NBS-7047B – Quantitative Finance

Posted: August 26th, 2021

Student Number: [Provide your student number here]
Module: NBS-7047B – Quantitative Finance
Academic Year: 2019/20	Semester: 2

SECTION 01 – CAPITAL BUDGETING

(Word count; excluding tables and references = 825)

Part (a)

Cost of capital

It refers to the required return that a capital budgeting investment or project should meet to make it viable for investment in a company. Such capital budgeting projects include purchasing new equipment and building a factory expansion unit. The cost of capital consists of a combination of cost of equity as well as debt used to evaluate the financial viability of a capital project by a firm and if the investors should invest in the project compared to its risk (Giambona et al.,794). Therefore, the cost of capital can either be the cost of equity when funded by the company’s equity and finances or the cost of debt when financed by debts. Several methods are used in capital budgeting analysis. These include net present value (NPV), payback period, and internal rate of returns (IRR). Thus, the three projects that are projects A, B, and C have a different cost of capital as they have different initial outlay, returns, and expected returns.

Part (b)

NPV Method

The net present value (NPV) metrics are used to plan investments and capital budgeting as well as forecasting by analysing the profitability of a proposed investment. It is expressed as the variation of the present value (PV) of the aggregate cash inflows with the PV of total cash outflows in a stipulated period (Marchioni et al., 365). The NPV is given by the summation of discounted cash inflows adjusted with the expected rate of return than less the initial outlay in the project (Marchionni et al., 365). NPV is calculated by use of the following formula;

Where- i= cash flow at period 1

            r= discount rate

            n= period

The decision rule on NPV holds that a project with a positive NPV should be adopted and implemented. The positive value shows that the forecasted or expected earnings from a project or a financial investment will exceed the projected costs in the present dollar. Therefore, a project that yields a positive NPV implies that it is profitable, while an investment with a negative figure will result in a net loss to a company (Marchionni et al., 365). Thus, only projects with a positive NPV figure should be implemented. Additionally, in mutually exclusive projects, the project or investment which has the maximum NPV should be considered by the company for investment.

            Furthermore, NPV accounts for the time value of money by modifying the expected future returns to the present value by use of a discount rate. This is because the present value of money is higher than in the same future returns due to inflation and earnings from alternative investment options (Marchionni et al., 365). Thus, a dollar earned in the future is less than the same present values. Table 1 below shows the cash inflows of project A, B, and C that is mutually exclusive.

Table 1: Cash inflows of project A, B, and C

Year	1	2	3	4	5	6
Project A	6000	6000	4000	4000
Project B	8000	6000	5000	4000	3100
Project C	9000	7000	6000	3000	2000	150

Table 2 below indicates the NPV of the three projects

Table 2: NPV for Projects A, B, and C

Project	NPV
A	8576.3
B	19049.9459
C	20616.8408

From the NPV calculations above, all three projects are viable for investment as they have positive return figures. Therefore, the company can adopt the three projects according to the availability of funds. However, since the three projects are mutually exclusive, the company can implement project C since it offers the highest expected returns based on the NPV analysis. Thus, if the company adopts project C, the expected value will increase by $20, 616.8408.

Part (c)

IRR Method

IRR is used to estimate the profitability of a proposed project or investment in capital budgeting. The IRR method of capital budgeting is a discount rate that makes the NPV of the total present value of cash flows in a particular investment equivalent to zero (Ma’aji et al., 101). Additionally, the calculations and formula used to get the investment’s IRR are the same as the NPV method.

Part (d)

Pitfalls of IRR Method

The IRR method encounters some problems in the valuation of capital budgets that may mislead the appraisal of projects. The main problems are multiple rates, multiple discount rates, and does provide for the time value of money by use of a discounted rate. Initially, the method is complicated, and calculating the IRR can give multiple rates, which may be negative, indicating the company is losing value, though not always the case (Ma’aji et al., 101). In the case of mixed cash flows, which have positive and negative signs, the IRR gives more than one inaccurate value. More so, only projects more significant than the opportunity cost of capital should be considered though IRR does not cater to discounted rates in the time value of money. Thus, the decision can either be using an annual discounted rate and making investment decisions or computing a weighted average IRR and using it in making investment decisions. Furthermore, the method is calculated by trial and error to make the NPV equal to zero. Hence, it is difficult to comprehend and compute.

References

Giambona, Erasmo, et al. “The theory and practice of corporate risk management: Evidence from the field.” Financial Management 47.4 (2018): 783-832. Academic Medicine, 4.1 (2018): 60.

Ma’aji, Muhammad M., and Casey Barnett. “CAPITAL BUDGETING PRACTICES AND RISKS ADJUSTMENT: PRACTICES AMONG CAMBODIAN COMPANIES.” ICMEBSS 2018 (2018): 101.

Marchioni, Andrea, and Carlo Alberto Magni. “Investment decisions and sensitivity analysis: NPV-consistency of rates of return.” European Journal of Operational Research 268.1 (2018): 361-372.

SECTION 02 – CAPITAL ASSET PRICING MODEL (CAPM)

(Word count; excluding tables, figures,and references =946)

Part (a)

CAPM metrics define the connection between the expected returns of an underlying asset, mainly stock and the systematic risk. CAPM is used for pricing risky financial securities, for providing the expected returns of the financial instruments when presented with the cost of capital and risk (Crowe, 1). CAPM is calculated using the formula below;

Where: ERi= Investment’s expected return

            Rf=Risk-free rate

            βi= investment’s beta

            (ERm​−Rf​)= Risk premium​

Logarithmic Rate of Return, Excess Logarithmic Return, and  Descriptive Statistics

The logarithmic rate of return is also referred to as a compounded return or force of interest. It is given by the natural log of the division of the current and previous day returns. Excess logarithmic returns refer to the gains achieved above the performance of the proxy in investment. The excess returns depend on the chosen investment returns comparison for evaluation. It is taken by comparing the risk-free rate with risk levels in the investment analysis (Porras, 45). The excess return can either be a positive or negative amount. A positive difference indicates that the investment exceeded the comparison returns, while a negative excess return shows the project or financial investment were underachieved.

Descriptive statistics are adopted in summarizing data in a meaningful technique for establishing specific patterns and trends from the data. The two types of descriptive statistics are measures of spread and central tendency. The statistical methods used in central tendency are mode, mean, and median, which show the central position of data distribution (Porras, 43). Whereas, measures of spread include quartiles, absolute deviation, standard deviation, range and variance that indicate how data is distributed on a group (Porras, 43). The mean of the three data samples of Alphabet Inc., Coca-Cola Company, and S&P 500 is given as the average of the monthly dividend by the adjusted closing prices. It is calculated by aggregating the monthly data divided by the data sets. The variance and standard deviation show how data is spread from each other. Standard deviation also indicates the stock volatility; thus, the higher the figure, the greater the investment risk (Porras, 45). Skewness in a data set shows the probability distribution asymmetry, of a random variable against its average or the mean. The skewness values can be expressed either as zero, positive, negative, or undefined (Porras, 49). Negative skew is left-skewed, and the curve leans on the right. On the other hand, positive skew is right-skewed, and the curve leans on the left. Skewness shows data distribution and the magnitude of deviation from the standard curve. Kurtosis measures financial risk, and it defines how the tail of distribution differs from the normal distribution tail. Kurtosis compares with normal distribution whose value is equal to three (Porras, 45). Thus, if the Kurtosis value is higher than three, it implies that it is heavily tailed. The number of observations is the sample size of a data set. Table 3 below presents the descriptive statistics for Alphabet Inc., Coca-Cola Company, and S&P 500.

            From the table below, the risk of the S&P 500 is the greatest as it has the highest standard deviation figure, followed by Alphabet Inc. and finally, the Coca-Cola Company. The values of all the data sets are not heavily tailed as the kurtosis value is less than three. The value of kurtosis is negative, implying that the distribution is flatter compared to a normal curve with an equivalent standard deviation and mean. 

Table 3: Descriptive Statistics

Descriptive statistics	Alphabet Inc.	Coca-Cola Company	S&P 500
Mean	$746.57	$36.78	$2,158.81
Median	$727.30	$37.02	$2,082.96
Variance	56,632.44	17.39	132,724.59
Std. Deviation	237.98	4.17	364.31
Skewness	0.40	0.33	0.31
Kurtosis	-0.97	-0.58	-0.70
Number of observations	72.00	72.00	72.00

Table 2 below presents the descriptive statistics for the algorithm returns for Alphabet Inc., Coca-Cola Company, and S&P 500.

Table 4:  Algorithmic Returns for Alphabet Inc.

Descriptive statistics	Alphabet Inc.	Coca-Cola Company	S&P 500
Mean	$0.01	$0.01	$0.01
Median	$0.01	$0.01	$0.01
Variance	0.00	0.00	0.00
Std. Deviation	0.06	0.04	0.03
Skewness	0.68	-0.49	-0.77
Kurtosis	1.19	-0.17	1.50
Number of observations	72.00	72.00	72.00

Part (b)

Price Evolution of Alphabet Inc. And Coca-Cola Company

From the annual standard deviation of the two companies, Alphabet Inc. has a greater investment risk as it has a higher standard deviation compared to Coca- Cola Company (Bakar and Sofian, 42). The Alphabet’s Inc. The standard deviation is 237.98 compared to the Coca-Cola Company’s standard deviation of 4.17. Figure I below shows the stock chart for the alphabet Inc. and Coca-Cola Company.

Part (c)

Logarithmic returns of Alphabet Inc. and Coca-Cola Company

From the computation of the descriptive statistics, the performance of the two companies is interdependent and have a normal distribution (Bakar and Sofian, 42). Both companies are performing well as they have figures that are close to each other from the descriptive logarithm statistics computed in table 2 above.

Part (d)

The two stocks are normally distributed

Based on the descriptive statistics for the logarithmic returns, the return series of the two stocks from Alphabet Inc. and Coca-Cola Company is normally distributed because they have features of standard distribution curves. Initially, the mean of the data set in both sample size in 0.01 and for a normal distribution should be zero (0) (Bakar and Sofian, 42). Furthermore, the standard deviation of the population is below one, a close characteristic in normal distribution whose standard deviation is equal to one. Additionally, the mean and mode of the data are similar just like the normally distributed population. Finally, the data variance is equal to zero on both populations.

Part (e) (excel)

Part (f)

Regression Analysis

It is a statistical method used to examine the relation of more than two variables under consideration (Bakar and Sofian, 42). It analyses the interdependence between dependent and independent variables.

Part (excel)

References

Bakar, N. Abu, and Sofian Rosbi. “High volatility detection method using statistical process control for cryptocurrency exchange rate: A case study of bitcoin.” The International Journal of Engineering and Science 6.11 (2017): 39-48.

Porras, Eva R. “Asset Price Dynamics, and Stochastic Processes.” Bubbles and Contagion in Financial Markets, Volume 2. Palgrave Macmillan, London, 2017. 1-51.

SECTION 03 – OPTIMAL CAPITAL ALLOCATION

(Word count; excluding tables, figures,and references = 913)

Part (a)

It is a characteristic of an efficient market, where capital is allocated in the most beneficial way to different parties involved. Therefore, it symbolizes an optimal capital distribution to the consumers in an economy, financial capital allocation in companies, and investment projects to investors (Malenko, 1750). Thus, it involves distributing and investing the organization’s financial resources in an ideal method to increase its efficiency and maximizing the profits to generate the maximum returns to the shareholders. An investor should consider the viability of investment options available in the market, the effects and risks, and allocate funds appropriately to generate the best overall outcome.  When evaluating the investment options, an investor considers the efficient frontier, capital allocation line (CAL), separation theorem, and optimal investor portfolio.

            In this case, the investor would like to make a financial decision between Alphabet Inc. (GOOGL) and Microsoft Corporation (MSFT). The investment budget is $1,000,000, and the client is a risk-averse investor with risk-aversion factor (A) equal to ten. Investment analysis is undertaken to propose the optimal capital allocation for the investment portfolio.

Alphabet Inc. (GOOGL) and Microsoft Corporation (MSFT) descriptive statistics

Descriptive statistics are used in data analysis to describe, summarize, or show data in a more meaningful way establishing specific patterns and trends from the data. It is mostly used in extensive data analysis, where the presence of the raw data is huge and unpresentable. However, descriptive statistics only describe a set of data and cannot be used to draw any conclusion from the analysis as they visualize what raw data is presenting (Kaur et al., 60). Thus, descriptive statistics enable data to be presented in a meaningful way allowing more straightforward data interpretation.

The mean of the two companies, Alphabet Inc. and Microsoft Corporation, is given as the average of the monthly dividend-adjusted closing prices. It is provided by the summation of all data divided by the number of data. The variance and standard deviation show how data is spread from each other (Kaur et al., 60). Standard deviation indicates the volatility of the stock, and the higher the amount, the greater the risk of an investment. Correlation shows how two sets of data are interrelated and the dependence on each other. Table 1 below shows that the two data sets from Alphabet Inc. and Microsoft Corporation have a strong positive correlation of 0.97. Covariance indicates the directional relationship between datasets, and the data below shows a strong relationship as it is a high value of 5563.56. Table 1 below shows the descriptive statistics, correlation, and covariance of Alphabet Inc. and Microsoft Corporation’s monthly dividend-adjusted closing prices for the period 1^st January 2013 to 31^st December 2018.

Table 5: Company Statistics

Descriptive statistic	Alphabet Inc. (GOOGL)	Microsoft Corporation (MSFT)
Mean	$746.57	$54.82
Variance	$56,632.44	$583.54
Std. Deviation	237.98	24.16
Correlation	0.97	0.97
Covariance	5563.56	5563.56

Part (b)

Investment Opportunity Set

It refers to all investments available that an investor can undertake in a given period. It is determined by the investor’s financial capability and types of financing, which can be equity, debt, personal savings, or capital venture (Camilleri et al., 30). Sharpe ratio is used to measure the investment’s performance in comparison to a risk-free asset, after making adjustments of risks of the underlying asset (Camilleri et al., 30). The ratio is the difference between the investment’s returns and risk-free returns, then divided by the investment’s standard deviation.

Part (c)

Efficient Frontier

The efficient frontier refers to a set of optimal portfolios selected to offer maximum expected return at the lowest level (Camilleri et al., 27). The investment returns are dependent on the investment portfolio. The standard deviation of the investment denotes the risk levels, and lower covariance between a data set in a portfolio denotes a low standard deviation and low investment risk (Camilleri et al., 30). The portfolios under the efficient frontier curve are sub-optimal as they will not give adequate returns with the level of risk. Hence, if a portfolio clusters to the right side of the frontier curve it implies it is sub-optimal. In this case, they have a greater risk level for the expected returns.

Part (d)

Minimum-Variance Portfolio as the Risky Portfolio

A portfolio of risky assets when combined results in thelowestattainable riskfor anexpected return as such a collectionhedges the risk of every investment (Camilleri et al., 30). However, investments in minimumvarianceportfoliosareindependentlyriskierthanthe entire portfolio.This is denoted by the Markowitz Portfolio Theory, where the investment’s volatility is used to replace the risk.

	Alphabet Inc.	Microsoft Corporation
Sharpe ratio	2.3804%	23.9857%

Part (e)

Capital Allocation Line (CAL)

It is a line that graphically portrays the risk-and-return of a financial asset or portfolio. It can be adopted to get the optimal portfolio (Camilleri et al., 30).

Part (f) (see excel)

Part (g)

1. Optimal Capital Allocation When, The Risk-Aversion Factor, Is 20

The optimal risky portfolio is given by when tangent of the efficient frontier curve.  It is optimal at this point as the CAL is highest at this point, thus generating maximum returns with the additional risk level. When the risk factor changes to 20%, the risk factor also changes by (20/10) = 2%.

References

Camilleri, Silvio John, and Ritienne Farrugia. “The Risk-Adjusted Performance of Alternative Investment Funds and UCITS: A Comparative Analysis.” International Journal of Economics and Finance 10.7 (2018): 23-37.

Kaur, Parampreet, Jill Stoltzfus, and Vikas Yellapu. “Descriptive statistics.” International Journal of Academic Medicine, 4.1 (2018): 60.

Malenko, Andrey. “Optimal dynamic capital budgeting.” The Review of Economic Studies 86.4 (2019): 1747-1778.

SECTION 04 – MARKET EFFICIENCY

(Word Count, Excluding References = 562)

The efficient market hypothesis (EMH) is also known as the efficient market theory. It is a hypothesis that claims that the stock prices is a reflection of all market information. Equally, it is impossible to generate consistent alpha (Rossi et al., 185). The theory expects that the investors trade the stocks at fair value on the stock exchange market, thereby making it hard to buy undervalued shares or sell shares at an inflated prices (Rossi et al., 185). Therefore, as per the theory, investors cannot outdo the overall market performance by using market timing and expert stock selection. Thus, investors can only generate higher returns through buying risky investments as the higher the risk on a stock or portfolio, the greater the returns. The theory proponents aver that investors will only benefit from low-cost investment options that are inactive portfolios. At the same time, the opponents on the other hand hold that investors can outperform the market since the shares are subject to deviation from their fair market value (Rossi et al., 187). Hence, the stocks cannot be purchased when undervalued or traded while overvalued as the theory holds that the only way investors can gain higher investment returns is through speculative investments that pose a high risk. There are three versions of EMH that are weak, semi-strong, and strong and vary according to their degree.

            The weak efficient market hypothesis argues that the present share prices replicate all historical information of stock in the market. Additionally, it states that there is no type of technical analysis or data valuation that can effectively be utilized to assist an investor make investment and stock trading decisions (Hamid et al., 1). The proponents of the weak EMH believe that when the fundamental analysis is utilized properly, undervalued and overvalued share prices can be determined. Hence, the investor’s research organization’s financial statements to improve the chances of making greater returns than the average market returns.

            A semi-strong efficiency hypothesis holds that investors cannot utilize fundamental or technical analysis to generate higher market returns. The reason is that all adequate information used to calculate the share’s current price is public (Mackey et al., 34) The companies’ financial information can be accessed through its annual reports on their websites, yahoo finance, NASDAQ or other stock exchanges. Thus, the advocates of this theory contend that only company information that is not available to the general public can assist investors to boost their chances and performance of getting returns above the general market profits.

            Subsequently, the strong market theory states that all financial information known and available to the public, as well as unknown information, account for the current share prices (Rossi et al., 185). Therefore, there is no other additional information that can offer investors an added advantage in the stock market. The supporters of this form of EMH suggests that an investor cannot generate investment returns above the normal market returns regardless of the research conducted or the information retrieved.  As such, arbitrageurs cannot identify and exploit the mispricing of securities in financial markets due to the efficient market hypothesis. Also, they cannot make abnormal profits above the market returns (Rossi et al., 185). Though there are anomalies of efficient market hypothesis such as price-earnings ratio, January effect, and neglected firm effects, it is still difficult for investors to outperform market averages in stock or portfolio investment.

References

Hamid, Kashif, et al. “Testing the weak form of efficient market hypothesis: Empirical evidence from Asia-Pacific markets.” Available at SSRN 2912908 (2017).

Rossi, Matteo, and Ardi Gunardi. “Efficient market hypothesis and stock market anomalies: Empirical evidence in four European countries.” Journal of Applied Business Research (JABR) 34.1 (2018): 183-192.