**Markowitz Portfolio Theory**

Harry Markowitz developed a theory, also known as **Modern Portfolio Theory (MPT)** according to which we can balance our investment by combining different securities, illustrating how well selected shares portfolio can result in maximum profit with minimum risk. He proved that investors who take a higher risk can also achieve higher profit. The central measure of success or failure is the relative portfolio gain, i.e. gain compared to the selected benchmark.

**Modern portfolio theory** is based on three assumptions about the behavior of investors who:

- wish to maximize their utility function and who are risk averse,
- choose their portfolio based on the mean value and return variance,
- have a single-period time horizon.

**Markowitz portfolio theory** is based on several very important assumptions. Under these assumptions a portfolio is considered to be efficient if no other portfolio offers a higher expected return with the same or lower risk.

- Investors view the mean of the distribution of potential outcomes as the expected return of an investment.
- Investors view the variability of potential outcomes about the mean as the risk of an investment. Variability is measured by variance or standard deviation.
- Investors all have the same holding period. This eliminates time horizon risk.
- Investors base all their decisions on expected return and risk. By connecting all the points of equal utility, a series of curves called the investor’s indifference or utility map is created.
- For a given risk level, investors prefer higher returns to lower returns, or for a given return level, investors prefer less risk to more risk.

By using risk (standard deviation – σ) and the expected return (R_{p}) in a two-dimensional space, following figure presents portfolio combinations available to the investor. Thus, each point within the space enclosed by points XYZ, represents a certain portfolio.

By analyzing the figure the conclusion can be drawn that in a new combination of securities the portfolio can be moved:

- upwards – which would imply higher returns with the same level of risk or
- to the left – this implies higher returns with less risk.

It can be noticed that the portfolios below the XY curve, unlike the portfolios on the curve, offer the investor the same return with a higher level of risk or a higher risk with less return, which is not acceptable to the investor. Investors tend to select the combination of shares that would position their portfolio on the XY curve, called the efficient frontier. If the portfolio does not belong to the frontier, the investor can improve the situation by changing the structure of the portfolio, i.e. by changing its content.

Investors will opt for the portfolio that best corresponds to their risk attitude. Those who are more risk inclined will select the portfolio on the efficient frontier, closer to point X, whereas the more risk averse will select the portfolio closer to point Y. It can be said that the Markowitz portfolio theory helps investors in the selection of the set of shares that will ensure a higher portfolio return with the desired level of risk (the tendency is to minimize risk and maximize return on investment).

**The Efficient Frontier**

Markowitz constructed what is called the **efficient frontier**. First, he combined all the stocks in the universe together into “two stock” portfolios. He observed that the risk-return line of each of the two stock combinations bent backwards toward the return (Y) axis. He then built “two portfolio” portfolios out of all the “two stock” portfolios. The risk-return line of these combination portfolios bent even further back toward the return (Y) axis. He kept combining stocks and portfolios composed of different weightings until he discovered at some point you get no more benefits from diversification. He called this final or “optimal” bent line the **efficient frontier**. The efficient frontier represents the set of portfolios that will give you the highest return at each level of risk.

The efficient frontier has a curvilinear shape because if the set of possible portfolios of assets is not perfectly correlated the set of relations will not be a straight line, but is curved depending on the correlation. The lower the correlation the more curved.

**Investor’s Utility Curves or Indifference Curves**

The utility curves for an individual specify the trade-offs he/she is willing to make between expected return and risk. These utility curves are used in conjunction with the efficient frontier to determine which particular efficient portfolio is the best for a particular investor. Two investors will not choose the same portfolio from the efficient set unless their utility curves are identical. In the following picture, l1 and l2 denotes investors utility curves.

Investor’s utility curves are important because they indicate the desired tradeoff by investors between risk and return. Given the efficient frontier, they indicate which portfolio is preferable for the given investor. Notably, because utility curves differ one should expect different investors to select different portfolios on the efficient frontier.

**Efficient frontier and Capital Market Line (CML)**

An efficient portfolio is one that produces the highest expected return for any given level of risk. Markowitz showed how to find the frontier of risk and returns for stocks. Only portfolios on the frontier are efficient. Sharpe added the riskless asset return and noted that returns on a line connecting r_{rf} and the tangency point on the efficient frontier was also “feasible” in the sense that portfolios consisting of some of the riskless asset and some of the market portfolio could be developed.

The introduction of a risk-free asset in the portfolio changes the Markowitz efficient frontier into a straight line. He called that straight efficient frontier line the Capital Market Line (CML), and he used indifference curves to show how investors with different degrees of risk aversion would choose portfolios with different mixes of stocks and the riskless asset. Investors who are not at all averse to risk could borrow and buy stocks on margin, and thus move out the CML beyond the tangency point. Since the line is straight, the math implies that any two assets falling on this line will be perfectly positively correlated with each other.

The optimal portfolio for an investor is the point where the new CML is tangent to the old efficient frontier when only risky securities were graphed. This optimal portfolio is normally known as the market portfolio.

Two rational investors could hold portfolios at different points on the CML. An extremely risk averse investor could hold only riskless assets, while someone who is not at all sensitive to risk but who wants to maximize expected returns could move on out the CML by buying stock on margin.

Note, though, that all rational investors would have a portfolio that is on the CML. So, a highly risk averse investor would not load up on low-risk stock, nor would a “risk taker” load up on highly risky stocks. Both would invest in the market portfolio and then increase or decrease risk by either buying the riskless asset or borrowing to buy more of the market portfolio.