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Section One: Construction of Optimal Security Portfolio and Computation of Portfolio Risk
Making appraisals for business investments needs an analysis of the market, the income per capita of the population within jurisdiction, and the rate of return of the investment. For instance, in a situation when a client needs to invest $1 million, it is essential to consider the average quarterly returns of the asset classes. In this regard, an analysis of the average quarterly returns on one hand and the datum for the standard deviation of the asset classes on the other could give lead to appraisal of the viability of the investment. Calculation of these quarterly returns puts into consideration the beginning and the ending market values. Consequently, the asset vales must be in accordance with the right stipulations of the business endeavour, which not only leads to acquisition of the right payback period but also gives the viability of the investment. The quarterly rate of return is obtainable through the formula:
TWRR= (EMV-BMV-CF)/ (BMV+.5CF),
where TWRR is the quarterly weighted rate of return, BMV is the beginning market value; EMV is the ending market value, and CF is the net cash flow. The net cash flow is helpful in calculation of the payback period, which is essentially the time which it takes for the business investment to mature.
Review of the Portfolio
For a proof of validity in an investment appraisal, it is fundamental that the risks of its investment are lowered. This stems from the fact that lowering the risk of investment adds the risk factors to the investment, which reduces the chances of vulnerability. The vulnerability of an investment portfolio resides from the submission of the asset classes without proper identification of the risky asset classes. From the above datum, it is clear that the quarterly rate of investment for asset classes like equities stands at 60%, while that for fixed interests stands at 52%. Consequently, the quarterly rate of investment in asset classes like the real estates should be at 50%, while that for the currency and commodity alternatives stands at 56%. This implies that the rate of quarterly returns is above average in all the asset categories, which proves the viability of the investment appraisal. On the other hand, the standard deviation in all the asset returns is calculated from the formula:
where E(R) is the expected value for the standard deviation. In a situation of over $1 million investment in different asset class categories, the vale for E(R) is obtained as being 0.8. This is from the summation: E(R) = (0.1)*(0.5) + (0.06)*(0.5) = 0.08, or 8.
Application of the excel software is useful in calculation of the correlation matrix for the asset categories in that is gives the viability of benefits from the diversification of the investment. In this regard, the correlation matrix is obtained as
The formula that is employed in the calculation of the viability of the business investment took into account the portfolio variance. By consideration, this calculation involved:
1. Specification of the portfolio variance(cell G10) to ensure that minimum requirement is met, which is 0.8.
2. Changing of the cells containing the portfolio weights in cells B1-B20
3. Due to the constraints of the target cells, the solver displays the summation results for the portfolio weights, which gives 1.
4. After generating the solution, the solver software then gives the resultant report for the final value of the return, which is the value in US dollars of risk factor.
Excel's covariance function gives a covariance matrix for the assets at varying levels, depending on the quarterly rate of return for each asset category. This is explained by the fact that the higher the rate of quarterly returns, the higher the value of the covariance matrix. The converse is also true: the lower the rate of quarterly returns for each individual asset category, the lower the value of correlation matrix. From the Excel datum, it is evident that the covariance matrix for each individual asset stands at $11. This is the turnover rate of return per unit time of investment into any asset category, which is essential in determining the minimum variance portfolio.
The minimum variance portfolio for this investment appraisal takes into consideration the assumption that there are no evident situations of short selling. By application of solver, the allocation weights, the expected quarterly returns, and the standard deviation of the returns for the minimum variance for the minimum portfolio give rise to 54.8%, which is lower than the simple average for each asset class category. This value is obtained from the formula STDEVP(L4:L11), in which the portfolio consisting of the same amount of investments the first four stocks ( cell L3) is lower than the simple average standard deviation of the 12 stocks (cell L4). By consideration, solver gives allocation weights, expected quarterly returns, and the standard deviation of the returns for the minimum portfolio that is above the required value for determining the viability of the investment appraisal. This leads to the consideration that the investment is viable, which results in a proposal for investment in one of the asset class categories.
Section Two: Modeling Futures Risk
The ten efficient portfolios depict a variation in the standard deviations in that the first three portfolios have incrementally larger standard deviations than the MPV value. For instance, the covariance between the preceding ten values has negative values, which shows that the simple average deviation is also a negative value. By consideration, the standard deviation for all the ten efficient portfolios (cell L3) is lower than the simple average standard deviation for the ten efficient portfolios (cell L4), which gives rise to -0.88. consequently. A plot with efficient frontier and the underlying assets on a chart with expected quarterly returns on the y-axis and the standard deviation of returns on the x-axis gives rise to a curve with an emphasis of the test levels being at the most expected value, which is 8%
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In another situation, where one holds 1 OZ gold and the aim is to hedge the holdings by selling six months future contracts, it is appropriate to test the viability of such an investment appraisal. By consideration, the risk of holding the gold portfolio lies in the payback period, which is essentially calculated from the cash flows of the investment. The risk of holding the gold is obtained as being -0.39, which is a probability that is lower than the expected value, which stands at 0.8. Consequently, an analysis of the market datum is a viable tool for computing the number of hedges that one can buy in order to minimize the risk of the hedged portfolio. This is computed from the price per index share for the commodity against delivery of the hedged portfolio. The table gives an analysis of these index shares, which is critical in determining the risk of holding the gold appraisal investment.
The spreadsheet for the risk-minimized portfolio for holding of the gold is a negative value. This implies that the risks of venturing into such an investment tend to vary from zero to a negative value. This also implies that such an investment is viable, since there are minimal chances of vulnerability to loses for the investor in question.
Obtaining of the negative value of the minimal risk portfolio involves the following calculations. By consideration, there was setting up of the target cell $E$12, then there was selection of the equilibrium signs. This gave rise to the request for specification of the target cells, which were changed to $E$1:$E$5. This change gave rise to the need for consideration of the constraint that the$E$5 equals to one (100%). Consequently, there were 4 additional constraints from the difference between the cells i.e. 5-1=4. The additional constraints had the requirement that each weighting in the range $E$1:$E$5 >zero, which implies that the investor has the jurisdiction of investing in only three of the available stocks among the four stocks of investment. Consequently, this implies that the number of investment stocks is above the average of the total number of the stocks, which proves viability of the business endeavor.
It is evident that there is an increase in the price per share index of the gold over the period of time. By consideration, the number of hedged portfolios is three, which is the time that it takes for the commodity to appraise by the expected value of 0.8.
Section Three: Option Pricing with Continuous Stock Returns Develop a Spreadsheet Model of the Black-Scholes Option Pricing Model.
The Black-Sholes option pricing model takes into consideration the stock prices, the strike prices, and the volatility, which is a model for determining the standard deviation of the investment. This is evident from the fact that the continuous returns are calculated from the stock analysis of both the current and future assets. The stock prices for the FTSE 100 Company stand at $10, while the striking price also stands at $10.00. These are the values that depict performance within the company, as determined for the past five years. These values are in agreement with the current price per share, which is at $10.00. Moreover, the current exercise price per share for the company stands at $10.00, while the price for the call option 10% stands at $0.01.
Review of the Portfolio
Consequently, the datum for the UK government bond for the latest 20 years is helpful in the computation of the average annual rate of interest. In the analysis, the average annual rate of interest is examined.
From the demographics, it is evident that the average annual rate of interest stands at 25%, which is in agreement with the standard values for interest rates in the global market. This value is obtainable from the moving averages of the yearly returns divided by the number of years. On average, the rate of interest return changes although the deviations do not culminate in a value that is below or above 25%. This gives rise to conclusive evidence that the annual rate of return of interest within the market share for the UK government is stable.
On the other hand, the call value for the UK government stands at 12%, which is half the rate of return of interest. By consideration, if all the factors are kept constant, the value of call is dependant on the time in proportion. This implies that the value of call is lowered when the time of call is increased, while the converse is also true: the value of call is increased if the time of call is reduced. This gives rise to the consideration that the more time is spent on a call, the lower the call value is. Consequently, the interest rate is proportional to the value of call, in that the higher the interest rate is, the lower the call value becomes. The converse is also true : a reduction in the interest rate leads to a consequent reduction in the call value.
The datum generated from the spreadsheet excels software shows that the call value is greater or equals to one, which is a representation of the whole 100%. This stems from the fact that the interest rate is proportional to the call value. Once the rate of interest tends towards infinity, the rate of the call value tends towards one. Consequently, if the interest rate tends towards zero, the call value tends towards infinity.
The rate of interest is proportional to the call value, in which an increase in the interest rates leads to an increase in the call value. The converse is also true: a reduction in the interest rates leads to a reduction in the call value.