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- The Orientation Note
- A Brief note on The Risk (Part 01)
- A Brief note on The Risk (Part 02)
- A Brief note on The Risk (Part 03)
- A Brief note on The Risk (Part 04)
- The Equity Curve
- The Expected Returns
- The Portfolio Optimization
- The Portfolio Optimization (Part 02)
- The Value at Risk
- The Position sizing for Active Trader
- The Position sizing for Active Trader (Part 02)
- The Position sizing for Active Trader (Part 03)
- The Kelly’s Criterion
- Understanding Trading Biases
- Understanding Trading Biases (Part 02)

Let's begin this chapter by flashbacking. Many of us can relate to the great rock and pop music produced around the world in the 70's. But economists and bankers saw things very differently about the 70's.

The United States of America was plunged into an economic depression by the global energy crisis of 1970. This led to high levels of inflation in the United States and increased unemployment. Perhaps this is why many people turned to music to make great music. The economy began to recover and improve only in the late 70's. The United States did the right thing and took the correct steps to improve the economy. This resulted in the economy of the United States being back on track by the late seventies and early eighties. Naturally, stock markets grew as well as the economy.

From the 1980s to mid-1987, markets rallied in unbroken fashion. This was one of the most successful bull runs in American history, according to traders. In August 1987, Dow reached an all-time record 2,722 levels. This represented a 44% increase over 1986. There were also signs of a stagnating economic environment. This is known as the'soft landing' or'soft landing', which is when the economy takes a break. The market began to take a break after August 1987's peak. There were mixed emotions in the months of August, Sept, and Oct 1987. Every small correction brought new long leveraged positions. There was also a lot of unwinding. The markets did not rally nor correct.

While all was going well on the domestic front with Iran bombing American super tanks stationed close to Kuwait's oil port, trouble was already brewing offshore. The October 1987 month was unique in financial market history. The sequence of events that took place during the 2 nd week in October 1987 is fascinating to me. There was too much drama and horror panting out across the globe.

- 14 th October 1987 (Wednesday), - Dow fell nearly 4% during that period, which was a record.
- 15 Oct 1987 (Thursday), - Dow fell another 2.5%. Dow was almost 12% lower than its August 1987 high. The other side of the world, Iran attacked an American supertanker stationed just outside Kuwait's oil port with a Silkworm missile.
- These two events caused enough panic and fear to spread throughout the global financial markets
- 16 October 1987 (Friday): London was enveloped by a sudden giant storm. Winds blowing at 175 kmPH caused blackouts throughout London, especially the southern area, which is the financial center. Officially, London markets were closed. Dow opened weak and fell nearly 5%. This created a global concern. The Treasury Secretary made a statement about economic concerns. This would only increase panic.
- 19 th October 1987 (
**Black Monday**) -- Markets began from Hong Kong and sliced off points like melting cheese. Panic reached London and finally the US. Dow experienced the largest ever drop with close to 508 (or 22.61%) being knocked off in a single day. This naturally attracted the Black Monday tile.

This dramatic turn of events was unprecedented in the financial world. This was one of the first 'Black Swan' events that made headlines. After the dust settled, a new breed was formed of Wall Street traders, who they called "The Quants".

Multiple repercussions were caused by the dramatic sequence of events that occurred in October 1987. Financial regulators were more concerned about the system-wide shocks and firms' ability to assess risk. If things were to shake up the financial sector again, financial firms were looking at the likelihood of a firm-wide survival. The theory predicted that October 1987 would not happen, but it did.

Financial firms are very likely to open speculative trading accounts across different geographies, with diverse counterparties and with various assets and structured assets. It is no easy task to assess risk at this level. This was what the company needed. They had to be able to calculate how much they could lose if October 1987 happened again. Quants, a new breed of risk managers and traders called themselves "Quants", developed sophisticated mathematical models that allowed them to evaluate and monitor real-time risk levels and monitor their positions. These people came with different doctorates, including those from finance, statistics, physics, mathematicians and traditional finance. Officially, firms recognized "Risk management" as an important layer of the system. Risk management teams were inducted into the "middle office" segment across Wall Street's banks and trading companies. All were working together towards the common goal of assessing risk.

Denis Weatherstone was then the CEO of JP Morgan. He commissioned the famous "4:15 PM" report. This report was a one-page document that gave Weatherstone a good idea of the total risk at the firm-wide levels. The report was due at his desk every day at 4:15 PM, 15 minutes after the close of the market. JP Morgan published the methodology in order to make it more accessible to other banks. JP Morgan eventually created a separate company called 'The Risk Metrics Group'. This group was later purchased by the MSCI group.

The report contained basically what is known as the "Value at Risk" (VaR), which is a metric that gives you an idea of the worst-case loss, even if it were to happen tomorrow morning.

This chapter is all about that. This chapter will focus on Value at Risk for your portfolio.

The concept of normal distribution is at the heart of the Value at Risk (VaR), approach. This topic has been discussed in Varsity multiple times. At this stage, normal distribution will not be explained. Let's just assume that you are familiar with what we're talking about. We will be discussing the Value at Risk concept. This is a quick and easy way to estimate portfolio VaR. This method has been used for several years and it works great for an equity portfolio that is "buy and keep".

Portfolio VaR is a simple way to answer these questions.

- What would be the worst possible portfolio loss if a black-swan event occurred tomorrow morning?
- What is the likelihood of the worst-case loss?

This is what Portfolio VaR does. These steps are simple and can be done in the following order:

- Identify the distribution of portfolio returns
- You can map the distribution to see if portfolio returns are 'normally distributed'.
- Arrange portfolio returns in ascending or descending order
- The last 95% observation can be viewed here
- Portfolio VaR is the lowest value in the last 95%
- The cumulative VaR (or CVar) is the average of the last 5 %

For a better understanding, let's apply this to our portfolio so far, and calculate its Value At Risk.

This section will focus on the two main steps involved in the calculation of the portfolio VaR. We will first identify the distribution of portfolio returns. We need to determine whether the portfolio returns are the normalized or direct returns. Remember that we already calculated the normalized return when we discussed the equity curve. This is the exact same thing.

**image 1**

These returns are found in the sheet titled EQ Curve. To calculate the Portfolio's Value at Risk, I have copied the portfolio returns to a separate sheet. The new sheet will look like this at this stage.

**image 2**

Our goal at this stage is not to determine the distribution of portfolio returns. We do the following:

**Step 1** Calculate the maximum and minimum returns from the time series (of portfolio return). This can be done using excel's '=Max() and '=Min() functions.

**image 3**

Step 2: Estimate the number data points. It is very easy to calculate the number of data points. This can be done using the '=count()' function.

**image 4**

You will see 126 data points. Please keep in mind that we are only looking at the last six months of data. You should run this exercise with at least one year of data. The purpose is to get the brief of concept out there.

**Step 3** - Bin Width

Now we need to create a 'bin array' in which the frequency of returns can be placed. Frequency of returns is a way to determine the frequency of a return. This allows us to answer the following question: "How many times have you seen a return of 0.5% in the last 126 days?" Calculate the bin width first.

Bin width = (Difference between max return and min return) / 25

Based on how many observations I have, I chose 25.

= (3.26% - (2.82 %))/25

=0.002431

**Step 4** Create the bin array

It is very simple: we begin with the lowest return, and then increase it by increasing the bin width. The lowest return would be -2.82, so that the next cell would contain this value.

= -2.82 + 0.024331

= - 2.58

This is the maximum return at 3.26%. We continue to increase it. This is the stage of the table.

**image 5**

Here's the complete list.

**image 6**

Now we need to calculate the frequency of return within the bin array. Let me first present the data and then explain what's going on -

**image 7**

To calculate frequency, I used the excel function '=frequency ()', to calculate it. The first row shows that only one observation had a return of -2.82% out of 126. There were zero observations that fell between -2.82% or 2.58%. There were also 13 observations between -2.82% and 2.58%. And so on.

We can calculate frequency by selecting all cells that are adjacent to Bin array and not deselecting them. This is how it looks.

**image 8**

Remember to press CTRL + Shift + Enter simultaneously, and not just enter. This will result in the frequency of the returns.

**Step 5** Draw the distribution

It is very simple. The bin array is where all the returns are located. Next to it is the frequency. This is the number of instances a return occurred. The frequency distribution can be obtained by plotting the frequency graph. Now, we need to visualise if the distribution looks normal (or a bell curve).

I can plot the distribution by selecting all frequency data and choosing a bar chart. This is how it looks:

**image 9**

It is obvious that what we see is a bell-shaped curve. Therefore, it is reasonable to assume that portfolio returns are normally distributed.

Once we have established that returns are distributed normally, we can calculate the Value At Risk. The rest of the process is straightforward. This involves reorganizing portfolio returns from ascending to descendant order.

**image 10**

To do this, I used Excel's sort function. I'll now calculate Portfolio CVaR and Portfolio VaR. This calculation is explained in detail below.

Portfolio VaR - This is the lowest value within 95% for an observation. There are 126 observations, which is 95% of 120 observations. Portfolio VaR, which is the minimum value of the 120 observations, is important. -1.48%.

I calculate the Cumulative VaR from CVaR by taking the average of remaining 5% of each observation.

The CVaR is -2.39%.

There may be many questions you have at this stage. Let me list them here with the answers.

- What was the purpose of plotting the frequency distribution for the portfolio?
- It is important to note that portfolio returns are usually distributed.

- Why is it important to check for normal distribution?
- If the data that we are studying has a normal distribution, then characteristics of normal distribution can be applied to it.

- What are the characteristics for normally distributed data?
- There are many, but it is important to note that 68% of data lies within one SD, 95% within 2 nd and 99.7% within the 3 rd. To learn more about the normal distribution, I recommend you read this chapter .

- What was the purpose of sorting the data?
- We have shown that the data set is usually distributed. Remember that we only care about the worst-case scenario. This means that if we sort it from the highest to the lowest, we can look at the returns more systemically.

- Why bother taking only 95% observation?
- Keep in mind that 95% of data falls within the 2 and standard deviations according to normal distribution theory. The portfolio's return is expected to be the sum of all 95% observations on any given day. The Value at Risk or the worst-case loss should be the value with the lowest 95% observation.

- What does the VaR value of -1.48% mean?
- It shows that the portfolio's worst-case loss is -1.49%. We can conclude this with confidence of 95%

- Is it possible for the loss to exceed -1.48%
- It can, and CVaR is a key factor. There is a chance of portfolio loss of -2.39% if there is an extreme event.

- Is it possible for the loss to exceed -2.89%
- It can, but it is very unlikely.

I hope you find the above discussion useful. I recommend that you apply it to your equity portfolio. You will be able to gain greater insight into your portfolio's position.

We've already discussed a lot of things about the portfolio and the risks associated with it. Now, we will discuss risk in relation to trading positions.

- Black Swan events are events with a low chance of occurring.
- Portfolios can suffer higher losses when a black-swan event occurs
- Value at Risk is one way to estimate the worst-case loss in the event of a black swan.
- The distribution of portfolio returns can be used to estimate the portfolio VaR.
- The portfolio's Value at Risk is calculated as the average of the last 5 % of observations.