Quantitative Finance Analysis - Term Paper Example

?Quantitative Financial Analysis Introduction Quantitative financial analysis techniques are widely used these days in order to assess the performance and viability of the past financial decisions and future investment opportunities respective. There are certain techniques that investigate the reasons, patterns and specific behaviors of a previous financial decision whereas some techniques are designed to judge whether a future investment opportunity is worthwhile for the entity to encounter. It is important to note that the influence of using highly specialized quantitative techniques is on an increasing trend mainly due to its ability to provide inferential statistics. The results are inferred in a highly precised and accurate manner, which lead the quantitative analysts to sort out the different aspects attached with the behavior of a financial decision. This report highlights various kinds of scenarios in which the specialized quantitative techniques are applied in order to sort out the behavior of the financial decision taken under that scenario. Specialized techniques such as regression, correlation, NPV, IRR, yield to maturity, annuity, etc are utilized to analyze the different scenarios. Part a Question 1 Net Present Value Background Information This question inquires about the viability of an investment opportunity such that the opportunity requires $10,000 to be invested today. The cash inflows from this opportunity will be derived in such a way that $500 will be received after one year, $1,500 after two years and $10,000 after ten years. With reference to the particular technique of Net Present Value (NPV), the viability of this opportunity is asked whether the opportunity is attractive if interest rates are 6% and 2% respectively. Theoretical Background Net Present Value (NPV) is the technique, which mainly works on the concepts of time value of money. According to time value of money, the money received in future time does not have the same worth, had that money received today (Brigham et al, 2010 pp. 380-81). If that future money is brought back today, it would have lesser worth. NPV is the technique that works on the basis of cash flows such that the initial investment is deducted from the discounted cash flows. The resulting answer provides the estimated amount of benefit earned or loss incurred in case of opting the investment opportunity. In case if the NPV is found to be positive, it means, that the investment opportunity is financially viable and hence should be accepted. In case of negative NPV, the investment opportunity is not up-to-the-mark and it should be rejected. Computation Years 0 1 2 10 Investment (10,000)   Cash Inflows 500 1,500 10,000 Discount Factor (6%) 1.0000 0.9434 0.8900 0.5584 Discounted Cash Flows (10,000) 471.70 1,334.99 5,583.95 NPV (2,609.36)       Years 0 1 2 10 Investment (10,000)   Cash Inflows 500 1,500 10,000 Discount Factor (2%) 1.0000 0.9804 0.9612 0.8203 Discounted Cash Flows (10,000) 490.20 1,441.75 8,203.48 NPV 135.43       Interpretation From the above stated results, it can be observed that if the interest rate is set to be 6% to discount the later coming cash inflows, it would result in negative NPV of $2,609.36. Since the negative NPV states that the investment opportunity is not attractive enough to be accepted, therefore it should be rejected. On the other hand, if the interest rate is trimmed to only 2%, it will generate positive NPV of $135 from the same cash flows. This positive NPV reflect that the investment opportunity is financially viable and it should be accepted. Question 2 Present Value Annuity Background Information This question pertains to annuity such that a house is to be purchased on mortgage basis. The actual cost of house is $350,000. However, the buyer is willing to pay $50,000 as down payment. For the rest of $300,000 the buyer wants to pay the amount along with the interest payments in next 30 years at 7% interest per annum. The buyer wishes to pay loan in 30 equal installments, which should include principal amount of the loan as well as the interest payments. Theoretical Background Since the question requires annuity payments to computed. An annuity is the amount of payment in which in which equal installments are paid at a fixed rate of interest for a certain time such that there should not be any delay in the payments (Brigham et al, 2010 pp. 141-43). The amount of annuity payment carries both the principal amount as well as the amount of interest. If the principal is invested or received today, it would be called as present value annuity and hence the equal annuity payments for future will be computed carrying both the interest and principal payments. If the annual payments are deposited or received constantly for a specific future time and at that time, those payments would be summed up, it would be called as future value annuity. Computation Cost Price 350,000 Down Payment 50,000 Present Value Annuity 300,000 Annual Payments ? No. of Years 30 Discount Factor 7% Present Value Annuity Factor 13.2776 Present Value Annuity = PVAF x Annual Payments 300,000 = 13.2776 x Annual Payments Annual Payments = 22,594.44     Interpretation From the above computations, it can be noted for a period of 30 years, the loan payment of $300,000 will be paid in equal installments of $22,594.44. This installment amount will carry both the interest payment as well as the principal payment. Question 3 Retirement Plan Background Information This question requires the quantitative analyst to design the retirement plan of a person who can invest $5,000 per year for the next 30 years such that he is currently 35 and wants to retire at 65 and thus planning to invest $5,000 each year until he retires. After his retirement i.e. at the age of 66, he wants to receive a single installment every year until he turns 90 i.e. for next 25 years. Interest rate to be applied on yearly deposits is 8%. No. of installments to be deposited would be 30 and no. of installments to be received will be 25. Theoretical Background This retirement question basically, deals with annuity such that in the first part, the concepts of future value annuity are applied and in the second part, present value annuity is used. Future value annuity is used in order to accumulate the equal yearly installments of $5,000 for 30 years so that, that summed amount would be later used as a present value in order to withdraw an equal installment for the next 25 years. That equal installment will be computed based on the present value sum, which effectively is the future value sum of the preceding 30 installments. Computation Future Value Annuity ? Annual Payments 5,000 No. of years 30 Interest rate 8% Future Value Annuity Factor 113.2832 Future Value Annuity = FVAF x Annual Payments Future Value Annuity = 113.2832 x 5,000 Future Value Annuity = 566,416     Present Value Annuity 566,416 Annual Payments ? No. of Years 25 Discount Factor 8% Present Value Annuity Factor 11.5287 Present Value Annuity = PVAF x Annual Payments     566,416 = 11.5287 X Annual Payments     Annual Payments = 49,130.95     Interpretation This question is split into two parts such that future value annuity is calculated in first part and then that value is used as a present value in order to compute the equal installments in the next part. It can be observed that from the first part that the person going to retire in 30 years’ time will be depositing $5,000 which would generate a sum of $566,416. Later on, that sum would be used by the retiree in order to withdraw $49,130.95 each year for the next 25 years until he reaches at the age of 90. Part b Question 4 Background Information This question focuses on the behavior of zero-rated bonds such that their yields, their curves, and the pattern of their curves are to be discussed. In the given question, four zero rated bonds are given such that they have the expiries ranging from year 1 to year 4. The one, which has the least maturity, has the highest price (face value) and the one, which has the longest maturity, has the lowest price. Theoretical Background Zero rated bonds are those bonds, which do not pay any coupon payments. These bonds can be bought or sold at the discounted price, which is called as the price of the bond (Jaffe and Ross, 2004 pp. 106-07). These bonds are matured at their face value and the return on these bonds can be computed by deducting the purchase price from the maturity price. In this way, the return cannot be computed like the other bonds, which have coupon payments attached with them. The following is the formula, which is used to compute the return of zero rated bonds. Yield of a zero rated bond = (Future Value/Present Value) ^ (1/n) – 1 Computation No. of Bonds 0 1 2 3 4 Present Value 100 95.51 91.05 86.38 81.65 Future Value 100 100 100 100 100 No. of Years 0 1 2 3 4 Yield to Maturity 0.00% 4.70% 4.80% 5.00% 5.20% Yield Curve Interpretation From the above the computations and yield curve, it can be observed that yield to maturity has an increasing trend with respect to the maturity of the zero rated bonds. The longer the maturity of the bond, higher is yield to maturity of that bond. In this way, the yield curve shows an upward sloping curve, which reflects the increase in the yield to maturity as the term to maturity increases. Question 5 Background Information This particular question relates to find out the price of the bond, which pays 7% coupon rate. This bond is under consideration to be issued for a 30 years period. There are two possibilities that if Andrew Industries manages to keep its rating as A graded, the respective yield to maturity applied on its bond would be 6.5%. However, in case if it remains unable to sustain its a-rated grading and it falls to BBB rating, the yield to maturity applied to its bond would be 6.9%. This question requires the calculation of the prices of bonds under both A and BBB rating. Theoretical Background Bonds provide returns in both the forms of coupon payment as well as capital gains to be earned on the prices of the bond. The prices of the bonds are computed by discounting all the cash inflows associated with the bond (Jaffe and Ross, 2004 pp.107-08). The discount factor is the yield to maturity, which is in fact the market interest rate given for that particular bond or the entity. In case, if the entity issues the bond at a coupon rate higher than yield to maturity, the price of the will be higher than its face value and would be trading at premium. On the other hand, if the bond has the coupon rate lower than yield to maturity, the price of the bond will be dropped from the face value and it would be trading at a discounted price. Following is the formula, which is used to calculate the price of a bond: Price = Coupon Payments/ (1 + Yield to Maturity) n +Redemption Value/ (1 + Yield to Maturity) n Computation Face Value 1000 Coupon Rate 7% Yield to Maturity 6.5% Price ? Settlement Date 1-Jan-11 Maturity Date 1-Jan-41 No. of Years 30 Price of Bond 1065.293 Face Value 1000 Coupon Rate 7% Yield to Maturity 6.9% Price ? Settlement Date 1-Jan-11 Maturity Date 1-Jan-41 No. of Years 30 Price of Bond 1012.535 Interpretation From the above two tables, it can be observed that in case of having A rating or BBB rating, the price of the bonds is greater than its face value. The major reason for this is that yield to maturity for both the ratings are lower than the coupon rate of 7%. Hence, at 6.5% and having A rating, the price of the bond would $1,065.293 and at BBB rating having yield to maturity of 6.9%, the price of the bond would be $1012.53. Question 6 This question relates to the development of the new software such that it has to be investigated whether the new software is viable for the entity or not. Non-conventional or random based cash flows are provided in the scenario such that in the beginning $500,000 is to be paid as an upfront payment. Development costs, which are expected to be earned, are $450,000 per annum for the next three years. At the final stage, $900,000 will be paid the developer. This question requires calculating IRRs of this opportunity and asks about the attractiveness of this opportunity. Last part of the question requires the final payment to be made should be $1 million and inquires whether the opportunity is still beneficial for the entity. Theoretical Background Since the question requires the use of IRRs, therefore the orthodox technique of IRR cannot be applied because the cash flows are non-conventional. Non-conventional cash flows are those in which cash inflows and outflows occur at random basis unlike conventional cash flows in which an outflow is followed by the inflows (Shim and Seigel, 2008 pp. 187-88). Multiple IRR is the technique, which is generally applied in case of non-conventional cash flows such that two or more IRRs are calculated and then compared with the cost of capital. NPV profile is designed on chart on which the cost of capital and IRRs are placed on the x-axis and NPV remains on y-axis. If NPV remains positive within the range of IRRs as compared to cost of capital, the opportunity is feasible and should be accepted, otherwise it should be rejected. Computation Year Cash Flow 0 500000 1 -450000 2 -450000 3 -450000 4 900000 IRR 8.53%   31.16% Interpretation Multiple IRRs computed are 8.53% and 31.16%. From the above graph, it can be observed that at 10% cost of capital, NPV of the project is positive which implies that opportunity is active and it should be accepted. In case if the final payment of $1 million is made, in that case multiple IRRs cannot be computed because outflows will exceed the inflows and thus no return can be obtained. Part c Question 7 Effect of Bad Data on Regression Analysis Results Background Information The given scenario presented in the question highlights the effect of data quality, which is used for the regression analysis. Dependent variable presented in the question is Neptune’s Return on Portfolio whereas the independent variable included in the analysis is the Return on a Commodity Index. The junior analyst erroneously entered some incorrect observations due to which the correlation coefficient presented a very strong relationship between the dependent and independent variable. Instead of entering 7.21% and 6.49%, she made a mistake and entered the data observations as 72.10% and 64.9%. This error made a huge difference especially in terms of the correlation between the two data series such it reached at 0.996 which is approximately equal to 1 suggesting that both the variables are almost perfectly associated with each other. However, upon highlighting this error, the correlation results are found to be much lower and more realistic and reached at 0.824, which shows a strong association between the returns of portfolio and commodity index, but still they are not perfectly correlated. This scenario requires the reasons to be investigated which caused this much differences in the coefficient results between the two variables. Theoretical Background Correlation coefficients indicate the association between the data series or variables included in the model. Data series should be linear in nature so that the appropriate association between the variables can be observed. In general, if the data series have the values, which present a bit similar nature, then a strong association can be found which is depicted by a higher correlation coefficient. Conversely, if the data series included in the model shows a random behavior and do not reflect any specific pattern, then it can be concluded that there is very less or no association at all in the data series. At times, there comes certain issues in the data series, which can cause severe problems in correlation results and they adversely affect the true results of the analysis. For instance, the inclusion of “influential points” or “outliers” can deteriorate the overall results and harm the intent of the purpose of the analysis. In case of a model where there are only two data series included, and each data series is found to have an outlier such that both the outliers are very much close to each other, the overall results of correlation can be significantly jeopardized. Both these extreme values or influential points or outliers can eat out the effect of other observations included in the model and in essence, will pose their own relationship. In this way, the ultimate results obtained will not reflect the behavior of all the data points included in the model, rather it will show those results which are contaminated by the effects of influential points of outliers. Discussion and Explanation In the given scenario, there are two influential points that have adversely affected the results of regression and coefficients. Outliers having the value of 72.10% and 64.90% in Return on Portfolio and Return on Commodity Index have deteriorated the results of the correlation such that correlation coefficient reached at 0.996, which shows a perfect association found between the two variables. After reviewing the whole data series, the corrected data series included the observations of 7.21% and 6.49% in both the data series, the effect of outliers is eliminated, and the genuine results are obtained which show the correlation coefficient to be 0.824, which seems to be a better and more realistic association between the two returns. Briefly, it can be concluded that in case of the effect of bad data in the form of outliers, can really sabotage the regression result especially in those cases where there found to be perfect correlation. In those cases there is no point left in applying regression as the both the independent and dependent variables show exactly similar patterns and hence the purpose of finding out the degree of relationship between the two variables is hampered. Question 8 Cross-sectional Variation in number of Quantitative Analysts Background Information This question involves the use of regression analysis on number of quantitative analysts required with respect to different criteria such that Market Capitalization (MCAP), Debt-to-Equity Leverage (DTL) and Membership in FTSE (FTSE). There are seven sub-parts covered under this question, which are considered on individual basis. Part a Problem This part requires to consider two companies such that both the companies have the same debt-to-equity ratio of 0.75 but one of them has the market capitalization of $100 million and the other one has $1 billion. It is required to compute that how many more analysts are needed by one company with respect to the other one. Solution This problem can be sorted out by utilizing the following regression equation in which the number of analysts is regressed by market capitalization and debt-to-equity leverage. Number Analyst = b0 + b1 (MCAP) + b2 (DEL) +Error Following are the equations in respect of both the companies: Number Analyst (Company 1) = -0.2845 + 0.3199(100) – 0.1895(0.75) = 31.56 or 32 analysts Number Analyst (Company 1) = -0.2845 + 0.3199(1000) – 0.1895(0.75) = 319.47 or 319 analysts Result From the above results, the company having the market capitalization of $1 billion will have 287 more analysts covering that stock as compared to the company, which has a market capitalization of $100 million. Part b Problem This part requires analyzing the p-value attached with the coefficient of Debt-to-Equity Leverage (DTL). The respective p-value obtained from the DEL coefficient is found to be 0.00236. In this part, it is required to explain and interpret the meaning of the 0.00236 p-value. Solution p-value refers to the probability value assigned to the coefficient of a particular variable such it is investigated whether the respective variable is statistically significant enough to predict the dependent variable at a given confidence level. For instance, at p-value of 0.05, it can be said there is only 5% chance that the variable will produce the random or unrealistic results but there is 95% chance that the results produced by the variable will be realistic. Result For this particular case, the p-value of 0.00236 of DEL variable, so it can be concluded that there is only 0.236% chance that the variable DEL may not explain the independent variable. It can be ensured with more than 99% confidence level that the results produced by DEL will have a significant impact upon the independent variable thus indicating that the variable DEL is highly significant to the overall model. Part c Problem This part asks to include a new variable named as Membership in FTSE 100 (FTSE) in the existing model. The reason behind its inclusion is that more analysts cover those companies, which are included in the FTSE 100 index. In this part, the equation covering this dummy variable as well as other two existing variables i.e. MCAP and DEL. Equation Following is the equation covering dummy variable FTSE as well as the other two existing variables i.e. MCAP and DEL: (Number Analysts) i = b0 + b1 (MCAP) i + b2 (DEL) i + b3 (FTSE) i + Error i Part d Problem This part requires the establishment of appropriate null and alternative hypotheses in respect the dummy variable FTSE. It is required that hypotheses should be developed by taking two-sided test of significance. Solution In regression, mostly the hypotheses are developed on two-sided basis, which denote that whether the independent variable is significant enough to predict the dependent variable. Following are the hypothetical hypothesis in both the notation and statement form: Y = b0 + b1 (X) Null Hypothesis (H0) = There is no impact of X on Y (b1=0) Alternative Hypothesis (H1) = There is an impact of X on Y (b1?0) Result Under the given scenario, following are the most appropriate null and alternative hypotheses for FTSE in respect of two-sided test of significance. (Number Analysts) i = b0 + b1 (MCAP) i + b2 (DEL) i + b3 (FTSE) i + Error i Null Hypothesis (H0) = There is no impact of Membership in FTSE 100 on Number of Analysts (b3=0) Alternative Hypothesis (H1) = There is an impact of Membership in FTSE 100 on Number of Analysts (b3?0) Part e Problem This part requires whether the null hypothesis should be rejected given at 0.05 level of significance, in a two-sided test of significance. Null hypothesis in respect of all the three variables are provided and by considering their coefficients, standard errors and t-statistics, it is investigated whether these variables are statistically significant in estimated the dependent variable. Solution In the absence of the p-values, the alternative way of finding whether a particular variable is statistically significant in explaining the dependent variable, the t-statistics criteria is used (DeFuscoe et al,2007). For a two-sided test of significance, the degree of freedom is first computed and then it is checked at a given level of significance. After that the value of t-statistics computed is compared with its critical values (from t-table) in order to find out whether the value lies in the area of acceptance of rejection. Result Since there are 500 observations (n=500) and 3 explanatory variables (k=3), therefore, degree of freedom will be n-k-1 i.e. 500-3-1= 496. By looking at 496 degree of freedom given at a 0.05 level of significance in t-table, it can be noted that ±1.965 is the critical value. So any computed or observed value of t-statistics within this ±1.965 range would be accepted and if the values go beyond this region, then the null hypothesis should be rejected. If the individual variables are taken into consideration, then MCAP has the observed t-statistics of 13.8639 and FTSE has 4.5898, which are greater than +1.965. It means that the null hypotheses for both of these variables should be rejected at a 0.05 level of significance. Similarly, DEL has the observed t-statistics of -3.0082 which is also smaller than -1.965 and hence the null hypothesis in respect of this variable should be rejected as well. Part f Problem This part requires calculating the number of analysts that would be covering a company, which has the debt-to-equity ratio of 2/3, and market capitalization of 10 billion. Specifically, if company is included or not included, both these scenarios are to be applied in estimating the number of analysts required covering such company. Solution Following is the regression equation in which the specific data pertaining to the given company would be entered in order to compute the number of analysts required. (Number Analysts) i = b0 + b1 (MCAP) i + b2 (DEL) i + b3 (FTSE) i + Error i Company having the FTSE 100 membership = -0.0075 + 0.2648 (10000) -0.1829(2/3) + 0.4218(1) = 2648 Company not having the FTSE 100 membership = -0.0075 + 0.2648 (10000) -0.1829(2/3) + 0.4218(0) = 2648 Result Even though the dummy variable FTSE is turned out to be the significant variable in model, however, it did not contribute too much because the inclusion or exclusion of the company has either 1 or 0. So in case of a company having membership in FTSE100 index, it will contribute only 0.4218 times which is quite negligible. In short, the number of analysts required to cover the company under consideration will remain same irrespective of its inclusion in membership of FTSE 100. Part g Problem This is the last part of this comprehensive question, which requires analyzing the reasons due to which the coefficients of MCAP have the discrepancy. In first analysis, regression coefficient was estimated to 0.3199 but in the last analysis, it resulted as 0.2648. It is required whether there is inconsistency in the results. Solution In multiple regression analysis, when new variables are added or existing variables are omitted from the model, it affects the significance of the other remaining variables in the model. All the variables jointly predict the behavior of the dependent variable thus contribute their share accordingly. In the same way, their level significance changes as other variables are included or excluded from the variable. Result In the light of above discussion, it can be concluded that the inconsistencies observed in both the results of MCAP are due to the inclusion of a new variable FTSE, which changed the respective significance of the existing variables. Conclusion This report highlighted the use of different quantitative techniques such as regression analysis, correlation, hypothesis testing, NPV, IRR, annuity, yield to maturity and others in order to assess the different aspects of the financial matters inquired under a scenario. In ‘part a’ of this report, investment opportunities are analyzed through the use of NPV, annuity and retirement plans. ‘Part b’ of the report emphasized upon the yield to maturity of the zero rated bonds, price of a long-term bond and multiple IRR function. The main emphasis of this report is place on ‘part c’ where different elements of regression and correlation are analyzed and discussed in order to find out the patterns and behaviors of the past performance of various returns and other specified variables. References Baker, H. Kent . and Martin, Gerald S., 2011.Capital Structure and Corporate Financing Decisions: Theory, Evidence, and Practice. New York: John Wiley & Sons. 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