6
B
y now we know that there’s more to risk than the volatility of individual assets. In this chapter we’ll look into the factors that determine volatility and the role they play in the risk of a portfolio. And while doing all that, we’ll end up redefining the concept of risk. We’ll also explore further the role of correlations (or covariances) in portfolios, and briefly discuss the benefits of international diversification.Total risk v. systematic risk
When discussing risk in Chapter 3 we stressed that, unlike the concept of return, which is easy to define, the concept of risk is far more slippery. That’s one of the reasons why a few chapters of this book are allocated to discuss it from different points of view. Having said that, we have argued that a possible way to think about risk is as the standard deviation of the asset’s returns. And, under some conditions, that is by far the most widely accepted way to think about risk.
What are the conditions? Basically that we consider the asset not as a part of a portfolio but in isolation. Yes, risk does depend on the context and that should come as no surprise. Giving darts to a monkey and setting him lose on the street may be dangerous, but giving him the darts in a crystal cage may be less so.
Let’s think about volatility for a minute. We see stock prices changing all the time. Have you ever wondered why? Of course we can think of a million reasons, but let’s try to fit all factors into two boxes. In one, let’s put all the factors that are specific to the companies behind the stocks. You know, a new CEO, the introduction of a new product, the departure of a well-known executive, a competitor’s release of a better technology . . . you get the picture. These and many others are all idiosyncratic factors that originate in the company (or perhaps the industry) and affect the company’s stock price (and perhaps that of its competitors).
Having said that, there are of course many reasons why a company’s stock price may fluctuate that have to do with factors unrelated to the company.
Think, for example, about macroeconomic events, such as changes in interest rates, in expected inflation, or in the expected growth of the economy, to name but a few. Or think about political events, such as presidential elections. Or think, more generally, about events that affect the economy as a whole. These and many others are economy-wide factors that affect the stock price of all companies, at the same time, and in the same direction (though not necessarily in the same magnitude).
Let’s give names to these two boxes before we go on. The idiosyncratic events that originate in the company are usually referred to as unsystematic factors; the economy-wide events exogenous to the company are usually called systematic factors. Most of the time we should have little trouble placing most of the events that affect stock prices into one of these two boxes. Which is another way of saying that volatility is determined by systematic and unsystematic factors. Or, put differently, total risk is the sum of systematic (or market) riskplus unsystematic (or idiosyncratic) risk.
Diversification again
In the previous chapters we discussed several reasons for which diversification is beneficial for investors. We’ll discuss now yet another angle. Suppose we (unwisely) have invested all our money in one stock. Then the risk of our (one- asset) portfolio will be fully determined by the total risk of this stock, that is, by the standard deviation of its returns.
Now suppose we decide to add one other stock to our portfolio. Then the risk of our (now two-asset) portfolio should decrease. Why? Formally, because as long as the correlation between the two stocks is not exactly equal to 1, then the risk of our portfolio will be lower than the weighted average of the risks of the assets in our portfolio. (In other words, as long as the correlation is lower than 1, we will obtain somebenefits from diversification.)
Intuitively, this happens for two reasons. First, because as we add another stock to the portfolio, the importance (weight) of each stock decreases. In a two- stock portfolio, events that affect the price of either stock only partiallyaffect the risk of the portfolio. Second, because unless all the factors that affect one company also affect the other (at the same time, in the same direction, and in the same magnitude), the two stocks will not move exactly in sync. As long as that happens, we will obtain some diversification benefits.
What happens as we add more and more stocks to our portfolio? The same thing over and over again. That is, the more stocks we have, the more likely it becomes that the very many idiosyncratic factors that affect the price of the stocks in our portfolio cancel each other out. As information about idiosyncratic events flows into the market, the negative impact on some stocks will tend to be averaged out by the positive impact on others. In a fully diversified portfolio, the whole impact of unsystematic events vanishes and we’re left bearing only systematic risk. In other words, diversification is a way of reducing (and, at the limit, eliminating) unsystematic risk.
Exhibit 6.1 displays a graphical representation of the preceding arguments.
Note that the rate at which risk falls decreases as we increase the number of stocks, each additional stock reducing risk a bit less than the previous one.
Eventually, adding more assets to the portfolio will reduce risk by a negligible amount, and that is when we have achieved a fully diversified portfolio. In that situation, all the unsystematic risk has been diversified away and we’re left bearing only the systematic risk. It should come as no surprise, then, that unsystematic risk is sometimes referred to as diversifiablerisk, and systematic risk as undiversifiablerisk.
Why we can’t diversify away the systematic risk should be obvious. As we discussed above, these are economy-wide factors that affect all companies, at the same time, and in the same direction. In other words, there’s no escaping from the impact of these events. (International diversification, which we discuss below, may do the trick, though.)
EXHIBIT 6.1
Limits to diversification
More on systematic risk
It should be clear from the previous discussion that systematic risk puts a limit on the benefits of diversification. That is, diversification enables us to reduce risk but never to eliminate it. How much risk we can diversify away, however, is an empirical question with a different answer across markets and over time. (As
Risk
Number of stocks Systematic risk
a very crude estimate, consider that an investor fully diversified in US stocks may bear some 25% of the volatility of the average stock.)
Much the same could be said about the number of stocks we need in order to achieve a fully diversified portfolio. Estimates vary widely across markets and over time. Some argue that full diversification in the US could be achieved with a careful selection of as few as 10–15 stocks, though others argue that the number is closer to 30 stocks. (And others think that the number is no less than 300.) Again, it’s rather pointless to entertain an answer to this question without a specific market or point in time in mind.
It should also be clear now why finding stocks with a negative correlation is virtually impossible in practice. Although unsystematic factors may push different stocks in different directions, the systematic factor pushes all of them in the same direction. This induces a positive correlation among all stocks in the market, from which the limit on the benefits from diversification follows.
Now, how do we assess an asset’s risk if instead of being considered in isolation we consider it within a diversified portfolio? The math of it is less than trivial and the intuition less than great, but the bottom line is this: the risk of a stock that is part of a diversified portfolio is measured by the contribution of the stock to the risk of the portfolio, which can be assessed in absolute or in relative terms. The absolute contribution is measured by the covariance between the stock and the portfolio, and the relative contribution by beta.
Like Sylvester Stallone speaking Chinese again? Fear not, an example’s on the way.
An example
Panel A of Table 6.1 shows the returns of three companies, Apple, Home Depot (HD), and Procter & Gamble (P&G) over the years 1994 to 2003. According to our measure of (total) risk, the standard deviation of returns, Apple is the riskiest of the three (SD= 85.9%) and P&G the least risky (SD= 19.9%).
Let’s now form an equally weighted portfolio of these three stocks by investing one-third of our money in each company. We know that the expected return of this portfolio is given by the weighted average of returns, that is, (1/3) · (0.300) + (1/3) · (0.256) + (1/3) · (0.174) = 24.3%. But right now we’re more interested in its risk, for which we need standard deviations and covariances.
The standard deviations are displayed at the bottom of Table 6.1, and the variances (the square of the standard deviations) and covariances in panel A of Table 6.2.
TABLE 6.1
Apple HD P&G
(%) (%) (%)
1994 35.2 16.9 11.3
1995 –17.3 4.3 36.8
1996 –34.5 5.4 32.2
1997 –37.1 76.9 50.5
1998 212.0 108.4 15.9
1999 151.2 69.0 21.6
2000 –71.1 –33.3 –27.1
2001 47.2 12.1 3.1
2002 –34.6 –52.7 11.5
2003 49.2 49.0 18.5
AM 30.0% 25.6% 17.4%
SD 85.9% 47.6% 19.9%
We know that the risk of any portfolio is given by the sum of all the elements in the variance–covariance matrix, which contains all the relevant variances, covariances, and weights. (Take a look at Chapter 4 if you don’t remember.) That is precisely what is shown in panel B of Table 6.2. The 0.0820, for example, is calculated as (1/3) · (1/3) · (0.7379); the 0.0338 is calculated as (1/3) · (1/3) · (0.3038); and so on. The sum of these nine elements yields the variance of the portfolio (0.1905), and the square root of this number yields its standard deviation, 43.6%.
TABLE 6.2
Panel A Apple HD P&G
Apple 0.7379 0.3038 0.0075
HD 0.3038 0.2265 0.0439
P&G 0.0075 0.0439 0.0396
Panel B Apple HD P&G
Apple 0.0820 0.0338 0.0008
HD 0.0338 0.0252 0.0049
P&G 0.0008 0.0049 0.0044
Panel C Apple HD P&G
Sum 0.1166 0.0638 0.0101
Proportion 0.61 0.34 0.05
Note that the risk of the portfolio, 43.6%, is lower than the weighted average of the risks, which is (1/3) · (0.859) + (1/3) · (0.476) + (1/3) · (0.199) = 51.1%.
This reduction in risk is the result of diversification. In other words, when we put these three stocks together in a portfolio, part of their unsystematic risk vanishes and the risk of the portfolio is lower than the weighted average of the individual risks. It then follows, as a mathematical necessity, that each stock is contributing to the risk of the portfolio lessthan its total risk.
Let’s look at the numbers, but, for convenience, instead of focusing on the standard deviation of the portfolio let’s focus on its variance (0.1905). The row labeled ‘Sum’ in panel C of Table 6.2 is the vertical sum of the rows in panel B.
(For example, 0.0820 + 0.0338 + 0.0008 = 0.1166.) Each of these numbers represents the absolute contribution of each stock to the risk of the portfolio.
In other words, if we add up these numbers, we obtain the risk of the portfolio (0.1166 + 0.0638 + 0.0101 = 0.1905).
The row labeled ‘Proportion,’ on the other hand, is simply made up of the numbers in the row above divided by the variance of the portfolio. (For example, 0.1166/0.1905 = 0.61.) Each of these numbers represents the relative contributionof each stock to the risk of the portfolio. In other words, if we add up these three numbers we obtain 1 (0.61 + 0.34 + 0.05 = 1). These numbers suggest that Apple, HD, and P&G contribute 61%, 34%, and 5%, respectively, to the risk of our equally weighted portfolio.
That is the way we measure the risk of a stock that is part of a portfolio: by its (absolute or relative) contribution to the risk of the portfolio. Now for the names. Each number in the row labeled ‘Sum’ is the covariancebetween each stock and the portfolio; each number in the row labeled ‘Proportion’ is the beta of each stock relative to the portfolio. (We’ll define formally and discuss the beta of a stock in the next chapter, but for the time being note, as the discussion above suggests, that it’s obtained from the covariance between the stock and the portfolio divided by the variance of the portfolio.)
A brief digression on covariances
Most investors, perhaps for no good reason, tend to think of risk as volatility usually measured by the standard deviation of an asset’s returns. However, as we just discussed, that is not the proper measure of risk of an asset in a portfolio, particularly when the portfolio is properly diversified. Put differently, the larger the number of assets in a portfolio, the less relevant the total risk of each asset (measured by its standard deviation), and the more relevant the
asset’s contribution to the risk of the portfolio (measured by the covariance between the asset and the portfolio).
The example above illustrates why this is the case. And there are two other quick ways of reinforcing this idea. First, think that as the number of assets in a portfolio grows, the number of covariances grows much faster than the number of variances. In a two-asset portfolio, we have two variances and two covariances; in a 20-asset portfolio we have 20 variances and 380 covariances; in a 100-asset portfolio we have 100 variances and 9,900 covariances. Which do you think will have more impact on the risk of the portfolio, those few variances or those very many covariances?
Second, let’s make a couple of assumptions that are not really needed to get to the final conclusion; we’ll make them just to get the point across more easily.
Assume, first, that all the nassets in a portfolio have the same variance (let’s call it V); second, that the covariance between any two assets in the portfolio is the same (let’s call it C); and third, that we invest an equal amount in each of the n assets (which makes all weights equal to 1/n). Then, the risk of this portfolio, measured by its variance (Varp), is given by
(6.1)
If the expression above looks a bit complicated, just give it some thought. We have nassets, nvariances, n2– ncovariances, the weight of each asset is 1/n, all the variances are the same (V), and all the covariances are the same (C). Got it?
OK then, now think what happens as we increase the number of assets in the portfolio. For a very large n, the expression (1/n) · V tends to 0, and the expression (1 – 1/n) · C tends to C. In other words, as the number of assets grows, the risk of the portfolio is largely determined by covariances and largely independent of variances.
A brief digression on international diversification
So far we have implicitly been thinking of portfolios of stocks within a given market. That’s why, after obtaining a fully diversified portfolio that eliminates all the unsystematic risk, we are left bearing the (undiversifiable) systematic risk of the market of our choice. But think about it: why end there?
Varp= n·
(
n1)
2· V+ (n2– n) ·(
1 n)
2· C=(
1 n)
· V+(
1 –1n)
· CSuppose we have a fully diversified portfolio of US stocks and we’re therefore subject to the systematic risk of the US economy. Now, there’s no reason to think that the factors that affect Japanese stocks should be perfectly correlated to those that affect US stocks. In fact, there are good reasons to think otherwise.
(Think of events in the US that would have no impact on Japanese stocks, and events in Japan that would have no impact on US stocks.) Which means that if we now add Japanese stocks to our portfolio, the risk of our portfolio should fall.
That’s good; but again, why end there?
What if we now consider European stocks? Same story. As long as the factors that affect European stocks are not perfectly correlated to those that affect American and Japanese stocks (and of course they never are), European stocks should provide diversification benefits. In other words, the risk of our portfolio should fall again.
Can you see where we’re going? In the same way that within a market we add stocks to our portfolio until we are fully diversified in that market, we’re now adding international stocks (or markets) to our portfolio until we are fully diversified internationally. And, needless to say, the volatility of a fully diversified portfolio of international stocks would be lower than that of a fully diversified portfolio of stocks in any one country. (As a very crude estimate, consider that an investor fully diversified in international stocks would bear some 10% of the volatility of the average stock.)
Now for the bad news, which you probably expected anyway. No matter how many stocks from how many countries we include in our portfolio, we will never be able to eliminaterisk; we’ll only be able to reduceit. In other words, even the world market is subject to systematic factors (wars, international crises, oil prices . . .) that will prevent us from eliminating completely the volatility of our portfolio.
One final comment. Note that we have discussed diversifying across stocks within a market, and diversifying across international equitymarkets. However, a good diversification strategy does not have to be restricted to stocks only. The portfolio may (perhaps even should) contain other assets, such as bonds, derivatives, real estate, and more. In short, put your eggs in different baskets, and make sure that the baskets come in many different colors and sizes.
The big picture
However volatile an asset might be, we don’t have to bear all its risk. That’s one of the mean reasons why we diversify. But diversification does not eliminate
risk; it merely reducesrisk. How much we can reduce risk and how many stocks make up a fully diversified portfolio, in turn, are empirical questions that depend on the time and place we ask the question. In both cases, the estimates vary widely across countries and over time.
Unlike an asset in isolation, the risk of an asset within a diversified portfolio is measured by the contribution of the asset to the risk of the portfolio. This contribution can be assessed in total terms by the covariance between the asset and the portfolio, or in relative terms by the asset’s beta. And this relative contribution to portfolio risk is, as we’ll discuss in the next chapter, the proper measure of risk in the most widely used asset pricing model.
Excel section
There is no new Excel material in this chapter; all the magnitudes discussed have already been covered in the Excel sections of previous chapters.
Challenge section
We’ll do only one thing in this Challenge section: we’ll explore how the risk of the portfolio changes as we add more and more stocks to it. In order to make the task manageable, let’s make the following two assumptions:
(1) All assets have a standard deviation of 30%; and (2) the covariance between any two stocks is 0.02. Note that the first assumption implies that all assets have a variance of 0.09. The second assumption, in turn, implies that the correlation between any two stocks is 0.5.
■ Open an Excel spreadsheet and input in cells A1, B1, and C1 the labels
‘Number of stocks in the portfolio,’ ‘Variance of the portfolio,’ and
‘Standard deviation of the portfolio,’ respectively.
■ In cells A2 through A31 input the numbers 1 through 30, representing the number of stocks in the portfolio.
■ In cell B2 input the expression ‘=(1/A2)*0.09+(1–1/A2)*0.02’ for the variance of the portfolio. Then copy this expression all the way down through cell B31.
■ In cell C2 input the expression ‘=sqrt(B2)’ and copy all the way down through cell C31.