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Module 9 Risk Management Trading Psychology

Chapter 3

3.1 – Variance

In the previous chapter, we touched upon the topic of expected return, continuing on it, we will understand the concept of ‘Portfolio variance’. Portfolio Variance helps us understand the risk at a portfolio level. I’m hoping you are familiar with ‘Standard Deviation’ as a measure of risk. We have discussed standard deviation multiple times in the previous modules (refer to Click women fall new product model show at the mouth of the womens fish large blue golden sands fine highheel shoes Blue odVckZepl
). I’d suggest you get familiar with it if you are not already. While we can easily measure the risk of a single stock by calculating its standard deviation, calculating the risk of a portfolio is a whole different ball game. When you put a few individual stocks together and create a portfolio, it becomes a different animal altogether. The agenda for this chapter is to help you understand how to estimate risk at a portfolio level.

However, before we proceed, we need to understand the concept of Variance and Covariance. Both Variance and Covariance are statistical measures. Let’s deal with the Variance first.

The variance of stock returns is a measure of how much a stock’s return varies with respect to its average daily returns. The formula to calculate variance is quite straight forward –

Where,

σ = Variance

X = Daily return

µ = Average of daily return

N = Total number of observation

Note, the variance is measured as sigma squared; I will not get into the reasons for this as the explanation is quite complex and we could digress. For now, I’d request you to be aware of the fact that variance is sigma squared. Anyway, calculating variance is quite simple, I’ll take a simple example to help us understand this better.

Assume the daily return for a stock for 5 consecutive days is as below –

Day 1 – + 0.75%

Day 2 – + 1.25%

Day 3 – -0.55%

Day 4 – -0.75%

Day 5 – +0.8%.

In this case, the average return is +0.3%. We now need to calculate the dispersion of daily return over its average return, and also square the dispersion.

We now sum up the dispersion squared to get 0.0318000%. We divide this over 5 (N) to get the variance i.e

0.0318000% / 5

σ= 0.0063600%.

So what does this number tell us? It gives us a sense of how the daily returns are spread out from the average expected returns. So you as an investor should look into the variance to determine the riskiness of the investment. A large variance indicates that the stock could be quite risky while a small variance can indicate lesser risk. In the above example, I would consider the variance high, since we are looking at just 5 days worth of data.

Now, here is something you may be interested in knowing. Variance and standard deviation are related to each other by the following simple mathematical relationship –

Square Root of Variance = Standard Deviation

We can apply this to the example above and calculate the 5-day standard deviation of the stock,

%

~ 0.8%

which is the standard deviation a.k.a. the volatility of the stock (over the last 5 days). Anyway, at this point, I want you to be aware of Variance and what it really means. We will eventually plug variance along with covariance into the portfolio variance equation.

3.2 – Covariance

Covariance indicates how two (or more) variables move together. It tells us whether the two variables move together (in which case they share a positive covariance) or they move in the opposite direction (negatively covariance). Covariance in the context of stock market measures how the stock prices of two stocks (or more) move together. The two stocks prices are likely to move in the same direction if they have a positive covariance; likewise, a negative covariance indicates that they two stocks move in opposite direction.

I understand covariance may sound similar to ‘correlation’, however, the two are different. We will discuss more on this further in the chapter.

I guess calculating the covariance for two stocks will help us get a grip on understanding covariance better. The formula to calculate covariance of two stocks is as follows –

Where,

Rt S1 = Daily stock return of stock 1

Avg Rt S1 = Average return of stock 1 over n period

Rt S2 = Daily stock return of stock 2

Avg Rt S2 = Average return of stock 2 over n period

n – The total number of days

In other words, you can calculate the covariance between two stocks by taking the sum product of the difference between the daily returns of the stock and its average return across both the stocks.

Sounds confusing? I guess so.

Let us take up an example and see how we can calculate the covariance between two stocks.

For the sake of this illustration, I’ve selected two stocks – Cipla Limited and Idea Cellular Limited. To calculate the covariance between these two stocks, we need to work around with the above formula. We will resort to good old excel to help us implement the formula.

Before we proceed, if you were to guess the covariance between Cipla and Idea, what do you think it would be? Think about it – two large corporate, similar size, but in two completely unrelated sectors. What do you think would be the covariance? Give it a thought.

Anyway, here are the steps involved in calculating covariance in excel (note, although there is a direct function in excel to calculate covariance, I’ll take the slightly longer approach, just to ensure clarity) –

Step 1 – Download the daily stock prices. For the purpose of this illustration, I’ve downloaded 6 months data for both the stocks.

Step 2 – Calculate the daily returns for both the stocks. Daily returns can be calculated by dividing today’s stock price over yesterday’s stock price and subtracting 1 from the result of this division

Step 3 – Calculate the average of the daily returns

Step 4 – Once the average is calculated, subtract the daily return by its average

Step 5 – Multiply the two series calculated in the previous step

Step 6 – Sum up the calculation made in the previous step. Take a count of the number of data points. You can do this by using the count function in excel and giving any of the fields as the input array. I’ve used the count on the dates here.

Step 7 – This is the final step in calculating the covariance. To do so, one needs to divide the sum by count minus 1 i.e (n-1). The count, in this case, is 127, so count-1 would be 126. Sum calculated in the previous step was 0.006642. Hence, covariance would be

= 0.006642/126

= 0.00005230

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the excel sheet.

As you can see, the covariance number is quite small. However, that’s not the point here. We only look at whether the two stocks share a positive or negative covariance. Clearly, since the two stocks share a positive covariance, it means that the returns of the two stocks move in similar directions. It means that for a given situation in the market, both the stocks are likely to move in the same direction. Note – covariance does not tell us the degree to which the two stocks move. The degree or magnitude is captured by correlation. The correlation between Idea and Cipla is 0.106, which indicates that the two stocks are not tightly correlated.

By the way, here is something very interesting fact. The mathematical equation for correlation between two stocks is as follows –

Where,

Cov (x,y) is the covariance between the two stocks

σ= Standard deviation of stock x

σ= Standard deviation of stock y

Note, the standard deviation of a stock is simply the square root of the variance of the stock. Here is a task for you – we have calculated the correlation between Idea and Cipla using the direct excel function. Can you confirm the accuracy by implementing the formula?

Anyway, in the case of building a stock portfolio, do you think a positive covariance is good or bad? Or rather do portfolio managers desire stocks (in their portfolio) which share a positive covariance or they don’t? Well, portfolio managers strive to select stocks which share a negative covariance. The reason is quite simple – they want stocks in the portfolio which can hold up. Meaning if one stock goes down, they want, at least the other to hold up. This kind of counter balances the portfolio and reduces the overall risk.

Now, think about a regular portfolio – it will certainly contain more than 2 stocks. In fact, a good portfolio will contain at least 12-15 stocks. How would one measure covariance in this case? This is where things start getting complicated. One will have to measure covariance of each stock with all the other stocks in the portfolio. Let me illustrate this with a 4 stocks portfolio. Assume the portfolio is like this –

In this case, we need to calculate the covariance across –

Note, the covariance between stock 1 and stock 2 is the same as the covariance between stock 2 and stock 1. So as you can see, 4 stocks require us to compute 6 covariances. You can imagine the complexity when we have 15 or 20 stocks. In fact, when we have more than 2 stocks in the portfolio, the covariance between them is calculated and tabulated using a ‘Variance – Covariance Matrix’. I would love to talk about this now, but I guess, I’ll will keep it for the next chapter.

Stay tuned for more!

Key takeaways from this chapter

Module 9

Chapters

says:
March 16, 2017 at 3:41 pm

I have calculated variance of stock using daily return of last 1000 days. Then I calculated Standard Deviation using square root of variance.Then I calculated Standard Deviation using standard deviation method. But values are not matching. Do you know why? e.g I took “NSE:ITC” everyday close value of past 1000 days. So from variance calculations I am getting “0.000487” and from Standard Deviation method I am getting “0.1788465”. Am I doing anything wrong calculations?

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January 26, 2017 By Contributor

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The destinies of cultures can be read in games.

–Roger Callois, Man, Play, and Games

Before it was stolen, patented, and sold to the Parker Brothers, Monopoly was “ The Landowner’s Game ,” a Georgist propaganda piece meant to illustrate the unfair behavior of the landowning class. The game accomplished this by setting up rules and fictions (game mechanics ) that generate a reliable system behavior (game dynamic) which produced the intended experience (aesthetic): That aesthetic, frustration , has disrupted family game nights for decades. The dynamic is familiar to nearly anyone who has played it: those who manage to own more property have the money and power to be better insulated against chance, and those who don’t are likely to lose even more. The game spirals out as losers are burnt down to nothing and winners become even more powerful. Winners might find the game fun. Losers are deliberately irritated by a slow, nearly unavoidable death. In theory, the game mechanics could be adjusted to produce a ‘smoother’ outcome for more players, but it was never the point. It wouldn’t be “Monopoly” otherwise.

Monopoly’s rude feedback loop, illustrated in Hunicke et al., MDA: A Formal Approach to Game Design and Game Research.

Games are living things. In multiplayer experiences, the community of players can enforce rules, eject jerks, and adapt the game to fit their collective needs through precedent setting, apologetics, etc.- I often have to ask about “house rules” when getting into the nitty-gritty of Monopoly – what do you do with Public Parking in this house? Are there housing limitations? The manual has “definitive” answers but people sometimes have their own canon.

Often, on the first tests of a newly designed multiplayer game, the mechanics won’t generate the intended system behavior because humans are tricky. Even if the system gets the major kinks worked out over time, we can still anticipate the occasional flouting of the letter of the rules (i.e. cheaters) and, often even worse, the flouting of the spirit of the system (i.e. spoilsports, who are technically not so much cheaters as the game-world’s apostates or psychopaths).

Play is apparently a very old behavior- certainly older than humans, as we see animals imitate aggression for sport or personal training or pecking order or maybe for no purpose at all. Structured games are younger, likely as old as civilization- some philosophers believe that it is the basis of civilization altogether. Despite these broad claims by some, ‘game design’ largely refers to the design of a very narrow set of artifacts: usually videogames (and sometimes board games), usually designed by a specific person or group and constructed for use with a narrow set of technologies to be sold for profit. I hope to convince you that, far beyond even Monopoly, there is a world of play and games that is much broader than definitions including ‘fun’ or ‘non-seriousness’, and that there exist frameworks for thinking about the way that games are built and maintained by play communities.

Return to footnote 7 referrer

Total - Private households by household size - 100% data

Household size refers to the number of persons in a private household.

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Total - Population aged 15 years and over by marital status - 100% data

For more information, refer to the Census Dictionary: Marital status .

Return to footnote 9 referrer

Total - Census families in private households by family size - 100% data

Census family is defined as a married couple (with or without children of either and/or both spouses), a common-law couple (with or without children of either and/or both partners) or a lone parent of any marital status, with at least one child living in the same dwelling. For more information, refer to the Census Dictionary: Census family structure .

Return to footnote 10 referrer

Total number of census families in private households - 100% data

'Census family structure' refers to the combination of relatives that comprise a census family. Classification on this variable considers the presence or absence of: married spouses or common-law partners and children.

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Total - Private households by household type - 100% data

For more information, refer to the Census Dictionary: Household type .

Return to footnote 12 referrer

One-census-family households - Without children in a census family

The household includes a single couple (without children). Other persons related or unrelated to the couple may also be present.

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One-census-family households - With children in a census family

The household includes either a couple with children or a lone parent and his or her children. Other persons related or unrelated to the census family may also be present.

Return to footnote 14 referrer

Total - Knowledge of official languages for the total population excluding institutional residents - 100% data

Knowledge of official languages refers to whether the person can conduct a conversation in English only, French only, in both languages or in neither language. For a child who has not yet learned to speak, this includes languages that the child is learning to speak at home.

For more information on language variables, including information on their classifications, the questions from which they are derived, data quality and their comparability with other sources of data, please refer to the Languages Reference Guide, Census of Population, 2016 .

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