One of the most basic procedures is that of frequency analysis.  In fact without this technique you won’t get very far in understanding any of the procedures behind substitution cryptanalysis. So let’s try and briefly explain what’s behind this technique in an effort to expand our mathematical education.

Cryptanalysis relies on the fact that all letters of a specific language have specific characteristics or personalilties.  To the ordinary observer all the letters might look fairly similar, but to the analyst they will know the specific traits and characters of each individual letter.  

GJXV EHRT NUCOT WMOHY UWGK HWIE

GJXV EHRT NUCOT WMOHY UWGK HWIE

GJXV EHRT NUCOT WMOHY UWGK HWIE

GJXV EHRT NUCOT WMOHY UWGK HWIE

GJXV EHRT NUCOT WMOHY UWGK HWIE

GJXV EHRT NUCOT WMOHY UWGK HWIE

GJXV EHRT NUCOT WMOHY UWGK HWIE

GJXV EHRT NUCOT WMOHY UWGK HWIE

For others this is the attraction of mathematics – it is a world that has its own immutable laws and its own internally coherent logic. Math seems neater and more precise than the real world.

However, this characterization as math being distinct, separate and autonomous from nature is a very much mistaken one. Math obviously originated as a means of dealing with amounts. This is the common way to present math problems to kids – you have 10 apples and 5 people  – how many apples can each person eat? From this example it can be seen that math allows us to symbolize amounts of actual things in numbers and make calculations. The next step is to substitute unknown amounts for letters (algebra) and to use math equations to work out the unknown quantity.

Math is fundamental to the world we live in. Math is the foundation behind scientific equations. These equations allow engineers to turn science into technology that can benefit our lives. The same is true for medicine. We wouldn’t have any of the life saving drugs we do have without maths and chemistry.

Man’s greatest achievement has been to make models of the world and through understanding these models to manipulate reality. These models whether computer simulations, scientific theories or chemical equations are based on maths and cannot exist without math. It is not wrong to describe math as abstract, but it is very wrong to consider math as separate and irrelevant to reality. Indeed for anyone living in a city what they see out of their window in the product of math and applied math.

Further interest:

Engineering, Science and Math

To start, think about your favorite bottle of wine.  Or more simply, think about any of the 90 point wine clubs which exist only to find those exact bottles.

The real question is how winemakers are able to craft wine which is of that high quality. There is, without a doubt more science involved than you might think.  Most of us believe that a winemaker walks through a vineyard throughout October and tastes grapes.  When the grapes taste just right they are picked and then left alone to ferment and create the wine we drink on a nightly basis.

While that certainly happens in some places-at most wineries these days winemaking is more science than art.  Wineries employ full time chemists (they call them enologists, but they’re nothing more than chemists) who test the sugar levels of grapes before the winemaker becomes involved at all.

I think we all know that math is incredibly important, but did you know how much math really went into that bottle of wine you’re enjoying with dinner?

Leonardo da Vinci was not only an outstanding artist he was also a draftsman, an inventor and a mathematician . He looked for the mathematical formulas that underlay beauty. It is da Vinci that bought the idea of the golden ratio to many people’s attention. It is found throughout nature and is one of the most fundamental principles of design. The golden ratio seems inherently ‘right’. It can be expressed in math as:

From this formula you can get a rectangle that can inform the design of tables, whiteboards and also the spacing of sections on a website.

Another mathematical aspect to design can be found in the Fibonacci sequence of numbers:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 etc.

This sequence is derided from adding the last two numbers together to get the next number in the sequence. Thus 1 +2 =3. 2+3 =5. 3+5=8 and so on. It is interesting to note that the bigger the numbers get the closer their relationship conforms to the golden ratio.

The Fibonacci sequence is particularly useful for graphic design. It gives us the correct proportions between font sizes and also the relative values for column sizes. For website design it is good to start with a content width of 987 pixels to make it easy to calculate the relative sizes of columns.

Finally, modern design has been greatly influenced by fractals. These are mathematically governed images that appear to magnify at each iteration. It is like the zoom on a camera; only with fractals the image quality never becomes blurred. Designers often have fractals moving and /or  rotating with each iteration to produce striking effects.

It is clear from these 3 examples that much of the best design is mathematically based. The more we can use math in design the easier it will be on the eye. For website design maths can be a powerful to grab the user’s attention.

A motif in music is introduced and then it is repeated and altered. The setting up of expectation of repetition in music and then fulfilling or dashing that expectation explains the dynamic of music. It is a series of patterns essentially that is behind the aesthetics of all music from chamber music to death metal.

This sets up an interesting question: is it possible to write music that is totally absent of pattern? This is different to random music – rather music that has patterns but no pattern is repeated. To do this math is needed.

John Costas was hired by the US Navy to solve the problem with the sonar ping. For the ping to work a sound with patterns where no pattern was repeated was needed. He solved the problem by using the Prime Number Theory developed by Evariste Galois. Each component of the ping was mathematically related to the next but no pattern was repeated.

The speaker in the talk (Scott Rickard) used the work of Costas that borrowed from Evariste Galois to write a piece of music without repetition. It should be the ugliest piece of music ever written as it is devoid of repeating patterns.

In a sense this music realizes the dream of the composer Schoenberg in his movement called ‘the emancipation of the dissonance’.

After listening to the talk and musical piece on TED it is worth reading the comments
(http://www.ted.com/talks/lang/en/scott_rickard_the_beautiful_math_behind_the_ugliest_music.html) as it details some of the key points of what is pattern? How does the mind perceive pattern? How is math converted to music?

Risk Management Course: Answers to the Group Assignment

Question 1

a. Graph 1.1 shows expected cash flow to Biopur in six months if the firm remains in a long position.

Graph 1.1

b. Graph 1.2 shows the gain or loss Biopur makes in six months if the forward contract matures.

Graph 1.2

c. Graph 1.3 shows the hedged cash flow of Biopur as it enters the short forward position, against the unhedged cash flow and the gains or losses at the different exchange rates.

 Graph 1.3

d. Graph 1.4 shows the payoffs at maturity against the exchange rates if Biopur goes long on a US$ put option.

Graph 1.4

e. Graph 1.5 shows more specific what in graph 1.4 already is represented if Biopur adopts this position. In this graph the put option premium is also taken into account.

Graph 1.5

f.

Arguments for strategy represented by b:

• Eliminate all its risk without spending any cash at initiation of the contract

• It will eliminate almost all possible risk

 Arguments against strategy represented by b:

• Counterparty default risk exists

• When the spot price (in 6 months time) increases, the firm will encounter a forward loss, because the forward contract has eliminated the upside potential

Arguments for strategy represented by d:

• unlimited gains against a small a premium

Arguments against strategy represented by d:

• Small loss due to the premium paid which cannot be spend or invested otherwise

Question 2

 Average annualized portfolio return for bad credit loans: 0.35*6.704%+0,4*8.072%+0,20*-3.594% + 0,05* 4%= 5,0564%

Portfolio volatility: [ Var]^1/2 = [w^T ?]^1/2

w^T =

? =

With Excel function MMULT (array 1, array 2) we get the volatility: 0.2060 or 20,6%

b.

ßEuropa = [Covariance(Europe; US)/ Variance (US)]

=COVAR(B2:B523,D2:D523)/VAR(D2:D523)

= 0.647580322

ßJapan = [Covariance(Japan; US)/ Variance (US)]

= COVAR(C2:C523,D2:D523)/VAR(D2:D523)

= 0.370865963

c.

E(r) = Rf+ß* [(Rm)-Rf]

Rf = 0.04

R(m)= 0.08

ß_Europa = 0.6476

ß_Japan = 0.3709

Table 2.1

d.

1 Month Loss

Portfolio volatility: [ Var]^1/2 = [w^T ?]^1/2

w^T =

? =

With Excel function MMULT (array 1, array 2) we get the monthly volatility: 0.058279 (5,8279 %) or € 5827,90

 Expected Monthly Return_portfolio= 0.4*0.621%+0.35*0.516%+0.2*-0.276%+0.05*4%*0,25= 0,0042357 (0,42357%) or € 423,57.

x = € 95000

E(x) = € 100423,57

With Excel we find with the formula Normdist(95000;100423,57;5827,90;TRUE) that the probability of losing 5000 or more in one month is 0,1760 (17,60%).

 3 Months loss:

3 Months Volatility_portfolio = [Montly Volatility_portfolio (calculated by a) x √3] = 0,1009422 (10,09422%) or € 10094,22.

Expected 3-Months Return_portfolio= [(Expected monthly Return_portfolio ) {calculated by a} x 3 ] = 0,42% x 3 = 0,0127071 (1,27071%) or € 1270,07.

x = € 85000

E(x) = € 101270.07

With Excel we find with the formula Normdist(85000;101270,07;10090;TRUE) that the probability of losing 15000 or more in three months is 0,0535 (5,35%).

e.

Probability that the value of the portfolio in one year will be between 105000 and 70000 = Probability(105000) – Probability(70000)

Probability (105000) = Normdist(105000;105056,39;20597,77;1) = 0,4989 or 49,89%

Probability (70000) = Normdist(70000;105056,39;20597,77;1) = 0,0444 or 4,44%

Probability (70000 < Value_portfolio < 105000 ) = 0.4974 or 45,45%.

f.

Table 2.2

Quantiles Normal distribution		VaR1
1 percent:	2.33	42.936,40
5 percent:	1.65	28.929,92
10 percent	1.28	21.340,71

¹VaR = -[5,0564% - %quantile*20.6%] * Portfolio value

g. 1) 3-monthly VaR before and after the sale

Before: σ =10,09422% => VaR(5%) = -[1,27071% - 1.65* 10,09422%]*100000= € 17926,18

After: σ_portfolio changed because of different weights:

w^T =

? =

Portfolio volatility: [ Var]^1/2 = [w^T ?]^1/2

With Excel function MMULT (array 1, array 2) we get the monthly volatility: 0.0549798 (5,49798 %) or € 5497,98

3 Months Volatility_portfolio = [Monthly Volatility_portfolio x √3] = 0.09522786 (9,522786%) or € 9522,79.

Also changes the expected return:

Expected 3-month return becomes: 0.35*3*0,516% + 0.36*3*0,621% + 0.24*3*0,276% + 0.05*0.25*4% = 1,2621%

VaR (5%) AFTER SALE= -[1,2621% - 1.65* 9,522786%]* 100000 = € 16974,65

IMPACT on VaR = value at risk decreased with 17926,18 – 16974,65 = 951,52 because the expected return of for Japan is lower and the

2)

VaR impact of trade: -(E(Ri)-E(Rj)) * ?w * W + (ßip – ßjp) * 1.65 * Vol (Rp) * ?w * W = -(-3*0,276% – 3*0,621%) * 0.1 *100000 + (0.370865963-0.647580322)*1.65*0,1009422*0,1*100000 =

The probability that a random variable following the standard normal distribution takes a value lower than -1.65 is 5 percent.

Question 3

a. 1 April: forward price on 3 month T-Bills of face value € 1

1 July: delivery

30 September: pays € 0.97

1 April: spot price of bills with same face value maturing on June 30^th = 0.968 and maturing on September 30^th = 0.936

[A_{pril 1}(September 30) – A_{pril 1}(July 1)* F] = 0

F = ₀P₂ / ₀P₁ =A_{pril 1}(September 30) / A_{pril 1}(July 1)

F = 0.936 / 0.968 = 0.9669

At this forward price the replicating portfolio for the long position has the same value as the forward contract and there is no arbitrage opportunity.

2. Yes, there is an arbitrage opportunity at these prices because the replicating portfolio for a long forward position is worth (0.936 – 0.968*0.97)= -0.00296 Taking this long position in the replicating portfolio and a short position in the forward contract will gain a cash flow of 0.00296. Also shown in table 3.1 below.

Table 3.1

Positions on April 1		Cash flow on April 1	Cash flow on September 1
Short forward position		0	0.97 – PJuly1 (September 30)
Replicating Long Forward	Long Bill maturing on September 30	-0.936	PJuly 1(September 30)
	Borrow present value of F	0.968*0.97 = 0.93896	-0.97
Net Cash flow		0.00296	0

Question 4

• 6 month interest rates in the Netherlands and USA; 4.5% and 7% / annum

Current spot price; $ 1 / € 0.9

Futures price contract deliverable in 6 months: $ 1 / € 0.91

a. Arbitrage-free currency forward price:

S_t *e^{-r^FX *i} – Fe^{-r * I} = 0

F = S_t e^{(r –r^FX) * i}

F = 0.9*e ^{(0.07-0.045)*.}

Logbook loans are a product where you put your car down as collateral for a loan, and due to them being so easy to get they are becoming increasingly popular as folks struggle to make ends meet. Logbook lenders prey on borrowers who are desperate for cash, and don’t have time to read the small print on the documents they are signing. Do a little bit of math before you commit to one of these products and you’ll see that they will do nothing but get you into trouble.

The main problem with logbook loans is the interest rate charged by lenders, which can be up to 450%. It might feel good to get your hands on that £1500 when you really need it, but you’ll end up repaying a total of £4100 over the course of the next year and a half.

Before you take out any sort of loan from any lender make sure you do the figures so you know what you are getting into.

The ability to apply math to everyday situations is not something that comes naturally to most people.  Rather, it is necessary to practice a commonsense approach to educate yourself on the necessary topics.  A comprehensive fiscal and mathematical vocabulary is essential for navigation of the complex legal and fiduciary environment that we find ourselves in today.  Without adequate preparation, hard work and dedication can go to waste.  In understanding everything there is to know about financial responsibility, bankruptcy attorneys learn to see all sides of an issue.  They are well-equipped to deal with the harsh realities of fiscal difficulties, yet they understand that at the heart of every distressed business or individual, there are families that are depending on fair disposition of circumstances.

Behind the numbers there are people who need solutions that work so that they can live their lives, not under the looming shadow of impending disaster, but in the light of hope for the future.  By assessing your financial portfolio and making statistically informed decisions, it is possible for even the most seemingly hopeless financial obstacle to be overcome.  Fully understanding the nature of debt requires complex analysis and calculation of debt tables, interest rate formulas, percentages, statistics and much more, along with a complete knowledge of the governmental policies and procedures.  Only bankruptcy attorneys are equipped with the math and other skills necessary to fully assess your financial situation and how to fix it.

When I was last searching for  houses to rent I came across a beautiful property and fell in love right away. It had everything I wanted. A Playroom for the kids, a swimming pool for the wife, and a fire pit for me.  All these thing we first read about in Skymall and, of course, wanted for ourselves. I drew the line at the replica sarcophagus coffee table.

We progress through many stages of meetings, after one month we were adjusting closing costs when my broker informs me,  the current owners were adjusting the price bye $5000 for furniture in the home. I made the argument a two part response. Firstly, I said I do not want your furniture. Secondly, it is inconceivable that your bed and sofa were worth $5000. I countered with the offer of individual pricing per item totaling somewhere around $1500. When they scoffed I had no problem walking away from the deal
Another example is when my brother purchased his very first home. It had to do with bills and receipts of previous tenants’ electricity and gas bills. When my brother moved in, he was billed immediately for a year is negligent payments. It took three weeks of e-mails and threatening letters to reach a resolution that did not cost my brother anything but his time. Very luckily for him. If a mathematics-based system could surplant the often arbitrary system we have today we would be much better off.

Anyone would like to know how much interest he or she has to pay on a mortgage or a loan. You would also want to keep track of the amount of interest to pay over a given period of time if you are borrowing money. The simplest formula to use when calculating interest rate is P x(1 + r/n)^ Yn where P = principal, r = rate, n = number of times per year for interest compounding and Y = the number of years for the loan. Sometimes, it can be challenging to manually calculate the interest on a loan and you can also make mistakes. Instead of calculating manually, you can use a compound interest calculator to calculate your compound interest.

With the compound interest calculator, you can know how much you shall have saved after a certain number of years and you can also know how much of your final balance is due to make more interest for you. The calculator is very easy to use. You just need to enter the needed information in the fields specified. You should enter your original deposit in the principal box, your APY in the Rate box and the length of time in the Years box. After inputting the information as required, you can then hit on calculate to get the results. You can either select to calculate your rate using months or years.

The compound interest calculator can be very important to both bankers and those holding savings accounts. Whether you are a lender or a borrower, you will find this tool very useful when it comes to keeping track of how much money you are earning or how much money you would have to pay on a loan. It is an easy to use tool that does not need any special skills. You can find the software online. You can do all your calculations just sitting in front of your computer.

Financial management is something that some people take for granted and it can be very crucial when it comes to financial stability. You might be losing a lot of money without actually understanding how if you cannot track your savings properly. Compound calculators will make things very easy even for dummies. They can also be very useful when you need to determine a loan program to get if you are looking for a loan. When shopping around, you can use this calculator to compare the interest rates between the different lending companies before selecting which one to get.