Asset Allocation: How to Build an Efficient Portfolio

In our discussion of robo-advisors, the first thing that we want to discuss is asset allocation, which is one of the key concepts in finance, and it’s the main thing that a robo-advisor is going to do for us. Asset allocation refers simply to how much of your money you’re going to put into different classes of securities. So that is to say, how much of your money you’re going to put into broad groups of securities like stocks and bonds, as opposed to each individual stock or each individual bond. Therefore, this is more about types of assets than about individual assets.

So it’s thinking about financial instruments that conform to a specific type as opposed to thinking about a specific stock like say, Apple stock or Microsoft stock. Examples of asset classes include domestic stocks, so those equities that are traded in the United States, United States Treasury bonds, emerging market stocks, so stocks that are trading in emerging markets such as Mexico or the Philippines and so forth, and real estate.

The principles of finance suggests to us that we ought to determine an allocation to meet our financial goals. So when thinking about a particular investing strategy, we want to think about what we want to achieve when we set that particular strategy, and then think about an allocation that is going to help us to best meet those goals. Examples of these types of problems include things like how much to save for retirement, or how much to save for children’s education, and so on and so forth. So how do we go about determining an allocation? The first thing that we need to think about according to the principles of finance is what expected return do we need to meet our goals?

That is to say, what rate of return do we think that we need in order to achieve whatever goal it is that we have set out to do. To take an example, let’s suppose that we have $500 to save per month over the next 40 years. We’d like to have a million dollars at the end of those 40 years. So again, this is a typical retirement saving example where we know how much we have to save, we know how long we’re going to save it for, and we have a goal as to what the amount is that we need to have at the end of our saving period. What return do we need to earn to make this happen?

This again, is a very typical finance problem and the answer to it comes from the time value of money problem from principles of finance where we think to ourselves, if we invest at some rate of compound interest, how much are we going to have when we’ve finished investing? The future value that we’re looking for here is a million dollars, the amount of the payment is $500, the number of payments that we’re going to have is $480. The question is, what return do we need to achieve in order to get to this goal? The answer turns out to be one-half of one percent per month. This can be easily achieved in a financial calculator or in Excel.

So when we are trying to solve this problem or think about this particular kind of goal, we know that we need to find an investment or an asset allocation that’s going to return to us half a percent per month or approximately six percent per year.

The other consideration when we’re thinking about an asset allocation is that we usually think that there’s a trade-off between the return that you can expect and the risk that you have to bear. So for a higher expected return, we expect that you’re going to have to bear more risk. We usually quantify that risk with a concept called the standard deviation, the variance or the volatility, all refer to roughly the same concept. As an example, let’s think about stocks and treasury bills. So equities are usually thought of as relatively high risk investments that have high expected returns.

On the contrary, treasury bills, which are paid by the United States government and promise an expected return, are usually thought of as being relatively low-risk and in fact usually thought of as being default free because it is expected that the US Treasury will not default on its obligations. Asset allocation is all about reaching your goals. So again, that one million dollars for retirement, for example, with an expected return and risk that you personally are comfortable with. To get a little bit of a sense of the risk and return of different asset classes, this particular table shows us for five different asset classes what returns have averaged over the past several years and what their standard deviation is.

I’ve listed the different asset classes according to their average return. You can see that there’s a broad relation between the average return that we realize on the different asset classes and the risk involved in the asset class or its standard deviation. This relation isn’t perfect, but it doesn’t have to be the case that an asset with the highest average return always has the highest standard deviation. So starting from US residential real estate, you can see that as abroad class, residential real estate has had an average return of 1.6 percent per year with a standard deviation of 5.82 percent per year.

This may come as a surprise to many investors as we hear a lot about really large returns to investing in residential real estate. But when we take a look at US real estate as a whole, the return on investing in real estate is actually not especially high, it just happens to be very high in some marquee areas. As I mentioned above, treasury bills are usually thought of as the safest asset, and you can see here that they have an average return of 4.73 percent. They do have some standard deviation, but this occurs because interest rates change over time and during different times we can invest at treasury bills at a different rate.

The next average return comes to treasury bonds, which again are considered fairly safe and relatively risk free, with an average return of about 7.5 percent and a standard deviation of about nine percent. US Corporate Bonds come next, and then the riskiest and highest returning asset class is US stocks, which have an average return of about 11.5 percent and a standard deviation per year of about 18 percent. Again, what these numbers are trying to show to us is that there’s generally some kind of a trade-off between the return that we can get on a particular asset class and the amount of risk that we have to bear. There’s another concept that’s important when thinking about asset allocation, and that’s diversification.

One of the biggest insights of finance theory is that holding multiple asset classes produces what we call a diversified portfolio. What this means in simple terms is that sometimes when one asset class has bad returns, another may have good returns. So we may be in a situation, for example, in which bonds have good returns, but stocks are having relatively bad returns. In 1952, Harry Markowitz showed that investors can reduce the risk of their portfolios through diversification, and this is part of what earned him a Nobel Prize. In order to quantify the degree to which an asset class can help diversify a portfolio, we need one more piece of information, which is the correlation of the returns amongst the asset classes.

Again, here is a sample of what historically the correlation between different assets have been. Again, I have the same five asset classes that I had before, real estate, treasury bills, T-bonds refers to treasury bonds, C bonds to corporate bonds, and stocks. As we look across the first row, we can see that one of the advantages to real estate is that it has a relatively low correlation with other asset classes. The correlation always ranges between minus one, which essentially says that when one asset class goes up, the other asset class goes down, the positive one, which says that when one asset goes up, the other asset class goes up.

You can see with real estate that relative to bills and treasury bonds, it has a negative correlation. This essentially means that it’s diversifying relative to those securities. It has virtually no correlation with corporate bonds and a slightly positive correlation with stocks. I won’t go through every single number on this particular table, but a couple of places to highlight are, for example, the correlation between treasury bonds and corporate bonds. If you look in the fourth column and the third row, you can see that the correlation between corporate bonds and treasury bonds is about 0.7.

This is because the level of interest rates is one of the major determinants of the return on both corporate bonds and treasury bonds, and so as interest rates go up and down, corporate bonds and treasury bonds tend to move together in terms of their price. The other one to highlight is the correlation between corporate bonds and stocks, which is about 0.47. What we’re seeing here is that, since both of these securities are issued by corporations, they tend to be positively correlated. So when corporations are doing well, the returns on their bonds do relatively well and the returns on their stocks do relatively well.

We use this information, the expected return, the standard deviation, and the correlation to form what we call efficient portfolios. We use these different building blocks, expected returns, standard deviations, and correlations to determine what we call efficient portfolios. An efficient portfolio is a portfolio where we determine a target expected return for our allocation. So again, thinking back to the retirement example, suppose that our expected return is 0.5 percent, so we want to make sure that we earn at least 0.5 percent per month.

What we do is we think about all of the different possible portfolios that we could make up of the different asset classes that we are considering, and then find the one amongst those that has the lowest standard deviation. This turns out to be a relatively straightforward math problem which we won’t go into in this particular video, but there’s always a solution to tell us that there’s a single portfolio that’s going to have the lowest standard deviation for any target expected return. What is this windup looking like? Well, here what I’m going to do is first plot the low side of the expected returns and standard deviations of the different asset classes that I showed you before.

On the x or horizontal axis, I have the standard deviation of returns annualized, going from zero to about 25 percent, and on the y-axis I have the average or mean of the returns annualized. Again, I have the five different asset classes plotted as red points on this particular graph. What you should be able to see is that there’s a generally upward relationship between the mean return, so the expected return and the standard deviation or the risk.

Again, treasury bills have the lowest risk and so you can see those in the lower southwest part of the graph with an expected return of a little bit less than five percent, and stocks are the most volatile and highest expected return asset class with an average return of about 11 percent and a standard deviation nearing 20 percent. We construct an efficient frontier, as I mentioned, by trying to find all of the different portfolios that have the lowest risk for a given target expected return or a given target mean. The blue line that I’ve now superimposed on this graph, is the efficient return for this set of assets.

Each point on that blue line represents a different portfolio that gives you a unique trade-off between expected return and standard deviation. We call this the efficient frontier because it’s a limiting case of how low our standard deviation can go or how high our expected return can be. Essentially, there is no possibility of forming portfolios that lie to the northwest of that particular blue line. We just simply can’t form portfolios with a combination of standard deviation and mean return to fit those profiles. So if you look at this particular graph, you can see that there is no way to find a portfolio that has a standard deviation of five percent and a mean return of 10 percent.

Some combinations are simply not feasible and the efficient frontier tells us the best trade-offs for a given expected return that we can get in terms of standard deviation. So how do we choose amongst these different efficient portfolios? According to economic theory, what we should do is find an efficient portfolio that suits our risk tolerance. What do we mean by risk tolerance? Risk tolerance is an idea that simply says, “How much of a loss are you willing to bear in a bad case scenario?” Financial advisers typically assess this with a questionnaire with a number of different questions that includes things like, how much do you already have saved?

The idea here is that if you already have a lot saved, you can bear a larger risk than if you’re someone who doesn’t have a lot saved. How long do you have to save? Again, if you have a very long time to save, you have more time to make up for any given loss than you do if you don’t have very long to save. This is one of the reasons why we usually advise investors to have a safer portfolio as they are nearing retirement than when they’re first starting to work. Finally, how much of a loss would you take before selling?

This is a bit of a subjective question, but gives a financial adviser an idea of what situations would cause you to liquidate your risky portfolio and therefore take a loss and suffer the consequences.

We encapsulate this idea in a concept from economics called utility. Utility is a concept that describes how we trade off between two things. So as an example, if we’re trying to decide between wine and beer, utility is a concept that is supposed to tell us how much we like beer relative to liking wine. In our case, it’s the tradeoff between standard deviation and expected return. So as we mentioned, we tend to like expected return because it gives us more money at the end of the day, and we dislike standard deviation because it’s a measure of risk which we’re attempting to avoid.

We want to find the tradeoff between expected return and standard deviation that’s going to give us the highest utility. How do we go about doing this? Well, again, I’m going to come back to our plot of the efficient frontier. Just to refresh, our x-axis is the standard deviation of returns, which reflects the amount of risk that we have to take on, and our y-axis is the mean of returns, which tells us something about what we expect to get from investing. Again, the red dots are the loci of expected returns and standard deviations for the five different asset classes we mentioned before. The blue line is the efficient frontier.

So the set of portfolios that have the lowest risk for a given expected return. The green line that I have now plotted on this particular graph is a utility or an indifference function. Again, as I mentioned before, we’re trading off between expected returns and standard deviation. So what the green line is trying to represent is combinations of standard deviation and expected returns that we are indifferent among.

That is to say, we’re just as happy to take a position in the upper right corner of the graph where we have a standard deviation of about 20 percent and an expected return of 15 percent, as we are to take a position in the lower part of the graph where we have an expected return of five percent and a standard deviation of zero percent. So again, just to reiterate, all points on that green line are more or less the same to us. Utility tends to increase to the northwest. So again, this green line that I’m plotting right now has a specific functional form.

It basically says that 0.05 is equal to the expected return minus 1.5 times 5 times the variance of the return or the standard deviations squared. Now, most of those numbers don’t mean a whole lot all by themselves. Again, this is just a way of quantifying how much we like a particular trade off. Let’s look at this utility function just a little bit more closely. We can see that the utility function is increasing in the expected return. So again, the higher the expected return, all things equal, the higher the utility. If we increase the expected return keeping the standard deviation the same, we’re going to increase the quantity on the left-hand of the equals sign.

On the other hand, the more we increase the standard deviation, the more we’re going to decrease that utility. So again, holding the expected return constant, if we increase the standard deviation, we’re going to wind up decreasing the amount on the left-hand side of the equation. The number on the left-hand side of the equation, 0.05, doesn’t really have an intuitive meaning. The main reason that we use it is to compare amongst different tradeoffs. So what we can say is that if a tradeoff gives us an answer of 0.1, we like it better than if it gives us an answer of 0.05. But other than that, it doesn’t have a real interpretation.

What I’m showing you here is a portfolio where we’ve actually maximized the utility function that I showed you before. Utility here is 0.07. So again, as I mentioned, it gives us a higher utility than we had in the previous slide. Again, that 0.07 doesn’t mean a whole lot, except to tell us that we prefer portfolios that happen to be on the green line to portfolios that happened to be on the green line from the previous slide. The other feature of this particular graph that you can see is that the green line touches the blue line in only one spot. That indicates to us that we’re at a point of maximum utility.

Any higher utility would push us further to the northwest on this particular graph. As mentioned before, portfolios to the northwest of the blue line are simply not accessible to us. So what we’re trying to find when we’re trying to maximize utility is a point on the blue line that pushes our utility as far to the northwest as possible. Gives us the best possible tradeoff between risk and return. That particular portfolio has weights that are given on the right, it’s invested 60 percent in corporate bonds and 40 percent in stocks. So this illustration suggests to us an example by which we could determine a portfolio allocation that’s going to give us the maximum utility that we desire.

We entered into this discussion talking, however, about risk tolerance. Again, risk tolerance is an idea that tells us how we trade off expected return and standard deviation. How much risk are we actually willing to tolerate? This graph is a replication of the previous graph where the green line represents a maximized utility function for a particular individual with a portfolio that’s 60 percent in corporate bonds and 40 percent in equities. The purple line is a maximized portfolio for a different individual who has a higher risk tolerance. If we look at the two functions, we can see that they look very similar. They’re the expected return minus 1.5 times some number times the variance, which is the standard deviation squared.

That some number, five in the case of the green line and two in the case of the purple line, reflects our risk tolerance. An investor with a risk tolerance of two has a higher risk tolerance than an investor with a risk tolerance of five. So as a result, our investor with the risk tolerance of two is going to choose a different optimal portfolio than our investor with a risk tolerance of five. Our investor with a higher risk tolerance with the coefficient of two is going to choose a portfolio that’s different than our investor with the lower risk tolerance or a coefficient of five. In fact, the weights to their optimal portfolio are provided to the right of the graph.

You can see that it puts 33 percent of the portfolio into corporate bonds and 67 percent of the portfolio into stocks. If we think about what we know about expected returns and standard deviations, we know that stocks have higher expected returns and higher standard deviations than do corporate bonds. So it makes sense that our investor with a higher risk tolerance would allocate more to stocks and less to bonds than our investor with a lower risk tolerance. Just to sum up, asset allocation is a key concept in finance. In any finance course that you take is going to spend a considerable amount of time talking about how we should allocate our assets amongst different asset classes.

What we want to do in an asset allocation problem is find an allocation that’s consistent with a tolerance for risk. Again, this is a bit of a subjective concept. What we want to do is determine how willing we are to bear a particular risk and find an allocation that we’re comfortable with. Traditionally, this approach has been performed by financial advisors who will sit down with you, ask you a number of different questions, and then come up with an asset allocation that’s consistent with the tolerance that you have for risk.

What we’re going to talk about in the next video is the question of, do we really need to have an advisor to do this, or is this a process that can somehow be automated?