Building a Portfolio Using Markowitz Principles

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Investment Analysis & Portfolio Management (FIN630)

Lesson # 35

PORTFOLIO SELECTION

Building a Portfolio Using Markowitz Principles:

To select an optimal portfolio of financial assets using the Markowitz analysis; investors

should;

1. Identify optimal risk-return combinations available from the set of risky assets being

considered by using the Markowitz efficient frontier analysis. This step; uses the

inputs from, the expected returns, variances, and covariances for a set of securities.

2. Choose the final portfolio from among those in the efficient set based on an

investor's preferences.

Using the Markowitz Portfolio Selection Model:

Even if portfolios are selected arbitrarily, some diversification benefits are gained. This

results in a reduction of portfolio risk. However, to take; the full information set into

account, we use portfolio theory as developed by Markowitz, Portfolio theory is nonnative,

meaning that it tells investors; how they should act to diversify optimally. It is based on a

small set of assumptions, including;

1. A single investment period; for example, one year.

2. Liquidity of positions; for example, there are no transaction costs.

3. Investor preferences based only on a portfolio's expected return and risk; as

measured by variance or standard deviation.

Efficient Portfolios:

Markowitz's Approach to portfolio selection is that an investor should evaluate portfolios on

the basis of their expected returns and risk as measured by the standard deviation. He was

the first to derive the concept of an efficient portfolio, defined as one that, has the smallest

portfolio risk for a given level of expected return or the largest expected return for a given

level of risk. Rational investors will seek ethcient portfolios, because these portfolios are

optimized on the two dimensions of most importance to investors, expected return and risk.

The basic details of how to derive efficient portfolios? In brief, based on inputs consisting

of estimates of expected return and risk for each security being considered of well as the

correlation between pairs of securities, an optimization program varies the weights for each

security until an efficient portfolio is determined. This portfolio will have the maximum

expected return for a given level of risk or the minimum risk for a given level of expected

return.

Selecting an Optimal Portfolio of Risky Assets:

Once the efficient set of portfolios is determined using the Markowitz model, investors must

select from this set the portfolio most appropriate for them. The Markowitz model does not

specify one optimum portfolio. Rather it generates the efficient set of portfolios, all of

which, by definition; are optimal portfolios (for a given level pf expected return or risk).

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Investment Analysis & Portfolio Management (FIN630)

In economics in general and finance in particular, we assume investors are risk averse. This

means that investors, if given a choice, will not take a "fair gamble," defined as one with an

expected payoff of zero and equal probabilities of a gain or a loss. In effect, with a fair

gamble, the disutility from the potential loss is greater than the utility from the potential

gain. The greater the risk aversion; the greater the disutility from the potential loss.

ALTERNATIVE METHODS OF OBTAINING THE EFFICIENT FRONTIER:

The single-index model provides an alternative expression- for portfolio variance, which is

easier to calculate than in the case of the Markowitz analysis. This alternative approach

can be used to solve the portfolio problem as formulated by Markowitz--determining the

efficient set of portfolios. It requires considerably fewer calculations.

The Single - Index Model:

William Sharpe, following Markowitz, developed the single-index model, which relates

returns on each security to the returns on a common index. A broad market index of

common stock returns is generally used for this purpose. Think of the S&P 500 as this

index.

The single-index model can be expressed by the following equation:

Ri = αi + βiRM + еi

Where;

Ri = the return (TR) on security i

RM = the return (TR) on the market index:

αi = that part of security i's .return independent of market performance

βi = a constant measuring the expected change in the dependent variable, Ri given a

change in the independent variable, RM

еi = the random residual error;

The single-index model divides a security's return into two components: a unique part,

represented by ai and a market related part represented by βiRM. The unique part is a micro

event, affecting an individual company but not all companies in general. Examples include

the discovery of new ore reserve, a fire, a strike, or the resignation of a key company figure.

The market related part, on the other hand, is a macro event that is broad based and affects

all (or; most) firms. Examples include a Federal Reserve announcement about the discount

rate, a change in the prime rate, or an unexpected announcement about the money supply.

Given these values, the error term is the difference between the left-hand side of the

equation, the return on security i, arid the right-hand side of the equation, the sum of the two

components of return. Since the single-index model is, by definition, equality, the two sides

must be the same.

Selecting Optimal Asset Classes--the Asset Allocation Decision:

The Markowitz model is typically thought of in terms of selecting portfolios of individual

securities; indeed, that is how Markowitz expected his model to be used. As we know,

however, it is a cumbersome model to employ because of the number of covariance

estimates needed when dealing with a large number of individual securities.

An alternative way to use the Markowitz model as a selection technique is to think in terms

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Investment Analysis & Portfolio Management (FIN630)

of asset classes, such as domestic stocks, foreign stocks of industrialized countries, the

stocks of emerging markets, bonds, and so forth. Using, the model in this manner, investors

decide what asset classes to own and what proportions of the asset classes to hold.

The allocation of a portfolio's funds to classes of assets, such as cash equivalents,

bonds, and equities

The asset allocation decision refers to the allocation of portfolio assets to broad asset

markets; in other words, how much of the portfolio's funds are to be, invested in stocks, in

bonds, money market assets, and so forth. Each weight can range from zero percent to 100

percent. Asset allocation is one of the most; widely used applications of modern portfolio

theory (MPT).

Examining the asset allocation decision globally leads us to ask the following questions:

1. What percentage of portfolio funds is to be invested in each of the countries for

which financial markets are available to-investors?

2. Within each country, what percentage of portfolio funds is to be invested in stocks,

bonds, bills, and other assets?

3. Within each of the major asset .classes, what percentage of portfolio funds is to be

invested in various individual securities?

Many knowledgeable market observers agree that the asset allocation decision is the most

important decision made by an investor. According to some studies, for example, the asset

allocation decision accounts for more than 90 percent of the variance in quarterly returns for

a typical large pension fund.

The rationale behind this approach is that different asset classes offer various potential

returns and various levels of risk, and the correlation coefficients between some of these

asset classes may be quite tow, thereby providing beneficial diversification effects. As with

the Markowitz analysis applied to individual securities, inputs remain a problem, because

they must be estimated. However, this" will always ¥e a problem in investing, because we

are selecting assets to be held over the uncertain future.

The Impact of Diversification on Risk:

The Markowitz analysis demonstrates that the standard deviation of a portfolio is typically

less than the weighted average of the standard deviations of the securities in the portfolio.

Thus, diversification typically reduces the risk of a portfolio--as "the number of portfolio

holdings increases, portfolio risk declines.

Systematic and Nonsystematic Risk:

The riskiness of the portfolio generally declines as more stocks are added, because we are

eliminating the nonsystematic risk, or company-specific risk. This is unique risk related to a

particular company. However, the extent of the risk reduction depends upon the degree of

correlation among the stocks. As a general rule, correlations among stocks, at least domestic

stocks and particularly large domestic stocks, are positive, although less than 1.0. Adding

more stocks will reduce risk at first, but no matter how many partially correlated stocks we

add to the portfolio, we can riot eliminate all of the risk. Variability in a security's total

returns that is directly associated with overall movements in the general market or economy

is called systematic risk, or market risk, or non diversifiable risk. Virtually all securities

have some systematic risk, whether bonds or stocks, because systematic risk directly

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Investment Analysis & Portfolio Management (FIN630)

encompasses interest rate risk, market risk, and inflation risk.

After the non systematic risk is eliminated, what is left is the non diversifiable portion, or

the market risk (systematic part). This part of the risk is inescapable, because no matter how

well an investor diversifies, the risk of the overall market cannot he avoided.

Investors can construct a diversified portfolio and eliminate part of the total risk, the

diversifiable or non market, part. As more securities are added, the non systematic risk

becomes smaller and smaller, and the total risk for the portfolio approaches its systematic

risk. Since diversification cannot reduce systematic risk, total portfolio risk can be reduced

no lower than the total risk of the market portfolio. Diversification can substantially reduce

the unique risk of a portfolio. However, we cannot eliminate systematic risk. Clearly,

market risk is critical to all investors. It plays a central role in asset-pricing, because it is the

risk that investors can expect to be rewarded for taking.

The Implications of the Markowitz Portfolio Model:

The construction of optimal portfolios and the selection of the best portfolio for, an investor

have implications for the pricing of financial assets. Part of the riskiness of the, average

stock can be eliminated by holding a well-diversified portfolio. This means that part of the

risk of the average stock can be eliminated and part cannot. Investors need to focus on that

part of 'the risk that cannot be eliminated by diversification, because this is the risk that

should be priced in the financial markets.

The relevant risk of an individual stock is its contribution to the-riskiness of a well-

diversified portfolio. The return that should be expected on the basis of this contribution can

be estimated by the capital asset pricing model.

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