Quantitative Finance and Financial Market - Desklib

This article provides solved assignments on Quantitative Finance and Financial Market. It covers topics like t-test, Chi-square test, regression analysis, efficient frontier plot and more. The article includes tables, graphs and equations to explain the concepts. The assignments are solved with step-by-step explanations and interpretations. The article is relevant for students pursuing finance courses in colleges and universities.

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Quantitative Finance and Financial Market
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1QUANTITATIVE FINANCE AND FINANCIAL MARKET
Table of Contents
Answer 1..........................................................................................................................................2
Answer 2..........................................................................................................................................3
Answer 3..........................................................................................................................................5
Answer 4..........................................................................................................................................7
Answer 5..........................................................................................................................................9
Answer 6........................................................................................................................................11
Answer 7........................................................................................................................................14
References:....................................................................................................................................19

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Answer 1.
We are applying Two-sample t-test with unequal variances to verify the hypothetical
proposition about median incomes of men and women.
1. a)
The hypotheses:
Null hypothesis (H0): The difference of median incomes for men and women who are buying
homes is less than or equal to $10000.
Alternative hypothesis (HA): The difference of median incomes for men and women who are
buying homes is greater than $10000.
Level of significance: 0.05.
Calculated t-statistic: 0.006039.
Critical t-statistic: t (395, 0.05) = 1.966003.

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Decision-making: 0.006039<1.966003. Hence, there is an absence of significance of the test of
equality of means (De Winter 2013). The null hypothesis cannot be rejected with 5% level of
significance. Similarly, the alternative hypothesis could be rejected.
Interpretation: The difference of median incomes for men and women who are buying homes is
not greater than $10000. Therefore, it is 95% obvious that the ‘median salary’ for men is just
$10000 higher than ‘median salary’ for women.
Conclusion: The median income of males to buy homes is more than $10000 larger than that of
females.
1. b)
The additional assumptions required to validate the results of the hypothesis test are-
 The two variables are continuous or non-discrete.
 The two samples follow normal distributions.
 Both samples are ‘independent’ in nature.
 The two samples are simple random samples from their respective populations. Every
data in the population has an equal probability to be selected in the sample.
Answer 2.
The Chi-square test is applied to test the association between two categorical variables
that are ‘Employment sector’ and ‘Investment preference’. Failing to establish the association of
two categorical variables, Chi-square test proves the independence between these two variables.

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2. a)
The hypotheses:
Null hypothesis (H0): ‘Employment sector’ and ‘Investing preference’ are independent (no
associated) to each other.
Alternative hypothesis (HA): ‘Employment sector’ and ‘Investment preference’ are associated to
and dependent to each other (McHugh 2013).
2. b)
The Chi-square statistic with 1 degrees of freedom is calculated to be 8.36. It creates the
p-value 0.0038. The calculated p-value is less than assumed ‘Level of significance’ (5%).

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Therefore, null hypothesis could be rejected (Sharpe 2015). Correspondingly, alternative
hypothesis is accepted with 95% probability.
It could be interpreted that ‘Employment sector’ and ‘Investing preference’ are
independent to each other. That is, the types of preferences of investment depend upon the types
of employment sector.
2.c)
The p-value for the hypothesis testing of the Chi-square statistic ( χ2 ( 0.05 ,1 )) is found to
be =0.003838762.
Answer 3.

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The least square regression equation is-
Y = β0 + β1∗X
Here, Y = dependent variables, X = Independent variable, β0=intercept of the model , β1=
slope of the least square regression model.
The dependent variable (Y) is the actual balance and the independent variable (X) is the
computer-generated account balance in this regression model.
3. a)
The least square regression equation-
‘The actual balance (Y)’ = 5.372965458 + 0.94658618* ‘The computer-generated account
balance (X)’.
3. b)
If the computer-generated account balance (X) was 100, then the actual account balance
as confirmed by the accountant is-
The actual balance (Y) = 5.372965458 + 0.94658618 * 100 = 5.372965458 + 94.658618 =
100.0316.
3.c)
The regression equations of upper and lower 90% confidence limits are-
‘The actual balance (YUpper)’ = 12.915787 + 0.996054 * ‘The computer-generated balance’.
‘The actual balance (YLower)’ = -2.169856016 + 0.897118312 * ‘The computer-generated
balance’.
While the computer-generated balance for Timothy Jones is 100, then
‘The actual balance (YUpper)’ = 12.915787 + 0.996054 * 100 = 12.915787 + 99.6054 = 112.52.

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‘The actual balance (YLower)’ = -2.169856016 + 0.897118312 * 100 = 87.54.
The 90% interval estimates are = (87.54, 112.52).
3. d)
The average upper and lower 90% confidence limits of predicted actual balance when all
the computer-generated balances are same is given by- (101.81, 128.37).
It means that predicted average actual balance of the investors lies in the interval of
$101.81 to $128.37 with 90% probability.
Answer 4.

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The multiple linear regression equation is-
Y = β0 + β1 ¿ X1+ β2 ¿ X2 +…+ βn ¿ Xn
Here, Y = dependent variables, X1, X2…, Xn = Independent variables,
β0=intercept of the model , β1 ,… βn= Slopes of the multiple regression model.
The dependent variable is ‘Price ($)’ and the independent variables are ‘Square Footage’
and ‘Type’.
4. a)

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The regression equation to predict the selling price for residences-
‘Price ($)’ = 69630.83 + 90.37203 * ‘Square Footage’ -3629.5 * ‘Type’ (Montgomery, Peck and
Vining 2012)
4. b)
The first slope parameter (β1) (90.3703) is positive. It interprets that the independent
variable ‘Square Footage’ has positive association with the dependent variable ‘Price ($)’.
Therefore, for higher sizes of real estate in terms of square footage, the price also gets higher.
The second slope parameter (β2) (-3629.5) is negative. In this case, the independent
variable is a binary variable with two levels 0 and 1. ‘1’ is ‘condominium’ and ‘0’ is a ‘single-
family home’. The negative value of slope refers that for higher value (1=condominium) of the
predictor ‘Type’, the dependent variable ‘Price ($)’ decreases. Similarly, for lower value
(0=single-family home) of the predictor ‘Type’, the dependent variable ‘Price ($)’ increases.
In summary, β1 and β2 show that for greater square footage and single-family home, the
price of the real-estate enhances.
4. c)
The regression equation that describes the relationship between selling price and square
footage of
 ‘Condominiums’-
‘Price ($)’ = 69630.83 + 90.37203 * ‘Square Footage’ -3629.5 * 1
= (69630.83 – 3629.5) + 90.37203* ‘Square Footage’
= 66001.33 + 90.37203 * ‘Square Footage’
 ‘Single-family home’-
‘Price ($)’ = 69630.83 + 90.37203 * ‘Square Footage’ -3629.5 * 0
= 69630.83 + 90.37203 * ‘Square Footage’

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4. d)
The F-statistic indicates the relation between two slope values (β1 and β2). The F-statistic
is 5.821765 with p-value 0.01186. The p-value is less than 0.05. Therefore, it could be stated that
slopes of selling-price (β0), slopes of types of real-estate (condominium and single-family home)
(β1) and square footage (β2) are unrelated to each other (Ngo and La Puente 2012). Hence,
selling-price and the square footage are different between ‘condominiums’ and ‘single-family
homes’.
Answer 5.
5. a)
The graph depicts the trend of number of new clients generated for four quarters for four
years. Overall, the numbers of new clients generated is highest in third quarters of chosen years
and lowest in second quarters of these years. Overall, the highest clients are generated in 2007

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and lowest in 2006. Individually, most of the new clients are generated in 3rd quarter of 2007 and
least clients are generated in 1st quarter of 2006.
Therefore, it could be said that the number of new clients generated increases after 1st
quarter of the year and then again decreases in 3rd and 4th quarter.
5. b)
The first quarter is denoted by the dummy variable (S1 =1, S2 = 0, S3 =0), the second
quarter is denoted by the dummy variable (S1 =0, S2 =1, S3=0), the third quarter is denoted by
the dummy variable (S1 =0, S2 = 0, S3 =1) and the fourth quarter is denoted by the dummy
variable (S1 =0, S2 =0, S3 =0). The transformation is essential for multiple regression equation.

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The multiple linear regression equation is-
Y = β0 + β1 ¿ X1+ β2 ¿ X2 +…+ βn ¿ Xn
Here, Y = dependent variables, X1, X2…, Xn = Independent variables,
β0=intercept of the model , β1 ,… βn= Slopes of the multiple regression model (Kaiser and
Maravall 2012).
The multiple linear regression equation using quarterly data as dummy variables (S1, S2,
S3 and S4) is given by-

13QUANTITATIVE FINANCE AND FINANCIAL MARKET
‘Number of new clients generated’ = 220 + 1.25* ‘Period’ + 6.5* ’S1’ -15.25* ‘S2’ + 21.75*
‘S3’.
5. c)
The positive value of ‘Period’ refers that for greater period, the predicted number of ‘new
clients’ increases. ‘S1’ and ‘S3’ are also positive. It refers that for the shift of first period to
second period and the shift of third period to fourth period, the number of new clients increase.
On the other hand, the negative value of ‘S2’ refers that for the shift of second period to third
period, the number of ‘new clients’ decreases (Adhikari and Agrawal 2013).
5. d)
The quarterly forecasts of new number of clients of 2010 and 2011 are-

14QUANTITATIVE FINANCE AND FINANCIAL MARKET
Answer 6.
6. a)
Three types of stocks are undertaken in the calculation that are ‘Stocks’, ‘Bonds’ and
‘Commodities’. The aim is to find the standard deviations and expected returns for various
combinations of the stocks and thereafter the ‘Efficient Frontier’ plot.
With the help of the given data, the expected returns and standard deviations of all
portfolio combinations using 10% as the smallest unit of the combination are given in the
following table-
Portfolio Stocks Bonds Commodities Expected Standard

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no. proportion proportion proportion return deviation
1 1 0 0 10.00% 147.00%
2 0.9 0.1 0 9.30% 134.80%
3 0.8 0.1 0.1 8.80% 127.07%
4 0.8 0.2 0 8.60% 123.26%
5 0.8 0 0.2 9.00% 131.51%
6 0.7 0.3 0 7.90% 112.58%
7 0.7 0.2 0.1 8.10% 116.13%
8 0.7 0.1 0.2 8.30% 120.38%
9 0.7 0 0.3 8.50% 125.28%
10 0.6 0.4 0 7.20% 103.04%
11 0.6 0.3 0.1 7.40% 106.23%
12 0.6 0.2 0.2 7.60% 110.23%
13 0.6 0.1 0.3 7.80% 114.93%
14 0.6 0 0.4 8.00% 120.27%
15 0.5 0.5 0 6.50% 94.96%
16 0.5 0.4 0.1 6.70% 97.70%
17 0.5 0.3 0.2 6.90% 101.33%
18 0.5 0.2 0.3 7.10% 105.76%
19 0.5 0.1 0.4 7.30% 110.90%
20 0.5 0 0.5 7.50% 116.65%
21 0.4 0.6 0 5.80% 88.76%
22 0.4 0.5 0.1 6.00% 90.90%
23 0.4 0.4 0.2 6.20% 94.04%
24 0.4 0.3 0.3 6.40% 98.08%
25 0.4 0.2 0.4 6.60% 102.91%
26 0.4 0.1 0.5 6.80% 108.43%
27 0.4 0 0.6 7.00% 114.54%
28 0.3 0.7 0 5.10% 84.85%
29 0.3 0.6 0.1 5.30% 86.27%

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30 0.3 0.5 0.2 5.50% 88.77%
31 0.3 0.4 0.3 5.70% 92.27%
32 0.3 0.3 0.4 5.90% 96.66%
33 0.3 0.2 0.5 6.10% 101.82%
34 0.3 0.1 0.6 6.30% 107.65%
35 0.3 0 0.7 6.50% 114.04%
36 0.2 0.8 0 4.40% 83.55%
37 0.2 0.7 0.1 4.60% 84.15%
38 0.2 0.6 0.2 4.80% 85.89%
39 0.2 0.5 0.3 5.00% 88.71%
40 0.2 0.4 0.4 5.20% 92.50%
41 0.2 0.3 0.5 5.40% 97.15%
42 0.2 0.2 0.6 5.60% 102.55%
43 0.2 0.1 0.7 5.80% 108.58%
44 0.2 0 0.8 6.00% 115.15%
45 0.1 0.9 0 3.70% 84.98%
46 0.1 0.8 0.1 3.90% 84.73%
47 0.1 0.7 0.2 4.10% 85.64%
48 0.1 0.6 0.3 4.30% 87.65%
49 0.1 0.5 0.4 4.50% 90.71%
50 0.1 0.4 0.5 4.70% 94.70%
51 0.1 0.3 0.6 4.90% 99.52%
52 0.1 0.2 0.7 5.10% 105.05%
53 0.1 0.1 0.8 5.30% 111.19%
54 0.1 0 0.9 5.50% 117.84%
55 0 1 0 3.00% 89.00%
56 0 0.9 0.1 3.20% 87.96%
57 0 0.8 0.2 3.40% 88.03%
58 0 0.7 0.3 3.60% 89.20%
59 0 0.6 0.4 3.80% 91.43%

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60 0 0.5 0.5 4.00% 94.65%
61 0 0.4 0.6 4.20% 98.76%
62 0 0.3 0.7 4.40% 103.64%
63 0 0.2 0.8 4.60% 109.21%
64 0 0.1 0.9 4.80% 115.36%
65 0 0 1 5.00% 122.00%
6. b)
The plot of efficient frontier is given below-
80.00% 90.00% 100.00% 110.00% 120.00% 130.00% 140.00% 150.00% 160.00%
0.00%
2.00%
4.00%
6.00%
8.00%
10.00%
12.00%
Efficient Frontier
Risks
Returns
(Source: Created by author, Bailey and Lopez de Prado 2012)
Answer 7.
In the 17th century, the brilliant mathematician Carl Friedrich Gauss invented normal
distribution that is further also known as ‘Gaussian distribution’. Most of the study of statistics

18QUANTITATIVE FINANCE AND FINANCIAL MARKET
takes help from Gaussian distribution that supports markets, prices, probabilities and other
applications. The normal distribution is a bell-shaped curve with mean and variance parameters.
Most of the random real-life situations are assumed to be normal distribution. Because of the
radial symmetric shape of the Gaussian distribution, a ‘Gaussian copula’ at all do not have any
kind of dependence (Degiannakis, Filis and Kizys 2014). Such kind of financial market
applications are often used. Today in the world of economic globalization, international trading
and global governance are the most granted conception of monetary policy and financial
integration. The Gaussian analytical technique is helpful in this matter.
The Financial crisis started from 2002 was ended in 2009 with a huge supply of funds
looking for ‘low-risks’ as well as ‘liquid investment’ (Rich 2013). Banking sectors of east Asian
countries, western countries and European countries got affected because of the bad performance
of oil industry, production industry and stock market. Business intelligences used complex
models to estimate multivariate probabilities in Gaussian approach for the transformation of
shape and actual loss of the distribution (Rubin and Rubin 2013). In real applications, such kinds
of procedures are found to be trouble-shooting and unreasonable from the view point of smart
investor. It seduced the economic dealers in distinguishing the credit risk and fortified the
volatile risks in the inappropriate places. Gradually, it started to give the mathematical support in
case of superb disaster scenarios by evolving the break down.
Role of Gaussian approach in pre and post-financial crisis:
After the ending of financial crisis of 2007-09, many participating reasons were
discovered and one of them was ‘Complex mathematical modelling’. The ‘Complex
mathematical modelling’ is heavily relying up on gaussian probability distribution. The Gaussian
model is a statistical tool which is utilized to price and manage the risk to use complex financial
products. The Gaussian distribution would be more investigated in the in the crisis situation and
discussed suitable Gaussian model. Also, multivariate normal distributions of credit risk
generated by the Gaussian approach, fail in capturing default ‘Clustering’ in crisis situation
obligated in a short period of time. Transformation of crisis situation could be recovered using
particular credit rating, stochastic dynamics and time-varying approaches. Even though the
association between debt and equity varies with market situations, it has a negative effect on the

19QUANTITATIVE FINANCE AND FINANCIAL MARKET
stability of situations. However, all the propositions against usage of gaussian approach are not
always true.
Application of Normal model in case of Trades:
Traders are able to measure the closing prices as difference from the mean. Note that,
more volatility is observed due to a greater difference between actual value and suggested mean.
The gaussian distribution is the starting of understanding the ‘Market probabilities’. This
approach later led to time series concepts and complex models like ARMA model, ARCH model
and GARCH model (Wu, Chung and Chang 2012). As the significant advantage of normal
model, a shortcoming in capturing the ‘risks’ in the market share would not be observed. The
occurrences due to usage of normal distribution in marketing strategy and trades caused so much
beneficiary. For these reasons, people are inventing mathematical and statistical processes for the
trades and financial markets. Not only that focusing on the long-term marketing approach, the
market interdependencies and integrations are enhanced. The trend analyzing technique does not
explicitly verify the contagion and consequences.
The actual market prices do not match with theoretical computations. The entire exercise
received trading prices from the view point of sales and accounting purposes. The Gaussian
models in crisis period proved to be unjustified. The firms were losing money in additional risky
condition and thereby, the level of losses has threatened the financial system. However, the
trading communications between companies were proved to be the main predictors at the time of
‘International crises transmission’ (Farmer et al. 2012).
Beneficial for Non-volatile Market:
The bell-shaped curve is the typical pattern of the normal distribution. Skewness and
kurtosis are the data deviated from a normal pattern of any data set. Skewness measures
asymmetry in the distribution of stock market, trading data and interest rates. An application of
kurtosis and skewness analyze to determine the volatility of any portfolio while variation is
observed in case of interest rates. Standard deviation (SD) might measure the volatility and
expected return of performances. Lesser variation might indicate lesser risk for a stock and
greater volatility might mean a higher uncertainty level (Winkelbauer 2012). Market traders are
able to measure closing prices from the average as it is isolated from the mean. The standard

20QUANTITATIVE FINANCE AND FINANCIAL MARKET
deviation would be able to measure the deviation of financial values. A larger deviation between
predicted and actual parametric measures causes greater standard deviation and hence greater
volatility (Vlastakis and Markellos 2012). Prices that deviate far away from the average cause
the advantage of the situations. It is known to all that trade in a small range is the reason for
breakout (Anand et al. 2013).
Helpful in Crisis situation:
There is a complain about the fact that wide usage of the Gaussian model and the
dramatic enhance in market evaluation caused immense losses when the markets did not behave
in the way of indicated ideas. It has been widely observed that usage of normal probability
distribution functions on financial random variables and depends on structural definition. Most
of the financial problems are developed primarily with portfolio theory using the stable functions
of Normal distribution. Financial markets and assets are more properly accomplished with the
help of normal probability distribution. It is a certain mathematical fact that the multivariate
Gaussian distribution is not capable of tail dependencies utilized in enhancing quantity in
financial markets. Market volume along with the usage of Gaussian model is pretty much
beneficiary for banks, traders, rating agencies, stock markets and funding organizations.
Practitioners would be better to utilize the correct ideas in place of those for which shortcomings
would be clearer, quantifiable and really good during the crisis situation (Carlson, Duygan-Bump
and Nelson 2015).
The risk management model in crisis situation grows importance of complex financial
models. The recognition of model risk emerges the model validation contributing failures in risk
management model. As a result, a substantial improvement in the risk management model, the
enhanced challenges are being regulated using increased reliance (Sornette 2017). Constant
volatility, market instability, enhanced dynamic volatility models are the vibrant natures of crisis
situation (Garcia-Appendini and Montoriol-Garriga 2013). Evolution of ‘Risk management
model’ expands usage of model, identification of risk of the model, model validation and higher
trust on modelling of capital regulation.
Utilization and failure in ‘Economic capital model’:

21QUANTITATIVE FINANCE AND FINANCIAL MARKET
Normal probability approach helps to explain the most complicated and complete model
to discuss association of assets. The Gaussian model enhances a statistical convenience that
merely provides an approximation of compound association. Model development is a
complicated and vulnerable process with which the blue-print of rigorous validation procedure
could be structured. The well-known model failure in pre-crisis situation would be disappointing
as per ‘Economic capital model’ to evaluate the needed capitals (Beber and Pagano 2013). These
models are expected to be subjected to a rigorous and formal a ‘Risk management program’. The
normal distribution is subjected the proper model validation. The risk managing model and
curriculum needs a framework that includes proper policy making and managing attention with
the help of Gaussian approach (Wachter 2013).
Establishing the Contagion theory newly:
The 2007-09 crisis was unique economical hazard from the point of view of septicity
perception that affected the global marking and economic scenario. In post-crisis situation,
economical credibility gradually cleared out the disaster scenario of ‘Double great depression’.
The volatility index (VIX) was rapidly unstable for a long crisis period (Bekaert and Hoerova
2014). The Gaussian model is properly reformed and applied enabling a business that grew far
rapidly and vastly. The extensive studies have shown how the financial crisis has changed the
impact level from East Asian countries and Russia to Western world.
Firstly, the Gaussian approach referred the contagion by comparing the cross-market
associations among the countries and continents to stabilize the economic shock. Secondly, the
direct approach of normal probabilistic process distinguishes the contagion theories. The
changed prediction of after-crisis situation is the reflection of an association of Gaussian model
and transformed marketing techniques. As a result, the information is processed quickly into
financial relative processes. The financial market is estimated in post-crisis situation based on
potential explanatory variables with normality measures. The pre-crisis stable period is a
comparative attempt of statistical and mathematical approach (Claessens, Tong and Wei 2012).
Utilizing log-returns retrieved from the daily closing values, the Gaussian model calculates
competitor market values. It could be ventured that differences in the market integration might
depict a number of explanatory factors. The advanced and simple trading Gaussian modelling
techniques are enhancing the trading associations and the strength of linkages within

22QUANTITATIVE FINANCE AND FINANCIAL MARKET
organizations of certain sector (Hubrich and Tetlow 2015). This logical and statistical approach
coordinates and supports the responsibility for socioeconomic sciences. It led the global
economy to come back and recover in the post-crisis situation.
As a summary it could be said that, usage of complex multivariate model might be a
significant cause for global crisis situation. However, the role of normal probability approach is
not denying. This approach has brought radical change in the financial and economic zone. It is
known to all that market conditions at the time of global financial crisis produces the financial
disaster. The linear, non-linear or asymmetric dependence between several manufacturing and
trading sectors are retrieved from the approach of Gaussian correlation. The criticism of any
particular Gaussian model should not be credited as a reason of financial crisis keeping in mind
the contribution of Gaussian model in economic enhancement and transformation. Hence, the
criticism is proved to be entirely unjustified. The future policy makers and analyst must keep
their reliance upon sophisticated mathematics and analytics with no doubt.

23QUANTITATIVE FINANCE AND FINANCIAL MARKET
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24QUANTITATIVE FINANCE AND FINANCIAL MARKET
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