Question

(a) In terms of Markovitz model, explain, using words and graphs, how an investor goes about identifying his or her optimal portfolio. What specific information does an investor need to identify this portfolio?  

(b) Briefly explain why the efficient set must be concave?  

(c) In what significant ways does the Arbitrage Pricing Theory (APT) differs from the Capital Asset Pricing Model (CAPM)?  

(d) “Multi-factor models better explain the return generating process” Elaborate.  

Answer 1

(a)

MARKOWITZ'S MODERN PORTFOLIO THEORY

In 1952, Harry Markowitz presented an essay on "Modern Portfolio Theory" for which he also received a Noble Price in Economics. His findings greatly changed the asset management industry, and his theory is still considered as cutting edge in portfolio management.Modern portfolio theory (MPT) is a theory on how risk-averse investors can construct portfolios to optimize or maximize expected return based on a given level of market risk, emphasizing that risk is an inherent part of higher reward. It is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk. It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return. It uses the variance of asset prices as a proxy for risk.

There are two main concepts in Modern Portfolio Theory, which are;

• Any investor's goal is to maximize Return for any level of Risk

• Risk can be reduced by creating a diversified portfolio of unrelated assets

Other names for this approach are Passive Investment Approach because you build the right risk to return portfolio for broad asset with a substantial value and then you behave passive and wait as it growth.

MAXIMIZE RETURN - MINIMIZE RISK

Let's briefly define Return and Risk. Return is considered to be the price appreciation of any asset, as in stock price, and also any Capital inflows, such as dividends. Risk is evaluated as the range by which the asset’s price will on average vary, known as Standard Deviation.

In a practical application of Markowitz Portfolio Theory, let's assume there are two portfolios of assets both with an average return of 10%, Portfolio A has a risk or standard deviation of 8% and Portfolio B has a risk of 12%. As both portfolios have the same expected return, any investor will choose to invest in portfolio A as it has the same expected earnings as portfolio B but with less risk.

DIVERSIFIED PORTFOLIO & THE EFFICIENT FRONTIER

Risk, as we have seen above, is a welcomed factor when investing as it allows us to reap rewards for taking on the possibility of adverse outcomes. Modern Portfolio Theory, however, shows that a mixture of diverse assets will significantly reduce the overall risk of a portfolio. Risk, therefore, has to be seen as a cumulative factor for the portfolio as a whole and not as a simple addition of single risks.

Assets that are unrelated will also have unrelated risk; this concept is defined as correlation. If two assets are very similar, then their prices will move in a very similar pattern. Two ETFs from the same economic sector and same industry are likely to be affected by the same macroeconomic factors. That is to say, their prices will move in the same direction for any given event or factor. However, two ETFs (Exchange Traded Funds) from different sectors and industries are highly unlikely to be affected by the same factors.

This lack of correlation is what helps a diversified portfolio of assets have a lower total risk, measured by standard deviation than the simple sum of the risks of each asset. Without going into any detail, a bit of math might help to explain why.

Correlation is measured on a scale of -1 to +1, where +1 indicates a total positive correlation, prices will move in the same direction par for par, and -1 indicates the prices of these to stocks will move in opposite directions.

MARKOWITZ EFFICIENT FRONTIER

The concept of Efficient Frontier was also introduced by Markowitz and is easier to understand than it sounds. It is a graphical representation of all the possible mixtures of risky assets for an optimal level of Return given any level of Risk, as measured by standard deviation.

The chart above shows a hyperbola showing all the outcomes for various portfolio combinations of risky assets, where Standard Deviation is plotted on the X-axis and Return is plotted on the Y-axis.

The Straight Line (Capital Allocation Line) represents a portfolio of all risky assets and the risk-free asset, which is usually a triple-A rated government bond.

Tangency Portfolio is the point where the portfolio of only risky assets meets the combination of risky and risk-free assets. This portfolio maximizes return for the given level of risk.

Portfolio along the lower part of the hyperbole will have lower return and eventually higher risk. Portfolios to the right will have higher returns but also higher risk.

(b)

Any portfolio that set as concave the Efficient Frontier is considered sub-optimal for one of two reasons: it carries too much risk relative to its return, or too little return relative to its risk. A portfolio that lies below the Efficient Frontier doesn’t provide enough return when compared to the level of risk. Portfolios found to the right of the Efficient Frontier have a higher level of risk for the defined rate of return.

At every point on the Efficient Frontier, investors can construct at least one portfolio from all available investments that features the expected risk and return corresponding to that point. A portfolio found on the upper portion of the curve is efficient, as it gives the maximum expected return for the given level of risk.

The Efficient Frontier offers a clear demonstration of the power behind diversification. There’s no singular Efficient Frontier, because investors can alter the number and characteristics of the assets to conform to their needs.

(c)

APT may be informative over the medium to long term, but are not considered to be accurate in the short term. The CAPM, on the other hand, is a snapshot, and appears to be more accurate in the short term than it is in the long term (Bodie et al, 2012).

The APT focuses on risk factors rather than assets, so it has an advantage over the CAPM in that it does not have to create an equivalent portfolio to assess risk.

The CAPM assumes that there is a linear relationship between the assets, whereas the APT assumes that there is a linear relationship between risk factors. This means that where there no linear relationship exists, the models are unable to adequately predict outcomes.

However, both the CAPM and the APT make relatively unrealistic assumptions in that assets are freely available and desirable, there are no costs incurred in the acquisition of assets and that all investors tend to think alike and come to the same conclusions. This seems intuitively contradictory, as the most successful investors are likely to be those who are able to spot potential which has remained unnoticed by the market as a whole. Indeed, when all investors do think alike, a ‘bubble’ can be created which inflates the asset price and downplays the risks inherent in the asset (Blanchard & Watson, 1982). In this circumstance, assessing the risk of an asset based on the mood of the market is likely to be far more risky than can be predicted by either the CAPM or the APT. Theoretically, therefore, it could be argued that using a CAPM or APT analysis is likely to increase the propensity for ‘bubbles’ to emerge, as they are using static predictions of behaviour by investors.

This is compounded by the subjective decisions made by analysts creating risk projections (e.g. Levy, 2012): although it may be professionally desirable for analysts to consider levels of risk in a rational and objective fashion, it is unlikely that they have no preferences or particular areas of expertise – or areas where they lack knowledge – and this will impact on the validity of the results of mathematical projections. That is, the calculation is only as good as the analyst who is choosing the factors to be included in it.

Therefore, although the CAPM and APT are useful as rule-of-thumb heuristics of the market as it currently operates, they are both static models which use a limited number of factors (Krause, 2001) to predict risk in a highly complex market. Although they are based on mathematical principles, they are subjective in that the analyst performing the calculation has the freedom to decide which factors are relevant in each particular case.

(d)

Multi-Factor Model

A multi-factor model is a financial model that employs multiple factors in its calculations to explain asset prices. These models introduce uncertainty stemming from multiple sources. CAPM, on the other hand, limits risk to one source – covariance with the market portfolio. Multifactor models can be used to calculate the required rate of return for portfolios as well as individual stocks.

CAPM uses just one factor to determine the required return – the market factor. However, the market factor can be split up even further into different macroeconomic factors. These may include inflation, interest rates, business cycle uncertainty, etc.

A factor can be defined as a variable which explains the expected return of an asset.

A factor beta is a measure of the sensitivity of a given asset to a specific factor. The bigger the factor, the more sensitive the asset is to that factor.

A multifactor appears as follows:

Ri=E(Ri)+βi1F1+βi2F2+⋯+βikFk+eiRi=E(Ri)+βi1F1+βi2F2+⋯+βikFk+ei

Where:

RiRi= rate of return on stock ii

E(Ri)E(Ri)= expected return on stock ii

βikβik= sensitivity of the stock’s return to a one unit change in factor kk

FkFk= macroeconomic factor kk

eiei= the firm-specific return/portion of the stock’s return unexplained by macro factors

The expected value of the firm-specific return is always zero.

Return Generating Model

A return generating model is one that can provide investors with an estimate of the return of a particular security given certain input parameters. The most general form of a return generating model is a multi-factor model. The multi-factor model in its simplest form is the single index model, a common implementation of which gives the market model.

Return-Generating Functions

for a share of common or preferred stock, a bond, a real estate investment, or other investments, this holding period return (HPR) is computed as follows.

Holding Period return, rt =

[(Price change during the holding period) + (Cash income received during the holding period, if any)]/

(Purchase price at the beginning of the holding period)

or equivalently,

rt = [(Pt - Pt-1)+dt]/ (Pt-1)

where rt, is the rate of return, or holding period return (HPR); Pt represents the market price at time t; and dt stands for cash dividend income, or other source of income received during the holding period.