Modern Portfolio Theory and Index Funds

Modern Portfolio Theory and Index Funds

My introduction to Modern Portfolio Theory was through the Financial Economics course I took while studying for my master’s degree in the USA. I am a curious person. I have been curious about modern finance for a long time, and when the opportunity came my way, I took it. The story of Modern Portfolio Theory begins with an article published in 1952 by Harry Markowitz. Let me add that Markowitz later won the Nobel Prize in economics with this theory he developed. Modern Portfolio Theory is a solution to a mathematical optimization problem that maximizes expected return given the level of risk. The answer to this problem shows that a risk-averse investor can create a portfolio that maximizes the expected return for each risk level. The curve formed by these portfolios is called the efficient frontier.

Modern Portfolio Theory and efficient frontier

Measuring risk

The chart above shows that the standard deviation on the x-axis measures the risk. Those familiar with mathematics and statistics know that standard deviation is the square root of variance. Variance, on the other hand, is a value that indicates the deviation from the mean in a sample. The extent to which the actual rate of return deviates from the rate of return you expect (the average value) defines the risk. There are objections to this definition at the academic level. Still, this blog is not the place for this topic. We now have the opportunity to quantify and compare the rate of return and risk.

Systematic risk

Under Modern Portfolio Theory, the risk falls into two categories: systematic and unsystematic. The theory suggests that we can eliminate unsystematic risk by creating a portfolio or, in colloquial terms, a basket. In other words, if you invest in a single stock, you assume both systematic and non-systematic risks specific to the company. If you create an extensive enough portfolio, even if a firm performs poorly, the return on your portfolio will not be affected much. Thus, a rational investor eliminates unsystematic risk by creating a diversified stock portfolio.

Systematic risk, on the other hand, is a risk that affects the whole market and cannot be eliminated by building a portfolio. For example, let’s say the Government has increased corporate tax from 20% to 30%. The shares of all operators traded in Borsa Istanbul are affected by this risk. As an investor, we cannot eliminate this by creating a portfolio. Systematic risk; Non-diversifiable risk is also called volatility or market risk.

Finding the least risky portfolio

Modern Portfolio Theory says that a portfolio that minimizes the risk of a risk-averse investor lies on the efficient frontier. This portfolio is technically called the minimum variance portfolio. In other words, you have no chance to lower the risk by creating a different portfolio. Modern Portfolio Theory has one more important thing to say. If you want more returns, you have to take more risks. In other words, you must move away on the efficient frontier (to the right on the x-axis) to create a portfolio that provides higher risk and higher expected returns.

Application of modern portfolio theory

Let me share an old work I did several years ago for those curious about the application of Modern Portfolio Theory. I’ve been following stocks for years but didn’t buy them because I found their prices high. In 2018, the stock market’s value dropped to historic levels. It was time for me to enter the stock market. Last year, I searched for the answer to the question of what should be the share weights in a portfolio that minimizes risk based on some of the stocks I am interested in. If interested, you can do similar exercises in Excel using the Solver add-in.

Returning to work, I first downloaded the historical price data of stocks from Yahoo Finance and calculated the monthly rate of return. I then calculated the mean, the standard deviation, of these rates of return. As seen in the figure, Tüpraş has the highest average rate of return (1.6% = 0.02) and the lowest standard deviation (9.9%). On the other hand, Sigma is the variance-covariance matrix created with the monthly return rate data of the shares. We use it while minimizing the risk of the portfolio. Below the figure is the share weights we are looking for, which minimizes the risk. Below that is the portfolio’s expected return (E[Rp,m = 0.9%]) and its standard deviation (SD[Rp,m = 8%]. The portfolio’s standard deviation is lower than the standard deviation of all individual stocks at 8%!

Portfolio weights that minimize risk

More returns more risk

While calculating the portfolio weights that minimize the risk, we may have a specific rate of return that we aim for. We can also calculate portfolio weights that minimize risk under this target rate of return constraint. For the sample app, I set my monthly rate of return target as 1.2%. As can be seen from the figure below, portfolio weights change significantly under this constraint. For example, the weight of Tüpraş increases from 36% to 45%. More importantly, while the expected return rate of the portfolio increased from 0.9% to 1.2%, the portfolio’s risk, that is, the standard deviation of the expected rate of return, rose to 8.2%. In summary, we have to take more risks for more rewards.

Portfolio weights that minimize risk under a certain rate of return target

Does Modern Portfolio Theory work?

Photo by Roberto Júnior on Unsplash

Professional fund managers apply Modern Portfolio Theory in their daily business life. So, does Modern Portfolio Theory work in real life? Knowing some of the theory’s assumptions is necessary: (1) Investors maximize their profits. (2) Investors are rational and avoid unnecessary risk. (3) All investors have the same information. (4) Taxes and transaction costs are not taken into account. (5) Investors do not have the power to influence market prices. (6) Investors can borrow as much as they want at a risk-free rate of return. As you can imagine, some of these assumptions are invalid in real life.

Famous value investor Warren Buffet says that this is just a theory. For success in investing, the investor must rely not only on mathematics but also on his own skill and effort. However, I still think the main implications of the theory are valid. So you can reduce your investment risk with a well-diversified portfolio.

Capital Asset Pricing Model

Another essential analytical technique based on the Modern Portfolio Theory is the Capital Asset Pricing Model (CAPM). Those who developed this model also won the Nobel Prize in finance. CAPM is a technique used to determine the price of a portfolio or a single stock in a portfolio. Putting it verbally without diving into formulas, the CAPM says that a stock’s risk premium equals its market risk premium times beta.

The risk premium shows how much a stock’s expected return is more than a risk-free investment’s return. Beta is the sensitivity of a stock’s expected risk premium to the market risk premium. In other words, beta represents the systematic risk of individual security relative to the market. If beta equals one, we can interpret the stock’s risk as similar to the market risk. If it is more significant than one, it is risky compared to the market, and if it is less than one, it is less risky than the market.

Security Market Line

If we graph the CAPM, we get the security market line (SML). The X-axis shows beta, that is, risk. The y-axis shows the expected rate of return. On the other hand, SML offers a portfolio’s expected return-risk (beta) level that includes all securities in a market. As we mentioned earlier, the beta of SML is equal to one. As seen from the chart, CAPM says that increasing the expected return on investment can only be possible by taking more systematic risks.

What does SML do in practice?

Photo by Chris Liverani on Unsplash

In fact, SML is an investment evaluation tool derived from CAPM. We can use SML when deciding whether to include a stock in our portfolio. When we evaluate the return – beta values of the individual stock on the SML chart, we can interpret that this stock is undervalued (cheap) if it is above the SML line. And overvalued (expensive) if it is below the SML line. We can also use SML to determine which of the two stocks with similar rates of return is less risky.

What are we going to do with these techniques?

Don’t worry. I don’t recommend using these techniques actively. I don’t use it. After all, I am not a full-time professional portfolio manager. However, knowing what these theories mean is essential to being an informed investor. Let me summarize the key points in a few sentences: (1) By creating a sufficiently diversified portfolio; we can eliminate unsystematic risk without reducing the expected rate of return. (2) More systematic risk is required for a higher rate of return.

Modern Portfolio Theory implies

Creating a portfolio of all stocks to eliminate unsystematic risk may be an extreme solution. Instead, creating a portfolio that mimics market indices (such as S&P500 or Bist100) may be more practical. As a matter of fact, index investing as a passive investment strategy is quite popular today. Investments with this method outperform 95% of mutual funds actively managed by professional portfolio managers in the long run. So unless you’re a Peter Lynch or Warren Buffet, this seems your best option. In addition, it does not require deep financial knowledge for the investor. It saves the investor from the problem of stock selection. The savings rate is the most crucial variable for financial independence in the short and medium term. Therefore, using index investment and high savings rate together is the key to success.

Can it be done in Turkey?

One of the problems in Turkey is that there are no financial products (such as Vanguard S&P500 index funds or ETFs) that mimic the low-cost index as in the USA. Of course, there is also the problem of the low historical return of our stock market. Due to the poor management of the economy and underdeveloped financial institutions, Borsa Istanbul has a cycle of sharp ups and downs. Long-term investment does not seem very profitable.

In my opinion, in two different ways, we can approach the solution to the problems in Turkey. We can apply active portfolio management by taking advantage of the ups and downs in the stock market and targeting medium-term earnings. However, I find it difficult for the small investor to achieve this. Even with sufficient knowledge and experience, it takes a lot of time. After all, we all work in salaried jobs. The second solution is to invest abroad and Vanguard S&P500 index fund etc. It may be to purchase similar developed/emerging country ETFs with funds. Here, too, transaction costs become essential. The temporary solution I found was to create a portfolio consisting of Eurobonds issued by the Treasury and 4-5 stocks. Then I added low-cost ETFs to the portfolio once the portfolio reached a specific size and the rates of return decreased.

See you next post.


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