An Introduction To Portfolio Management

Published Date: 02 Nov 2017

One of the major advances in the portfolio management is that investors are unable to obtain their optimal portfolio by simplily combining a large number of securities. Instead, investors should consider the relationship among securities to get the optimal portfolio. In this section. we explain some basic concepts in the portfolio theory first.

Some background assumptions

Before presenting the portfolio theory, we introduce some general assumptions for the theory. One basic assumptions in the portfolio theory is that investors maximize the returns of the total sets of investment for a specified risk level. It also assumes that investors are risk averse. Which denote that the investor will choose the asset with the lower level of risk when two assets have the same rate of return.

Markowitz portfolio theory

In the 1960s, the investment community discussed the risk, however, there is no specific measure for it. Investors need to quantify the risk in order to build a portfolio model. The basic portfolio model was proposed by Harry Markowitz, who derived the expected rate of return for a portfolio of assets and an expected risk measure. Markowitz obtained the variance of the expected rate of retrun of assets was a good measure for quantifying the risk under some assumptions. Moreover, he derived the formula that was used to calculate the variance. The formula not only showed the significance of diversifying investment to reduce the total risk of the portfolio but also denoted how to diversify effectively. The Markowitz theory is based on the following assumptions on the investor behaviors:

1 Investors choose each portfolio by the probability distribution of expected returns over some holding period

2 Investors aim to maximize the expected utility for one period, and the utility curve shows the diminishing marginal utility of wealth

3 Investors use the variability of expected return to estimate the risk of the portfolio

4 Investors select a portfolio by the expected return and risk, so the utility are the function of expected return and variance(or standard deviration) of returns solely.

5 For a given expected return level, investors prefer less risk to more risk. Likewise, for a specified level of risk, investors prefer higher expected returns than lower returns.

Under these assumptions, a portfolio assets or a single asset is considered to be optimal if no other portfolio assets or other asset can offer higher expected returns with the same

or less risk or less risk with the same or higher returns.

Measures of risk

One of the most popular measures of risk is the variance or the standard deviration of the expected returns. The more away from the expected returns , the more uncertainty of the future expected returns. Another measure of risk is the range of returns. A large range of returns from highest to lowest denotes the greater uncertainty of future expected returns. Moreover, semivariance( deviations below the mean returns) is also used to quantify risk. These measures of risk assume that investors aim to minimize the damage from returns that is less than the sepecified rate.

We will use the variance or the standard deviation of returns though there are a large of number measures of risk. It is due to the fact that this measure is intuitive, it is a wide accepted risk measure and it has been used in most of the portfolio models.

Expected rates of return

Expected rates of return for a portfolio of investments is the weighted average of the expected rates of return for the individual investments in the portfolio. The wights are the percentage of total value for the individual investment.

Therefore, the formula for the expected rates of return E(r) is:

E(r) =

Where indicates the wight of an individual asset in the portfolio

denotes the expected rate of return of asset i

Variance(Standard deviation) of returns for an individual investment

The variance or the standard deviation of an individual investment E(r) is

Variance==

Where is the probability of rate of return

Standard deviation is the square root of the variance.

Standard deviation= =

1 Introduction:

Portfolio selection is one of the most significant problems in investment management. The first paper in the field go back to the "Portfolio Selection" of Markowitiz( 1952), which became the pioneer work of modern portfolio theory. The main idea of the theoretical framework is using the variance to quantify the risk, and then build risk - return analysis framework on base of it. Classic portfolio theory is built on the assumption that investors make decisions clearly understanding all information, especially financial assets parameters (such as the expectation and the variance of the return on assets). However, this hypothesis is difficult to set up. Investors often do not know the exact values ​​of the parameters to make investment decisions, which requires to estimate the unknown parameters. Therefore, the estimation error will bring the estimated risk of the portfolio. Therefore, in recent years, the research focus shifted to how to solve the estimated risk in the investment portfolio and the optimal portfolio.

According to the special nature of the problem, we are trying to find a solution - Bayesian analysis method. Bayesian methods may be attractive. First, it can use priori information for quantifying the interests. Secondly, it accounts for the estimation risk and uncertainty of the model. Third, it is easily implemented with acceptable numerical algorithms.

In this method, the mean and variance of the yield parameter is assumed to a random variable rather than a fixed value. Forecast distribution of security rate of return that investors use to make investment decisions, should be adjused accordingly. Therefore, the high sensitivity of the mean - variance model can be solved by Bayesian analysis method. Bayesian approach, the forecast distribution of security rate of return only depends on the history of the sample data, which improves the stability of the model

I introduce the mean variance model firstly, then the two popular Bayesian approach: diffuse prior and conjugate prior are discussed. In addition, the empirical examples about are given. Finally , I get the conclusions.

2 Mean variance model

The mean variance model is regarded as one of the basic theories in financial economics. It is a single period model that provides investors the optimal tradeoff between expected return and risk. It solves the portfolio problems by using two parameters: expected return(mean return) and risk(variance of return).

Assumptions

Markowitz made the following assumptions when developing the model.

All investors aim to maximize utility.

Only one period investment is considered.

Investors are risk averse.

Investors select a portfolio by the mean and the variance.

There are no taxes and transaction costs. And all securities can be divided into any size.

Actions of investors do not affect prices.

Correlations between assets are fixed and constant forever

For a standard portfolio problem of m assets with expected returns(i=1,2,…,m). And they are independently across time as normal, with variance . Let be the weight that the portfolio value invested in the asset i, which sums to 1. And let = (,…,), w= and 1=.Negative weights as well as weights that are greater than 1 are allowed, corresponding to short selling. The expected return and variance of the portfolio are = and =w respectively. Therefore the portfolio selection problem can be written as:

Min =w

Subject to =

1=1 (1)

for all levels of expected portfolio return above that of the minimum variance portfolio. Investors select a particular portfolio investment on Markowitz’s efficient frontier in maximizing the expected utility. This portfolio is defined as the optimal portfolio with the optimal weights , expected return and variance .

The first constrain denote the expected return is specified. There are two constrains and we introduce two Lagrangian multipliers and to get the Lagrangian function:

L(w, ,)=w- 2(-)-2（ 1-1）

The optimal condition is w-2-21=0 （2）

-2=0 (3)

1-2=0 (4)

Assume is reversible, from equation 2) , we can obtain:

W=+1) (5)

Put (5) in (3), we can obtain:

+1=B+A= （6）

Put (5) in (4), we can obtain:

1+11=A+C=1 (7)

And A=1,B=,C=1,D=BC-

Since is reversible, B,C>0 and =B(BC-)>0

So D=BC->0 ,from (6), (7), we can obtain

=(C-A)

=(B-A)

In equation (5), we can get

W=(C-A)+ (B-A)1)

And the corresponding variance is

=w=(+1))= +1=+

=(C-2A+B)= + (8)

Equation (8) is equal to (9)

is the expected return and is the corresponding variance.

In ( dimensions, equation (9) is the standard form of hyperbola whose center point is (0, and axis of symmetry are =0 and =. We only consider the part in the first quadrant as the standard deviation() is greater than 0.

a

-

From the picture above. the point a is the vertex of hyperbola in the first quadrant. Also as can be seen, this point has the lowest variance. So the optimal solution in this case is:

This approach pioneered by Markowitz, may be viewed as a major breakthrough in modern finance theory. It is suggested that portfolio management may be a science with a strong foudantion. Finally, linear risk return relationship has been widely used in the modern finance theory. The researchers have undertaken a lot of analysis to relax its restrictive assumptions. However, a maintained assumption that investors know the true value of parameters(expected return and variance) is still assumed in most of portfolio analysis, the estimation error or the management error is still ignored.

Under the assumption that is identical independent normal distributed, the sample mean and covariance matrix and by using equations below:

(4)

And (5)

Putting (4) and (5) into the equations (2) and (3) to obtain the final results.

The calculation methods above are known as the plug in methods.

The plug in method regard as the estimated values and as the real values, and thus ignore the errors, which may lead to better portfolio choice existing. Best and Grauer also said the optimal portfolio choice obtained by the plug in method is sensitive to the values of parameter especially the mean value. Increasing or decreasing several samples may have a small influence on values of parameter estimators, but the small change may lead to the optimal portfolio solution changing. In addition, investors invest most of their assest in the portfolio with high expected return and the low variance when the optimal portfolio is obtained by this method. However, this type of assets tend to have higher estimation error compared to other assets. Therefore, the optimal portfolio choosen by this method usually performs bad and is not stable.