# Markowitz's Portfolio Theory — Formulae, Assumptions, Limitations [Worked Examples]

by **Jack Bodeley** on *October 12, 2021*

##### Markowitz Portfolio Theory is the seminal theory for which the renowned American Economist, Harry Markowitz (1927– ), was awarded a Nobel Prize in Economics in 1990. It theorizes the ability of investors to diversify away unsystematic risk by holding portfolios consisting of a number of different shares.

Markowitz developed the theory in the early 1950s. He formalized the concept of portfolio diversification by showing quantitatively why and how portfolio diversification reduces portfolio risk to an investor.

His theory was the first to develop a specific measure of portfolio risk and derive the expected returns and risk for a portfolio. This organized the existing thoughts and practices into a formal framework.

## Portfolio Theory Analysis

Markowitz Portfolio Theory is based on the expected returns and risk characteristics of securities. *It is, in essence, a theoretical framework for analyzing risk-return choices*.

To identify an efficient portfolio—

- On one hand, investors can specify an expected portfolio return and minimize the portfolio risk at the specified level of return through diversification.
- On the other hand, investors can specify a portfolio risk level they are willing to assume and then maximize the expected return on the portfolio for this level of risk.

### What is a Portfolio?

A portfolio is a group of assets (e.g. projects, investments or securities) held by an investor with the objective of maximizing return for a given level of risk.

A firm is generally faced by two types of risks:

*Financial risk*— risk resulting from the use of debt in a capital structure. Debts accrued fixed finance charges (i.e. interest). Financial risk is measured using the gearing ratio.*Business or operating or total risk*— risk resulting from fluctations in the company's expected earnings due to the nature of business environment in which it operates.

### Systematic v Unsystematic Risk

*Systematic or market or non-diversifiable risk* refer to the variations in returns of securities due to factors which affect all firms adversely. *Unsystematic or unique or diversifiable risk* refer to variations in returns of securities due to factors that are unique to a specific investor or firm.

**Systematic risk cannot be eliminated with diversification of a portfolio whereas unsystematic risks can be offset against each other.**

Examples of systematic risks include natural catastrophes, wars, inflation and so and so forth. Examples of unsystematic risks include lawsuits, strikes, marketing programs, contracts and so on and so forth.

### The Envelope

Markowitz's starting point is to construct what is known as the *envelope curve* which represents a set of portfolio choices available to investors when investing in different combinations of risky securities. This is represented by region AMEGH in the illustration below.

An investor can build a portfolio with risk and return characteristics anywhere in this envelope by holding different combinations of risky securities in differing proportions.

### The Efficient Frontier

Whereas investors can build a portfolio with characteristics anywhere in the envelope, *the rational investor will only invest in a portfolios on the efficient frontier arc*, represented by BME.

**It is called the efficient frontier because all portfolios on this arc are superior to (i.e. more efficient than) all other portfolios within the envelope curve—assets on the efficient frontier give either the maximum return for a given level of risk, or the minimum risk for a given level of return.**

Suppose, for instance, we compare portfolios A and N on the boundary of the envelope curve, both bearing the same level of risk.

It is apparent that portfolio N offers a higher return without taking on additional risk. Portfolio N is said to dominate portfolio A. In fact, all portfolios on the arc between A and E are dominated by portfolios on the arc BME, and so cannot be regarded as efficient.

## Factors that Influence the Efficiency of a Portfolio

Generally, two factors influence a portfolio:

*The number of securities forming a portfolio*— several studies have indicated that around 20-25 well selected securities will form an efficient portfolio. This is generally considered to be the saturation mark.*The relationships between returns of securities forming a portfolio*— a relationship between any two securities can be either*positive*or*negative*.

## Portfolio Theory Formulae

### CoVariance ($CoV$)

**For diversification to work, there must be a negative relationship between securities.** Securities are said to have a negative relationship if a given economic factor impacts their performance in opposite directions.

The relationship between returns of securities is measured using CoVariance:

$$CoV_{ab}=\sum^{n}_{i = 1}(R_a-\bar{R_a})(R_b-\bar{R_b})(P_i)$$

where;

- $R_a$ and $R_b$ — actual returns of assets $a$ and $b$, respectively
- $\bar{R_a}$ and $\bar{R_b}$ — expected returns of assets $a$ and $b$, respectively
- $P_i$ — probability of event $i$

#### Interpretation of CoVariance

CoVariance will be negative for negative relationships and positive for positive relationships.

To measure the strength of the relationship, a standardized measure called *correlation coefficient*, $p$ (read as rho), is used ranging between negative 1 and positive 1 i.e. $-1\le p \le+1$, thus:

Value of $p$ | Interpretation |
---|---|

$p\lt0$ | Duly hedged porfolio can be formed |

$p=0$ | Risk can be reduced through portfolio formation |

$p\gt0$ | No risk reduction possible through portfolio formation |

### Portfolio Return ($\bar{R_p}$)

The expected return of the portfolio is equal to the weighted average of the returns of the individual assets making up the portfolio. The weights are the proportions of each asset in the portfolio.

$$\bar{R_p}=E(R_p)=\bar{R_1}W_1+\bar{R_2}W_2+...+\bar{R_n}W_n=\sum^{n}_{i = 1}R_iW_i$$

where;

- $\bar{R_p}~ or~ E(Rp)$ — expected return of the portfolio
- $\bar{R_1}$, $\bar{R_2}... \bar{R_n}$ — expected returns of securities 1, 2 through to $n$
- $W_1$, $W_2... Wn$ — proportions of securities 1, 2 through to $n$ in the portfolio

### Portfolio Risk ($\delta_p$)

The risk of a portfolio is estimated by it's standard deviation and variance. It depends on:

- CoVariance
- Risk of each investment in the portfolio
- The weight of each investment in the portfolio

In a *two-asset* case:

$$\delta_p^2=W_1^2\delta_1^2+W_2^2\delta_2^2+2W_1W_2CoV_{12}$$

where;

- $\delta_p^2$ — variance of the portfolio
- $W_1$ and $W_2$ — proportion of assets 1 and 2, respectively
- $\delta_1$ and $\delta_2$ — standard deviations of returns of assets 1 and 2, respectively

## Assumptions of Markowitz's Portfolio Theory

### Assumption #1 — Investment Alternatives are Representative

### Assumption #2 — Investors are Risk Averse

### Assumption #3 — Investors Estimate Risks with Excpected Returns

### Assumption #4 — Investors Maximize One-period Expected Utility

## Limitations of Portfolio Theory

There are several problems associated with trying to applying portfolio theory in practice, some of which are summarized below:

### Limitation #1 — Borrowings are Never Risk-free for Individuals and Firms

It is unrealistic to assume that investors can borrow at the risk-free rate. Individuals and firms are not risk-free and will therefore not be able to borrow at a risk-free rate — borrowings will reflect their higher level of risk.

### Limitation #2 — Identifying the Market Portfolio is Problematic

There are problems associated with identifying the market portfolio. It generally necessitates an accurate assessment of the risk and return of all risky investments and their corresponding coefficient. This is non-trivial in practice.

### Limitation #3 — Building the Market Portfolio can be Cost Prohibitive

Once the make-up of the market portfolio is identified, it will then be expensive to build. For smaller inverstors, particularly, the transaction costs and other associated costs can be limiting.

One way smaller investor's can get round the cost prohibitiveness is by buying a stake in a large diversified portfolio such as a unit trusts, investment trusts or index tracker funds.

### Limitation #4 — The Market Portfolio Changes Over Time

The market portfolio will change over time *due to shifts in in both the risk-free rate of return and the envelope curve* and hence the efficient frontier.

## Worked Examples

### Example 1.1

Assume that an investor has capital for £1,000,000 which he wishes to invest in two assets, A and B, in equal following proportions. You are further provided with the following information:

State of Economy | Probability | Return of A | Return of B |
---|---|---|---|

Down | 0.2 | -0.2 | 0.5 |

Average | 0.5 | 0.18 | 0.18 |

Up | 0.3 | 0.5 | -0.2 |

- What is the mean return (or expected return) of asset A?
- What is the coefficient of variance of A?
- What is the mean return (or expected return) of asset B?
- What is the coefficient of variance of B?
- What is the covariance of the two-asset portfolio?
- What is the standard deviation of the two-asset portfolio?
- What is the expected return of the portfolio?
- What is the correlation coefficient of the two-asset portfolio?

#### Solution 1.1

State of Economy | Probability $P$ | Returns of A $R_A$ | $\bar{R_A}$ | $R_A-\bar{R_A}$, $\bar{R_A}$=0.2 | $(R_A-\bar{R_A})^2$ | $P(R_A-\bar{R_A})^2$ |
---|---|---|---|---|---|---|

Down | 0.2 | -0.2 | -0.04 | -0.4 | 0.16 | 0.032 |

Average | 0.5 | 0.18 | -0.09 | -0.02 | 0.0004 | 0.0002 |

Up | 0.3 | 0.5 | 0.15 | 0.3 | 0.09 | 0.027 |

Total | 1.0 | 0.2 | 0.0592 |

- What is the mean return (or expected return) of asset A? $$\bar{R_A}=0.2$$
- What is the coefficient of variance of A? $$CoV_A = \frac{\delta_A}{\bar{R_A}}=\frac{\sqrt{0.0592}}{0.2}\approx1.2166$$

State of Economy | Probability $P$ | Returns of B $R_B$ | $\bar{R_B}$ | $R_B-\bar{R_B}$, $\bar{R_B}$=0.13 | $(R_B-\bar{R_B})^2$ | $P(R_B-\bar{R_B})^2$ |
---|---|---|---|---|---|---|

Down | 0.2 | 0.5 | 0.1 | 0.37 | 0.1369 | 0.0274 |

Average | 0.5 | 0.18 | 0.09 | 0.05 | 0.0025 | 0.0013 |

Up | 0.3 | -0.2 | -0.06 | -0.33 | 0.1089 | 0.0327 |

Total | 1.0 | 0.13 | 0.0614 |

- What is the mean return (or expected return) of asset B? $$\bar{R_B}=0.13$$
- What is the coefficient of variance of B? $$CoV_B = \frac{\delta_B}{\bar{R_B}}=\frac{\sqrt{0.0614}}{0.13}\approx1.9061$$
- What is the covariance of the two-asset portfolio?

State of Economy | Probability $P$ | $(R_A-\bar{R_A})(R_B-\bar{R_B})(P)$ |
---|---|---|

Down | 0.2 | (-0.4)(0.37)(0.2) = -0.0296 |

Average | 0.5 | (-0.02)(0.05)(0.5) = -0.0005 |

Up | 0.3 | (0.3)(-0.33)(0.3) = -0.0297 |

Total | 1.0 | -0.0598 |

$$CoV_{AB} = -0.0598$$ 6. What is the standard deviation of the two-asset portfolio?

Because the investor wishes to invest in equal proportions in the two assets, the weights are 0.5 each.

$$\delta_p^2=(.5)^2(.0592)+(.5)^2(.0614)+(2)(.5)(.5)(-.0598) = 0.00025$$ $$\delta_p=\sqrt{0.00025}\approx0.0158$$ 7. What is the expected return of the portfolio? $$\bar{R_p}=(.2)(.5)+(.13)(.5)=0.165$$ 8. What is the correlation coefficient of the two-asset portfolio? $$P_{AB}=\frac{CoV_{AB}}{(\delta_A)(\delta_B)}$$ $$P_{AB}=\frac{-.0598}{(\sqrt{.0592})(\sqrt{.0614})}\approx-.09919$$