She does this while ensuring that the investors get the best possible returns at the lowest possible price volatility of these stocks. She, therefore, adjusts the weight of the underlying securities to ensure that the investment portfolio matches the GMV portfolio.

To make better decisions, Jenny ideally needed more accurate information, which is tough to gather when certain volatilities are considered for specific stocks. However, her clients were largely happy with Jenny’s decision because though she focused on reducing the overall portfolio volatility, she tried to deliver the maximum possible return under the given circumstances.

#### Example #2

Suppose Harry follows a global minimum variance portfolio strategy. His portfolio comprises two assets, A and B. Asset A weighs 30% of the portfolio, while Asset B weighs 70%. If the standard deviation of returns of Asset A is 18% and that of Asset B is 10%, determine the overall variance of the portfolio considering that the covariance of returns of both the assets is 0.0049, and the expected returns of assets A and B are 12.75% and 9.90%, respectively.

**Solution**:

Given that:

Particulars | Asset A | Asset B |
---|---|---|

Weights | 30% | 70% |

Expected Returns | 12.75% | 9.90% |

Standard Deviation | 18% | 10% |

Covariance | 0.0049 | 0.0049 |

σ_{P}^{2}= 0.32 * 0.182 + 0.72 * 0.12 + 2 * 0.3 * 0.7 (0.0049) = 0.09 * 0.0324 + 0.49 * 0.01 + 0.002058 = 0.002916 + 0.0049 + 0.002058 = 0.004974

Portfolio standard deviation = √0.004974 = 0.07, i.e., 7%

Now, in this case, the portfolio risk is lower than the individual risk profiles of assets A and B, which were 18% and 10%, respectively. Therefore, it is the most favorable investment opportunity among the options available to Harry.

### Global Minimum Variance Portfolio vs Minimum Variance Portfolio

The global minimum variance portfolio and the minimum variance portfolio are theories of portfolio selection based on investors' varying risk appetites and return expectations. The differences between them have been listed in the table below.

Basis | Global Minimum Variance Portfolio | Minimum Variance Portfolio |
---|---|---|

Definition | It is the most favorable composition of risky assets out of all the efficient portfolios, representing minimal variance or riskiness when seen in light of the expected returns. | It is a range of various portfolio compositions that are considered efficient based on the correlation of their underlying assets' risk-return profiles. |

Comprises | A bunch of risky assets that have the lowest volatility or variance makes such portfolios. | All the optimal portfolios with different sets of high-risk assets suitable for investing are seen here. |

Depiction on Graph | It appears as a point on the left-most boundary of the efficient frontier. | It is represented as a minimum variance frontier or curve that depicts all the efficient portfolios. |

Risk-Adjusted Returns | It is the most optimal risk-adjusted return portfolio. | The risk-return profiles of the various portfolios change along the efficient frontier, and an investor can select a composition of assets that suits their risk-return appetite. |

Relationship | A GMV portfolio is necessarily an MVP. | All MVPs are not GMV portfolios. |

### Frequently Asked Questions (FAQs)

**1. What is the global minimum variance portfolio formed from?**

The global minimum variance portfolio usually comprises all those risky assets that have a high risk or return volatility but provide diversification benefits when held together in specific proportions or weights. Volatility contributes to variance, and volatilities, asset returns, and correlations are all considered while building such portfolios.

**2. What is the significance of the global minimum variance portfolio?**

The various advantages of a GMV portfolio are discussed below:**•** It ensures the best possible combination of high-risk assets such that the overall portfolio risk is mitigated or controlled.**•** It helps achieve expected returns in proportion to the risk-aversion level of an investor.**•** It can be adjusted with time for better performance as the risk and return profile of the assets fluctuate. Additionally, volatility and asset correlation in a portfolio also fluctuate.

**3. Is the global minimum variance portfolio efficient?**

Yes, the GMV portfolio is the most efficient point out of all the optimal portfolios lying on the efficient frontier since it depicts the collection of assets that offer expected returns at the least possible risk-taking. There can be no risk mitigation beyond this point, given the expected returns.

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