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Ace your Portfolio Theory exam - MPT and CMT explained (28:32)
Videos are available at one organized Quant 101 Playlist on YouTube (opens in a new browser window).
Welcome. Today's goal is to get our arms around two important theories from a half century ago, to define terms that come up on test day and to see what aspects are practical.
I'm Paul, and no matter how many times I've seen portfolio theory whether in college or for CFA test preparation I'm always a bit fuzzy, especially when trying to balance the theory and its many unrealistic assumptions with what goes on in the real world.
So here we will walk through both Modern Portfolio Theory and Capital Market Theory together focusing on the concepts so you can ace your exam and at least be dangerous when it comes to speaking about the theory.
Here we are roughly in the middle of a 30 tutorial series called Quant 101. This specific tutorial is for both the casual observer on YouTube and for those who, by Chapter 3, worked with a real data set long enough to plot a Portfolio Possibilities Curve on a risk and return scatter plot in Excel using real historical data.
If you would like the full transcript of the video, including where you can download a data set, the first link in the video description goes straight to it. There you will other information about this free financial modeling tutorial series.
So with that, let's look at our plan for today.
First, we will make the connection between the curve charted on the risk and return plot with its source data, stock returns.
Second, we jump to concepts associated with Modern Portfolio Theory and work by Harry Markowitz.
Third, we add on top of that Capital Market Theory developed a decade later.
Fourth, we take a step back and cover what happens in the non-academic world at Institutional asset management firms today.
And in our next episode we will reinforce statistical concepts before moving on to portfolio optimization and advanced Excel skills.
As mentioned, we walked through the creation of a risk and return plot with a curve that depicts possible portfolios, and here we will expand on that.
There will be a lot going on here, so try to focus on each line, curve and dot as we introduce it.
To start out we're working in two dimensions here, so start with a line plotting an x-axis and y-axis.
On the x-axis we plot risk and sometimes scholars use variance here but we prefer the standardized version of variance called standard deviation as it is more interpretable.
Risk can't be negative by definition so our x-axis starts at zero.
On the y-axis we plot return and three points are important here. First, the calculation uses the average arithmetic return method.
Second, to back up a moment, here we try to be clear on timeframes used in financial modeling and they apply to both return and risk. When looking back we call this the historical timeframe to be clear. Up to this point we have focused exclusively in the past. You can think of what sits on the Returns tab as a whole slice of data for this historical period.
The academic framework shifts to the present and for this it is customary to use data from the past to see what is built into market expectations. That is where the expected timeframe comes into play, which will be the focus of later tutorials in the series.
Finally, active portfolio managers are in the business of forecasting future stock returns that differ from market expectations, or else they would be passive index managers. So the forecast period refers to the return side as well as the risk side. Typically investors subscribe to data from third-party risk model providers for risk forecasts.
So when looking back the returns on the y-axis can certainly be negative depending on recent stock market performance. Stocks definitely post negative returns for short periods of time, even for periods as long as 3 and 5 years. But when we set expectations and forecasts it is more common to predict positive rates of return for stocks, or you would likely just invest in cash.
The last point is on the curve. Earlier we called it a parabola, but technically it is a hyperbola, and here we focus on the top half.
In previous tutorials we saw how this was created so let's take a little mathematical mystery out of this and retrace our steps a bit and review portfolio math for the two-stock case. This will help us understand the theories as we transition to thousands of stocks.
Of course, these concepts are easier after working with real data, and in the last tutorial we charted portfolios and created the Portfolio Possibilities Curve. We altered weights in 10% increments to two stocks and monitored changes in portfoio return and saw the free benefit of risk reduction from diversification to stocks with less than perfect correlation. Meaning a 50/50 portfolio produced an average return at the midpoint between the return on Microsoft and eBay, yet the risk wasn't on that line segment, it was at a position well to the left, or with lower risk. So there we quantified diversification.
To do so we had to work through the formula for portfolio risk discussing the nuances of the covariance matrix, simplifying the confusing formula notation to basic English. We saw how the covariance matrix, stores risk data and sits behind the scenes in optimization programs and risk analyzers.
Before that, for the y-axis we nailed down the calculation of portfolio return which is simply the weighted average of returns using the arithmetic method. In Chapter 2 we covered other material on stock return calculation methods and procedures.
And in likely the most important tutorial of the whole series we mastered the calculation of stock risk, calculating a variance and standard deviation for each individual stock and a covariance and correlation for each pair of stocks.
All of this is based on a small subset of data that sits on the Returns data tab.
So that gives you a little roadmap to follow if anything here for you is fuzzy.
As an aside, where I think people in the Technology industry have an advantage over those coming from a background in Finance, at least from my experience in creating tutorials for Linux, Python and web development, is that students with a Technology background are more comfortable grabbing data and investing their time playing with it.
This helps people learn and remember, so I encourage you to do that, because some of the math behind all of this theory is easy to forget. Struggling through it with examples will pay huge dividends down the road.
Now let's say goodbye to our 2-stock case for good and move to the more realistic scenario in a multi-asset world, leaning on portfolio theory to take us there and our first stop is in the 1950s with Modern Portfolio Theory.
With that, let's start with a basic question, "What is the 'Market'?"
For now we are thinking of the concept of the 'Market' the way a scholar would see it. People often have a difficult time explaining what it is, because it has so many different contexts.
In the United States, many people think of the 'Market' as the broad stock market, like the S&P500 or the 30-stock Dow Jones Industrial Average. For others it may be the S&P 1500 which includes small-cap companies, or the Wilshire 5000 which is even more broad.
As we look around the globe, individuals in Germany may think of the 'Market' as the DAX, in France the CAC 40, in the UK the FTSE 100 Index. In Japan, Hong Kong and China, the 'Market' may be the Nikkei 225, the Hang Seng and Shanghai Composite, respectively.
These broad equity benchmarks are country-specific, so to others who think more globally, there is the MSCI World, S&P Global 1200 or the Dow Jones Global Index.
Now, as we start to head into the realm of theories about the 'Market' and participants, those in academia see the 'Market' differently. Think about it, why should the 'Market' be focused on equities?
As we broaden out the definition, the 'Market' takes a life of its own. Rather than narrowly limiting it to equities, as many do, here we need to also consider bonds, real estate, options, art, collectibles and commodities? In theory they are all risky assets available to investors and are included in the academic definition of the 'Market'.
Now before I dive too deeply, we should bring this all back to the advancements made in the 1950s. Prior to that time, investors were picking stocks using Fundamental and Technical Analysis. Little work was done on the topics of risk, volatility and risk aversion so there was no scientific work disseminated about how to quantify diversification. So setting weights to stocks in practice was ad-hoc, arbitrary and likely rules-based. Like creating a portfolio by equally-weighting 20 stocks, for example.
Then Harry Markowitz, who studied at the University of Chicago, wrote a 15-page paper called Portfolio Selection, which was published in The Journal of Finance in 1952.
He brought his expertise from physics, philosophy and applied mathematics to analyze stock market behavior. His focus was on the creation of a portfolio, rather than the prevailing research at the time which was focused on the discounting of cash flows to determine the value of individual stocks.
So the 'rulers of the day' included fundamental analysts, like John Burr Williams, Benjamin Graham and David Dodd, as covered in our review of stock market history earlier in this series.
Okay, back to the advancements on portfolio construction. Markowitz postulated that expected return was "a desirable thing" and variance of return was "an undesirable thing." We saw this. A portfolio in the top left is better than one in the bottom right. The trade-off between the two was his maxim, or hypothesis, for investor behavior.
He also postulated that there exists a diversified portfolio which is preferable to all non-diversified portfolios. This is where the concept of the 'Market' came into play.
Later, in 1959, he published a book titled Portfolio Selection: Efficient Diversification of Investments which built on his original research.
With the two works, Markowitz was credited with the concept called Modern Portfolio Theory that separates the portfolio allocation decision from the security selection decision. He also made popular the measure of variance, and related standard deviation, as the standard measures of risk.
So if you were wondering why I took you through portfolio return and risk, rather methodically, I wanted you to have a good background as we head to portfolio theory and beyond. So don't blame me, blame Harry Markowitz.
Next, the Portfolio Possibilities Curve, like the one created in the last tutorial represents the edge, or border, of this hyperbola. Here is the boundary of portfolio combinations that show risk at different levels of expected return.
I'll say that again. It shows the portfolios with the minimum risk at different levels of return. I didn't complete the hyperbola because investors wouldn't, in theory, invest in portfolios with lower return at any specific level of risk. Had I drawn the complete curve, then this would represent the whole Portfolio Possibilities Curve.
Now the opportunity set, or feasible set, represents all of the possible portfolios, with all combinations of risky assets drawn here with green dots (see video). For two stocks the portfolios sit on the edge of the hyperbola, as we go to three stocks and beyond, then the dots, or portfolios with all possible combinations of weights to each stock fill the hyperbola.
We can't draw them all because there are an infinite number of dots in theory. Think about all of the weights to all of the assets you could possibly hold. Now we are talking about a world with a nearly unlimited number of assets.
Another point is that we are talking about this in the expected return and risk sense. We have been working with historical data up to this point in Excel, but the theory refers more to the expected timeframe.
In other words, what is built in to market expectations? Of course no one really knows for certain, because the 'Market' is made up of millions of participants and expectations can be, especially if you watch the financial news media, a moving target every day.
So, to summarize, the opportunity set is the aggregate of all portfolio combinations contained with this imaginary hyperbola.
The Minimum Variance Portfolio, or MVP, sits at the tip of the hyperbola, is the lowest risk portfolio that can be constructed from the entire combination of all portfolios.
The Efficient Frontier is the top half of the hyperbola above the MVP. Think about what it represents. It is the set of portfolios that maximizes return at any given level of risk. So, theoretically, these are the best possibilities of all return-to-risk combinations.
Remember, everyone wants to be in the top left? Slide up to the right and as you increase risk it shows what return can be offered. As you slide to the left, it offers the highest return as risk decreases.
It might be helpful to think of the Efficient Frontier as a supply curve, meaning this is what can be offered to investors.
So if there is a supply curve, then there must also be a demand curve. You will often hear this referred to as a utility curve or indifference curve.
This upward sloping convex curve represents another theoretical topic that attempts to define the trade-offs investors make with respect to the return they get for the amount of risk they assume. Basically, what an investor wants, return, versus what the investor is willing to give up, which is risk.
Each investor has her own utility curve. A more risk-averse investor would have one that shifts to the left. A risk-seeking investor has a curve that shifts to the right. That is the whole curve, not just along the curve.
So the utility curve represents the decision-making process for one investor to assume the risk of a portfolio. Where that curve touches the Efficient Frontier is the location where the investor is willing to assume the risk, for the expected rate of return. Voilà!
We can see how close the investor is to making the decision to invest as we slide along the dark green curve to the right, the portfolio risk increases materially but return barely increases. So for the investor to have the same utility, or satisfaction, she would require a higher rate of return as show by her light green curve. No match is made.
Conversely, as you move left on the dark green curve the return offered by the marketplace isn't high enough for the investor, despite the lower risk. Again, no match is made. Only at the point where the utility curve touches the Efficient Frontier does the investor agree to the terms, and we now have what is called an Optimal Portfolio for that one investor.
The Optimal Portfolio represents the portfolio on the Efficient Frontier, meaning the one with the highest return-to-risk trade-off, that offers the best utility given the specific investor's tolerance for risk. Again, this is the point where both curves intersect.
So for the next decade this was the prevailing theory. Keep in mind this corresponded with the beginning of the adoption of computers. So before this, average returns and stock variances for hundreds or thousands of companies had to be calculated by hand, which wasn't very practical.
So what we can done easily here in our spreadsheet just couldn't be done prior.
As covered in an earlier tutorial on stock market history we saw that shortly thereafter consulting firms sprung up to bring these services to large pools of assets in the Institutional space at the time, pension funds. A new industry was developed and remains with us today.
Meanwhile scholars were working on a new and equally impactful Capital Market Theory and the Capital Asset Pricing Model, or CAPM.
CAPM is an asset pricing theory that establishes an expected return on a stock based on a component of time, the risk-free rate, plus the market price of risk, or beta multiplied by the expected market return.
CAPM is also used in corporate finance to compute a company's cost of equity which is a component of a firm's weighted average cost of capital (WACC).
We will more fully explore CAPM later in the series.
I am making a distinction here between Modern Portfolio Theory and Capital Market Theory because, we need another advancement to take us from building a portfolio to a theory of how the addition of a risk-free asset, in a sense, changes the Efficient Frontier from the top half of the hyperbola, as we discussed, to a straight line from the risk-free rate to what is now called a Market Portfolio. It's a big challenge, so let me explain.
We will talk about assumptions in a moment, but CMT builds on MPT in that it is assumed that all investors are efficient investors, meaning they follow the assumptions of the MPT.
A Risk-Free investment is one that provides a guaranteed rate of return for a yield or what in this context we call the risk-free return. A proxy for this can be a short-term government fixed income security, like a Treasury-bill, but could also be a bank deposit, like a CD. Often we just call this cash.
Yes, the prevailing interest rate on a Treasury-bill comes with no risk meaning it doesn't go down in value, so it has zero variance and zero standard deviation.
Now in real life, we know that there is always some element of risk, so bear with me as everything here comes from a theoretical perspective.
To make a point let's look at the chart again. You can see that at point rf1, there is no risk, so the x-variable risk is zero. Let's assume the interest rate at point rf1 was 1%, and then extend a line from that point to the Efficient Frontier so now we have a line from the y-axis and it meets at a point labeled M1. To gain an understanding of what this line means, think of it as a plank, the higher you go up the plank the greater the expected risk, and greater the expected reward, or return.
The investor now selects a portfolio that is a combination of only two choices, first is the Risk-Free asset and second is the M1 portfolio, which we call the Market Portfolio. The allocation between the two determines the expected risk and return of the portfolio. Don't worry too much about scale here, just follow the concept.
Now we have a combination of risky assets touching the hyperbola that represents the most suitable portfolio of all risky assets, and a line connecting it with the y-axis which is risk free. So a client can select a portfolio that matches her risk tolerance along the dashed line from rf1 to M1. So instead of a curve formed by the Efficient Frontier, it changes to a flat line. If we were to draw the utility curve now, we would do so on this line for each investor.
Let's introduce the Market Portfolio.
The Market Portfolio is the one Optimal Portfolio based on consensus estimates that all risk-averse investors use to exceed the Risk-Free return. We should take that one step further to say that, it is a passive portfolio with weights allocated to assets based on their market value.
So, in theory, again all investors, should invest along the line between rf1 and M1 on this chart. The midpoint is the portfolio with 50% invested in T-bills and 50% in the Market Portfolio, which we could assume is an index of equities.
Now assume the risk free rate moves up to rf2, the Market Portfolio now moves from M1 to M2 and the line that forms between the two is less steep than the first line. However, at the time, this line is the highest sloping line available.
You wouldn't meet at Portfolio A because you'd get more bang-for-your-buck at portfolio M2, meaning the slope of the line would be greater, rise over run remember, and you always want to maximize the slope of that line. The theory assumes we are all efficient and rational investors.
One last point to remember is that you would expect risky assets to have a higher expected return than the risk-free return or you wouldn't bother investing in them. Right? Especially since you know the risk-free return ahead of time and for risky assets you're not sure of future returns and could easily lose value in the short run.
This line is given the name Capital Market Line. Earlier, when we discussed the utility curve, it was a curve that intersected with the Efficient Frontier. Now that the Efficient Frontier in a sense turns into straight line, we can think of the Utility Curve intersecting with the line and this determines the investor's allocation.
If you think about it, now there is a linear relationship between expected return and standard deviation. It is as simple as that, and an added benefit, of having a straight line, is that the math gets a whole lot easier.
Additionally, think about what is going on when you own a Risk-Free asset like a T-bill. You are lending money to the government, right?
Using the midpoint example again, for half of the portfolio the investor is lending. For the other half the investor buys the Market Portfolio, in an index fund for example, which in turn owns stocks, also known as equity of corporations. So here you are lending to corporations for a higher expected rate of return.
If you noticed, the Capital Market Line does not stop at M2. It keeps going, and when it does, instead of lending money, the investor is borrowing money at the Risk-free rate to invest in equities. I'll stop there with that because, in my experience, leverage can take us on unnecessary detours.
The term Separation Theorem, or Mutual Fund Separation Theorem, refers to the distinction between the two decisions an investor must make.
First, the decision to invest in the Market Portfolio, and second the financing decision, meaning is borrowing and leverage involved, or lending in the form of the Risk-Free asset.
Advancements here simplified the investment process not only for Institutional investors, but also for many Professionals to help Individuals find their tolerance for risk and select an appropriate portfolio made up of non-risky assets and risky assets.
A common term for this is asset allocation, which refers to finding the client's spot on the plank that matches the investor's tolerance for risk. Generally the asset allocation decision goes deeper to select different types of bonds, domestic and international stocks, real estate and the like, but that is something to tackle at a later time.
Risk aversion is another term used in this context. It is another way to think about the investor's preference for risk. An investor who is given a choice between two assets with the same expected return would select the less-risky asset. Portfolio theory assumes that all investors are risk averse. Again, in theory.
To offer a live example of how this impacts investors on whole, think about what the Federal Reserve in the US and other central bankers worldwide, who after the Great Recession in 2009 left short-term interest rates near zero for about 8 years.
Think about this from the standpoint of this theory. If you created an rf3 at basically zero, what would that do?
It would make the Capital Market Line steeper which would offer investors more bang-for-their-buck when considering risky assets. In a sense, the Fed forced investors to walk up the plank, assuming that if asset prices increased then through the trickle-down effect it would stimulate the economy, which was their goal in the first place.
Of course this is theory, but in hindsight now we can see how asset prices globally were pushed up because of central bank actions. Stocks, real estate, collectibles, bonds, you name it, they rose in value and risky assets became less volatile as a result. Now that central banks have been successful and are in a tightening cycle, interest rates are rising, which gives risk-averse investors a place to earn a yield without taking risk.
So there's a case for why the theory matters.
On the slope of that line, the formula called the Sharpe Ratio quantifies it. It is named after William Sharpe, who is primarily credited with Capital Market Theory, and the Capital Asset Pricing Model, or CAPM.
We have a whole Chapter devoted to CAPM, so we will pick this back up later.
Let's move on to Step 4 and discuss how these theories are made practical.
Listed below are the assumptions associated with Modern Portfolio Theory. I mention this because we are talking about the theory of portfolio management and theories are often postulated with assumptions everyone knows are false.
Scholars need to set variables constant so they can focus on the theory. I won't go over each one here but take some time to think about how unrealistic these are in the real world.
If you would like, let me know, and I can circle back with an addendum tutorial on the assumptions, but for now, let's just read through them, know they're here and come back to them at a later time.
Investors:
And:
That said, what is practical is how firms and investors have taken lessons from MPT and CMT to incorporate the scientific aspects to managing portfolios. Practitioners use three elements today, regardless of investment strategy, meaning they don't only apply to Quants. So regardless of how a manager selects stocks for a portfolio the following three functions are performed by Institutional investment managers.
These are all elements we have and will spend time exploring with real data during this tutorial series.
By way of summary, we covered a lot of ground here as we moved from our 2-stock example to the multi-asset world. We covered the 'Market' and advancements by Nobel-price-winner Harry Markowitz.
Portfolio theory took us to the development of risk measures, the trade-off between return and risk and the establishment of the Efficient Frontier.
Utility curves helped us understand the investor's viewpoint and then we added a risk-free asset which changed the return and risk equation from the top half of a concave hyperbola to a linear equation from the risk free asset to the Market Portfolio. We will keep working with these concepts and eventually build an optimizer that maximizes the ratio of return over risk. So what you see here we will work through together in a spreadsheet, so it sticks.
In the meantime, you have a fairly comprehensive review so you can ace your exam.
In the next episode we will explore correlation and regression on our road to moving from the historical timeframe to the expected and forecast timeframes.
Also, this is one of the presentations I'm using to teach a University course so leave a note and join the conversion. Please forward a link to your friends who may be in the same position as you.
Good luck with your exam and have a nice day.
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