Proofiness - Limit of sin(x)/x

This is my proof for the limit of [sin(x)/x] as x approaches 0 equals 1. I recommend that you print out the image, and mark it as you read the proof, this should help you in understanding the proof better, and make it 

easier to follow along.

Given: A unit circle on the cartesian plane, with circle A. Hence line segments AB and AD are 1 unit in length.

Construction:
1) BC is a prependicular through point B onto the X-axis. Consequently angle BAC lies in the first quadrant.
2) A line segment joining points B and D, hereinafter referred to as line segment BD.
3) A tangent to circle A at point D, marked by line DE.
4) Extend line DE and AB and mark their point of interesection as point F

To prove: limit [sin(x)/x] as x approaches 0 equals 1.

Proof:
Assume, angle BAC is equal to x degrees.
Now, in triangle BAC:
sin(x) = BC/AB = BC/1 (As circle A is a unit circle, and AB is its radius)

sin(x) = BC - I

Now, in triangle AFD:
tan(x) = FD/AD = FD/1 (As circle A is a unit circle, and AD is its radius)

tan(x) = FD - II

Area of triangle ABD = 1/2 x (BC) x (AD) = 1/2 x (sin(x)) x (1) = 0.5sin(x) - From statement I

Area of sector ABD = 1/2 x Angle BAC x (AD) = 1/2 x (x) x (1) = 0.5(x) - Area of a sector

Area of triangle AFD = 1/2 x (FD) x (AD) = 1/2 x (tan(x)) x (1) = 0.5 tan(x) - From statement II

Now, we can observe that:

Area of triangle AFD > Area of sector ABD > Area of triangle ABD

0.5tan(x) > 0.5(x) > 0.5 sin(x)

Multiply throughout by 2

tan(x) > x > sin(x)

Now, we know that pi/2 > angle BAC = x > 0

Hence, tan(x) > 0 | x > 0 | and sin(x) > 0

Using laws of positive numbers and inequalities:

1/tan(x) < 1/x < 1/ sin(x)

Multiply throughout by sin(x)

1/cos(x) < sin(x)/x < 1

Now, limit of 1/cos(x) as x approaches 0 is 1 and limit of 1 as x approaches 0 is 1

Thus, by using the squeeze theorem for continuous functions

limit of [sin(x)/x] as x approaches 0 is 1

QED

The Financing of Old Electronics

Recently, I had my finance exam. It was my first brush with finance in my academic career and I must say I’m head over heels in love. Finance is simply put, figuring out the present value of future cash flows. It is quite distinct from accounting, which functions more on the basis of recording of financial activity and its appropriate presentation. Finance is more concerned with pricing of securities of various natures, it is related to a firm’s ability to generate value, and most importantly a firm’s ability to finance its operations. I learnt about topics such as Net Present Value, Discounted Cash Flow Analysis, Weighted Average Cost of Capital, and my favourite Capital Asset Pricing Model. Most people assume that finance means the stock market, and trading, and hustling of Wall Street. While that may be true, finance does involve sale and purchase of securities on the stock market, and it does involve long hours, it has more to it than meets the eye. Finance is also about how a firm wishes to finance itself, it is about a firm figuring out the ideal capital structure for its new project, and about a firm’s manager’s quest to correctly predict if a project is profitable for a firm. As a person who is interested by economics, I am now looking at a career path that combines a healthy dose of finance and economics, a sort of financial economics mixture, sounds exciting and I’m looking forward to it. Finance also involves a lot of mathematics, financial mathematics is a branch by itself, and it used for everything from game theory based firm decision making, to simple stuff such as figuring out your mortgage payment.

Today, for the sake of this article I’m going to discuss the something that most people have, a portfolio. You see person’s portfolio is sort of like a bag, in which the investor puts all the securities, he, or she has purchased, and hopes that generate solid returns for him, or her. However, there are certain factors that one must keep in mind while choosing securities to place in his or her portfolio. To begin with, a portfolio must be well diversified. By diversification, I mean investing in securities provided by firms in various sectors, in various industries, of various sizes, and if possible located in various countries. By doing so, an investor hedges himself, or herself, from facing significant losses if one company or a certain sector faces a downturn. Therefore, it is advisable to pick securities of various natures, such as equity trading funds, corporate bonds, stocks of various firms, stocks of different natures such as preferred versus common, and perhaps if possible treasury bills and long-term government bonds. By diversification it is implicitly implied that investors must pick securities that are negatively correlated to one another, to maximize their diversification power. For example, an investor who buys one gold stock and one automobile stock is better diversified than an investor who simply chooses to purchase a gold stock and a silver stock. The reason being, that while a gold stock faces a positive return in recessionary times, an automobile stock faces positive returns in boom times. Hence, an investor holding both securities will make a decent return in either circumstance. Meanwhile, an investor holding only gold and silver stocks will make an absolute killing in recessionary times, but may struggle to pay the bills in boom times. Hence, he, or she, is not well diversified.

Another important aspect regarding to a portfolio is understanding risk. Finance classifies risk as the variability in returns of a security. This definition makes sense, because the higher the variance in returns of a security the more risky it is. Each security faces two sources of risk, unique risk, and market risk. Now, unique risk is the risk that is “unique” to the firm, or entity offering the security, whereas the market risk, is the risk that is faced by all securities all over the market. As an example, consider an Air Canada stock, the firm’s stock individually faces these two sources of risk. The stock faces unique risk because the firm is susceptible to strikes due to breakdowns in union negotiations hence adversely affecting its operations and bottom line. This risk is unique to Air Canada alone, whereas market risk is the possibility that Air Canada will be unable to fill its airplanes if the economy dips into another recession. This risk is faced not just by Air Canada alone but by all companies operating in Canada. Now, investors are never considered to be compensated for unique risk. Why? Because it is assumed that all investors are rational and will diversify enough to hedge themselves against unique risk. Hence, it is assumed that if an investor holds Air Canada stock, he will also hold stocks of firms whose employees are not going on strike, or are actually generating positive R&D. Therefore, for a diversified investor, unique risk is considered to be eliminated. Therefore, by default, investors are compensated only for their exposure to the market risk.

Now, a firm can be exposed to market risk in varying proportions. For example, a food company is less prone to variance in sales due to economic forces as food is a staple is needed in all economic circumstances. However, a company such as Tiffany’s is highly prone to variance in sales due to macroeconomic pressures as very few people can afford luxuries in a recessionary market, as opposed to a growing one. Therefore, each firm is exposed to a different level of market risk, some more than others. However, it is not easy to estimate the market risk faced by a firm. For example, which company is more exposed to market risk, General Electric, or Starbucks? Thankfully, we don’t need to make judgement calls when calculating a firm’s market risk. We possess a quantitative measure called the Beta of a security that shows the systematic (read market) risk, faced by a firm. It is obtained by plotting data points of the firm’s stock, or security’s, price versus the movement of the market index as a whole. The Beta is obtained by plotting the line of best fit through these data points. As some of you may have identified, this is nothing more than simple linear regression, and Beta is nothing more than the coefficient of X that illustrates the slope of the line of best fit.  Therefore, a company whose security possess a high beta means that company is prone to market movements, and a small change in the market index is reflected by a large change in the security’s price on the market. These stocks are called aggressive stocks. Conversely, there exist firms whose stocks fluctuate very little with market movements, and are termed defensive stocks. Now with regards to a firm’s portfolio, the Portfolio Beta or P(b) can be measured by taking the weighted average of all the Betas of the securities in a portfolio. Moreover, the portfolio risk can be measured by conducting a simple calculation involving the variance, weights, and correlations between securities in a portfolio. The formula is long and tedious and is not stated for the purposes of this discussion.

Moving on, there exist securities such as T-bills, or treasury bills. These securities are debt offerings provided, and guaranteed by the government. Ideally, treasury bills are considered to be completely risk free, because they face no unique risk, as long as the government is a stable organization, and they face no market risk because the return on the security is fixed irrespective of market conditions. Hence, these forms of securities are considered to be completely risk free and have a Beta of 0. They are considered to be the gold-standard for returns, and any firm’s expected returns, judged by investors are based on the returns offered by treasury bills of equal duration. Furthermore, a quick note that must be made for the purposes of this discussion is that the Beta of a market portfolio (market index) is always 1. Investors use the risk free rate of return, the market’s average rate of return, and a stock’s Beta to judge the rate of return they can expect from a stock. If you understand this article so far, congratulations you have just learnt the Capital Asset Pricing Model! CAPM, is a simple equation created by William Sharpe, for figuring out the appropriate return on securities based on the risk free rate of return, the securities inherent risk (represented by the Beta), and the rate of return provided by the market in general. To expand, consider the Air Canada example. If the Treasury bills offer a 5% rate of return on an annual deposit, and the annual rate of return provided by the market is around 13%, and the Beta of Air Canada stands at around 1.5. The rate of return expected by investors on the Air Canada stock stands at:

E(ROR) = 5+[1.5 x(13-5)] = 15%

Thus, investors who are exposed to 1.5 times the market risk, and have an option of investing risk free at 5%, will want a return of 15% on their Air Canada stock to compensate for the risk they are taking on. Therefore, the portfolio’s rate of return is obtained by taking a weighted average of these expected rates of returns of various securities held in the portfolio.

The CAPM has worked as an indicator of returns expected by investors, but it does explain investor sentiment fully, and works on the assumption that all investors are rational, and well diversified, which again not always true. Thus, it is a strong arrow in the quiver worn by financial managers, but has its own shortcomings.

The last topic for discussion would be the Security Market Line, or SML as it commonly referred to financial circles. The SML is a line that plots the Beta and Expected rate of return of various securities. The two points that decide the SML are the rate of return on risk free treasury bills, and the rate of return on market indices (market portfolio), with their Betas being fixed at 0 and 1 respectively.  The SML indicates if a security is underpriced, or overpriced. If Air Canada with a Beta of 1.5 provided a rate of return of only 13%, it would be overpriced, and would plot below theSML. No investor would consider buying Air Canada, as its rewards do not match its inherent risk, and there exist other securities that provide higher returns for the same level of risk. In this case, the price of the stock would fall until the rate of return rises to 15%, and coincides with the SML. The converse holds true for underpriced securities.

And with that, we bring to an end our little journey into the world of finance, and portfolio investing. There were many more things learnt this semester, and if I’m in the mood, I shall discuss them further on here.

Ramble: For today’s ramble I would like to discuss old electronics. I sometimes feel that I’m the only person on this planet who wants to go in reverse when it comes to technological change. I really enjoy the feel, touch and texture of old electronic items. To me, they carry some sort of a personality. I wish I had an old electric typewriter, or even a manual one for that matter. They seem friendlier than the glowing light, and silent hum of laptops. Not to mention, they NEVER run out of battery, and make you search for a power outlet in the middle of a class. I also would love to have a record player, records these days are cheap, and record players are not. I can get a nice vinyl for around 50 cents, but a record player is rarely to be found, and if I do discover one, its très expensive. Lastly, I would love to have an old walkman. I know they are completely out of line when compared to iPod’s, and such, but I like the idea of playing music on a cassette, rather than downloading it off the internet. Although, I must admit downloading of the internet is cheaper. I suppose old electronics have a story to tell, like how a 30-year old record would recite the tale of his previous owner turning him on, and making him play some Jazz tunes, as the owner, and his wife danced on the patio on a warm summer night. Imagine, a typewriter regaling the tales of how he was furiously pounded on by the soft fingers of a secretary as she took down the letter being dictated to her by her boss in the late 80’s. It’s sexist, I know, but in reality most women did not break the corporate glass ceiling until the late 90’s. Anyways, these are my thoughts. Until next time, be nice, and if you can’t be nice, be careful.

Merry Monetary Policy :)

A belated Merry Christmas to everyone, hope Santa got you tonnes of presents, from the latest flat screen telly, to that book you always wanted. Of course, who can forget the candy canes melting in the cups of hot chocolate, as one sees the crystalline structures of water (read: snowflakes), deposit themselves on your window ledges, and rooftops? Alas, the holidays are around, time for happiness, joy, and cheer. Moreover, Happy Boxing day so get your cars drive to the nearest mall, and spend more money AFTER Christmas gifts on things that are not really that darn cheap after all. Trust me, a 70% off at Tommy Hilfiger, still means forking out some wads of cash, some people call me a damper on Boxing Day, others call me cheap, I agree with both. Nonetheless, I digress I must refocus on something else I wish to blog about today. C’est la politique monétaire, or as most English speakers know, monetary policy. Monetary policy is a little tool used by our modern day government, in particular, by the Central Bank, to regulate interests, and affect economic activity within a nation.  Interest rates affect economic activity through affecting capital inflows and outflows from a nation, and by regulating levels of business investment, consumer expenditure, and even governmental spending. As I explain monetary policy, I’ll explain the jargon in more detail, which should assist in understanding of the topic. The central bank has a balance sheet, think of it as a form of record of all the assets and liabilities that the bank posses. The assets usually include bonds that the bank owns, and issues to other chartered banks, and loans issued by the bank, that function as a receivable for the central bank. Moreover, the bank has foreign assets, in the form of reserves (reserves are usually held in the form of foreign currency, or gold). Traditionally, the US dollar is used as a reserve currency; although most banks keep a mix of the US dollar, Euro, and the Japanese Yen, and a small ratio of other nations’ currencies. The other side consists of liabilities, in the form of bank reserves that other chartered banks deposit at the central bank, and currency in circulation (the dough in your purses), whose valuation is again the responsibility of the central bank.  Now, let’s see how monetary policy works:

Consider a country like Canada, whose central bank (Bank of Canada), hereinafter referred to as BOC, has an interest rate of 0.25%. Now, consider another country, such as France, with its central bank (European Central Bank), hereinafter referred to as ECB, with the interest rate at 0.25% as well.

To begin, consider a scenario where Mark Carney feels that the global economic recovery is weak, and needs some stimulus to allow business activity to gain strength, and resurrect the fragile economy. He might order that the interest rate be lowered to a measly, 0.05%, a drop of 2000 basis points. This is of course a huge drop, and is rarely ever seen, but for the sake of our understanding this shall suffice. A drop in Canadian interest rate means that, Canadian chartered banks begin to offer lower interest rates on securities, and the treasury bills offered by the BOC see a drop in yield as well. This prompts Canadian investors, and international investors to pull money out of Canada (capital outflow from Canada), and deposit it in ECB’s treasury bills, and French chartered banks, which earn a higher interest rate (capital inflow into France). This means that the Canadian money supply increases as money locked up in savings accounts is pulled out, and is shuffled abroad. Leading to people’s need to convert Canadian dollars to Euros; thus, there are more Canadian dollars to go around. Moreover, a drop in interest rates means that the Canadian people are willing to borrow a lot more money at the cheap interest costs. This borrowing of money leads to increases in consumption, and businesses find it easier to finance investment. Therefore, the lower interest rate means more people willing to use their credit cards, and businesses rushing to the bank to take out loans to build more factories, which will hire more people and reduce unemployment. However, this virtuous loop is not infinitely cyclical and does come to an end, as the markets rebalance. Of course, there are other nuances and intricacies in the aforementioned paragraphs that I have skipped, and I will explain them in more detail when I post about monetary policy again. This post is just meant to function as a quick crash course into interest rates and their effect on our economy, via monetary policy.

Conversely, consider the scenario where Mark Carney increases interest rates, as he feels that inflation is threatening the economy. This would lead to capital inflows into the Canadian economy, and capital outflows from France. The French currency, Euro’s supply increases, and Canadian dollars are locked away in interest earning instruments. Thus, business activity is dampened, and inflation is averted. Voilá, the world of monetary policy! So the next time you pick up the telegraph, or le monde, or the financial times and see an interest rate change announcement, or story. Read it, and feel proud that you understand one of the fundamental instruments of macroeconomic policy.

Ramble: Usually, after a serious post similar to the one above I love to ramble out random thoughts in my head.  I’m going through a phase where I’m fascinated by the European culture, geography, politics, and immigration laws. I know that the last ones seem weird, but immigration laws float my boat. Did you even know that in India, certain places exist where in order to protect the local culture the government does not allow foreign nations to enter those regions of the country? Not only are foreign nationals not allowed, Indian citizens who live in other states, are not allowed to visit those regions. Regions off limits include Arunachal Pradesh, Mizoram, Andamman and Nicobar Islands, and other little bitty crannies of India. Hmmm, what else? I’m reading a book on Fermat’s Last Theorem, and the quest for its proof that was ended by Andrew Wiles. It is an interesting read, and has a section devoted to female mathematicians. I’m sorry if I sound ignorant, but there are not many female mathematicians that come to my head when I try to think of mathematical breakthroughs. I’m sure that there are quite a few female mathematics professors in the modern world, but very few mathematicians, with two X chromosomes, have been documented for their achievements in mathematics. Nonetheless, my scrimmaging through lead to an interesting find: Danica McKellar, the girl who played Winnie Cooper in the Wonder Year, is actually a holder of a degree in mathematics summa cum laude from UCLA! She has published various math exercise books, and has a whole theorem dedicated to her, check out the link for more info. (http://www.nytimes.com/2005/07/19/science/19math.html?_r=1) I was quite pleased to find this, after years of imagining that mathematicians were people with weird OCD’s who sat in rooms and worked on numbers all day (sort of like me), it was refreshing change to find a mathematicienne who looks like this! :D