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The first lecture was basically to introduce the topic and to use some simple examples to make it easier to understand how Numerical methods can be used in real situations.
Mathematical models are essential in solving many Engineering problems and these models can lead to the need for mathematical procedures that include differentiation, non-linear equations, integration, simultaneous linear equations and differential equations etc. These mathematical procedures can be used to solve many kinds of problems exactly if you understand how to apply them however numerical methods are more likely to be used to solve them approximately. 
DCF is a method that is used for valuing a project, company or asset using the idea of the time value of money. The future cash flows are basically estimated and then they are discounted in order to find the present values. The DCF method is usually used in investment, real estate development and many other things. 
To calculate the discounted cash flow the following formula is used:
DPV is the discounted present value of the future cash flow
FV is the nominal value of a cash flow amount in a future period
i is the interest rate
d is the discount rate
n is the number of years before the cash flow occurs
The example used to explain how to use the formula in the lecture was:
John Doe buys a house for $100,000, three years later he expects to be able to sell the house for $150,000.
The interest rate needed in order for him to achieve $150,000 after three years would be:
So it can be seen that the interest rate value that he needs in order to achieve the property value that he wants in three years is 14.47%.
But since three years has passed since the purchase and the sale of the house it means that the cash flow from the sale has to also be discounted. So if when he bought the house the 3 year US Treasury Note rate is 5% per annum the present value would actually be:
There are some risks involved with doing this like we are assuming that the $150,000 is John's best estimate but as it's about house prices nothing can be guaranteed and so the outcome could be different than what is expected.
When DCF analysis is usually done is the net present value after discounting the future cash flows is positive. If it is negative it would actually mean money is being lost even though there would appear to be some nominal profit being made.
There are usually four DCF methods that are used today:
- Equity- Approach
- Flows to equity approach
- Entity- Approach
- Adjusted present value
- Weighted average cost of capital
- Total cash flow
Time is money
Interest: "The cost of borrowing money, usually expressed as a percentage per year."
Interest can be applied to situations like if a person takes a loan from a bank, or if someone opens a savings account the bank will pay interest. However due to the economy at the moment it may not have a major effect on the amount as the interest rate is 0.5%. This is the lowest it has been in a long time.
When money is borrowed from a bank lender interest has to be paid which is basically a fee for borrowing the money. And so by the end of one year it will be:
P is the principle; the money borrowed
r is the interest rate
For n years it would be:
This is similar to the Binomial expansion; in mathematics, the binomial theorem is an important formula giving the expansion of powers of sums.
So using the binomial series the expansion of the series above is
For example if P=100, r=8%, and n=2 years by inserting the numbers into both formulas we should receive the same value.
This is the formula for calculating compound interest and it can be applied to a situation of when you deposit money in a savings account as you are lending the bank money and so interest will have to be paid to you.
So it can be said that money is time dependant.
The usual assumption is that interest is paid once a year but interest can be paid more frequently. It can make a big difference on the outcome:
- If a £1000 is invested at 8% after one year:
- But for quarterly compounding:
- And if it's done monthly:
For the £1000 value used there really isn't much difference between the yearly and weekly compounding only a few pounds. However if the original value was bigger if it was in the hundred thousands or millions then whether the compounding is done weekly or yearly would make a huge difference. 
It can be seen that if the compounding frequency is less frequent than the payment frequency then it has little or no practical value. As the payment has already been made and the balance is reduced and there is nothing to compound. That's why the graph below is asymptotic and can be seen to start levelling off at 1083.287. 
Plotting the values from the frequent compounding formula in MathCAD the results can be seen below.
Also if the compounding is more frequent than the payment then so does the effective rate but more slowly.
As k tends towards infinity the effective rate will tend toward zero.
[Reference for the paper as a whole] 
Next Lecture: Focus on Binomial expansion and research gamma function.
- A. Kaw. Introduction to Numerical Methods. Available: http://numericalmethods.eng.usf.edu
- Wikipedia. Discounted Cash Flow. Available: http://en.wikipedia.org/wiki/Discounted_cash_flow
- I. dictionary. Interest. Available: http://www.investordictionary.com/definition/interest.aspx
- Wikipedia. Binomial Theoram. Available: http://en.wikipedia.org/wiki/Binomial_theorem#cite_ref-0
- D. University. The Maths Forum. Available: http://mathforum.org/dr.math/
- J. H. Webb. Have we caught your interest? Available: http://18.104.22.168:8080/LectureNotes/Interest.pdf