## Project return Report.

### Mathematically** Modeling:**

Consider a portfolio with two financial assets; the investor’s selection might be based on projected returns and variances, which are the two necessary characteristics for defining a normal distribution. The variance shows the level of risk, whereas the expected value or return represents the weighted average rates of return.

Assume that the type has a utility function.

Any reasonable investor would aim to increase or decrease utility. That is, greatest predicted return at a given risk level or least risk at a given return level. If we suppose that in an investment portfolio, only two risky assets, A and B, are available for evaluation. Because there are two assets in the portfolio with distinct quantities, x-(1-x). It is simple to determine the functional relationship between risk and return.

In this case we have,

Where R_{p} is the portfolio return x_{A }and x_{B }are the portfolio weights.

The Portfolio weights sum to one

x_{A}and x_{B } = 1

By taking expectation of equation

The mean portfolio return is found to be

Consequently the portfolio variance is

And the portfolio standard deviation is

**Mathematical Formulation of Minimum Risk Two-Asset Portfolio Mix:**

Let

**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. Therefore, we simplify Equation (10) as:

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 and, the sign of the slope in Equation (17) depends on the denominator. Notice that the slope is vertical at some point and portfolio which produce this point is known as minimum variance portfolio.

**Methodology: **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 expressed as:

Where:

R = rate of return.

n = number of returns.

The stock return in any time period is given as

Where:

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

### Conclusion:

According to the research, two of the three assets in the portfolio, Guinness and First Bank, are efficient ideal assets, whereas Cadbury is the lone inefficient asset. Guinness had the lowest risk, at 4.268, while Cadbury had the highest risk, at 11.538. The Global Minimum Variance Portfolio, with an expected return of 0.131 and a variance of 3.397, was found to be an appropriate and suggested aid in determining the best portfolio. Because Guinness has the greatest utility value of 0.031, the utility function test revealed that it is an efficient optimal asset and the best firm for investment. Even if other assets deteriorate in value during a recession or serious economic downturn, Guinness Nigeria Plc’s asset will be able to give some protection from a severe loss. It should be noted that the research is not designed to entice investors to buy in Guinness Nigeria Plc, but rather to examine and explain how investors might choose the best asset from a three-asset portfolio mix…..

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