Education - Business, Finance & Quantitatives

Mortgage calculations

2006-04-10T09:01:00.000Z

A mortgage is taken out for 80000$. It is to be paid> by annual instalments > of 5000$ with the first payment being made at the> end of the first year that > the morgage was taken out. Interest of 4% is then> charged on any outstanding > debt. Find the total time taken to pay off the> morgage.

Answer: (It may help to followw this solution with a pen and paper)

[This first part is the Long Answer]

Let us denote the mortgage amount taken (principal) as P=80000

Let annual payments (in arrears) be given by C = 5000

Interest rate is given by r= 4% (annual rate) - or 0.04

let n be the number of years.

A step-by-step approach will be taken to address the problem.

At the end of the first year n=1:
Principal amount will have grown to P(1+r) e.g 80000(1+0.04) =83200

Payment made at the end of first year is C or 5000

Thus, amount left to pay is P(1+r)-C (or 83200-5000 = 78200)

we will not need numbers again until we are ready to solve the problem

End of second year n=2:
Interest is added to principal again
Thus new principal is (P(1+r)-C)*(1+r) = P(1+r)^2-C(1+r)

payment of C is again made.
Thus principal remaning is P(1+r)^2-C(1+r)-C

End of third year n=3:
Interest is again added to principal
Thus new principal is {P(1+r)^2-C(1+r)-C}*(1+r)=P(1+r)^3-C(1+r)^2-C(1+r)

payment of C is again made
Thus principal remaning is P(1+r)^3-C(1+r)^2-C(1+r)-C

You may see a pattern emerging - which will be illustrated next

We now extend this to year n

After the end of n years and the 'last' payment of C
The principal remaining is [P(1+r)^n]-[C(1+r)^(n-1)]-[C(1+r)^(n-2)]-...-[C(1+r)^2]-[C(1+r)]-[C]

notice the use of brackets to separate the terms.
Since this is the 'last' payment, the principal remaining must be zero.

Thus
P(1+r)^n = C(1+r)^n-1+...+C (note the terms in the equation above) ***

The terms on the right are as follows:

C, C(1+r),C(1+r)^2,...,C(1+r)^n-1 (note that there are n terms)

This is effectively a GEOMETRIC PROGRESSION (GP) with first term equal to C
and constant ratio equal to 1+r

Remembering that the sum of a GP is a(d^n-1)/(d-1)
where a is the first term, d is the constant ratio and n is the number of terms, this becomes for this series:
For our case, a=C and d=1+r

C([(1+r)^n]-1)/(1+r-1) = (C/r)([(1+r)^n]-1)

going back to equation *** above, we can rewrite it as
P(1+r)^n = (C/r)([(1+r)^n]-1)

By doing some algebra (1+r)^n= C/(C-Pr)

giving r= [C/(C-Pr)]^(1/n) - to be used to find the appropriate rate given everything else or
n= log [C/(C-Pr)]/log(1+r) ****
which is what we need for this problem.

for this example
n = log[5000/(5000-80000*0.04)]/log 1.04
n=26.049 years

[The short answer]

This problem is a constant payment mortgage.

The approach here is to recognise that a constant payment at regular time over a time interval is an annuity.

Then present value (PV) of an annuity of constant payment C starting in the next period is given by:

PV =(C/r)[1- (1/(1+r)^n)] note the brackets

For this mortgage, the present value would be equal to the mortgage amount, P

therefore P = PV above.

Solving for n would give you the same result as equation **** above
(The reader is expected to work the algebra to verify the result.

Basics of Logarithms

2006-01-23T11:01:00.000Z

Logarithms are a very useful tool in mathematics, especially when solving problems involving indices.

For example, how do we solve y^2 = 25. One way is to raise the left side and the right sides of the equation to 1/2, giving y = 25^(1/2) and realising that this is the square root of 25, thus y = +5 or -5. Another way is to use logarithms (log) or natural logarithms (Ln).

First is to attempt to define what a log is. First every logarithm must have a base, and the base techically can be any positive number, but the base of 10 is most common. Definition - The logarithm to base 10 of any number [log(10)] is the index(or power) to which 10 must be raised to give that number. To explain this definition, let us use log(10)10 as an example. Since 10^1 = 10, the log(10)10 (said as log to base 10 of 10) is 1. likewise, log(10)100=2 as 10^2 = 100. Hopefully, the definition makes sense. This definition can be extended to other bases.

A special and important case is the log(e) or Ln, also called natural logarithm which has very common usage in mathematics and Finance. the "e" is the number 2.718281828459.... and goes on like that. It is called the Euler number or Napier number. see :http://en.wikipedia.org/wiki/E_(mathematical_constant) to know a bit more.

Some basic laws:
If Log(a)x=y, then a^y = x - [This was used in defining log(a)]
Log(a)xy = log(a)x + log(a)y - some kind of addition rule
Log(a)[x/y] = log(a)x - log(a)y - some kind of subtraction rule
log(a)a =1
log(a)m = 1/[Log(m)a] - some kind of inverse rule
log(a)m = log(b)m/log(b)a - change of base rule
log(a)[m^y] = y log(a)m

These rules also apply to log(e) or Ln (natural logarithms). They can be combined to prove identities that involve logarithms.

Merger Analysis - Corporate Finance

2006-01-16T18:53:00.000Z

Mergers and Acquisitions are a very important part of the market for corporate control. Mergers, acquisitions and takeovers involve a change in shareholding structure of a company. A merger can be taken as the coming together of two companies, A and B to form a third company AB where the owners (shareholders) of AB are the shareholders of company A and company B in a form of joint ownership, a marriage of sorts. It is normally taken as two companies of relatively similar sizes coming together.

An acquisition can be taken as when shareholders of company A effectively buy-out (by some form of financial security like cash) shareholders of company B (the target) where the owners of the new entity AB are only the shareholders of company A. A takeover is effectively an acquisition, and these could be friendly, if the boards of the two companies A and B are happy to go ahead with the process, or hostile, if the board of company B (the target) are not welcome to the takeover. This would involve the an attempt to takeover the company by bypassing the board of the target company B, and going directly to the shareholders. A well laid out process for this happening exists in different countries according to law.

Why should a merger happen? The key reason here should be that the the new entity, AB, is more valuable than the combined values of A and B on their own. This can be mathematically expresses as:

AB > A + B

This is the sum being greater than the parts argument, or 2+2 = 5 put another way, on in a way I prefer to put it, synergy in the coming together. How can this be? Well, being bigger has advantages (the economies of scale or economies of scope argument), there may be complementary resources between the two firms, there may be redundant operations between the firms or some other interesting argument which can be postulated by the management of merging firms. On the other hand, it could all be "hubris" (hopefully not). Mergers, when complementary resources are present, seem to make most sense. Mergers, in economic terms could be classified as conglomerate, vertical, horizontal. A conglomerate merger is when two firms in unrelated businesses merge, a vertical merger is when firms a different levels of a supply chain merge (such as a supplier merging with firm being supplied) and a horizontal merger when two firms at the same level of the supply chain (i.e in the same line of business)merge.

Regression Analysis

2006-01-04T11:58:00.000Z

Regression analysis enables you to test the strength of the linear association between a dependent variable and one or more independent variable(s).

With one independent variable, you have simple linear regression and with 2 or more independent variables, you have multiple regression.

A simple linear regression may be if a company wants to estimate the effect that advertising has on its sales. This may not fully explain the change in sales, thus the company may want to find out the combined effect of advertising and price changes in a multiple regression.

Effectively, what you are creating is a model that explains the changes in the dependent variable based on changes in the independent variable. It so happens that if the model is correctly specified, it can be used to predict changes in the dependent variable (in the future) based on changes in the independent variables.

A regression model is of the form:

y=b0 + b1*x1 + b2*x2 + ... + bn*xn + e

where bo is the dependent variable and x1 to xn are the independent variables, b0 is called the intercept coefficient and b1-bn are called coefficients of the independent variables x1 to xn. When the equation is:

y=bo + b1*x1 + e

we have a simple linear regression. The 'e' term is an error term that affects the estimation.

In order to estimate these coefficients, a statistical package, such as Microsoft Excel, SPSS, Minitab, Eview etc. is required.

A recommended procedure for checking if a regression model is appropriate for your data is as follows:

(Note: to follow this properly, please request an Excel Spreadsheet titled test data by sending an email with request "test data spreadsheet" in subject line to tuition@baselineeducation.co.uk)

1. Plot the data in a scatter graph (i.e. y vs x). Does it look like there is a linear relationship between the y and x, or can you find a straight line that could go through the whole data and explain the association? I calling this using your eyes to analyse the data. This can also be done for a y, xi pair (for i=1 to n) for a multiple regression. This helps you to know which variables are likely to be important from the statistical analysis. However, a variable that does not look important may still be in combination with other variables in the multiple regression. In the test data spreadsheet, the variable x1 does not look important, but actually explains the rest of the variation in y in the exact data case, and is thus important in the regression. This is so because the x2 values are much larger than the x1 values.

2. Perform a regression analysis using your statistical package of choice. Given that most people have access to Microsoft Excel, this is a very handy package to use.

Microsoft Excel Note: To perform this analysis, you must install the "Analysis Toolpak" add-in. Open up Excel, from the top, select "Tools - Add-ins" and then check the box for "Analysis Toolpak.". I also recommend the "Solver Add-in" as well, but this is not for the statistics analysis.

To do this, the data needs to be in an Excel Spreadsheet. To run the regression with the Analysis Toolpak installed, you select "Tool>Data Analysis>Regression" and select the appropriate cells for the y and x data. The "help" facility of any package you are using may come in handy. Sucessfully doing this will give you a bunch of numbers relating to the regression.

Using the Test Data file, 3 regressions have been done shown in the Reg1, Reg2 and Reg3 sheets for Data Set1, Data Set2 and DataSet3. DataSets are on the Rawdata sheet. Data Set1 is based on the regression equation y=6+15x1+12x2, Data Set2 is based on the same equation but is not exact (i.e. errors have been introduced) while Data Set3 is based on the same equation but with more errors than Data Set2.

3. Check the validity of the overall regression

This is done using the "Significance F" number. For a perfect regression, Significance F should be zero. It is zero for Reg1, and very small for Reg2 and Reg3. Thus these regressions overall are good. The test is for whether all the coefficients of the independent variables, b1 to bn (b1 & b2 in these examples) could be zero. If they could be zero, then the dependent variables do not depend on the independent variables. The Significance F should be compared to a significant level to test the regression. For example if you want to check if the regression is significant at the 0.05 significant level, or 5% (this is called a p-value), then Significance F should be less than 0.05 which means that you have 95% confidence that the regression estimates are significant (i.e. non zero). A significance level of 0.01 requires that significant F is less than 0.01 and that you have 99% confidence in the regression estimates. 100-(Significance F*100) gives you the % confidence in the regression estimates. The p-value can be defined as the smallest level of significance at which a hypothesis can be rejected (definition adapted from Defusco,Mcleavey,Pinto & Runkle, Quantitative Methods for Investment Analysts). More on hypothesis testing later.

4. Interpret the Values of Coefficient of determination (cod) R^2 and the Correlation Coefficient (cc), R.

A perfect correlation will have a cod of 1 and a cc of 1. The closer the number is to 1 the better. The cod refers to the percentage change in the dependent variable explained by the independent variables in the regression model.

5.Interpret the p-values (or t statistic) for the coefficients of the independent variables and the slope.

The objective of this section is to check whether the coefficients of the slope and the independent variables are statistically significant. To do this, the p-value for each one can be used. Rememeber that we are testing whether the coefficients are equal to zero (termed the null hypothesis). A p-value less than 0.05 (at the 5% significant level) means that at the 95% confidence level, we reject the hypothesis that the coefficients are zero. Thus the p-value is an indication of how strongly we reject the null hypothesis we have, in this case that the coefficients are zero. We defined the p-value earlier as "the smallest level of significance at which a hypothesis can be rejected." Another more elaborate way to achieve the same result is using t-tests. For convenience, with an adequate sample size (number of data greater than 30), if the t-statistic is greater than 2, then you can reject the hypothesis that the coefficients are zero (ie. thay are non-zero) and conclude that the coefficients are statistically significant. Finally, another way to conclude this, output by statistical packages is the confidence intervals (CI) of the coefficients. If it includes zero at the significant level of the test, then it is not statistically significant. For example, of a coefficient has the value 0.5, but 95% CI of -0.5 or +1.5 which includes zero, then you can conclude that it could be zeor and the hypothesis that it is zero is not rejected. These three approaches, the p-value method, the t-test method, and the Confidence interval method should result in the same conclusion, with the p-value method and the t test methods recommended.

6.Run a final regression based on your interpretations and check again.

Once you have decided which coefficients are statistically significant and which are not, then you should run the regression again with the significant coefficients. This is the model that should be used for further work, such as forecasting or predicting outputs based on inputs.

This section will be updated over time, and comments are welcome. Please address any questions to tuition@baselineeducation.co.uk with subject line: hypothesis blog.

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