The Complex World of Financial Analysis

The financial world is a complex equation with hundreds of variables that dictate what actions people will take in pursuit to generate wealth from their savings. The majority of the world’s population have little or no knowhow in managing their life savings, and thus rely on the few who act as intermediaries between the savings of the masses and the financial markets where wealth is generated.

Why do I need Math for Finance?
This is a very legitimate question, and I will walk you through a real-life example and show how math helps to answer key questions at each stage of development.
I will start with a simple hypothetical scenario that I feel is not too far off from reality. Suppose you find out that Apple Corp is releasing a new line of consumer tech products that have been rumoured to change the face of the tech industry in the years to come. You’ve always been very attracted to their products and with this news you are even more excited than before. You also happen to have a small but descent sum of money dormant in your bank account and never really wrapped your head around what to do with it.

Is this perhaps the right time to invest in Apple stock?

A quick look at stock price chart above shows you that, had you invested money two years ago, you would have bought 1 unit of stock at roughly $96 and today it would be worth roughly $184. That’s more than a 100% increase, meaning your money would have more than doubled in just two years! Which bank, secure as it may be, can come even close to these kinds of return?
It seems like a no brainer: you like the company’s strategy, returns have been exceptional, and the alternatives pale in comparison. Unfortunately, at this point, many would simply phone their broker (or log in their online platform) and order some Apple stock without digging deeper into what’s beneath the hood. And this is fine, because it is your money after all. It could turn out great and you triple your savings in a few years. But if you are fund manager, you are managing other people’s savings, and your profession requires that you look under the hood — and thoroughly — else you may be liable to misappropriation and mishandling with long years behind bars.
Any stock climbing up a mountain might find the route a bit choppy and the amateur may run into a number of disappointments. Mathematics will not completely avoid these disappointments, but it will tell you how likely these disappointments are going to be and if they can be so severe as to wipe away a substantial share of your original savings even before you actually start seeing some positive results. You might tell me “that’s absolutely fine for me: I am a patient person and can take a few hits on the way to success.” My answer, having studied the psychology behind investing money, is very simple: Never, ever underestimate your potential to make quick, irrational decisions; especially if things are not going your way for a long period of time and there are some super attractive alternative investments just around the corner, enticing you to jump ship into the new glitzy stock!

Shape
It is clear that we need more information than simply looking at prices and reading about the company’s plans for the future. The second big question (and by far the most important one) is this: What is my risk? Even though there are long and technical answers to that question, I am going to try and give you the gist of what one can do with some very simple mathematical tools learned during a sound degree in finance.


We need some forecasting tools: something that will tell me “The return on your investment can be in the range of x to y % with this likelihood. The most important piece of machinery that you will be learning about in this course is our dear old Normal Distribution, pictured below.


This famous bell shaped can be found everywhere: biology, economics, finance, physics etc. It tells you, in a nutshell, that numbers around the average are the most likely and everything else is symmetrical around it.

Applying the normal distribution to my analysis of Apple Corp stock, I could have a road map telling how likely or unlikely certain gains and losses are in the future, based on past performance. Quite neat! Let’s see if our returns on Apple Corp stock (notice that I apply it on returns and not to prices) are roughly shaped in this manner.

Using some Excel tricks that you will learn, you can see that the chart above could be called a distant cousin to the Normal Distribution (nothing will ever be perfectly Normal in practice) and, for the sake of this discussion, we can assume that our returns follow a Bell-Shaped curve with the peak being close to 0% daily returns (not a surprise here because in practice you should not expect any money within the very short period of a day). In practice there are various statistical tests that one can do but they are beyond the scope of this exercise.

Relationships
So now that we have assumed that our Apple returns follow a normal distribution with average daily return of 0% we can start thinking about doing some forecasting. Can we create a tool which accepts an input and gives us an output for expected return? The answer to this is “yes!” and the technique (that you will also learn in class) is called Regression.
Before going into the details of what Regression is, I would like to discuss this question with you: Since Apple is a company that is part of the Tech industry, is it reasonable to think that what happens in the tech industry in general can have a major effect on Apple’s fortunes as a company? Intuitively we might think that the answer is “yes”. This is because every company out there is affected by both internal affairs (profitability, efficiency of personnel, technological breakthroughs, scandals and conflict etc) and also system-wide affairs (economy, inflation, tourism, media relations etc). It is, therefore, fair to ask by “how much” the Tech industry effects the fortunes of Apple. The Regression tool that you will learn in this course tackles exactly this problem. If some conditions on your data are satisfied (errors look like a bell-shaped graph etc) you can safely apply this tool and get some forecasts on future data points to a reasonable degree of accuracy.

I will use this technique here and try to find the strength of the relationship between the Tech industry and Apple. To do this, I will use the NASDAQ index, which is the Index following the top Tech companies in the US, and measure the strength of the linear relationship between NASDAQ index and Apple stock. In a nutshell, what regression does is it attempts to fit a straight line (called line of Best Fit) to a scatter plot formed between Apple Returns and NASDAQ Index Returns, just like the chart below:

Focusing only on the last 2 years of daily data and applying the in-built Excel Regression function I get the following relationship:

Notice that this has the classical Y = mX + C Straight Line format that you learnt at school. This is because we are fitting a line. The 1.05 is the gradient and the 0.000414 is the y-intercept. Notice also that since NASDAQ returns are my “X-Variable”, they are being used to predict the “Y-Variable”, in this case Apple Returns.

The interpretation is that 1% increase in NASDAQ returns leads to 1.05% increase in the returns of Apple. It is to be kept in mind that this discussion is very basic, and in practice things are more complicated. However, this example should be a good taste of the type of analytical tools that will be applied in a finance degree to take a “Scientific Approach” to the decision-making problem of Investment. The Above indicates that if we could find a trend for NASDAQ returns (being an index it is easier to find such a trend) we could then apply the above formula to forecast Apple Corp returns in the next day. In practice you would want to apply such an analysis to at least monthly or quarterly data to have more predictive power.

Trend
How do we find a trend? Another tool that you will be learning is known as the “Moving Average”. As its name implies, applying this technique is just a matter of finding a sequence of averages to your data, moving through time. This will give you an average that moves through time, and thus gives you an indication of the trend. I have used here something a little bit more sophisticated known as the Exponential Moving Average, also very simple to apply. This method allows me to fit a trend for NASDAQ Prices as in adjoining graph.


Notice that this is just a seven-month Snippet. Using this MA method I can calculate that the next price in the daily series is going to be $7,549, even though the price is currently $7,568 at the time of writing (I am using an index that tracks the NASDAQ called ^IXIC). This is because our moving average is picking up some negative trend that is developing as it is considering not just the current price, but the current trend. I repeat that this would be much more insightful if it were monthly or quarterly data, but for the basis of this discussion it will suffice. If I know the next price, I know the next return. If price drops to $7,549 tomorrow, that’s a predicted 0.3% drop in price.

Using our regression equation above we get that the predicted return for Apple Corp. tomorrow is 1.05*-0.3% + 0.000414 = -0.00274 or -0.3% as well. I am sure you appreciate that this could be simply due to daily random movements in the market and not an actual trend, and this is why the numbers would make much more sense if taken quarterly, so aspects like seasonality (effects of seasons) can be captured. Another important point to mention is that all of these predictions (even if done on a quarterly basis) are not infallible and are only correct up to a certain
probability level which, as you will later on learn, is known as “Level of Significance”.

Worst Case Scenario
One final question that any self-respecting investor should ask is “What is the worst-case scenario?” Obviously, the answer is “You lose everything,” but not investing because of such an ill-inspired answer would be like never getting outside of the house because you could get hit by a bolt of lightning. Luckily, there is a more sophisticated answer to that question and it involves a very important concept known as VaR or Value at Risk. This tool simply seeks to answer the following question: “What is my maximum loss from an investment, in a particular period of time and with a particular probability?” If you look at Apple’s history, there was one day during the dot com bubble of 2000 where the stock lost 75%! Can you imagine that? Waking up one day and losing 75% of your investment? Luckily this was a one-off and is very unlikely to happen often. So what is a realistic expectation of loss? To answer this question we need to learn a statistical tool known as Standard Deviation. This number gives you an indication of how your losses and returns are dispersed around the mean (which in our case for Apple was 0%) and it is vital to calculating the Value at Risk. So, for the sake of argument, let us say I want to know my worst case scenario for a 1 day loss with 95% probability. Luckily, we know that our returns are roughly Normally Distributed and we can the following VaR formula:

Using the in-built Standard Deviation Excel function I get that the 1 day Standard Deviation for Apple Corp. is 2.9%. Plugging these numbers in the above formula (let us say we invest $1000) we get:

1 day 95% VaR = 1000 * 0.029 * 1.645 which is approximately $50. This means that I am 95% sure that my losses in 1 day will be a maximum of $50, which is a far cry from the 75% loss that happened on that faithful day during the Dot Com Bubble crash of 2000!

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