AI for Trading Series №9: Portfolio Risks and Returns

Learn about the basics of portfolio theory, which are a key for designing portfolios for mutual funds.

Purva Singh
Dec 24, 2020 · 10 min read
Photo by janilson furtado on Unsplash

Till now in our ‘AI for Trading Series’, we have learned about the indices and ETFs, their applications in the real world and how they work on a transaction level. In this article, we will look at the risks and return properties of a collection of stocks.

For example, consider a scenario where you have done your research and have come up with a list of stocks you want to invest in. You have calculated the amount of money you have to spend and you are now ready to buy those stocks that are needed to construct your portfolio for an ETF. But the main question is

On a high-level, you may feel that you should invest in those stocks that you expect to have highest returns. But what if the prices of these stocks end up fluctuating the most, i.e. they entail the greatest risk. For a while, you main gain returns but then, you could lose all of it. So the main question is:

Let us look at some of the ways in which we can answer this question! 😄📈

Reducing Portfolio Risks

Diversification

Consider a scenario where you have done you research and have come to a conclusion that based on historical data, Company A belonging to a IT Sector is going to perform well in future. Based on this conclusion, you invest all of your money in Company A. Now, Company A does well for a while and then one fine day, it drops to half the price you purchased it for. Since you used all your money to buy the stock, you just lost half your money!!

What if you put half your money in Company A and the other half in Company B belonging to the Pharma sector. In this case, Company A could start doing well when it launches a new tech product, while Company B recovers from a failed drug trial. Also, Company B can start doing well after promising many research leads and Company A could struggle during management change.

Diversifying Portfolio. (Source: AI for Trading nano degree course on Udacity)

In this scenario, we have reduced our portfolio risks by diversifying our portfolio. But one question still prevails: can we reduce our risks indefinitely by spreading our money into more and more stocks?

If all the sources of risks are independent, we could in theory, reduce risk to zero , by spreading our money between more and more stocks. This would be the case if each company is only subjected to its own independent sources of risk and there are no risks common to all companies.

Idiosyncratic and Systematic Risks. (Source: AI for Trading nano degree course on Udacity)

Risks specific to individual companies are called idiosyncratic risk or specific risk. However, in real world, all companies are subjected to common sources of risk that affect the entire economy such as risks of inflation or recession. Risks that are attributable to market-wide sources are called market-risk or systematic risk. Next, we will see ways to quantify this risk and find some guidelines to allocate our money.

Portfolio Mean

Consider the above scenario, where you have invested in 2 companies: A and B. We will call Wa and Wb are the weights on assets A and B and these weights sum up to 1. Now, to calculate the mean and variance of our portfolio, we will need to think of future log returns as random variables. You can think of returns at a future time as being indexed by the variable i.

Portfolio Mean Formula

The total return of the portfolio in each scenario is defined as the weighted sum of returns of each individual asset.

Total Portfolio Return

The expected value of portfolio return is the weighted sum the individual stock’s expected returns.

Expected Portfolio Returns.

Portfolio Variance

Now, we will measure the total risk inherit to the portfolio. We will measure this risk with volatility or more specifically with portfolio variance (square of the volatility). The variance of portfolio can be represented as follows:

Variance of an individual asset.

We can write the formula for variance of an individual asset in terms of Covariance. A covariance is a measure of joint variability of two random variables. When stock A is above its average and stock B is also above its average, we can say that the two stocks are varying together or covarying and have a positive covariance.

Deriving the Portfolio Variance

Over here, we will see how variance of log return distribution is related to the covariance of stocks in your portfolio.

Derivation of Portfolio Variance

Portfolio Covariance

Covariance is the correlation between the two variables times each of their standard deviations. Correlation coefficient takes values between -1 and +1. Both correlation and covariance are measures of how much two variables vary together.

Covariance between two variables

Hence, the portfolio variance can be re-written as:

Portfolio Variance

Now, let’s see what happens when the correlation between stock A and stock B is +1 and -1.

  1. When Covariance Coefficient is +1

In this scenario, the correlation between our two-asset portfolio (stock A and stock B) is +1. The graph of two positively correlated stocks should be as follows:

Two positively correlated stocks. (Source: AI for Trading nano degree course on Udacity)

Now, we will place +1 in place of the correlation coefficient to get the portfolio variance when the stocks in our portfolio are positively correlated. The updated portfolio variance is as follows:

Portfolio’s Standard Deviation.

2. When Covariance Coefficient is -1

Now, lets consider a scenario when the correlation between the two stocks in our portfolio is -1. The graph of two negatively correlated stocks shoould be as follows:

Two negatively correlated stocks. (Source: AI for Trading nano degree course on Udacity)

Again, putting the correlation coefficient as -1 in the formula, we get:

Portfolio Standard Deviation for Negatively Correlated Stocks.

In case when the correlation between the two stocks in -1, we can get a perfectly hedged portfolio.

We can solve the below equation, to get a perfectly hedged portfolio:

Weights that drive Portfolio Variance to Zero

However, in reality, since every asset is affected by systematic risk, the correlation between two assets will never reach -1.

Covariance Matrix

Now, lets look back at the portfolio variance formula between two stocks.

Covariance Matrix.

Now that we know how to calculate portfolio variance in terms of portfolio weights and covariance matrix, let us use the Numpy library to calculate the covariance matrix, given the return series of a set of stocks. For this, we will leverage the numpy.cov method. The Jupyter notebook for the same can be found below:

The Efficient Frontier

Till now, we have learned about diversification and how to calculate portfolio mean and variance. Now, we will learn, what are the best ways to assign each stock in our portfolio. Before we dive into portfolio optimisation, let’s have a look at all the sets of portfolios.

Let us understand this question with an example. Suppose you want to invest $10,000 in a portfolio of 7 stocks. Obviously, you would want to do this in such a way where, you have the highest returns and the least risk (volatility). For this, you will assign weights to the stocks in your portfolio and experiment with multiple combinations of weights that will give you the maximum weights with the least risk.

Experimenting with Portfolio Weights. (Source: AI for Trading nano degree course on Udacity)

The figure shows just one of the many simulations we would do in order to get the perfect combination of return v/s risk. For example in the below table, we can see that both Scenario 1 & 2 have the same risk, but Scenario 1 has a higher expected return. So, we will prefer Scenario 1. Also, Scenario 4 has the highest expected return but it also has the highest risk. Scenario 4 is a bit unattractive portfolio as it is the one with the highest risk.

Portfolio Simulations. (Source: AI for Trading nano degree course on Udacity)

Now, if you plot all these simulations on a graph, i.e. plotting expected portfolio return v/s portfolio volatility, we will get a graph as below. Each dot represents a possible risk-return combination that can be generated by a portfolio of stocks. The x-axis is the volatility (risk) and the y-axis is the return.

Portfolio Risk v/s Portfolio Return Graph. (Source: AI for Trading nano degree course on Udacity)

Also, if you carefully notice the graph, all the dots on the upper boundary perform significantly better than the dots below the boundary line. This is the Efficient Frontier. The efficient frontier is the upper boundary of the set of possible portfolios. Portfolios on this boundary have the maximum achievable return for a given level of risk. Any portfolios above the frontier are unachievable.

The Efficient Frontier. (Source: AI for Trading nano degree course on Udacity)

The Capital Market Line

A risk-free asset is an investment instrument that entails absolutely no risk or uncertainity. In theory, if you invest in such an instrument, you receive a guaranteed rate of return called the risk-free rate. In reality, entirely risk-free rate doesn’t exist since all the investments carry a certain level of risk. However, in practice, people normally refer to the rate of return on a three-month treasury bill as the risk-free rate.

Let us consider a risk-free asset (x-axis=0) and a market portfolio. Now this new portfolio would be a weighted sum of risk-free asset and the market portfolio. In the figure below, if you chose the green dot as your market portfolio and the red dot as your risk-free asset, then the line between them represents the potential portfolios that you can construct with the two assets.

Capital Market Line. (Source: AI for Trading nano degree course on Udacity)

Now, in order to achieve the best return for a given level of risk, we would chose, a market portfolio that allows us to draw a straight line starting from the risk-free asset that just touches the top of the efficient frontier.

Now, in order to calculate the expected return of the portfolio consisting of the risky portfolio and the risk-free asset, we calculate the formula for the capital market line, because each point on this line gives the return as a function of a possible combination of the risky portfolio and the risk-free asset.

Expected Return of Portfolio consisting of Risk-free Asset and Risky Portfolio.

The Sharpe Ratio

The Sharpe Ratio is the ratio of reward to volatility. It’s a popular way to look at the performance of an asset relative to its risk.

Sharpe Ratio.

The numerator of the Sharpe ratio is called the excess return, differential return as well as the risk premium. It’s called “excess return” because this is the return in excess of the risk-free rate. It’s also called the “risk premium”, because this represents the premium that investors should be rewarded with for taking on risk. The denominator is the volatility of the excess return.

The Sharpe Ratio allows us to compare stocks of different returns, because the Sharpe ratio adjusts the returns by their level of risk.

Annualized Sharpe Ratio

The Sharpe Ratio depends on the time period over which it is measured, and it’s normally annualized. The formula for calculating annualized Sharpe Ratio is given as follows:

Annualized Sharpe Ratio

Here, 252 represents the number of trading days in a year.

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Purva Singh

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Hi! I am a tech enthusiast currently working on leveraging language technologies to solve financial use-cases! View my work here: https://purvasingh96.github.io

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