Did we ask what is risk?
Problems with the standard deviation
One step in the right direction
The semideviation
A brief digression on the semideviation
The VaR
The big picture
Excel section
Challenge section
T
here is little controversy about how to measure returns. When it comes down to assessing risk, however, views on how to measure it differ widely. But one thing is for certain: although investors associate risk with negative outcomes, the widely accepted and widely used risk measures we discussed so far don’t. In this chapter we’ll discuss a relatively new but increasingly accepted way of assessing risk that aims to capture only the impact of those negative outcomes.Did we ask what is risk?
We sure did. Twice in fact. And we gave two different answers. First we argued that when we consider an asset in isolation we can think of risk in terms of volatility, measured by the standard deviation of returns. Later we argued that when the asset is part of a diversified portfolio, the unsystematic risk gets diversified away and the systematic risk that remains is measured by beta (which captures the contribution of the asset to the risk of the portfolio, or the asset’s sensitivity to fluctuations in the market).
Now, if you really think about it, there’s something inherently wrong with the standard deviation as a measure of risk. Consider an asset with a mean annual return of 10%, and assume that as time goes by this asset delivers returns of 20%, 45%, and 30%. Would that make you unhappy? Would you view this asset as risky because it tends to deviate aboveits mean? Certainly not. And yet, each of these deviations above the mean contributes to increasing the standard deviation.
Note that the standard deviation treats an x% fluctuation above and below the mean in the same way; that is, in both cases this measure of risk increases by the same amount. But investors obviously don’t feel the same way about these two fluctuations. They usually consider deviations above the mean as good and those below the mean as bad. Shouldn’t then a good measure of risk capture this asymmetry? (There is no special reason for which deviations should be measured with respect to the mean. In fact, different investors may be interested in measuring deviations with respect to different parameters. More on this below.)
There exist several measures of risk that isolate and measure the downside of assets. In this chapter we’ll focus on two: one is the counterpart of the standard deviation but in a downside risk framework; the other is a measure of the worst expected outcome under some specified conditions. But before we get into details, let’s think a bit harder about what’s wrong with the standard deviation, perhaps the most widely accepted measure of risk.
Problems with the standard deviation
Take a look at Table 9.1, which in the second column displays the annual returns (R) of Oracle between 1994 and 2003. As the next-to-last row shows, the arithmetic mean annual return (AM) during this period was a healthy 46%.
And as your eyes can tell you without resorting to any measure of risk, Oracle treated its shareholders to quite a rocky ride.
TABLE 9.1
Year R R–AM (R– AM)2 Min(R– AM, 0) {Min(R– AM, 0)}2
(%) (%) (%)
1994 53.5 7.4 0.0055 0.0 0.0000
1995 44.0 –2.0 0.0004 –2.0 0.0004
1996 47.8 1.7 0.0003 0.0 0.0000
1997 –19.8 –65.9 0.4341 –65.9 0.4341
1998 93.3 47.2 0.2231 0.0 0.0000
1999 289.8 243.7 5.9398 0.0 0.0000
2000 3.7 –42.3 0.1790 –42.3 0.1790
2001 –52.5 –98.5 0.9708 –98.5 0.9708
2002 –21.8 –67.8 0.4603 –67.8 0.4603
2003 22.5 –23.5 0.0554 –23.5 0.0554
Average 46.0% 0.8269 0.2100
Square Root 90.9% 45.8%
We know how to easily calculate a standard deviation of returns in Excel, but let’s take the long road here. The third column of the table displays the difference between each annual return and the mean annual return, for example 7.4% = 53.5% – 46.0%. The fourth column displays the square of these numbers, for example 0.0055 = 0.0742. The average of these squared deviations from the mean is the variance (0.8269). And the square root of the variance is the standard deviation of returns; in this case, 90.9%. Note that this standard deviation is over four times higher than the historical standard deviation of the S&P500 (around 20%), which would make Oracle a very risky stock. Right?
Not so fast. Take a look at the numbers in the fourth column. All those numbers are positive, which means that each and every one of these observations adds to the standard deviation. In other words, every annual return, regardless of sign or magnitude increases this widely accepted measure of risk (unless it is exactly equal to the mean). In fact, the largest number in this fourth column (the one that contributes to increasing the standard deviation the most) is that for 1999 when Oracle delivered a return of almost 290%. Now, if
you had held Oracle during 1999, by the end of the year would you be happy or unhappy? Would you count this performance against Oracle as the standard deviation does?
One step in the right direction
Tweaking the standard deviation so that it accounts only for deviations below the mean return is not difficult. The fifth column of Table 9.1 shows the lower of each return minus the mean return or 0. In other words, if a return is higher than the mean (and therefore the difference between the former and the latter is positive), the column shows a 0; if, on the other hand, the return is lower than the mean (and therefore the difference between the former and the latter is negative), the column shows the difference between the two.
In 1994, Oracle delivered a 53.5% return, which is higher than the mean return of 46%; therefore the fifth column shows a 0. In 1995, however, Oracle delivered a 44.0% return, which is 2 percentage points lower than the mean return of 46%; therefore, the fifth column shows the shortfall of –2%. If you compare the third and the fifth columns you will notice that when a return is lower than the mean both columns show the same number. When a return is higher than the mean, however, the third column shows the difference between these two numbers and the fifth column shows a 0. Finally, note that this fifth column shows only negative numbers and zeros but no positive numbers.
The last column of Table 9.1 shows the square of the numbers in the fifth column. As the next-to-last row shows, the average of these numbers is 0.2100, and, as the last row shows, the square root of this number is 45.8%. What does this number indicate? It has a simple and intuitive interpretation: it measures volatility but only belowthe mean. This obviously looks like a step in the right direction. We don’t ‘punish’ Oracle for its deviations above the mean; we do it only when it deviates below this parameter.
Now, is there anything special about the mean return? Isn’t it possible that some investors would be interested in measuring volatility below the return of the market? Or volatility below the risk-free rate? Or volatility below 0? Or, more generally, volatility below any given number that they may consider relevant?
The semideviation
That is exactly what one of the two measures of risk we’ll discuss in this chapter intends to capture. The downside standard deviation of returns with respect to a benchmark returnB (SSDB)is formally defined as
(9.1)
where B is anybenchmark return relevant to an investor, T is the number of observations, and t indexes time. Although it lends itself to some ambiguity, equation (9.1) is, for the sake of simplicity, usually referred to as the semideviation of returns, which is the name we’ll use from now on.
Let’s think a bit about equation (9.1), which is really not as complicated as it seems to be. It requires us to do the following. (1) In every period we calculate the difference between the return for the period and the benchmark return B;
(2) in every period we take the lower of the return minus Bor 0; (3) in every period we square the numbers in the previous step; (4) then we take the average of all the numbers in the previous step; and (5) we finally take the square root of the number in the previous step.
Take a look at Table 9.2, where we consider again the returns of Oracle during the 1994–2003 period, as well as three different benchmark returns. The third column performs the five steps outlined above and does so with respect to a benchmark equal to the mean return of 46%. (This column is identical to the last column of Table 9.1.) The fourth column does the same with respect to a benchmark equal to a risk-free rate (Rf) of 5%. And the last column does the same with respect to a benchmark of 0%. Finally, the last row shows the semideviation with respect to the three benchmarks.
How should we interpret these numbers? Each semideviation measures volatility below its respective benchmark. Note that, because the risk-free rate of 5% is below Oracle’s mean return of 46%, we would obviously expect (and find) less volatility below the risk-free rate. Similarly, there is even less volatility below 0.
Now, if you’re finding that a volatility of 21.5% below a risk-free rate of 5% (or a volatility of 19% below 0 for that matter) does not spark your intuition, you’re not alone. That’s why the semideviation is best used in two contexts: one is in relation to the standard deviation and the other is in relation to the semideviation of other assets.
SSDB=
√
(1/T) ·Σ
Tt=1{Min[(Rt– B), 0]}2TABLE 9.2
Year R {Min(R– AM, 0)}2 {Min(R– Rf, 0)}2 {Min(R – 0, 0)}2 (%)
1994 53.5 0.0000 0.0000 0.0000
1995 44.0 0.0004 0.0000 0.0000
1996 47.8 0.0000 0.0000 0.0000
1997 –19.8 0.4341 0.0617 0.0393
1998 93.3 0.0000 0.0000 0.0000
1999 289.8 0.0000 0.0000 0.0000
2000 3.7 0.1790 0.0002 0.0000
2001 –52.5 0.9708 0.3304 0.2754
2002 –21.8 0.4603 0.0718 0.0475
2003 22.5 0.0554 0.0000 0.0000
Average 46.0% 0.2100 0.0464 0.0362
Square root 45.8% 21.5% 19.0%
Take a look at Table 9.3, which shows the semideviations with respect to the arithmetic mean of each stock (SSDAM), with respect to a risk-free rate of 5%
(SSDRf), and with respect to 0 (SSD0) for both Oracle and Microsoft during the 1994–2003 period. The semideviations for Oracle are the same as those in Table 9.2. The mean return of Microsoft during this period (not reported in the table) was 39.8%.
TABLE 9.3
Company SD SSDAM SSDRf SSD0
(%) (%) (%0 (%)
Oracle 90.9 45.8 21.5 19.0
Microsoft 49.8 39.3 23.1 21.1
Note that, although Oracle is far riskier than Microsoft as measured by their standard deviations, their semideviations tell a different story. First, note that although the volatility of Oracle below its mean is about a half of its volatility (0.458/0.909 = 50.4%), the same ratio for Microsoft is almost 80% (0.393/0.498
= 78.9%). In other words, given the volatility of each stock, much more of that volatility is below the mean in the case of Microsoft than in the case of Oracle.
Of course, it is still the case that the semideviation of Oracle is larger than that of Microsoft. But recall that the mean return of Oracle (46%) is also higher than that of Microsoft (39.8%). In fact, it’s perhaps more telling to compare semideviations with respect to the same benchmark for both stocks. If we
compare, for example, the semideviations of Oracle and Microsoft below the same risk-free rate of 5%, we find that the downside volatility is higher in the case of Microsoft (23.1% versus 21.5%). And if we do the same comparison but with respect to 0, we also find that Microsoft exhibits higher downside volatility (21.1% versus 19.0%).
A brief digression on the semideviation
It should be clear from our previous discussion that a key advantage of the semideviation over the standard deviation is that it considers only the downside volatility that investors view as harmful. Volatility above the mean, which investors view as desirable, does not increase the semideviation but does increase the standard deviation. Note also that, in this framework, volatility is no longer necessarily bad. In fact, volatility below the benchmark is bad, but volatility above the benchmark is good. Doesn’t this make sense?
In addition, the semideviation can be calculated with respect to any benchmark, not just with respect to the mean. This implies that different investors using different benchmarks may perceive the same asset as more or less risky depending on the benchmark they use. And, of course, different investors do have different benchmarks; after all, not all of them invest for the same reasons or have the same goals. Again, doesn’t this make sense?
We mentioned above that calling semideviation the downside standard deviation of returns may be a bit ambiguous. The reason is, as you probably suspected, that the word ‘semideviation’ indicates volatility on only one side of the benchmark but does not explicitly indicate which side (above or below).
However, it is usually implicit in the normal use of the word ‘semideviation’ that the deviations considered are belowthe benchmark.
Having said that, note that we can also calculate the upside standard deviation of returns, which measures volatility above the benchmark B. The steps needed to calculate this magnitude are identical to those outlined above for the semideviation except that in the second step we now need to take the higher of the return minus Bor 0. (Formally, we need to replace the ‘Min’ in equation (9.1) by a ‘Max.’) Try calculating the upside standard deviation of returns with respect to the mean, the risk-free rate, and 0 using the returns of Oracle in Table 9.2, and you should find that these are 75.8%, 97.4%, and 100.1%, respectively.
If you compare these three numbers with the semideviations reported in Table 9.2, you’ll notice that, for all three benchmarks, the upside semideviations
are higher. This indicates that there is more volatility above than below each of these benchmarks or, similarly, that Oracle is more likely to deliver returns above the benchmarks than below them. The fact that there is more volatility above than below the mean, in particular, indicates that the distribution of Oracle’s returns exhibits positive skewness, a characteristic that investors find desirable.
(The concept of skewness is discussed in Chapter 29.)
Finally, although equation (9.1) may seem complicated, calculating a semideviation in Excel is just a tiny bit more complicated than calculating a standard deviation. As we’ll see at the end of the chapter, the semideviation can be calculated in Excel in just one cell.
The VaR
It’s often important for investors or companies to have an idea of how bad adverse outcomes can really be. To answer this question, JP Morgan introduced in 1994 a measure called Value at Risk (VaR), which basically yields the worst expected outcome over a given time horizon for a given confidence level. (Don’t panic, we’ll explain.)
In order to calculate a VaR, two parameters have to be chosen. The first is a time interval, which can be any that is relevant for an investor or company. A bank, for example, may want to know its worst expected outcome on a daily basis in order to set appropriate capital requirements; a long-term investor, on the other hand, may be interested in the worst expected outcome on an annual or a five-year basis. The second choice is the confidence level (c), which will indirectly determine the probability that the outcome is worse than the calculated VaR (as we’ll see shortly). The most typical confidence levels are c= 95% and c= 99%.
Take a look at Exhibit 9.1, which depicts the probability distribution of a random variable X, which we could think of as returns, revenues, profits, or any other variable of our interest. Let’s assume that the variable is measured on a daily basis (the time interval), and let’s choose a 95% confidence level (hence, c
= 95%). There are two identical ways of thinking about the VaR in this context.
We could define it as the worst expected outcome, over one day, at a 95%
confidence level. Or, perhaps more telling, we could say that a daily outcome worse than the VaR will occur with a probability of 5%.
As Exhibit 9.1 shows, the VaR is a number on the horizontal axis, and is measured in the same units as the variable of interest. If returns are measured in percentages, then the VaR will be measured in percentages; if revenues or
profits are measured in dollars, then the VaR will be measured in dollars; and so forth. Note, also, that having chosen a confidence level c, (1 – c)% of the area of the distribution will be to the leftof the VaR.
EXHIBIT 9.1 VaR
Formally, the value at risk (VaR)is defined as
VaRc= x such that P(X≤x) = 1 – c (9.2)
In other words, this expression says that the VaR is a number x such that the probability that the variable Xtakes a value lower than or equal to xis equal to 1–c, where cis the chosen confidence level.
The calculation of the VaR is not necessarily trivial. As you can see from Exhibit 9.1, it implies the calculation of a number that leaves (1 – c)% of the distribution to its left. This, in turn, implies that we first need to characterize the distribution, and then calculate this probability (therefore having to calculate an integral, as discussed in Chapters 28 and 29). But don’t throw your arms in despair just yet. If the variable X follows a normal distribution, then calculation of the VaR is very simple indeed. In fact, under normality, the VaR is defined as
VaRc= AM– z · SD (9.3)
VaRc X (1 – c)%
where AMand SDdenote the (arithmetic) mean and standard deviation of the underlying distribution, and zis a number that comes from the standard normal distribution (discussed in Chapter 28). For the most widely used confidence levels, 95% and 99%, ztakes a value of –1.64 and –2.33, respectively.
Take a look at Table 9.4, which reports the mean monthly return and monthly standard deviation of four markets between January 1994 and December 2003.
The distribution of monthly returns of these four markets is approximately normal, so the assumption that sustains equation (9.3) approximately holds.
TABLE 9.4
France Italy Japan UK
(%) (%) (%) (%)
AM 0.9 1.0 0.0 0.7
SD 5.6 6.7 6.0 4.1
VaR95 –8.3 –10.0 –9.8 –6.0
VaR99 –12.2 –14.7 –13.9 –8.9
Recall that before calculating the VaR we need to select a time horizon and a confidence level. Let’s select then a monthly time interval (to make it consistent with the frequency of the returns in Table 9.4), and a confidence level of 95%.
Let’s also consider the French market, which between 1994 and 2003 delivered a mean monthly return of 0.9% with a standard deviation of 5.6%. The 95% VaR then is simply calculated as 0.09 – 1.64 · 0.056 = –8.3%. In other words, the worst expected outcome in the French market over one month at a confidence level of 95% is –8.3%. Or, perhaps clearer, in the French market the probability of a monthly losshigher than 8.3% is 5%.
What if we change the significance level to 99%? Well, the probability distribution of returns doesn’t change, so the mean and standard deviation remain the same. The only change in equation (9.3) is the value of z, which will now be –2.33. Therefore, the 99% VaR for the French market is calculated as 0.09 – 2.33 · 0.056 = –12.2%. In other words, in the French market the probability of a monthly losshigher than 12.2% is 1%. Note, obviously, that the higher the confidence level, the lower the VaR. Or, put differently, the lower the VaR, the more unlikely it becomes that scenarios worse than the VaR materialize.
Following the same steps you can calculate the rest of the VaRs displayed in Table 9.4. Note that, from these four markets, the worst expected losses are in Italy. Not surprisingly, in fact, Italy has the largest monthly semideviation of these four markets (4.6%, not reported in the table).
One final comment. Note that normality is an assumption that may or may not apply to the variable of our interest. That’s why it is important to test whether or not the distribution for which we need to calculate the VaR is normal. (The distribution of returns of Oracle, for example, is nowhere close to normal.) If it is not, then the calculation of the VaR is more complicated than indicated by equation (9.3). The idea of the VaR remains the same regardless of the underlying distribution, that is, we’re looking for a number that leaves (1 – c)%
of the area under the distribution to its left. But its actual calculation may change substantially depending on the type of distribution of the relevant asset.
The big picture
Most investors associate risk with negative outcomes. However, one of the most widely used definitions of risk, the standard deviation, does not. Downside risk is an increasingly popular alternative to traditional notions of risk. It captures the downside that investors want to avoid and not the upside that investors want to be exposed to.
The semideviation defines risk as volatility below a benchmark. This benchmark is determined by each individual investor or company, which adds to the plausibility of this measure of risk. The semideviation also highlights the fact that not all volatility is bad; only volatility below the benchmark. The VaR, on the other hand, provides investors with an idea of how bad adverse scenarios can be. It is easy to interpret, useful, and widely used in banks and financial institutions.
Excel section
The semideviation can be calculated in Excel in more than way. We’ll focus here on two ways, emphasizing the first, which is the easier of the two (it takes just one cell). Let’s introduce a command that simply counts the number of observations in a series. Suppose you have a series of returns in cells A1:A10.
■ To count the number of observations in the series simply type
‘=count(A1:A10)’ in cell A11 and then hit ‘Enter.’
In order to calculate the semideviation, let’s assume that we have entered the ‘count’ command in cell A11, where we therefore have the number of observations in the series (ten in our case). Let’s also assume that in cell A12