5
T
he idea that an asset’s risk can be thought of as the volatility of its returns measured by the standard deviation seems plausible, doesn’t it? Well, the problem is that it doesn’t extend well when assets are combined, among other reasons because the volatility of a two-asset portfolio is not equal to the sum of the volatilities of each individual asset. It’s a bit more complicated than that. But not that complicated. You most likely heard the expression ‘Don’t put all your eggs in one basket.’ Well, at the end of the day, this chapter may be as simple as that.Three hypothetical stocks
Let’s consider the returns of the three hypothetical stocks in Table 5.1. We know by now how to calculate their (arithmetic) mean return and standard deviation, which are also reported in the table. And we also know that, given those numbers, stock 1 (SD= 10.0%) is riskier than stock 3 (SD= 5.0%), which in turn is riskier than stock 2 (SD= 1.5%).
TABLE 5.1
Year Stock 1 Stock 2 Stock 3
(%) (%) (%)
1 25.0 21.3 32.5
2 5.0 24.3 22.5
3 22.5 21.6 31.3
4 6.0 24.1 23.0
5 17.5 22.4 28.8
6 4.0 24.4 22.0
7 31.0 20.4 35.5
8 5.5 24.2 22.8
9 24.0 21.4 32.0
10 4.0 24.4 22.0
AM 14.5% 22.8% 27.2%
SD 10.0% 1.5% 5.0%
Now, instead of thinking of each of these stocks individually, let’s think of combinations of them. Let’s combine, for example, stocks 1 and 2. So suppose that at the beginning of year 1 we had invested $1,000, 13% in stock 1 and the rest (87%) in stock 2. We know by know how to calculate, period by period, the return of that portfolio.
The return for the first period is given by (0.13)(0.250) + (0.87)(0.213) = 21.7%. If we calculate in the same fashion the returns of the portfolio in all
subsequent periods we will find . . . surprise! The return of the portfolio in each and every period is the exact same 21.7%.
Take a look at Exhibit 5.1, which plots the returns of stocks 1 and 2, as well as the return of the proposed portfolio (the dotted line). Although stocks 1 and 2 fluctuate from period to period, the return of the portfolio remains constant at the calculated 21.7% return. Magic? Not really. But before we discuss what’s going on, let’s think of another combination of these hypothetical stocks.
EXHIBIT 5.1
Perfect diversification
Suppose now that, at the beginning of the first year, we had split our money equally between stocks 1 and 3. Exhibit 5.2 plots the returns of stocks 1 and 3, as well as the return of this equally weighted portfolio (the dotted line). Pretty different picture, huh?
If we compare the two-stock portfolios in Exhibits 5.1 and 5.2, it is obvious that they are very different: the first has no volatility, though it results from the combination of two volatile stocks, whereas the second seems very volatile. If you’re wondering what is the main driver of the difference between these two portfolios, you’re asking the right question.
40 35 30 25 20 15 10 5 0
1 2 3 4 5 6 7 8 9 10
Stock 2
Portfolio
Stock 1
Year
Return (%)
EXHIBIT 5.2 No diversification
The correlation coefficient
So if we combine two stocks, what determines that in one case we end up with a portfolio that locks a fixed return, while in the other we end up with a very volatile portfolio? It all comes down to one parameter: the correlation coefficient.
This coefficient, which is also discussed in Chapter 27, measures the strength of the (linear) relationship between two variables. When the coefficient is positive the two variables tend to move in the same direction, and when it’s negative they tend to move in opposite directions. It can take a maximum value of 1 and a minimum value of –1, with these two extremes indicating a perfect linear relationship (positive in the first case and negative in the second).
For reasonably long periods of time, however, it is virtually impossible to find a negative correlation (or a correlation very close to 1) between two stocks within a market, or between two equity markets. We’ll explore why in the next chapter, but for now keep in mind that the empiricalvalues of the correlation coefficient are within a much narrower band than its theoretical extremes.
Back to our hypothetical stocks now. What’s going on between stocks 1 and 2? Simply that they exhibit a perfect negative correlation; that is, a correlation
40 35 30 25 20 15 10 5 0
1 2 3 4 5 6 7 8 9 10
Stock 2
Portfolio
Stock 1
Year
Return (%)
equal to –1. In such situations, a combination between two stocks that enables the investor to lock a return (and obtain a portfolio with 0 volatility) can always be found. But, however interesting this may sound, it has little or no practical importance. As mentioned above, it’s virtually impossible to find two stocks with a negative correlation, let alone with a correlation equal to –1.
What’s going on, in turn, between stocks 1 and 3? Pretty much the opposite.
They exhibit a perfect positive correlation (that is, a correlation equal to 1) and, in such situations, the risk of the portfolio is simply given by the average volatility of the two stocks in the portfolio (weighted by the proportion of wealth invested in each stock). In other words, in terms of risk reduction, there is nothing to gain by combining these two stocks. (In fact, the onlycase in which the risk of a two-stock portfolio is equal to the weighted average of risks is when the correlation between them is 1. In every other case, the risk of the portfolio is lowerthan the weighted average of the risks.)
If the goal is to reduce the risk of a portfolio, we should look for stocks with low correlations to each other. This is particularly important when the portfolio has few assets. Remember, the lower the average correlation across stocks in the portfolio, the larger the reduction of risk. That is, the larger the difference between the weighted average of risks (a situation in which nothing is gained by combining stocks, from a risk-reduction point of view) and the actual risk of the portfolio.
One final word on the correlation coefficient before we move on. Don’t think of it as a statistical magnitude with little practical importance. The correlation does in fact determine the extent to which risk can be reduced by combining stocks. Think for example of emerging markets as an asset class. Though emerging markets are very volatile, their correlation to developed markets is relatively low; hence, they may lower substantially the volatility of a portfolio of assets in developed markets. Something similar could be said, for example, about venture capital funds, which are very volatile but also have a low correlation to the market. In short, don’t underestimate the practical importance of the correlation coefficient.
Three views on diversification
So, what is diversification? It is simply the combination of assets into a portfolio with the goal of reducing risk. Having said that, beware of this popular definition. As we discuss below, diversification can be thought of in other ways, and its ultimate goal is a bit more complicated than just reducing risk.
And why do investors usually diversify? If the first answer that comes to your mind is ‘to reduce risk,’ you’re obviously right. Most people avoid putting all their money in one stock (or even in a few stocks) to avoid a situation in which the stock unexpectedly tanks and takes their whole portfolio with it. Think Enron, where many employees had over 90% of their pension money invested in Enron stock. That’s a lesson on diversification learned the hard way!
But diversification can be thought of in at least three other ways, all useful though usually less explored. Take a look at panel A of Table 5.2, which shows the returns of Disney and Microsoft during the years 1994 to 2003. During this period, Microsoft delivered a higher return with a higher risk than Disney, as the numbers in the last two rows show. The correlation coefficient between these two stocks (not reported in the table) is a very low 0.05, which points to potentially high diversification benefits.
TABLE 5.2
Panel A Panel B
Year Disney Microsoft xD xM Risk Return RAR
(%) (%) (%) (%) (%) (%)
1994 8.7 51.6 100.0 0.0 23.7 8.5 0.361
1995 29.0 43.6 90.0 10.0 22.2 11.7 0.526
1996 19.1 88.3 80.0 20.0 21.9 14.8 0.676
1997 42.9 56.4 70.0 30.0 22.9 17.9 0.781
1998 –8.5 114.6 60.0 40.0 25.1 21.0 0.838
1999 –1.7 68.4 50.0 50.0 28.2 24.2 0.858
2000 –0.4 –62.8 40.0 60.0 31.8 27.3 0.857
2001 –27.7 52.7 30.0 70.0 36.0 30.4 0.845
2002 –20.3 –22.0 20.0 80.0 40.4 33.5 0.830
2003 44.4 6.8 10.0 90.0 45.0 36.6 0.814
AM 8.5% 39.8% 0.0 100.0 49.8 39.8 0.798
SD 23.7% 49.8%
Let’s consider some combinations between Disney and Microsoft, such as those shown in panel B of Table 5.2. The first two columns show different portfolio allocations to Disney (xD) and Microsoft (xM), the next two columns the risk and return of the different portfolios, and we’ll get to the last column in a minute.
An interesting question would be: what is the combination between Disney and Microsoft that yields the lowest possible risk (that is, the MVP)? That’s very
simple to find in the two-asset case, as we saw in the previous chapter. By investing 82.9% in Disney and the rest (17.1%) in Microsoft, we would obtain a portfolio with a standard deviation of 21.8%, just slightly lower than the risk of the 80/20 portfolio in the Table (21.9%). The return of this portfolio, on the other hand, would be 13.9%.
Which brings us to a second reason for diversifying. If we’re happy holding Disney stock, we should be even happier to hold the MVP. This is simply because it enables us to lower our risk by 1.9% (= 23.7% – 21.8%) and to increase our returns by 5.4% (= 13.9% – 8.5%), both with respect to holding Disney by itself.
In short, here’s a second way of thinking about diversification: it may enable us to lower our risk and increase our returns at the same time.
Now, let’s go back to assuming that we’re happy holding Disney, which means that we accept the level of risk of this stock. Having said that, if someone offered us an asset with the same volatility but a higher return, wouldn’t we want it? Of course we would, and that is just what we can obtain through diversification.
Take a look at Exhibit 5.3, which shows the feasible set between Disney and Microsoft. These numbers, of course, follow from panel B of Table 5.2. Besides the MVP we already discussed, the exhibit highlights another portfolio, labeled A. Given the choice between putting our money in Disney or putting it in portfolio A, what would you choose?
EXHIBIT 5.3
Disney v. Microsoft, feasible set
45 40 35 30 25 20 15 10 5 0
20 25 30 35 40 45 50 55
Microsoft
Disney MVP A
Risk (%)
Return (%)
Portfolio A consists of an allocation of 65.9% in Disney and 34.1% in Microsoft and, by construction, has the same level of risk as Disney. However, this portfolio has a 19.2% return, which is 10.7% higher than Disney’s return. That’s quite a difference, and is as close to a ‘free lunch’ as we can get in financial markets.
Why would we ever hold Disney by itself if, at the same level of risk, portfolio A has a much higher return? Which brings us to a third way of thinking about diversification: it may enable us to increase our returns given an acceptable level of risk.
Now, it seems that convincing someone who’s happy holding Disney to diversify would be an easy task. But here’s a challenge: how would you convince someone who’s happy holding Microsoftto diversify? It doesn’t look as if we can pull off the same ‘trick’ as before. We cannot offer this investor a portfolio with lower risk and higher return, or one with the same level of risk but a higher return. Is it, then, that diversification is not beneficial for someone who’s happy holding Microsoft?
Another view on diversification: Risk-adjusted returns
Not really. Investors do not just care about the returns of their portfolio; they also care about its risk. In fact, what investors really care about is maximizing risk-adjusted returns. There are different ways of defining this concept (which we explore in Chapter 10), but for the time being let’s simply think about a ratio that divides return by risk. This is exactly what the last column of Table 5.2 shows: the risk-adjusted return (RAR) defined as the ‘Return’ column divided by the ‘Risk’ column.
Can you see now why, even if we’re happy holding all our money in Microsoft, it would be beneficial for us to diversify? Because we could increase the risk- adjusted return of our portfolio. In our case, the ‘best’ portfolio of those shown in Table 5.2 is a 50/50 split between Disney and Microsoft, simply because it has the highest RAR (0.858).
However, the portfolio with the highest possible RAR is not shown in the table. An investment of 46.1% in Disney and 53.9% in Microsoft would have an RAR of 0.859, just slightly higher than that of the 50/50 portfolio. The portfolio- optimization program discussed in Chapter 11 can find this optimal combination in the blink of an eye.
It will virtually never be the case that the highest RAR is found in a portfolio fully invested in one asset. Which brings us to a yet another way to see why
diversification is beneficial: it enables us to obtain the highest possible risk- adjusted returns. And, don’t forget, that’s the most we could ever ask of any investing strategy.
Note that this last angle on diversification ‘contains’ all the others. We had agreed that both the MVP and portfolio A were better than putting all our money in Disney. The former enabled us to lower our risk and increase our return (relative to holding just Disney), thus increasing risk-adjusted returns.
The latter enabled us to increase our return given the level of Disney’s risk, again increasing risk-adjusted returns. And a portfolio invested 46.1% in Disney and 53.9% in Microsoft is better than investing just in Microsoft because, again, it increases (in this case, maximizes) risk-adjusted returns. In short, when grandma told us not to put all our eggs in one basket, she was, as usual, wiser than we probably gave her credit for.
The role of mutual funds
At this point you should have little or no doubts about the benefits of diversification. But in case any doubts remain, just look around you. The number of mutual funds has exploded throughout the world. In the US, in fact, there are more funds than individual stocks. And a lot of that explosive growth has to do with the fact that mutual funds provide investors with easy and low-cost diversification.
Think about the obstacles that a small investor faces when trying to diversify his portfolio broadly. First, he would have to choose wisely among hundreds (in some markets thousands) of stocks. That is no small task, to be sure.
Then he would have to decide how many stocks to include in the portfolio.
That’s tricky. How many stocks a properly diversified portfolio should contain is something that must be determined for a given market at a given point in time, and the estimates may vary widely.
And if this investor’s capital is rather limited and he’d want to buy, say, some 20–30 stocks, he’d end up paying relatively high commissions. Much higher on a per-share basis, for sure, than the big boys in Wall Street.
Compare all that with buying shares in a mutual fund that aims to follow or outperform a benchmark of this investor’s choice. By buying shares in this fund, our investor solves the problem of choosing among hundreds or thousands of shares and the problem of how many different stocks to buy, and does all that at a relatively low cost. Just one share in a fund may represent ownership in hundreds of companies. When it comes down to diversification, it doesn’t get any better than investing through mutual funds.
The big picture
Most investors diversify their holdings, and they do so for a good reason: to lower the risk of their portfolios. But diversification is not just about risk reduction. At the end of the day, it is about achieving the ultimate goal of investors: the maximization of risk-adjusted returns.
A key magnitude in the process of diversification is the correlation coefficient.
Far from being statistical magnitude with little practical importance, this coefficient plays a central role in the proper selection of assets to be included in portfolios. And it’s also instrumental in properly assessing the risk of different assets.
If you’re still not convinced that diversification is the way to go, there are two things you can do. First, just look around you at the explosive growth of the mutual fund industry worldwide. And second, read the next chapter, where we elaborate on the benefits of diversification.
Excel section
There is no new Excel material in this chapter; all the magnitudes we have discussed were covered in the Excel sections of previous chapters. The calculation of the portfolio that maximizes risk-adjusted returns is not trivial and we need more advanced tools to handle it. In Chapter 11 we’ll discuss a portfolio-optimization program that will enable us to do that and more.
Challenge section
1 Consider the annual returns of the Norwegian and Spanish markets, both summarized by the MSCI indices (in dollars and accounting for both capital gains and dividends), displayed in panel A of Table 5.3.
Then calculate:
(a) The mean annual return of both markets.
(b) The annual standard deviation of returns of both markets.
(c) The correlation of returns between the two markets. (Is it high?
Low? What do you make of it?)
TABLE 5.3
Panel A Panel B
Year Norway Spain xN xS Risk Return RAR
(%) (%) (%) (%) (%) (%)
1994 24.1 –3.9 100.0 0.0
1995 6.5 31.2 90.0 10.0
1996 29.2 41.3 80.0 20.0
1997 6.7 26.2 70.0 30.0
1998 –29.7 50.6 60.0 40.0
1999 32.4 5.3 50.0 50.0
2000 –0.4 –15.5 40.0 60.0
2001 –11.7 –11.0 30.0 70.0
2002 –6.7 –14.9 20.0 80.0
2003 49.6 59.2 10.0 90.0
0.0 100.0
2 Given the weights for the Norwegian and Spanish markets in panel B of Table 5.3, calculate the risk, return, and risk-adjusted return (RAR) of those ten portfolios. Then:
(a) Make a graph of the feasible set.
(b) Calculate the MVP. Would you rather put all your money in Norway or in the MVP? Why?
(c) Would you rather put all your money in Norway, all your money in Spain, or all your money in a portfolio invested 40% in Norway and 60% in Spain? Why?