Question 01:

Because most investment decisions are decided in an environment of risk, the tools that help us make decisions with logic and regularity fall under the umbrella of probability. As per the definition probability is the chance that something will happen. Furthermore, it is the number of times an event happens out of many trials. Knowledge of modern probability and statistics is essential for the development of economic and finance theories and for the testing of their validity through robust analysis of real-world data. For example, probability and statistics could help to shape effective monetary and fiscal policies and to develop pricing models for financial assets such as equities, bonds, currencies, and derivative securities. The purpose of Finance is to deal with uncertainty that happens in the markets.  So, it comes as no surprise that people that work in finance choose models that integrate some random behaviour, and they use probability concepts to interpret the outcomes of those models. The importance of developing robust methods for such empirical analysis has become particularly important following the recent global financial crisis in 2008, which has placed economic and finance theories under the spotlight (Brown & Wing , 2015). Because the loss associate to making decisions without probability or incorrect probability can be astronomical, how compagnies manage to pick the best fit for their business?

 There are 3 main way to assign probabilities: classical, empirical and subjective. The classical view of probability is simple as it has equal odds of something happening. For example, if we simple look at the share price of apple there are two outcomes, the probability for the share price to up is 50% and same for the opposite. or if we look at the inflation index of a country, the probability of if to rise is half and the same for it to fall.

In contrast of the classical view, the empirical view is base on experiments. It is the relative frequency of the occurrence of an event and is determined by actual observation of an experiment. Consider an experiment, event A occurs “f” number of times if we repeat experiment n times (Empirical Probability Formula, 2019). It stated that the probability that an even occurs will always be between the number 0 and 1. For the empiric probability of 0 to be the chance that the event never occurs and 1 it will always happen. According to Investopedia, an empirical probability is closely related to the frequency of an event (Kenton, 2018). In order for a theory to be proved or disproved, empirical evidence must be collected. An empirical study will be performed using actual market data (Kenton, 2018). Even though one of the most used theory in finance CAPM is not 100% accurate a lot of empirical studies were necessary to arrive to the final conclusion.

The last type of probability is subjective which use personal judgment about the possibility of an event to occurs. It does not have formula and only rely on the person ju

In probability we can count three basic rules: the addition rule, the multiplication rule and the complement rule or subtraction rule. There are two formula attached to the addition rule, first one is the sum of the probabilities of two events that cannot happen at the same time occurs. The second one is the probability of two mutual exclusive events minus the probability of both happening. The formula for the first one is: P (A u B) = P(A) + P(B) and for the second one we have P (A ∪ B) = P(A) + P(B) – P(A ∩ B). because the probabilities of two event cannot happen at the same time is equal to 0, only the second formula is applied in reel life. For example, if we take a stock price of a company, the price can either go up or down, the same thing goes for the government expenditure of a country.

Multiplication law in probability applies to combination of events. The probability that Events A and B both occur is equal to the probability that Event A occurs times the probability that Event B occurs, given that A has occurred (stattrek, 2019). In some case A and B are independent variable, but the rule can also be used with dependent events, you just have to be more careful in finding P(A) and P(B). The formula is:

P (A ∩ B) = P(A) P(B|A)

b) construction of probability distribution of suburbs:

Using the data analysis tool of excel we have the descriptive state of the suburbs as follow:

The mean with the data analysis is equal to the one I found calculating with the table: 3.104762 using the “sumproduct function” of excel multiplying the X to P(x).

Question 2:

• Using the average and the standard deviation function in excel we found out that the mean for the home selling price is 221.1028571 (000’) or €221,102.8571 and the standard dev. 46.88055635. with the mean of the data being the average and the standard deviation the spread, the average price of the house selling price would be 221,102.8571 and 46.88055635 or 46,880.55635 is our far data is from the mean. In a simper way the mean is the where your data is, and the standard deviation is how far most of the points in your data are from the mean.
• For the answers look at the appendix.

Question 3:

1) The probability to have 5 traders leaving the company is:

P(A)= 5/10= 0.5, the probability of 5 persons leaving out of the 10 is 50%.

2) The probabilityP4 or less to have 4 traders or less leaving the company means:

• Case 1: 4 leave and 6 stay, P1=210 (0.05)4*(0.95)6 = 0.000964808 OR
• Case 2: 3 leave and 7 stay, P2= 120*(0.05)3*(0.95)7 = 0.010475059 OR
• Case 3: 2 leave and 8 stay, P3= 45*(0.05)2*(0.95)8 = 0.074634799 OR
• Case 4: 1 leave and 9 stay, P4= 10*(0.05)1*(0.95)9 = 0.315124705

P4 or less=P1+P2+P3+P4 = 0.000964808+0.010475059+0.074634799+0.315124705 =0.401199

3) Saying that 5% of traders leave the company each year means that we can consider this percentage like a Bernoulli distribution on traders with a probability of success 0.05:
– that means the probability of success (the trader leaves the company) Ps=0.05
– the probability of failure Pe=1-0.05 =0.95

In other cases, on a sample of 10 traders, to find the different probabilities, we use a binomial distribution, a probability distribution of the number of successes in a sequence of n independent experiments of a Bernoulli distribution. On n experiments, the probability to have k success (k traders leaving over n) is P=Cnk *(Ps)k*(Pe)n where Cnk is the binomial coefficient.

Question 4:

1. probability tree:

2. the expected return given Scenario 1: (0.9*0.55) +(0.8*0.45) = 0.855
3. the expected return given Scenario 2: (0.5*0.85) +(0.4*0.15) = 0.485
4. overall expected return: (0.855*.75) + (0.485*0.25) = 0.7625

Question 5:

Introduction

Looking back to 100, 500, 1000 or even 2000 years ago the way people were doing finance is not the same as today. A lot of changes came in and change the way we run things. With the introduction of mathematic in finance, different type of theories and models came in and opened more doors for the financial world. Investors took more risks, expand their portfolios and created new types of financial assets. (Morris, 2018) was saying about modern Finance that it is the general term he uses to cover Modern Portfolio Theory, Capital Asset Pricing Model, Efficient Market Hypothesis, and a few others. Most of those theories were bult on different assumptions using the different types of probabilities as foundation.  Modern financial theory’s foundation is, in great part, built on the premises of probability theory and, to be more precise, the normal or Gaussian probability density function (Charles & Scott , 2013). Same as (Charles and Scott, 2013) (Merton, 1994) in the Royal Society stated: “Just as in Bachelor’s option-pricing model of 1900, so the normal and binomial distributions are central to today’s mathematical models of finance”. The purpose of this paper is to look at the contribution of the binomial and normal in modern finance. The first part of this paper will deal with the binomial distribution, from the definition to how it is used in modern finance and limitations it can possible have. The second part will highlight the introduction of the normal distribution and what that brought to the world of finance. It will also look at the impact and limitations of the normal distribution. Do the two distributions fit what happens in real world?

The binomial law is among the oldest probability laws ever studied. Swiss mathematician Jakob Bernoulli, in a proof published posthumously in 1713, determined that the probability of k such outcomes in n repetitions is equal to the kth term (where k starts with 0) in the expansion of the binomial expression (p + q) n, where q = 1 − p. (Hence the name binomial distribution) (Routledge, 2019). But where the Bernoulli trial was only focus on one parameter (the probability of success), the binomial distribution is an extension of the Bernoulli and focuses on two parameters (probability of success and the number of random variables). As explained in the (Encyclopedia, 2019) in general, a binomial distribution is used to characterize the number of successes over a series of observations (or trials), where each observation is referred to as a “Bernoulli trial.” In the real world the binomial distribution is used for different events. A binomial distribution can be used to describe the number of times an event will occur in a group of patients, a series of clinical trials, or any other sequence of observations (Encyclopedia, 2019). In all of the cases the event has a dual variable: It either arises or it does not. For example, when patients are treated with a new drug, they are either cured or not; when a coin is flipped, the result is either a head or tail (Encyclopedia, 2019). The underlying assumptions of the binomial distribution are that there is only one outcome for each trial, that each trial has the same probability of success, and that each trial is mutually exclusive, or independent of each other (Kenton, 2018). Like other distributions, the binomial distribution can be described in terms of a mean and the spread, or variance, of values. However, the binomial distribution differs from the others. The binomial distribution is a common discrete distribution used in statistics, as opposed to a continuous distribution such as the normal distribution (Kenton, 2018). Let’s say you’re a bank, a lender, who knows to within three decimal places the likelihood of a particular borrower defaulting. What are the chances of so many borrowers defaulting that they’d render the bank insolvent? Once you use the cumulative binomial distribution function to calculate that number, you have a better idea of how to price insurance, and ultimately how much money to loan and how much to keep in reserve (MCFARLANE, 2014).  

To use the binomial distributions, you must meet the following three criteria:

• The number of trials or observations is fixed. In other words, to figure out the probability of an event happening you will have to perform the test a certain number of times.
• Each observation or trial is independent. In other words, none of your trials influence the probability of the next trial.
• The probability of success (tails, heads, fail or pass) is the equal from one observation to another.

In modern finance, there different financial assets that can be couple with the binomial distribution to maximum the profit. Analysts with a grasp of the binomial distribution have an additional quality set of tools at hand when determining pricing, assessing risk, and avoiding the unpleasant results than can accrue from insufficient preparation (MCFARLANE, 2014). As example of the application of the binomial distribution in modern finance we look at one of the financial assets. One of the products that can be used is “options”, an Option is a contract giving you the right to buy or sell an underlying security at a stipulated price. If the conditions on the parameters of the generalized Binomial distribution are satisfy, then the generalized Binomial distribution can be combined with financial terms to determine the option value. There are two assumptions that are necessary before we can use the binomial distribution to price options. First, the binomial approach assumes that trading takes place in discrete time, that is, on a period-by-period basis. Second, it is assumed that the stock price (the price of the underlying asset) can take on only two possible values each period; it can go up or go down.

Unfortunately, no mathematical theory is perfect, and each has insufficiencies and limitations.  Mathematical models provide a good mechanism to quantify some variables with single, trackable numbers. But due to the limitations of assumptions, models may fail (Seth, 2018). Which is why the normal distribution can be used to approximate the binomial distribution.

The normal distribution or also called the “Gaussian curve” is the most important and most widely used distribution in statistics. It delivers an interestingly simple picture of the world. When looking into its origin, we can trace it to Abraham de Moire, a consultant for gamblers to determine probabilities. It all started with him studying the probability distribution of coin flips and producing a mathematical expression. As an answer to this question he derived a bell-shaped distribution which is commonly referred as the normal curve. It was a big discovery as most of the phenomena in the world follow approximately a normal distribution. The normal distribution is used to help predict and adjust for a wide range of financial goals by optimizing financial decision-making by applying and graphically mapping financial data into a distribution set of variables (moneycation, 2019). the normal distribution is a great tool that helps financial analysts and/or investors in making a better financial decision based on the statistical information provided by the normal distribution. Normal distributions help determine financial relationships and trends. These trends and relationships can then be used for comparing financial products, forecasting financial outcomes, assessing risk, and predicting return on investment, estimating cost and demand among other things (moneycation, 2019). The characteristics of a normal distribution are as follow:

• The mean, mode and median are all equal.
• The curve is symmetric at the center (i.e. around the mean, μ).
• Exactly half of the values are to the left of center and exactly half the values are to the right.
• The total area under the curve is 1.

Two aspects dominate when you look at any investment: risk and return. Investors seek for the highest possible return on an investment with the lowest risk possible. The normal distribution quantifies these two aspects by the mean for returns and standard deviation for risk (Kenton, 2018). In the “real” world of financial investors — where risk averse agents mainly hold government bonds, a few equities and do not hold derivatives — the normal distribution still plays a lead role ( Michele, Giuseppe, & Stefano , 2005). Distributions of stock market returns are often presented as bell shaped curves. For example, a return distribution that contains returns realized during the financial crisis will be very different than one covering a different period (Finance, 2019). Distributions of monthly returns can look different than distributions of daily returns. standard deviation is often called volatility when it comes to stock returns. If situation where the returns are normally distributed, more than 99 percent of the outcomes are expected to fall within three standard deviations of the mean (Finance, 2019). Because of these characteristics of the bell-shaped of the normal distribution, analysts and investors can make statistical inferences about the expected return and risk of stocks.

Conclusion

Normality has been an accepted wisdom in economics and finance for a century or more. Yet in real-world systems, nothing could be less normal than normality. Normal distribution, which forms the basis of portfolio theory, may not necessarily apply to stocks and other financial asset price patterns (Seth, 2018). The unanimous conclusion states that financial returns are not normally distributed ( Michele, Giuseppe, & Stefano , 2005). That contradicts with the general assumption of the Markowitz (1952) in his portfolio theory that assumed returns are normally distributed. In 2008, for instance, the profit and loss distribution of Goldman Sachs looked more like an elongated “U” than the bell shape predicted by VaR (Alloway, 2012). The last 30 years are the proof that even though we have theories and the different way to prevent investors to lose on their investment or hedge it the best they can, the reel world has a different view on the matter. Different crisis happened in the last three decades: Latin America debt crisis in early 1980’s. market crash in 87, burst of U.S technology bubble 2000 – 2001 and others… more non normality can be seen in the reel world often than usual, and because most of the asset allocation team used the mean-variance theory they failed face the situation when it arises. Because the world work of modern finance works differently than the old theories, a change is need in order to incorporate the non-normality factors in the equation when selecting probable investments. As when we incorporate that we will reduce the volatility of most portfolio. That does not mean we will drop the binomial or normal distribution as they are still relevant to other observations, but another type distribution that will incorporate the changes that happened with modern finance is needed to improve portfolio efficiency.

Appendix

References

• Michele, C., Giuseppe, C., & Stefano , I. (2005). The Role of the Normal Distribution in Financial Markets. Berlin: Springer.
• Alloway, T. (2012, Apr 13). Modelling: Normal distribution is not always the norm. Retrieved from Financial Times : https://www.ft.com/content/67d05d30-7e88-11e1-b7e7-00144feab49a
• Brown, S., & Wing , W. K. (2015, Aug 25). Probability and Statistics with Applications in Finance and Economics. Retrieved from ncbi.nlm.nih.gov: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4562097/
• Charles , C., & Scott , H. (2013, September 06). Normal Distribution, Probability, and Modern Financial Theory. Retrieved from onlinelibrary.wiley: https://onlinelibrary.wiley.com/doi/abs/10.1002/9781118755761.ch8
• Empirical Probability Formula. (2019, Jan 01). Retrieved from probabilityformula.org: http://www.probabilityformula.org/empirical-probability-formula.html
• Encyclopedia. (2019, Jan 03). binomial-distribution. Retrieved from encyclopedia: https://www.encyclopedia.com/science-and-technology/mathematics/mathematics/binomial-distribution
• Finance. (2019, Jan 03). Why Do Stock Market Returns Look Like a Bell Shaped Curve? Retrieved from finance.zacks: https://finance.zacks.com/stock-market-returns-look-like-bell-shaped-curve-6458.html
• Kenton, W. (2018, Apr 29). investopedia. Retrieved from investopedia.com: https://www.investopedia.com/terms/e/empiricalprobability.asp
• MCFARLANE, G. (2014, Sep 22). Basics Of The Binomial Distribution. Retrieved from investopedia: https://www.investopedia.com/articles/active-trading/092114/basics-binomial-distribution.asp
• Merton, R. C. (1994). Influence of mathematical models in finance on practice: past, present and future. The Royal Society, 451-463 .
• moneycation. (2019, Jan 03). The use of normal distribution in finance. Retrieved from moneycation: http://www.moneycation.com/2011/02/use-of-normal-distribution-in-finance.html
• Morris, G. (2018, Feb 20). What is it with Modern Finance? Retrieved from stockcharts: https://stockcharts.com/articles/dancing/2015/02/what-is-it-with-modern-finance.html
• Routledge, R. (2019, 01 02). Binomial distribution. Retrieved from britannica: https://www.britannica.com/science/binomial-distribution
• Seth, S. (2018, Feb 01). Optimize your portfolio using normal distribution. Retrieved from Investopedia: https://www.investopedia.com/articles/investing/100714/using-normal-distribution-formula-optimize-your-portfolio.asp
• stattrek. (2019, Jan 03). probability-rules. Retrieved from stattrek: https://stattrek.com/probability/probability-rules.aspx