Quatitative analysis

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MAF759 Quantitative Methods For Finance 2009

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QUESTION NO 1

PART A

From the given case it can be seen that Tony has made a down payment of $300,000 for a

house that costs a total of $650,000, thus the remaining $300,000 he has borrowed so the

mortgage monthly repayments for the next 30 years have been calculated with the mortgage

formula as under: (Rosenthal, 2009).

M = P [ i(1 + i)n ] / [ (1 + i)n

- 1]

The data available can now be used in this formula to calculate the desired monthly payments.

DATA:

Principal repayment amount=P=A$350,000Loan Rate (per year) =I=8%Monthly Loan rate=J =I/(12*100) = 0.0067/monthLoan Term=L= 30 yearsMonthly Loan Term=N=12*30= 360 monthsMonthly Repayments=M= ?

MORTGAGE CALCULATION FORMULA:

M = P* J

1 (1+J ) ^-N

M= $350,000 * 0.0067

1-(1.0067) -360

M= $350,000 * 0.0067

0.9094

Monthly Mortgage Repayments = $2568.18

(Excel reference: Question 1 sheet 1)


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PART B

CF0 CF1 CF2 CF3 CF4 CF5 CF6 CF7 CF8

($300,000) $50,000 $50,000 $50,000 $50,000 $50,000 $50,000 $50,000 $50,000

We will use Net Present Value (NPV) and the Internal Rate Of return (IRR) as two methods tocalculate whether Tony should invest in the project or not.

Method 1: Net Present Value (NPV)

Years Cash Flows

0 ($300,000)

1 $50,000

2 $50,000

3 $50,000

4 $50,000

5 $50,000

6 $50,000

7 $50,000

8 $50,000

opportunity cost of capital 6.80%

NPV $892.86

Method 2: Internal Rate Of Return (IRR)

Years Cash Flows

0 ($300,000)

1 $50,000

2 $50,000

3 $50,000

4 $50,000

5 $50,000

6 $50,000

7 $50,000

8 $50,000

IRR 6.88%



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CONCLUSION:

It can be seen here that the NPV is positive and IRR is 6.88% so both methods do notcontradict and hence Tony would be benefited by investing in the project.


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QUESTION 2

PART A

The formula for continuously compounded growth is given has:

g = ln(FV/PV)

We have calculated the continuously compounded growth for the median sale price using this

formula and the calculations are in the attached excel sheet.


PART B

Variables can be classified into two main categories:

Quantitative variables.

Categorical (Qualitative) variables.

Quantitative variables:

A variable that has meaningful arithmetic operations involved in it are categorized as

Quantitative variables such as any type of yield, GPA, age or height of a person, different

indexes or temperature or any other data that has numeric data in it.

Categorical variables:

A variable that does not involve any type of arithmetic operations is a categorical variable, they

take one value out of many possible categories. Examples are different provinces, different

types of colours, different universities attended or different courses of study.

Now we will explain our variable according to the variable categories mentioned above and see

in which class they fall (Variable Types).


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Suburb falls in the categorical variable. As it doesnt have any numeric data involved in it, it only

has names of different suburbs under consideration. Furthermore, as it doesnt have any order

thus it can be sub categorized as nominal variable.

Suburb:

Postcode also falls under the categorical variable category as it doesnt contain meaningful

numeric data.

Postcode:

Price for both 05 and 06 falls under the quantitative variable category as price is a meaningful

numeric data which changes over time and is in a continuous process.

Price 05,06:

Growth is also a quantitative variable as it is measures the change in the two prices on a

continuous basis thus it contains meaningful numeric data in a continuous form.

Growth:

PART C:

To obtain the levels of Melbourne house prices in 05 and 06 we have calculated the bins andfrequencies (summery statistics) and then the Histogram in excel sheet attached.



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Summery Statistics is as follows:

PRICE 05

Intervals Bin Range Bin Frequency Rel. Frequency Cum. Frequency Cumulative %

(184.00 x < 240.00) 240 240 24 0.09023 0.09023 9.02%

(240.00 x < 297.00) 297 297 56 0.21053 0.30075 30.08%

353 353 50 0.18797 0.48872 48.87%

410 410 37 0.13910 0.62782 62.78%

466 466 32 0.12030 0.74812 74.81%

523 523 16 0.06015 0.80827 80.83%

579 579 11 0.04135 0.84962 84.96%

636 636 8 0.03008 0.87970 87.97%

692 692 9 0.03383 0.91353 91.35%

749 749 4 0.01504 0.92857 92.86%

806 806 7 0.02632 0.95489 95.49%

862 862 3 0.01128 0.96617 96.62%

919 919 2 0.00752 0.97368 97.37%

975 975 1 0.00376 0.97744 97.74%

1032 1032 1 0.00376 0.98120 98.12%

1088 1088 1 0.00376 0.98496 98.50%

1145 1145 1 0.00376 0.98872 98.87%

1201 1201 0 0.00000 0.98872 98.87%

1258 1258 0 0.00000 0.98872 98.87%

1315 1315 0 0.00000 0.98872 98.87%

1371 1371 2 0.00752 0.99624 99.62%

1428 1428 0 0.00000 0.99624 99.62%

1484 1484 0 0.00000 0.99624 99.62%

1541 1541 0 0.00000 0.99624 99.62%

1597 1597 0 0.00000 0.99624 99.62%

1654 1654 0 0.00000 0.99624 99.62%

1710 1710 0 0.00000 0.99624 99.62%

1767 1767 0 0.00000 0.99624 99.62%

(1767.00 x < 1823.00) 1823 1823 0 0.00000 0.99624 99.62%

(1823.00 x < 1880.00) 1880 1880 1 0.00376 1.00000 100.00%

More 0 0.00000 1.00000 100.00%

Total 266 1.00000



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PRICE 06

intervals Bin Range Bin Frequency Rel. Frequency Cum. Frequency Cumulative %

(184.00 x < 240.00) 240 240 21 0.07925 0.07925 7.89%

(240.00 x < 297.00) 297 297 51 0.19245 0.19245 27.07%

353 353 42 0.15849 0.15849 42.86%

410 410 40 0.15094 0.15094 57.89%

466 466 33 0.12453 0.12453 70.30%

523 523 21 0.07925 0.07925 78.20%

579 579 8 0.03019 0.03019 81.20%

636 636 13 0.04906 0.04906 86.09%

692 692 5 0.01887 0.01887 87.97%

749 749 6 0.02264 0.02264 90.23%

806 806 5 0.01887 0.01887 92.11%

862 862 5 0.01887 0.01887 93.98%

919 919 6 0.02264 0.02264 96.24%

975 975 2 0.00755 0.00755 96.99%

1032 1032 2 0.00755 0.00755 97.74%

1088 1088 0 0.00000 0.00000 97.74%

1145 1145 0 0.00000 0.00000 97.74%

1201 1201 2 0.00755 0.00755 98.50%

1258 1258 2 0.00755 0.00755 99.25%

1315 1315 1 0.00377 0.00377 99.62%

1371 1371 0 0.00000 0.00000 99.62%

1428 1428 0 0.00000 0.00000 99.62%

1484 1484 0 0.00000 0.00000 99.62%

1541 1541 0 0.00000 0.00000 99.62%

1597 1597 0 0.00000 0.00000 99.62%

1654 1654 0 0.00000 0.00000 99.62%

1710 1710 0 0.00000 0.00000 99.62%

1767 1767 0 0.00000 0.00000 99.62%

(1767.00 x < 1823.00) 1823 1823 0 0.00000 0.00000 99.62%

(1823.00 x < 1880.00) 1880 1880 0 0.00000 0.00000 99.62%

More 1 0.00377 0.00377 100.00%

Total 265 1.00000



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Histograms for price 05 and price 06 are as follows:

PRICE 05


PRICE 06



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PART D

Suitability of mean, median mode for the price 05 and 06 have been calculated in excel

attached sheet.

PRICE 05 PRICE 06Mean 416 441Median 356 376Mode 220 260(Excel reference: Question 2 sheet 2)

If we look at the mean, median and mode then we would most likely consider the median for

both the prices to obtain the suitability as for a large number of data mean is usually not suitable

whereas median is more suitable for a large data.

Mode on the other hand in our data set is Bimodal which says that the set of data underconsideration has two mode values.

Thus in our data set to determine the suitability of the mean, median and mode for price 06 we

can say that the median is more likely to be considered for price 06 rather than mode or mean.

PART E



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Calculated variance and skewness for both price 05 and price 06 is as follows:

PRICE 05

Variance 43287.743

Standard Deviation 208.057

Skewness 2.698

PRICE 06

Variance 53334.839

Standard Deviation 230.943

Skewness 2.616


The calculated data depicts that the variance which is the measure of volatility is higher for price

06 as compared to price 05, thus these price 06 could be more sensitive to changes than price

05.

ANALYSIS

Moreover, if we look at the currency tables provided above then we can see that the price 05

frequency is more skewed as compared to price 06. Here we have to compare the scenes withthe central location thus we can say that as the mean for both the prices is positive, the mean is

greater than median.


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PART F

The variance for price 06 is greater than price 05 and we have also calculated the standard

deviation mentioned in the above table to see the volatility of price changes and we can analyse

that the variations in price 06 is more than the price 05 and thus the prices in 06 is more

deviating from the mean as compared to price 05.


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QUESTION NO: 3

PART A

The formula that we have used to calculate the continuous compounded daily return for the pre

and post crises is as follows:

r = ln(FV/PV)

The calculated returns are in the attached excel sheet.

(Excel reference: Question 3 "pre-crisis" sheet 3)

We have obtained the mean and standard deviation for both pre and post crisis which is as

follows:

PART B:

Pre-Crisis:

Mean = -0.147

Standard deviation is calculated at 1%, 2% and 3% respectively from the mean and it comes out

to be as follows:

mean-1sd -1.68 mean-2sd -3.21 mean-3sd -4.74

mean+1sd 1.38 mean+2sd 2.91 mean+3sd 4.44

% of obs 74% % of obs 95% % of obs 99%



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Post-Crisis:

Mean = -0.101

Again for post-crisis the standard deviation is calculated at 1%, 2% and 3% respectively, from

the mean and it comes out to be as follows:

mean-1sd 0.20 mean-2sd -1.40 mean-3sd -3.01

mean+1sd 3.40 mean+2sd 5.00 mean+3sd 6.60

% of obs 77% % of obs 95% % of obs 99%

(Excel reference: Question 3 "post-crisis" sheet 3)

PART C:

To determine whether the distribution of pre and post crisis has thicker tail than a normal

distribution or not we have calculated the Excess Kurtosis which is as follows: Excel sheet

reference (?????

Pre-Crisis

Excess Kurtosis 2.290


Post-Crisis

Excess Kurtosis 16.187


CONCLUSION:

For a normal distribution the excess kurtosis is zero where as in both pre and post crisis theexcess kurtosis is greater than zero. Thus it can be depicted that the Tail for both pre and post

crisis is Thicker than a normal distribution.


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PART D:

The probability of the negative returns in pre and post crisis has been calculated in the attached

excel sheet. (Excel reference: Question 3 "pre-crisis", post-crisis" sheet 3)

Pre-Crisis

Probabi li ty of Negative Return 0.544785


Post-Crisis

Probabi lit y of Negative Return 0.568819


Analysis

From the probability of Negative returns obtained from both pre an post crisis it can be said thatalthough the post crisis should have been worse than pre crisis as post crisis has better

probability of getting good returns than pre crisis.

PART E:

Here we would consider different calculated data to state whether the daily returns for pre and

post crisis is different and which is better as far as investment is concerned:

DATA Pre-Crisis Post-CrisisProbability of Negative Returns 0.544785 0.568819Excess Kurtosis 2.290 16.187Skewness -0.214 1.870(Excel reference: Question 3 "pre-crisis", post-crisis" sheet 3)


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Analysis

From the above data it is evident that for both the pre and post crisis the Excess Kurtosis is

greater than zero so both have thicker tails than normal distribution and thus more peaked.

However, post-crisis is far more Excess Kurtosis than pre-crisis which clearly depicts that post-

crisis has more thicker and thus peaked tail than pre-crisis.

Moreover, looking at the skewness of pre and post crisis it is seen that skewness for pre-crisis is

negative which depicts that pre-crisis would not be beneficial for the investors as they would

incur low returns

Post-Crisis, on the other hand , has positive skewness which depicts that the investors would

definitely be benefited from the returns.


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QUESTION NO. 4

PART A:

OVERVIEW OF S&P 500 INDEX

The S&P Index has been working since 1957 in USA. It includes large publicly listed companies

that invest hundreds of billions of Dollars in USAs two largest stock markets, the New York

Stock Market and NASDAQ. Moreover, it is known as the best symbol of USA equities market

with more than US$1.53 trillion in indexed assets and covers approximately 75% of USA equity

market(Standard and Poors, 2009).

S&P 500 is a very important element of USA economy that provides cost efficient investment

portfolios. In addition, it is also a component of S&P global 1200.

It is managed and maintained by the S&P index Committee, includes index analysts as well as

the members of Standard and Poors economists, a division of McGraw-Hill. They follow the

existed laws and police to maintain the index. The Committee is only responsible to select

companies in the S&P 500 and it do not allow those who are not trading publicly or do not have

sufficient liquidity.

S&P 500 represents the market therefore those companies, are listed under and considered as

the leading companies in particular industry.

OVERVIEW OF GENERAL ELECTRIC (G.E)

General Electric (G.E.) is a multinational company with diversified technology, media and

financial services. It produces consumer appliances as well as financial services for their

customers. In addition, it is the market leader in all its sectors, technological infrastructure,

energy infrastructure, GE capital, NBC Universal and also one of the largest financial services

providers in the world. Almost 25% of worlds total electricity is produce by GE power generation

equipment and technology.

In 2008, GE was the 12th largest company in the world with total revenue of $183 billion, has

grown 13% as compare with 2003 (GE Annual Report, 2007) and 2nd largest by market

capitalization. In other words it is a symbol of American business towards the world; they

believe in to build strong leaders for its strategically implications. Moreover, GE is having the

worlds most efficient business people in its management team from last 125 years that

contributing to make it World Class Company as it is today.


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GE diversification gives strength to the company against any poor performance in its any

business segment. It has approximately 36,000 technologists from the glob that working for the

company to provide some superior solution for worlds toughest problems (General Electric,

2009).

GE has affected by these financial crises, its earning has been decreased by 12% to 4.361billion since 2008 from it continuous operating.

PART B

To calculate the required Time series graphs for both S&P 500 index and General Electric Co.

(G.E) we have calculated the Returns and Return minus mean for both the indices to furthercalculate the average and standard deviation of the indices in the attached excel sheet.


Time Series Graphs constructed are as follows:


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S&P 500 Index and its Returns:


General Electric (G.E) and its Return:



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PART C:

To test whether the average return for both the indices is relatively different from thesignificance level of 1% and 5% we will conduct a z-test.

As the sample here is too large (n=254) and the variance is unknown so for the z-test we would

use sample standard deviation (s) instead of known population standard deviation ().

H0: = 0 versus HA: 0

Note that the rejection point Condition for the two-tailed test is as follows.

At 5% Level:

if z >1.96 Reject.

if z 2.57 Reject.

If z < -2.57 Reject.

The formula for the z-test is given as follows:

(Excel reference: Question 4 sheet 4)ns

Xz

/

0=


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ANALYSIS

S&P 500 Index

Z-Test 1.275778306


General Electric Index:

Z-Test 1.6469938514


As from the z-test it can be seen that for S&P 500 index z=1.275778306 which satisfies the

above condition and does fall within the range specified. Thus we would accept the null

hypothesis at the 5% level and also at the 1% level of significance.

Likewise, for our company G.E index z=1.6569938514, which is also within the range specified

and thus the null hypothesis will be accepted at both 5% and 1% levels of significance.

PART D:

Too find the correlation between the two indices, the two different methods used here are

correlation matrix and the covariance.

Method 1

First, we have used the correlation matrix from data analysis tool in excel to find the correlation

between S&P 500 index and G.E index.

S&P 500 Index

Return G.E Return

S&P 500 Index Return 1

G.E Return 0.789580554 1


Thus the correlation matrix gives out the value as follows:

Correlation 0.789580554


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Now we will use covariance to find the correlation between S&P 500 index and G.E index by

using the covariance formula provided in the excel sheet.

Method 2

Covariance 0.000724636


ANALYSIS

As the correlation coefficient is greater than zero hence it shows a positive linear relation

between the two indices which means that if the A&P 500 index increases the G.E index would

also increase and vice versa.

PART E

From the results in part D, we have the correlation between S&P 500 index and G.E index

which is 0.789580554 thus to test whether we would accept the null hypothesis at 10% and 5%

significance level we will use the "t-test".

Here again the variance is unknown and the sample is too large (n=254). Note the conditions

that:

Critical value at 5% is 1.96

Critical value at 10% is 1.645

The formula given for t-test is as follows:

t= 20.42575657(Excel reference: Question 4 sheet 4)

21

2

r

nrt

=


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Analysis

The result from the t-test (t=20.42575657) depicts that the critical value at 5% significance level

is either greater than 1.96 or less than -1.96, thus at this level the null hypothesis will be

rejected.

Likewise, the critical value at 10% significance level is either greater than 1.645 or less than -

1.645 so the null hypothesis is rejected at this level as well.

Hence at both the levels there is no correlation between both S&P 500 index and G.E index and

null hypothesis is rejected.


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References

1. GE Annual Report, 2007, Invest and Delivery Every Day, retrieved 5th May 2009,

2. General Electrics, 2009, About Us, retrieve 6th May 2009,

3. Rosenthal, M, 2009, How to calculate mortgage payments, Foner Books,

retrieved 6th May 2009,

4. Standard and Poors, 2009, S&P 500; Overview, retrieve 6th May 2009,

5. Variable Types, retrieved 6th May 2009,
http://www.ge.com/ar2007/pdf/ge_ar2007_full_book.pdfhttp://www.ge.com/company/index.htmlhttp://www.fonerbooks.com/interest.htmhttp://www2.standardandpoors.com/portal/site/sp/en/us/page.topic/indices_500/2,3,2,2,0,0,0,0,0,2,1,0,0,0,0,0.htmlhttp://www2.standardandpoors.com/portal/site/sp/en/us/page.topic/indices_500/2,3,2,2,0,0,0,0,0,2,1,0,0,0,0,0.htmlhttp://www.oswego.edu/~srp/stats/variable_types.htmhttp://www.oswego.edu/~srp/stats/variable_types.htmhttp://www2.standardandpoors.com/portal/site/sp/en/us/page.topic/indices_500/2,3,2,2,0,0,0,0,0,2,1,0,0,0,0,0.htmlhttp://www2.standardandpoors.com/portal/site/sp/en/us/page.topic/indices_500/2,3,2,2,0,0,0,0,0,2,1,0,0,0,0,0.htmlhttp://www2.standardandpoors.com/portal/site/sp/en/us/page.topic/indices_500/2,3,2,2,0,0,0,0,0,2,1,0,0,0,0,0.htmlhttp://www.fonerbooks.com/interest.htmhttp://www.ge.com/company/index.htmlhttp://www.ge.com/ar2007/pdf/ge_ar2007_full_book.pdf

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