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This study is aimed at determining the optimal portfolio in a three-asset portfolio mix in Nigeria. The research employed majorly two empirical methodologies which were Matrix algebra and Lagrangian method of optimization. Matrix algebra was used to compute the various portfolio weights. Lagrangian method of optimization was useful in obtaining the global minimum variance and the efficient frontier of the portfolio. In order to arrive at the best asset in the portfolio that is expected to yield maximum expected return, the study employed the utility function test. The data used for the study were daily stock prices for First Bank Nigeria Plc, Guinness Nigeria Plc and Cadbury Nigeria Plc obtained from the Nigerian Stock Exchange for the period of January 2010 to December 2013 The result obtained from the analysis indicated that among the three assets chosen in the study, Guinness has the highest utility value of 0.031 with lowest risk of 4.268 and the investment opportunity point (
*μ,**σ*) which is (0.169, 2.065) lies on the Capital Market Line. The assets of Guinness and First Bank are located above the Global Minimum at point (
*μ,**σ*) which is (0.10, 1.84) and are said to be efficient assets with high expected returns and low risk. The study therefore concluded that First Bank and Guinness were the only efficient optimal assets in the three asset-portfolio mix and therefore, the preferred choice for every investor since they yielded a high return with minimum variance.

The world of investment is strikingly full of uncertainty as investors face the possibility of either making gain or losing a fraction or even sometimes their entire investment funds. Every investor is faced with the decision of either holding back his investible funds (particularly, if the magnitude of returns on his proposed investment in not known) or invest all the same assuming he is a risk taker. Most investors hold their baskets of investment in more than just one or two portfolio for the obvious reason of diversification which is intended to minimize risk on the investment. Accordingly, [

Harry Markowitz’s mean-variance portfolio model, which is the foundation of modern portfolio theory, assumes that all available information and expectations on future prices are contained in the current prices of assets, and thus treat future payoffs and returns as random variable. In simple terms, it can be assumed that the returns of an asset (say asset i), follow a Gaussian distribution in which the expected mean value of the returns, ^{2} capture all the information concerning the expected outcome, likelihoods and range of deviations from it [

In choosing or combining assets in a portfolio, it is important for investors to know the degree of covariance that exists between the assets. Covariance reflects the degree to which the returns of two securities vary or change together. For instance, assets that have a positive upward co-movement will tend to reduced returns, while assets that have a downward and negative co-movement tend to yield increase returns. Thus, upward and downward deviations from the expected return will and can be reduced through diversification since diversification has the ability to reduce the variance of expected even when both have the same magnitude of expected return. As the number of securities included in a portfolio increases, the importance of the risk of each individual security decreases whereas the significance of the covariance relationship increases [

Arguably, only very few investors have recognized the role of diversification in portfolio risk reduction, while others are ignorant about it. The idea of identifying the appropriate asset mix is the main role in determining the portfolio risk and return. For every portfolio investment strategy, overcoming the problems of selection and allocation of investment funds to the constituent assets are of major concern to many investors [

Allocation of investment capital in a three-asset portfolio mix constitutes a problem to potential investors, institutions and corporate organizations. Several investors in Nigeria often face the difficulty of how to allocate their capital to companies quoted in the stock market in order to maximize returns while minimizing risk as ultimate goal. Apart from investors facing the challenge of selecting the type of asset to invest in, they also lack technical knowledge on how to allocate their funds to the selected portfolio. Most available literatures have enunciated portfolio selection given two assets while ignoring the case of multi-assets mix in a portfolio. This study intends to show how to determine the optimal portfolio in a three-asset portfolio mix through understanding 1) how to determine the fraction capital that should be allocated to the various assets 2) how to obtain a maximum expected return at the most minimum risk 3) how to determine the optimal portfolio among all possible efficient portfolios and 4) how to determine the best asset in a three-asset portfolio mix. The research utilizes daily stock prices for the period 2010 to 2013 from the Nigerian Stock Exchange for three institutional investors, viz, First Bank Nigeria Plc, Guinness Nigeria Plc and Cadbury Nigeria Plc.

Several studies have been conducted on Portfolio theory due to its vital importance in finance literature though pioneered by [

Introducing the model for Portfolio selection and stating the two stages of Portfolio selection, [

Estimating utility by a function of mean and variance of return of 149 mutual funds, [

Investigating the optimal holding period (investment horizon) for the classical mean- variance portfolio model, [

While investigating optimal portfolio allocation in a world without treasury securities, Bomfim [

[

[

[

[

Following [

Risk squared is the variance of the portfolio and risk tolerance is a number from 0 through 100. It is difficult to define the concept of risk tolerance precisely because the level is set by financial situation or financial disposition and preference of an investor. The concept of risk tolerance depends on the behavioral pattern of investors. For instance, if an investor experiences an increase in wealth, he/she will choose to increase (or keep unchanged, or decrease) the fraction of the portfolio held in the risky asset if relative risk aversion is decreasing (constant, or increasing). Thus, the capacity to bear the risk of losses without being upset depends on financial ability and personal temperament. Thus, the best asset for investment would be the one from the efficient frontier that maximizes the utility.

Consider a portfolio containing two financial assets, the investor’s decision may be based on the expected returns and variances which are the two sufficient parameters in which normal distribution may be defined [

Assume that there is a utility function of the type

Any rational investors would seek to maximize or minimize the utility. That is, maximum expected return at a given level of risk or minimum risk at a given level of return. If we assume that there are only two risky assets, A and B, available for consideration in an investment portfolio. Since the portfolio contains two assets of different proportions,

In this case, we have

where

The portfolio weights sum to one

by taking expectation of equation (6)

the mean portfolio return is found to be

Consequently the portfolio variance is

and the portfolio standard deviation is

Let

The minimum Risk Portfolio is obtained by minimizing the portfolio standard deviation. Taking the first derivative of Equation (11) with respect to

To minimize

We make

Solving (11) for

The sufficient condition for

The degree of correlation between two assets A and B is used to determine the shape of the frontier within the portfolio. There are three assumptions about the correlation between two assets A and B.

1) Perfect positive correlation,

We simplify Equation (7) as:

When the correlation coefficient assumes the value +1 it means that all the points in the scattered diagram lie on the same straight line. Thus, the slope is positive which means that while the value of asset A is increasing, the value of B is also increasing or the value of A decreases as B decreases [

2) Perfect negative correlation,

We simplify Equation (10) as:

When the correlation coefficient assumes the value −1 it means that the slope is negative that is while one of the assets is increasing the other is decreasing [

3) Non-perfect correlation,

When the correlation coefficient takes the value zero, it means that there is no linear relationship between the assets A and B or that assets A and B are linearly uncorrelated [

Consider the mean and standard deviation in the relationship derived from the equations (9) and (16) respectively. We can write the slope of the frontier as:

If we assume that

The study utilized daily stock prices for First Bank Nigeria Plc, Guinness Nigeria Plc and Cadbury Nigeria Plc from January 2010-December 2013. The choice of assets was determined by their high level of stock returns among other stocks in their sub-sectors. Stock returns was calculated as thus,

Average rate of return

where:

R = rate of return.

n = number of returns.

The stock return in any time period is given as

where:

Previously, we had model a two-asset mix, this model is common in most literatures and books. However, it is not common to find a model for a three-asset mix portfolio. If we consider a portfolio containing three financial assets (A, B and C), recourse is taken to matrix algebra which can represent a lot of data by putting them into groups which are called rectangular arrays. It is assumed that investors may invest in a total of three risky assets.

Suppose

Suppose x represents the fraction of capital in asset i

The return for the portfolio

Taking expectations of both sides of the above equation we obtain

This may be re-written as

where

Variance of the portfolio is written as

It is observed that the variance of the portfolio return is dependent on three variance terms and six covariance terms. Therefore, the covariance terms are twice of the variance terms which contribute to the variance of the portfolio. Matrix notation can be used to simplify the portfolio algebra for easy calculation [

Let

The lagrangian for this problem is

and the first order conditions (FOCs) for a minimum are

The above conditions are made up of four linear equations in four unknowns. It can be solved to obtain the weights of global minimum. The four linear equations describing the first order conditions have the matrix representation.

or, more concisely,

where

The system (25) is of the form

where

Then the solution for z_{m} is

Portfolio weights

There are two methods of finding efficient portfolios [

Secondly, minimize the risk of portfolio for a given level of return. Let

The investor in practice prefers to embrace the target expected returns rather to target risk levels. Thus, the second problem in Equation (28) is most often solved.

To solve the constrained minimization problem (11), first form the Lagrangian function.

The lagrangian for this problem is

The first order conditions for a minimum are the linear equations.

The above conditions are made up of five linear equations in five unknowns,

The system above is of the form

where

Then the solution for z_{x} is

Portfolio weights

Analytical expression for a minimum variance portfolio can be used to show that any minimum variance portfolio can be created as a convex combination of any two minimum variance portfolios with different target expected returns. If the expected return on the resulting portfolio is greater than the expected return on the global minimum variance portfolio, then the portfolio is an efficient frontier portfolio. Otherwise, the portfolio is an inefficient frontier portfolio. Thus, to compute the portfolio frontier in

Proposition I: Formulation of a frontier portfolio using two efficient portfolios [

Suppose

and portfolio y solves

Suppose a is any constant and define the portfolio z as a linear combination of portfolios, x and y:

Then

1) The portfolio z is a minimum variance portfolio with expected return and variance given by

where

2) If

The study begins by examining the descriptive results from the three selected companies. The result presented in

The result for standard deviation (which is the measure for the riskiness of the asset) revealed that First bank Nigeria, Guinness and Cadbury Nigeria Plc had standard deviation of 3.11, 2.73 and 3.26 respectively. Thus, Cadbury Nigeria Plc had the least return on asset but with the highest standard deviation.

First Bank | Guiness | Cadbury Nig. Plc | |
---|---|---|---|

Mean | 0.182946 | 0.177362 | 0.016873 |

Standard error | 0.139003 | 0.121943 | 0.145815 |

Median | 0 | 0 | 0 |

Mode | 0 | 0 | 0 |

Standard deviation | 3.114431 | 2.732179 | 3.267054 |

Sample variance | 9.699685 | 7.464806 | 10.673643 |

Kurtosis | 35.714168 | 0.3697218 | 63.058922 |

Skewness | −3.735929 | −0.112548 | −4.857296 |

Range | 38.432755 | 19.487447 | 48.607188 |

Minimum | −30.816619 | −9.84400 | −43.728172 |

Maximum | 7.616136 | 9.844007 | 4.879016 |

Sum | 91.839378 | 89.035734 | 8.470669 |

S/N | ASSET | E[R] | VAR | Cov | PAIRS(I,J) |
---|---|---|---|---|---|

1 | First Bank | 0.182946969 | 9.699685504 | 1.099998071 | (1,2) |

2 | Guinness | 0.177362022 | 7.464806402 | 0.343980153 | (1,3) |

3 | Cadbury | 0.016873844 | 10.673643803 | 0.199779782 | (2,3) |

Portfolio weights | |||||
---|---|---|---|---|---|

x_{1} | x_{2} | x_{3} | Constraint | VAR(Rp) | |

0.293640514 | 0.404530008 | 0.301829478 | 1 | 3.397026204 | |

E[Rp,x] | 0.1305633 | ||||

SD(Rp,x) | 1.843102331 |

Target | Portfolio weights | |||||
---|---|---|---|---|---|---|

μ_{A} | x_{1} | x_{2} | x_{3} | Constraint 1 | Constraint 2 | VAR(Rp) |

0.1294 | 0.443565 | 0.556562 | 0.001282 | 1 | 0.1800009 | 4.7659114 |

E[Rp,x] | 0.18000098 | |||||

SD(Rp,x) | 2.18309674 |

Target | Portfolio weights | |||||
---|---|---|---|---|---|---|

μ_{A} | x_{1} | x_{2} | x_{3} | Constraint 1 | Constraint 2 | VAR(Rp) |

0.17736 | 0.4069278 | 0.533036 | 0.060035 | 1 | 0.169999 | 4.268093 |

E[Rp,x] | 0.1699999 | |||||

SD(Rp,x) | 2.0659364 |

Target | Portfolio weights | |||||
---|---|---|---|---|---|---|

μ_{A} | x_{1} | x_{2} | x_{3} | Constraint 1 | Constraint 2 | VAR(Rp) |

0.01687 | −0.052694 | 0.0116674 | 1.04102 | 1 | 0.0100000 | 11.53808 |

E[Rp,x] | 0.010000003 | |||||

SD(Rp,x) | 3.396775395 |

indicates Global Minimum, First Bank, Guinness, Cadbury and Capital Market Line (CML) drawn from the expected return of the efficient frontier. From the graph, Guinness lies on the capital market line with higher expected returns and lower risk at the point

point

The study has shown that among the three assets in the portfolio, two assets Guinness and First Bank are efficient optimal assets and Cadbury is the only inefficient asset in the portfolio. Guinness was observed to have the lowest risk of 4.268 while Cadbury had the highest risk of 11.538. The study affirmed that the Global Minimum Variance Portfolio provided a suitable and recommended aid in selecting optimal portfolio with expected return of 0.131 and variance of 3.397. The utility function test revealed that Guinness is an efficient optimal asset and the best company for investment since it has the highest utility value of 0.031. In period of recession or deep economic doldrums, the asset of Guinness Nigeria Plc will still have the ability to provide some protection from an extreme loss even when other assets vale depreciates. A careful note of caution needs to be explained here that the research is not intended to lure investors to invest in Guinness Nigeria Plc, but rather to examine and explain how investors can select an optimal asset given a three-asset portfolio mix.

Offiong, A.I., Riman, H.B. and Eyo, E.E. (2016) Determining Optimal Portfolio in a Three-Asset Portfolio Mix in Nigeria. Journal of Mathematical Fi- nance, 6, 524-540. http://dx.doi.org/10.4236/jmf.2016.64041