Making a case for The BASEL III The WritePass Journal

Making a case for The BASEL III Making a case for The BASEL III In the face of the impending global financial crisis, world leaders at G20 called for reform from legislators across the globe. From the reports assessing the causes of the crisis and proposing regulatory reform, a general consensus has emerged on two elements. First, all systemically important institutions, instruments and markets should be regulated, preferably under the umbrella of a consolidated supervisor in each jurisdiction. Second, reform will be implemented at a national, not at an international level. A new set of banking regulations, published in response to the financial crisis, to ensure banks can cope with a similar meltdown. The BASEL III was published in 2010 in response to financial crisis. BASEL III The aim of BASEL III was to ensure that banks, in future, would be able to withstand the sort of financial meltdown they have just endured. Two points are discussed in the answer and details are as follows: Capital: The new rules state that the minimum amount of capital that a bank must hold in reserve will rise more than threefold from 2% to 7%. Banks have eight years and, in some cases, 13 to comply. New rules aimed at reducing risks associated with counterparty transactions and introducing liquidity stress tests have also been drafted. Supporters insist a good balance has been struck between improving the Basel 2 framework which ignored liquidity and maintaining enough lending capacity to fuel a global economic recovery. This rule did not get tough on banks, in fact , banks like J.P Morgan that were best capitalised gained most and the share price of many banks rose in response. Lending:   Property experts had reckoned the new rules would make it even harder for banks to lend to property companies and investors banks would have less money to lend and their cost of capital would rise. However these rules were not so significant in the sense that most large, internationally active banks already meet the requirements Lloyds boasts a 9% capital ratio and Barclays 13%. However Banks attitude to property lending will be more affected by their terrible experience in the wake of the credit crunch. Stress tests will identify other less obvious connections between counterparties, for example where a bank has made loans to a number of unconnected counterparties who are all affected by the same underlying business risks. Stress tests were mainly carried out in USA. UK Regulations There have been substantial changes in the way the UK financial services industry is regulated, with potentially significant consequences for its consumers. The first change is a change in the Financial Service Authority(FSA)’s philosophy and its approach to supervision. The second, and more significant change, is the new UK coalition government’s plan to put in place a new financial regulation regime, that will see the FSA phased out in 2012, and its functions carried out by the Bank of England and a number of new regulators. Change in FSA’s philosophy and supervision The review of Northern rock shows that the supervision of risk regulations is flawed and   the supervision of risks in financial services remains the key. The ‘Light Touch’ regulations which was considered the key contributor to the financial crisis(Sants, 2010) is being replaced by intensive supervision. FSA is adopting a new approach to create the environment where risks are lessened and at the same time innovation and an increase in competition The idea of treating customers fairly (TCF) remains firmly on the regulator’s agenda for the supervision of firms. The regulator accepts that the TCF initiative has not delivered the outcomes that consumers deserve, largely as a result of its implementation in a non-neutral, reactive manner (Sants, 2009, 2010). It has consulted on ways to improve professionalism within the industry (FSA, 2009) and the way firms handle consumer complaints (FSA, 2010). It has also re-focused the TCF initiative towards making the retail market work better for consumers. The new financial regulatory regime to replace the FSA The new coalition government in the UK has decided to get rid of the FSA and to split its responsibilities between the Bank of England and a new financial services consumer protection agency (currently being referred to as the Consumer Protection and Markets Authority (CPMA)). In this new regime the Bank of England will be responsible for the overall financial stability of the UK financial system, in addition to its already existing responsibility for monetary policy. Europe Regulations The European Commission has been among the most active in proposing reforms after the global financial crisis. The Commissions proposals include new regulatory bodies at the European level, changes to the way financial institutions are regulated in the EU, and changes to the regulation of certain financial products. The Commission and the European Council have called for an enhanced European financial supervisory framework, which will be composed of two new bodies: the European Systemic Risk Council (ESRC) and the European System of Financial Supervisors (ESFS). The ESRC will be responsible for macro-prudential oversight; specifically monitoring and assessing potential threats to financial stability that arise from macro-economic developments and from developments within the financial system as a whole. It will not have any regulatory authority over financial institutions or markets. The ESFS will consist of a network of national financial supervisors working in tandem with three new European supervisory authorities: the European Banking Authority : which monitors banks, the European Insurance and Occupational Pensions Authority: which look after insurance and pension and the European Securities Authority: which look after market. These three new bodies will replace the existing Committees of Supervisors, known as Level 3 committees that advise the Commission. These proposals are designed to create a framework within which financial risk at the EU level will be supervised, and through which the actions of national supervisors may be coordinated. US Regulations The position in the US is complicated because of the division of responsibilities amongst various agencies. Recent US proposals would coordinate these agencies through the creation of the Financial Services Oversight Council, which would be composed of representatives from multiple agencies and chaired by the US Treasury Department. This body will also have a more formal role in the regulatory process. This approach is broadly similar to the position in the EU, in that a supervisor, comprising representatives from various bodies in the sector, will monitor risk on a macro-basis and act accordingly through those bodies. The interaction between the ESRC: and its US equivalent will be key to the success of both initiatives. How this will occur remains to be seen. Conclusion: BASEL III: Banks will have to hold a greater amount of high quality capital, which should make failure less likely and deposits more secure. Higher quality capital is more expensive, which might increase the interest rates which banks charge their borrowers. The higher quality capital should also make bank bail-outs less likely and therefore protect taxpayer funds. Regulatory capital adequacy only ensures that banks have enough capital to meet their obligations over the entire life of their business. It does not address whether they have enough readily available funds to be able to meet their obligations as they fall due. The liquidity coverage ratio is intended to ensure that the bank can pay its obligations falling due over the next 30 days. Regulations: UK, EU and US are moving on a variety of fronts to improve their financial regulatory systems in response to the financial crisis. The regulatory bodies have been active proposing major reforms and accelerating the implementation of measures that were already under way. There is still considerable uncertainty regarding the final shape of some of these initiatives. Some initiatives seem likely to create tension with member state governments.   On both sides of the Atlantic Ocean, the key challenge will be to ensure that steps towards reform reflect a global consensus reached in the G20 and to ensure that the move to tighten regulation of the financial system does not create unintended barriers for global financial institutions. References Basel 3 quick fix is neither Critics find little assurance that proposed standards will prevent banking failures. By Thomas Watson Financial services and consumer protection after the crisis by Folarin Akinbami Durham University, Durham, UK Daily Mail, 9th March 2011 Perspectives on Basel III. International Financial Law Review, 02626969, Nov2010, Vol. 29, Issue 9

The 36 Trig Identities You Need to Know

The 36 Trig Identities You Need to Know SAT / ACT Prep Online Guides and Tips If you’re taking a geometry or trigonometry class, one of the topics you’ll study are trigonometric identities. There are numerous trig identities, some of which are key for you to know, and others that you’ll use rarely or never. This guide explains the trig identities you should have memorized as well as others you should be aware of. We also explain what trig identities are and how you can verify trig identities. In math, an "identity" is an equation that is always true, every single time. Trig identities are trigonometry equations that are always true, and they’re often used to solve trigonometry and geometry problems and understand various mathematical properties. Knowing key trig identities helps you remember and understand important mathematical principles and solve numerous math problems. The 25Most Important Trig Identities Below are six categories of trig identities that you’ll be seeing often. Each of these is a key trig identity and should be memorized. It seems like a lot at first, but once you start studying them you’ll see that many follow patterns that make them easier to remember. Basic Identities These identities define the six trig functions. $$sin(ÃŽ ¸) = 1/{csc(ÃŽ ¸)}$$ $$cos(ÃŽ ¸) = 1/{sec(ÃŽ ¸)}$$ $$tan(ÃŽ ¸) = 1/{cot(ÃŽ ¸)} = {sin(ÃŽ ¸)}/{cos(ÃŽ ¸)}$$ $$csc(ÃŽ ¸) = 1/{sin(ÃŽ ¸)}$$ $$sec(ÃŽ ¸) = 1/{cos(ÃŽ ¸)}$$ $$cot(ÃŽ ¸) = 1/{tan(ÃŽ ¸)} = {cos(ÃŽ ¸)}/{sin(ÃŽ ¸)}$$ Pythagorean Identities These identities are the trigonometric proof of the Pythagorean theorem (that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides, or $a^2 + b^2 = c^2$). The first equation below is the most important one to know, and you’ll see it often when using trig identities. $$sin^2(ÃŽ ¸) + cos^2(ÃŽ ¸) = 1$$ $$tan^2(ÃŽ ¸) + 1 = sec^2(ÃŽ ¸)$$ $$1 + cot^2(ÃŽ ¸) = csc^2(ÃŽ ¸)$$ Co-function Identities Each of the trig functions equals its co-function evaluated at the complementary angle. $$sin(ÃŽ ¸) = cos({Ï€/2} - ÃŽ ¸)$$ $$cos(ÃŽ ¸) = sin({Ï€/2} - ÃŽ ¸)$$ $$tan(ÃŽ ¸) = cot({Ï€/2} - ÃŽ ¸)$$ $$cot(ÃŽ ¸) = tan({Ï€/2} - ÃŽ ¸)$$ $$csc(ÃŽ ¸) = sec({Ï€/2} - ÃŽ ¸)$$ $$sec(ÃŽ ¸) = csc({Ï€/2} - ÃŽ ¸)$$ Negative Angle Identities Sine, tangent, cotangent, and cosecant are odd functions (symmetric about the origin). Cosine and secant are even functions (symmetric about the y-axis). $$sin(-ÃŽ ¸) = -sin(ÃŽ ¸)$$ $$cos(-ÃŽ ¸) = cos(ÃŽ ¸)$$ $$tan(-ÃŽ ¸) = -tan(ÃŽ ¸)$$ Sum and Difference Identities These are sometimes known as Ptolemy’s Identities as he’s the one who first proved them. $$sin(ÃŽ ± + ÃŽ ²) = sin(ÃŽ ±)cos(ÃŽ ²) + cos(ÃŽ ±)sin(ÃŽ ²)$$ $$sin(ÃŽ ± – ÃŽ ²) = sin(ÃŽ ±)cos(ÃŽ ²) – cos(ÃŽ ±)sin(ÃŽ ²)$$ $$cos(ÃŽ ± + ÃŽ ²) = cos(ÃŽ ±)cos(ÃŽ ²) – sin(ÃŽ ±)sin(ÃŽ ²)$$ $$cos(ÃŽ ± – ÃŽ ²) = cos(ÃŽ ±)cos(ÃŽ ²) + sin(ÃŽ ±)sin(ÃŽ ²)$$ Double-Angle Identities You only need to memorize one of the double-angle identities for cosine. The other two can be derived from the Pythagorean theorem by using the identity $sin^2(ÃŽ ¸) + cos^2(ÃŽ ¸) = 1$ to convert one cosine identity to the others. $$sin(2ÃŽ ¸) = 2 sin(ÃŽ ¸) cos(ÃŽ ¸)$$ $$cos(2ÃŽ ¸) = cos^2(ÃŽ ¸) – sin^2(ÃŽ ¸) = 1 – 2 sin^2(ÃŽ ¸) = 2 cos^2(ÃŽ ¸) – 1$$ $$tan(2ÃŽ ¸)={2 tan(ÃŽ ¸)}/{1– tan^2(ÃŽ ¸)}$$ Additional Trig Identities These three categories of trig identities are used less often. You should look through them to make sure you understand them, but they typically don’t need to be memorized. Half-Angle Identities These are inversions of the double-angle identities. $$sin2(ÃŽ ¸) = {1/2}(1-cos (2ÃŽ ¸))$$ $$cos2(ÃŽ ¸) = {1/2}(1+ cos (2ÃŽ ¸))$$ $$tan2(ÃŽ ¸) = {1-cos(2ÃŽ ¸)}/{1+ cos (2ÃŽ ¸)}$$ Sum Identities These trig identities make it possible for you to change a sum or difference of sines or cosines into a product of sines and cosines. $$sin(ÃŽ ±) + sin(ÃŽ ²)= 2sin({ÃŽ ± + ÃŽ ²}/ 2) cos({ÃŽ ± - ÃŽ ²}/ 2)$$ $$sin(ÃŽ ±) - sin(ÃŽ ²)= 2cos({ÃŽ ± + ÃŽ ²}/ 2) sin({ÃŽ ± - ÃŽ ²}/ 2)$$ $$cos(ÃŽ ±) + cos(ÃŽ ²)= 2cos({ÃŽ ± + ÃŽ ²} / 2) cos({ÃŽ ± - ÃŽ ²}/ 2)$$ $$cos(ÃŽ ±) - cos(ÃŽ ²)= -2sin ({ÃŽ ± + ÃŽ ²}/ 2) sin({ÃŽ ± - ÃŽ ²}/ 2)$$ Product Identities This group of trig identities allows you to change a product of sines or cosines into a product or difference of sines and cosines. $$sin(ÃŽ ±) cos(ÃŽ ²)= {1/2}(sin (ÃŽ ± + ÃŽ ²) + sin (ÃŽ ± - ÃŽ ²))$$ $$cos(ÃŽ ±) sin(ÃŽ ²)= {1/2}(sin (ÃŽ ± + ÃŽ ²) - sin (ÃŽ ± - ÃŽ ²))$$ $$sin(ÃŽ ±) sin(ÃŽ ²)= {1/2}(cos (ÃŽ ± - ÃŽ ²) - cos(ÃŽ ± + ÃŽ ²))$$ $$cos(ÃŽ ±) cos(ÃŽ ²)= {1/2}(cos (ÃŽ ± - ÃŽ ²) + cos(ÃŽ ± + ÃŽ ²))$$ Verifying Trigonometric Identities Once you have gone over all the key trig identities in your math class, the next step will be verifying them. Verifying trig identities means making two sides of a given equation identical to each other in order to prove that it is true. You’ll use trig identities to alter one or both sides of the equation until they’re the same. Verifying trig identities can require lots of different math techniques, including FOIL, distribution, substitutions, and conjugations. Each equation will require different techniques, but there are a few tips to keep in mind when verifying trigonometric identities. #1: Start With the Harder Side Despite what you may initially want to do, we recommend starting with the side of the equation that looks messier or more difficult.Complicated-looking equations often give you more possibilities to try out than simpler equations, so start with the trickier side so you have more options. #2: Remember That You Can Change Both Sides You don’t need to stick to only changing one side of the equation. If you get stuck on one side, you can switch over to the other side and begin changing it as well. Neither side of the equation needs to be the same as how it was originally; as long as both sides of the equation end up being identical, the identity has been verified. #3: Turn all the Functions Into Sines and Cosines Most students learning trig identities feel most comfortable with sines and cosines because those are the trig functions they see the most. Make things easier on yourself by converting all the functions to sines and cosines! Example 1 Verify the identity $cos(ÃŽ ¸)sec(ÃŽ ¸) = 1$ Let’s change that secant to a cosine. Using basic identities, we know $sec(ÃŽ ¸) = 1/{cos(ÃŽ ¸)}$. That gives us: $$cos(ÃŽ ¸) (1/{cos(ÃŽ ¸)}) = 1$$ The cosines on the left cancel each other out, leaving us with $1=1$. Identity verified! Example 2 Verify the identity $1 − cos(2ÃŽ ¸) = tan(ÃŽ ¸) sin(2ÃŽ ¸)$ Let’s start with the left side since it has more going on. Using basic trig identities, we know tan(ÃŽ ¸) can be converted to sin(ÃŽ ¸)/ cos(ÃŽ ¸), which makes everything sines and cosines. $$1 − cos(2ÃŽ ¸) = ({sin(ÃŽ ¸)}/{cos(ÃŽ ¸)}) sin(2ÃŽ ¸)$$ Distribute the right side of the equation: $$1 − cos(2ÃŽ ¸) = 2sin^2(ÃŽ ¸)$$ There are no more obvious steps we can take to transform the right side of the equation, so let’s move to the left side. We can use the Pythagorean identity to convert $cos(2ÃŽ ¸)$ to $1 - 2sin^2(ÃŽ ¸)$ $$1 - (1 - 2sin^2(ÃŽ ¸)) = 2sin^2(ÃŽ ¸)$$ Now work out the left side of the equation $$2sin^2(ÃŽ ¸) = 2sin^2(ÃŽ ¸)$$ The two sides are identical, so the identity has been verified! Example 3 Verify the identity $sec(-ÃŽ ¸) = sec(ÃŽ ¸)$ The left side of the equation is a bit more complicated, so let’s change that secant into a sine or cosine. From the basic trig identities, we know that $sec(ÃŽ ¸) = 1/{cos(ÃŽ ¸)}$, which means that $sec(-ÃŽ ¸) = 1/{cos(-ÃŽ ¸)}$. Substitute that for the left side: $$1/{cos(-ÃŽ ¸)} = sec(ÃŽ ¸)$$ The negative angle identities tell us that $cos(-ÃŽ ¸) = cos(ÃŽ ¸)$, so sub that: $$1/{cos(ÃŽ ¸)} = sec(ÃŽ ¸)$$ Again, we know that $sec(ÃŽ ¸) = 1/{cos(ÃŽ ¸)}$, so we end up with: $$sec(ÃŽ ¸) = sec(ÃŽ ¸)$$ Identity verified! Summary: Trig Identities Solver You’ll need to have key trig identities memorized in order to do well in your geometry or trigonometry classes. While there may seem to be a lot of trigonometric identities, many follow a similar pattern, and not all need to be memorized. When verifying trig identities, keep the following three tips in mind: Start with the trickier side Remember that you can change both sides of the equation Turn the functions into sines and cosines What's Next? Wondering which math classes to take in high school? Learn the best math classes for high school students to take by reading our guide! Wondering whether you should take AB or BC Calculus? Our guide lays out the differences between the two classesand explains who should take each course. Interested in math competitions like the International Math Olympiad? See our guide for passing the qualifying tests.