We understand that financial economics is one of the few practical disciplines of economic science, due, among many more, to the development of financial markets in recent years, which is why the need to create financial models that help understand their behavior, and its analysis.
The present work aims to explain the pricing models, CAPM, Markowitz, Multifactorial model, which are widely accepted in the stock market as portfolio selection instruments.
We deal with the subject of efficient diversification, which are possible through the construction of optimal risky portfolios, which are the best combinations of risk and return, in other words the objective is to find the best asset allocation.
It is worth mentioning the importance of Harry Markowitz in Portfolio Theory, which is not having discovered diversification and its effects, since this was already experienced intuitively by many, but rather his greatest contribution, which was to develop the theoretical-analytical technique that serves to deduce the efficient frontier of risky assets.
Finally, in the application section a case is shown: Mexico, where a portfolio simulation is carried out and its efficient frontier is found, a result is also reached and that is that the behavior of the average yield of the stock market cannot be explained by a single factor, but by multiple factors, that is, portfolios are significantly influenced by macroeconomic variables.
BI-VARIED PARAMETRIC MODELS
BACKGROUND
The study on the trajectory of the share price and the relationship between risk and return has been the object of study for many years, for example it is known that in the 19th century a young Frenchman named Louis Bachelier discovered the Brownian movement trying to explain fluctuations in the price of shares on the Paris Stock Exchange, however his discovery about the probability distributions of the share price was not taken into account, until 1950, when Mandelbrot and Eugene Fama took their theory and supported by this they discovered their Since the variances of the returns of the actions are not constant over time and that also the price distributions do not follow a Gaussian distribution, and these are very important foundations in financial theory today.
One of the major problems of financial discipline was solved with the construction of the CAPM (Sharpe and Lintner 1964), that is, the quantitative problem of the relationship between risk and expected return.
The creators of the portfolio theory are Markowitz, Sharpe and Lintner, who worked independently, they worked based on an assumption, that the behavior of the agents depends on the performance and the variance.
On the other hand, F. Black, M. Jensen and Myron Scholes in 1972 studied derivatives and their prices.
As an alternative to the CAPM model, in 1976 the Arbitrage Pricing Theory (APT) emerged, which was introduced by Stephen A. Ross, the APT model has the potential to improve the weaknesses of the CAPM, with this model you need fewer and more realistic assumptions to be generated by a simple arbitration argument and its explanatory power is much better because it is a multifactor model.
It is necessary to mention that the generality of the APT is its strength and its weakness: on the one hand, the APT model allows choosing the factors to be used for a better explanation of the data, however it cannot explain the variability of the return on assets in terms of a limited number of factors that have been easily identified. Contrary to this the CAPM is intuitive and easy to apply.
Currently the academic world is divided into the defenders of the CAPM, the defenders of the APT and those who protest against both methods.
METHODOLOGY
It is important to point out a crucial assumption in the development of these bivariate parametric models.
MARKET EFFICIENCY HYPOTHESIS:
In an efficient market, price changes can never be predicted if they incorporate the expectations and information of all market agents. Regarding this matter, Eugene Fama mentions that a market is called efficient where prices fully reflect the information available.
We detail the main financial models, especially the bivariate parametric models, we have a small classification:
| MODELS OF
PRICE FIXING |
MODEL
PARAMETRIC BIVARIATE |
MODEL
PARAMETRIC MULTIVARIATE |
APT |
| MODEL | CAPM
MARKOWITZ |
MULTIFACTORIAL | PRICE SETTING BY ARBITRATION |
Of course there are more models referring to the estimation of the cost of capital, which we will not deal with for the moment:
- Discounted Cash Flow Model Dividend Discount Model Risk Premium Model
These parametric models (from the box) have great applicability, which depends on the degree of efficiency of the market to which it is going to be applied, efficiency is very important, and is closely related to complete information and information symmetry; The study of this work will be to develop bivariate parametric models and their effectiveness when predicting the expected performance of a title or portfolio.
In the final part of this work, a case will be presented, the Mexican case, we will see as a result that financial models are applied many times, without considering the main hypothesis that underlies each of these, this hypothesis is "market efficiency" Without this hypothesis, the results of research will lack theoretical foundation and interpretation.
MODELING-INVESTMENT PORTFOLIO We have as statistical instruments:
First of all we must classify the risk, we can find two types of risk:
SINGLE RISK: company-specific factors.
MARKET RISK: macroeconomic factors.
Here we pursue an objective and it is the selection of portfolios, the objective of the selection of portfolios is simple, maximize performance and minimize risks, below we see each case:
RISK-PROFITABILITY OF AN ASSET
A) PORTFOLIO WITH TWO RISKY ASSETS
It is about choosing the optimal allocation between two classes of securities: shares and bonds.
- Portfolio Performance
The rate of return is the weighted average of the rate of return of the securities with the investment ratios as weights.
- Portfolio expected return
It is the weighted average of the expected return of the component assets with the same portfolio proportion as the weights.
- Portfolio Variance Covariance
It is the evaluation of the degree to which the return tends to covariate, if the returns of the two assets vary inversely, then the covariance has a negative value, in clear terms, while one has a good performance the other asset has a bad performance.
- Interrelation Coefficient (Correlation)
As it is difficult to measure the magnitude of the covariance, we use the statistical technique of correlation, which is equal to the covariance divided by the product of the standard deviations of the performance of each fund:
Using this knowledge, we can form our portfolios, by doing this we will obtain our set of investment opportunities; However, what we want is to minimize the risk of the portfolio, to obtain that portfolio with minimal variance we must first discover the proportion in which we must invest each asset:
Proportion to invest in each asset to reach the minimum variance portfolio:
- Correlation equal to zero:
When there is no variation between the returns of the two assets, the proportions can be calculated as follows:
- Non-zero correlation:
When there is a variation between the returns of the two assets, the proportions are calculated:
- Perfect negative correlation:
The proportion in bonds that must be invested to bring the standard deviation to zero when we have the negative perfect correlation:
- Investment and mean variance criterion:
Following the line of study, under this approach, what investors want is for their portfolio to be in the farthest northwest point possible, depending on their aversion to risk, it will be any portfolio within the orange line.
- Effective diversification:
4.1.1) Correlation equal to zero:
4.1.1) Positive perfect correlation:
It is a special case where there are no gains from diversification, and where it is very difficult to see which portfolio is inefficient.
4.1.1) Negative perfect correlation:
As long as the correlation is less than 1, there will be diversification gains, it is even better in the case when the correlation is equal to -1.
Correlation and investment opportunity set:
B) OPTIMUM RISKED PORTFOLIO WITH AN ASSET WITHOUT RISK
It is about choosing the optimal allocation between three classes of securities: shares, bonds and Treasury bills; the difference with the previous analysis is that an asset without risk is included.
The way to calculate the profitability and the variance is the same, it is actually the same process developed above, with the difference that since we incorporate the asset without risk, we can graph here the capital allocation lines (LAC), until we obtain a LACTangent to the set of investment opportunities, this is the optimal risky portfolio.
- Expected Portfolio Return: Portfolio Variance:
It remains this way because it has both the risk-free asset and the risky ones.
Recalling the expected return result:
The slope of the LAC is the Sharpe Ratio also known as the “volatility reward ratio”:
The difference between the ratio values of two portfolios, for example between the minimum variance portfolio and the optimal risky portfolio, shows by what percentage the expected return increases for each percentage point increased in the standard deviation.
Proportion to invest in each asset to achieve the optimal risky portfolio:
- Correlation equal to zero:
When there is no variation between the returns of the two assets, the proportions can be calculated as follows:
- Non-zero correlation:
When there is a variation between the returns of the two assets, the proportions are calculated as follows in order to find the optimal portfolio:
All investors would very much like to position themselves above the set of investment opportunities, for example at the orange point, this result may be possible but we need more than just three assets to be able to increase the effect of diversification:
C) OPTIMUM RISKY PORTFOLIO WITH MANY RISKY ASSETS
The well diversified portfolio:
The largest portfolio is the market portfolio, where only systemic risk is found, below are some characteristics of this portfolio, and then we will solve the problem of how to position ourselves above the set of investment opportunities.
- Expected portfolio return: Portfolio variance:
We noticed that when there were only two titles the number of variances was equal to the number of covariances, however, now that we have a multitude of titles the number of covariances is significantly higher than the number of variances.
To exemplify the result of diversifying, there is an exercise on variance:
From this it follows that if Z were 1, the variance of the portfolio would be the variance of the market, while Z increases the variance of the portfolio it will depend more strongly on the average covariance, when Z has reached the maximum limit then the variance of the portfolio is equal to the average covariance, so we conclude that the average covariance constitutes the remaining risk after reaching the maximum diversification.
- Standard deviation
Now let's imagine three stocks, each curve represents the portfolio set formed by the stocks, the curve between a and b shows the set of risk and return combinations of the portfolios that can be formed with the two stocks, the curve that passes between b and c is also the set of portfolios formed by shares b and c, now, the curve between e and f represents all the portfolios that can be built from combining portfolios e and f, in the end we will see that it is a combination of the three actions: a, b and c.
We soon learn that incorporating a larger asset expands the set of investment opportunities to the northwest, where it is best and desired.
Diversification with one more asset:
In this way we can incorporate more and more assets and position ourselves even above this new set of investment opportunities.
The efficient frontier:
The analytical technique to deduce the efficient frontier of risky assets was Harry Markowitz's greatest contribution, which is about building the portfolios that are most directed to the northwest of the universe of values, and specifically this efficient frontier is also located above the minimum variance portfolio.
Over time the efficient frontier can fluctuate.
The Property of Separation:
It was proposed by the American economist James Tobin, this property implies that the choice of a portfolio can be separated into two different exercises, the first exercise is to find the optimal risky portfolio and the second exercise is about the allocation of assets, something that it depends a lot on the risk aversion of investors.
MODELING-CAPM
It is used to establish profitability under conditions of market equilibrium; this is a market equilibrium model.
ASSUMPTIONS OF THE MODEL
- Corporate stocks are valuable in the market Markets are competitive
Where
The profitability of a portfolio is its cost of capital.
diversified risk component risk-free systematic risk risk premium factor
ANALYSIS OF BIVARIATE PRICE SETTING MODELS FOR THE MEXICAN MARKET
NOTE: this application case is for a portfolio that involves more than two assets, therefore the theoretical reference developed above does not necessarily contain the same formulas but they do pursue the same meaning.
HYPOTHESIS:
The initial and final performance of the portfolios is defined, which are composed of assets of the largest companies that are listed on the Mexican stock market, what is desired is to know if the Markowitz method can be applied in the Mexican Stock Market, the hypothesis must be tested:
Ho: the Markowitz method can be applied in the Mexican stock market H1: the Markowitz method cannot be applied in the Mexican stock market ASSET YIELD:
The performance of an asset can be calculated in the following way (for a period):
Where is the price of the asset / portfolio at the beginning of the period, and is the price of the asset / portfolio at the end of the period.
According to Markowitz, the investor must select his portfolio taking into account its expected value and its standard deviation.
PERFORMANCE:
35 Mexican companies are evaluated from January 1994 to December 1999.
Televisa Apasco
Autrey
Cemex
CPO
Telmex
Alpha
Benavides HERDEZ
Gissa KOF
Peñoles
Maseca
GIB
Femsa
UBD
Cemex
Cemex
ICA
Tribasa Etc.
The optimal portfolio calculation will be made by the aforementioned method. To do this, first the performance of the shares is determined, that is, the percentage change in prices, in the case of Televisa, for example from January 3 to 10, 1994 it was:
The same is done but for the 35 actions, here I show a box with only 11 actions:
Here is the correlation:
As a particular case that we are taking, that of Televisa, its correlation:
This indicates that 76% of the variation in their performance is explained by the variation of the CPI, and 23% depends on other factors that are not specified in the equation.
Third, we calculate the beta:
And since we continue to use the Televisa case, we have:
As the beta of televisa is greater than one, it means that it amplifies the risk trend to the portfolio, this action is very risky.
Fourth we find the risks:
Total Risk = Systematic Risk + Non-Systematic Risk
To find the systematic and unsystematic risk we need to have calculated the correlation and the variance.
Systematic risk = =
Non-Systematic Risk = =
Analysis summary (with only 11 actions):
With these data, the efficient frontier can be easily generated.
THE EFFICIENT BORDER:
The theory indicates that from N assets infinite portfolios can be formed, however each investor will have an optimal portfolio, the group of all optimal portfolios is the efficient frontier, their optimal portfolio will be the one that:
1) Offer the maximum expected return for different levels of risk. 2) Offer minimum risk for different levels of expected return.
Determination of the Feasible Group:
Considering the 35 shares to be chosen by the investor, the proportions of the fund that the investor is willing to spend on each share are denoted, they are for each share respectively: Televisa, APASCO, Modelo, Cemex, Telmex L, Alfa Benavides, HERDEZ, GISSA, KOF, Peñoles, Maseca GIB, TVAZ CPO, Gruma, Ciel, Gmex, Autrey, Femsa UBD, ARA, Cemex A, Cemex B, GEO, ICA, Hogar, Tribasa, Carso, Desc A, Desc B, Desc C, San Luis CPO, Telmex A, Comerci UBC, Hilasal, Elektra, and Cifra, and knowing that the total sum:
If it is decided to invest all the money in Televisa shares, then the proportion that will be invested in the other shares will be 0.
The expected return will be calculated by the following formula:
Where is the expected return of the portfolio
Let's see for Televisa in the first case:
Now let's take a set of portfolios, we obtain their expected return according to the proportions invested in each asset:
This process can be represented in the following algebraic form:
To facilitate the calculation of the deviation, a useful tool can be used, which is the matrix of variances and covariances (presented for the 10 assets below):
The main diagonal of the matrix is made up of the variances of each asset, for example if I want to calculate the standard deviation of a portfolio made up only of Televisa, then I take cell 1,1 and apply the square root:
The risk of this portfolio would be 6.9%.
If we do the same for the other portfolios mentioned above:
We are going to graph the expected return against the standard deviation, and we will obtain the efficient frontier, in the figure the points inside the curve are individual assets and inefficient portfolios, where M is the portfolio of minimum variance, it is also observed that the asset risk-free is 0.382.
Any other portfolio that the investor might consider will be dominated by one that is on the efficient frontier, since that is where higher returns or minimal risk are expected.
PORTFOLIO COMPOSITION:
The Elton, Gruber, Paltman algorithm is used: this algorithm assumes that the performance of the assets can be described by the market model, to determine the composition we build a table with the above data:
As additional data it is known that: the variance of the CPI = 0.225 and that the risk-free rate is equal to 0.382%.
We start the algorithm by deducing the Sharpe ratio (also known as the Volatility Reward Ratio):
The algorithm recognizes that the optimal risky portfolio is reached with the highest, and is tangent to the set of investment opportunities. The algorithm identifies the optimal portfolio that maximizes the value of in 5 steps:
1st calculation of the Treynor rate:
And these values are ordered in descending order:
2nd from the top of this table, we calculate the values, this value is the cut-off point, that is, the assets that exceed this point will be selected as part of the T portfolio, and those that do not exceed will be discarded:
3rd, the difference between the RVOLi values is evaluated and, we note that up to the sixth row the Treynor rate is greater than, this indicates that there is a cut-off point for = 0.52, which means that whenever the assets have a ratio of Rewards for volatility greater than 0.42 have a weight different from 0 in portfolio T.
4th, the Z values are calculated to set the weights for each asset, this using the following equation:
For DescA:
5 ° With these results we can calculate the portfolio components by:
The sum of all these is necessarily equal to one.
For Desc A:
RESULTS COMPOSITION OF THE PORTFOLIO
From this we can build our optimal portfolio: THEORETICAL PORTFOLIO PERFORMANCE
For this work, the real returns of each of these assets were also collected, for that reason we calculated the REAL PORTFOLIO PERFORMANCE:
While the real performance of the portfolio oscillates between both positive and negative values, those of the model have a lower variation and are positive for the period analyzed. Therefore, it concludes that this model under the Markowitz method is not capable of following the trend of the real portfolio performance.
The model obtained by the Markowitz method is not good enough to be applied as a mechanism to determine the performance of a portfolio.
Real and predicted returns by the Markowitz model, (1998-1999)
CAPM MODEL
The characteristic of the CAPM is that in equilibrium it prohibits any asset from having a weight equal to zero, its justification rests behind the separation theorem where it is indicated that risk is independent of the return-risk preferences of each individual. The efficient frontier
The efficient frontier of the CAPM is known as the Capital Market Line (lmc). Any portfolio that involves the market portfolio and a risk-free asset lending or borrowing must fall below the CML, although in some cases it could fall very close to it.
The slope of the LMC curve is given by:
Taking the previous exercise as a reference, the efficient portfolio T obtained an expected return of -7.78 and a variance of 0.475:
The first component is the risk-free asset (the reward for waiting and not taking risks), the second component is the slope (it is the reward per unit of risk taken).
The Asset Market Line:
The LMC is the balance between risk and return for efficient portfolios, however we can also calculate the balance between risk and return of a risky asset:
As the slope is positive, the more its covariance of the asset increases, the larger its prices, if its covariance decreases, its price will also decrease, this relationship is known as the Asset Market Line:
You can get one based on the beta:
Determination of Portfolio Performance
The performance of individual assets is determined, the following table shows data necessary to continue the analysis
The slope of the asset market line:
We observe that if it meets the equation of the asset market line.
We will check with an example for this we take Televisa, a company that has a beta
equal to 1.15:
The following table shows the expected return for the shares using our model obtained and the real performance of the shares (televisa):
If we focus on the cumulative deviation from expected returns, it presents a real value of 1786, only during 1998 and 1999.
The following table shows the deviations calculated for each of the actions, but during the entire period of analysis, and not only 1998 and 1999:
The actions that have the highest deviations of 5000, for example, have a symptom that the model is not working correctly, however for the other actions it is working correctly.
Now we are going to use the model obtained previously, but this time we are going to find its variance, the result is that the total deviation of the investment portfolio in the analyzed period is 758. To see how good this deviation is, it will have to be compared with the other models.
Results obtained:
By far the results of the CAPM model are superior to those obtained by the Markowitz model, as we have just seen, the CAPM does forecast the portfolio returns with positive and negative values, although the results of these are still lower than the real ones. By obtaining a deviation of 758 less than that of Markowitz, this confirms that there is less chance of failing in our result to reach the real performance.
Actual performance and predicted performance:
In addition to the above, the covariance and the correlation calculated between the actual and the estimated performance is 9.7 and 0.69 respectively, figures that show the best performance of the classic CAPM, compared to the Markowitz methodology. On the other hand, this last figure indicates that the CAPM supports 69% of the behavior of the real return of the portfolio, so we can affirm with sufficient statistical evidence that the CAPM in the Mexican Stock Market presents a high explanatory level.
REFERENCES:
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Efficient Frontier (Bodie Kane Markus): A graph representing a set of portfolios that maximizes the expected return at each level of portfolio risk.
Highest Possible Capital Allocation Line
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