What is this all about?
Long-term returns
Long-term risk: Volatility
Long-term risk: Shortfall probability
Time diversification and mean reversion
Forecasting target returns
The big picture
Excel section
Challenge section
F
inally we come to the end of the first part. But we can’t really finish without having at least a brief discussion about the long run. You see, it turns out that the investment horizon is a critical variable when making investment decisions, and although financial advisors do factor it carefully into their advice, most books ignore it almost completely. This one won’t.What is this all about?
Answer, fast, which asset is riskier, stocks or bonds? If you’re like most people, the first answer that came to mind was stocks. But why? Probably for two reasons. Most investors tend to read a lot more about the stock market than about the bond market, and therefore are more aware of the jumps and volatility in the former than in the latter. In addition, and perhaps more importantly, risk is often thought of as the volatility of annual returns. In the US, the annual standard deviation of the stock market is around 20% and that of the bond market around 10%; hence, the usual perception of relative risk.
There are, however, at least three problems with this perception. First, as we’ll discuss below, there are two different ways of defining long-term volatility, and they give contradicting results. Second, as we have discussed in previous chapters, volatility is not the only way of assessing risk, and again different risk measures may yield contradicting results. And third, as we’ll see, for long investment horizons the data tells a different story.
Long-term returns
Let’s get this straight: in the long term, the compounding power of stocks trouncesthe compounding power of bonds. There you have it. There is really no question about it. Want some evidence? In his fantastic book Stocks for the Long Run(3rd edn, 2002, McGraw-Hill), Jeremy Siegel reports that $1 invested in the US stock market in 1802 would have turned into $8.8 millionby the end of 2001. In comparison, the same dollar invested in bonds would have turned into $13,975. Now, thatis a difference!
Take a look at Table 12.1, which displays arithmetic (AM) and geometric (GM) mean annual returns, in both nominal and real terms, for 16 countries and the world market during the period 1900–2000. Real returns are nominal returns net of inflation and capture changes in purchasing power.
TABLE 12.1
Stocks Bonds
Nominal Real Nominal Real
Country GM AM GM AM GM AM GM AM
(%) (%) (%) (%) (%) (%) (%) (%)
Australia 11.9 13.3 7.5 9.0 5.2 5.8 1.1 1.9
Belgium 8.2 10.5 2.5 4.8 5.1 5.6 –0.4 0.3
Canada 9.7 11.0 6.4 7.7 5.0 5.4 1.8 2.4
Denmark 8.9 10.7 4.6 6.2 6.8 7.3 2.5 3.3
France 12.1 14.5 3.8 6.3 6.8 7.1 –1.0 0.1
Germany 9.7 15.2 3.6 8.8 2.8 4.7 –2.2 0.3
Ireland 9.5 11.5 4.8 7.0 6.0 6.7 1.5 2.4
Italy 12.0 16.1 2.7 6.8 6.7 7.0 –2.2 –0.8
Japan 12.5 15.9 4.5 9.3 5.9 6.9 –1.6 1.3
Netherlands 9.0 11.0 5.8 7.7 4.1 4.4 1.1 1.5
S. Africa 12.0 14.2 6.8 9.1 6.3 6.7 1.4 1.9
Spain 10.0 12.1 3.6 5.8 7.5 7.9 1.2 1.9
Sweden 11.6 13.9 7.6 9.9 6.2 6.6 2.4 3.1
Switzerland 7.6 9.3 5.0 6.9 5.1 5.2 2.8 3.1
UK 10.1 11.9 5.8 7.6 5.4 6.1 1.3 2.3
US 10.1 12.0 6.7 8.7 4.8 5.1 1.6 2.1
World 9.2 10.4 5.8 7.2 4.4 4.7 1.2 1.7
Average 10.3% 12.7% 5.1% 7.6% 5.6% 6.2% 0.7% 1.7%
Source: Adapted from Triumph of the Optimists: 101 Years of Global Investment Returns, by Elroy Dimson, Paul Marsh, and Mike Staunton. Princeton University Press, New Jersey, 2002
The evidence is shockingly clear: the mean return of stocks, geometric and arithmetic, nominal and real, is higher than the mean return of bonds in every country and in most cases by a substantial margin. The difference between the compounding power of both assets can be viewed from different angles but here’s an interesting one. Take a look at the last line, which shows averages across the 16 countries, and note that the mean annual compound real return of stocks is 5.1% and that of bonds 0.7%. These figures imply that by investing in stocks purchasing power would double in just under 14 years. In bonds? It would take only 98.5 years.
Long-term risk: Volatility
However impressive the differential compounding power of stocks and bonds may be, unfortunately none of us have a 200-year investment horizon. In fact, for most investors the long run is a 30-year period at most, and often much shorter than that. Whatever the length of the investment horizon, though, the interesting question is, how does risk evolve with it. In other words, how does risk evolve as the holding period increases from 1, to 5, to 10, to 30, or to any number of years?
To be sure, this question doesn’t have an undisputed answer. As a matter of fact, its answer is very controversial. Even if we agreed that the proper way to capture risk is with the standard deviation of returns (and that’s a big if), the controversy would not end there. To see why, take a look at Table 12.2, which is based on an asset with a standard deviation of annual returns of 17%, roughly equal to the annual standard deviation of the US stock market between 1871 and 2003. How should we assess risk for, say, five-year holding periods?
There are at least two ways: one is with the cumulativestandard deviation and the other with the annualized standard deviation. Let’s focus on the former first.
TABLE 12.2
Holding period Annual Cumulative Annualized P(R< 0%) P(R< 3%) P(R< 5%)
(Years) % % % % % %
1 17.0 17.0 17.0 30.6 37.0 41.3
5 17.0 38.0 7.6 12.9 22.9 31.2
10 17.0 53.8 5.4 5.5 14.7 24.4
15 17.0 65.8 4.4 2.5 9.9 19.8
20 17.0 76.0 3.8 1.2 6.9 16.4
30 17.0 93.1 3.1 0.3 3.4 11.5
100 17.0 170.0 1.7 0.0 0.0 1.4
The cumulative standard deviation
Recall how we calculate a standard deviation of annual returns. As discussed in detail in Chapter 3, all we need to do is to calculate annual returns over the relevant time period and then calculate the standard deviation of those returns.
If we did that for the (nominal) annual returns of the US market between 1871 and 2003, we would obtain roughly 17%. This suggests one way of calculating the cumulative standard deviation of five-year returns: calculate returns for
every five-year period between 1871 and 2003, and then calculate the standard deviation of those five-year returns.
Actually, there’s a shortcut. (This shortcut requires the assumption of independent returns, which basically means that returns are uncorrelated over time.) For any holding period or investment horizon T, we can calculate the cumulative standard deviation of returns (CSD)simply as
(12.1)
where SD denotes the annualstandard deviation of returns. For example, the cumulative standard deviation of five-year returns is 0.17 · (5)1/2 = 38.0%, as shown in the third column of Table 12.2.
Now, as is obvious from equation (12.1) and the third column of Table 12.2, the longer the holding period T, the larger the cumulative standard deviation.
This is the intuition. If we consider a sample of daily returns and calculate its mean and standard deviation, both numbers would be fairly small, largely because daily returns are fairly small. If we do the same for monthly returns, then both the mean and the standard deviation would be larger, simply because monthly returns are larger than daily returns. If we do the same for annual returns, then both the mean and the standard deviation would be larger, simply because annual returns are larger than daily and monthly returns. If we do the same for five-year returns . . . Get the picture?
As we increase the length of the period for which we calculate returns, the returns themselves increase and so do their mean and standard deviation.
Therefore, as we increase the holding period from 1 to 5, to 10, to 20, or more years, risk (measured by the cumulativestandard deviation) also increases. Or, viewed from another angle, swings in the capital invested on any asset would typically be larger over 20 years than over 10 years, over 10 years than over 5 years, over 5 years than over 1 year, and so forth. This is, in short, the intuition behind the cumulative standard deviation as a measure of long-term risk.
The annualized standard deviation
That sounds like a plausible story, doesn’t it? Great. Now let’s change it! Exhibit 12.1 is based on (nominal) returns for the US stock market between 1871 and 2003. Each bar shows the maximum and minimum return that could have been obtained during each holding period considered. To illustrate, if we consider
CSD= SD ·
√
Tevery one-year return between 1871 and 2003, in the best year we would have obtained 56% and in the worst we would have lost 42.5%. If we consider every five-year holding period instead, in the best we would have obtained a 28.6%
mean annual compoundreturn (that’s 28.6% on top of 28.6%, on top of 28.6%, on top of 28.6%, on top of 28.6%), and in the worst we would have lost money at a mean annual compoundrate of 11.1% (that’s –11.1%, on top of –11.1%, and so on).
EXHIBIT 12.1
Holding-period returns: Best v. worst
Can you see the pattern in the picture? The spread between the best and the worst in one-year holding periods is very large; in any given year, just about anything can happen. But as we increase the holding period to five years, the spread between the best five-year mean return and the worst five-year mean return decreases substantially. If we increase the holding period to ten years, the spread decreases even more. Essentially, what the picture shows is that as we increase the holding period, the mean annual compound return tends to converge to its long-term average (9%).
This intuition can be formally captured by the annualized standard deviation of returns (ASD), which is given by
(12.2)
Return (%)
Holding period (years) 80
60 40 20 0 –20 –40 –60
1
–42.5%
56.0%
5
–11.1%
28.6%
10
–1.5%
19.3%
20
3.0%
17.4%
30
5.0%
13.4%
ASD= SD
√T
where again SD denotes the annual standard deviation of returns and T the holding period. For example, the annualized standard deviation of five-year returns is 0.17/(5)1/2= 7.6%, as shown in the fourth column of Table 12.2.
Note that, as is obvious from equation (12.2) and the fourth column of Table 12.2, as the investment horizon increases, risk (measured by the annualized standard deviation) decreases. Or, put differently, the longer the holding period, the lower the dispersion around the long-term mean annual compound return.
(Incidentally, note from Exhibit 12.1 that the mean annual compound return of the worst 20-year and 30-year holding periods in the US were 3% and 5%, respectively, both positive and higher than inflation.)
There you have it. Two plausible stories that yield opposite results. Perhaps you can now see why this is a controversial issue. But we won’t leave it just like that.
We’ll look at this issue from yet another angle, and perhaps a clearer picture will emerge.
Long-term risk: Shortfall probability
When investing in any asset, an investor may be interested to know how likely is the asset to underperform a given benchmark. The investor, for example, may be interested to know how likely is the asset to deliver negative returns, or returns below inflation, or returns below a risk-free asset, or below any other benchmark he may consider relevant. This likelihood is usually known as the shortfall probability; that is, the probability that an asset falls short of a benchmark return. The interesting question is how this shortfall probability evolves as the holding period increases.
Take a look at the last three columns of Table 12.2, which display the shortfall probabilities of the US stock market with respect to 0%, to an annual rate of inflation of 3%, and to an annual risk-free rate of 5%. We’ll discuss how to calculate these numbers later in the chapter; for now, just focus on the numbers themselves.
Note that all these probabilities decrease steadily as we increase the holding period. There is roughly a 31% probability of obtaining a negative return in any given year, but the chances of obtaining a negative mean annual compound return over ten years are less than 6%. Similarly, although the probability of obtaining less than the mean annual rate of inflation (3%) is 37% in any given year, the probability of obtaining less than a mean annual compound return of 3% over 20 years falls to under 7%. Finally, although there is roughly a 41%
probability of obtaining less than the mean annual risk-free rate (5%) in any given year, the probability of obtaining less than a mean annual compound return of 5% over 30 years falls to under 12%. The message is clear: whatever the benchmark, the probability of falling short of it decreases steadily as we increase the investment horizon. (For this statement to be strictly true, the benchmark must be lower than the asset’s mean compound return.)
The previous numbers are estimations based on the distribution of US (nominal) stock returns between 1871 and 2003. But the historical data itself has an interesting story to tell. Take a look at Exhibit 12.2, based on data discussed in Stocks for the Long Run, which shows the proportion of periods in which stocks underperformed bonds during the years 1871 to 2001. If we consider all the one- year periods between 1871 and 2001, stocks underperformed bonds in basically four periods out of ten. Perhaps you consider this number surprisingly high, but remember that the annual volatility of stocks is roughly twice as high as that of bonds. And note that this exhibit is based on whether one asset outperformed the other, but not by how much. In other words, if stocks outperformed bonds by 20%
in any given period, and bonds outperformed stocks by 1% in the next period, this exhibit would count one win for each asset.
EXHIBIT 12.2 Shortfall probability
Return (%)
Holding period (years) 45
40 35 30 25 20 15 10 5 0
1
39.7%
5
26.0%
10
17.6%
20
4.6%
30
0.0%
Now look at what happens as the holding period increases. If we consider five-year investment horizons, stocks underperformed bonds in only one period out of four; in ten-year holding periods, in less than one period out of five; in 20- year holding periods, in less than one period out of twenty. And in 30-year periods? Surprise! There has been no30-year period in the (1871–2003) history of the US markets in which stocks underperformed bonds.
Now, of course some investors can theorize and forecast probabilities as much as they’d like. But it should still be rather comforting for most stock investors to know that in 30-year periods stocks neverunderperformed bonds, and that even in 20-year periods such underperformance occurred in less than one period out of twenty.
Time diversification and mean reversion
The idea behind the annualized standard deviation as a measure of long-term risk is that, as the investment horizon increases, the dispersion around the long- term mean compound return decreases. In other words, the longer the holding period, the more likely an asset is to deliver its long-term mean compound return. The idea behind the shortfall probability as a measure of long-term risk, in turn, is that, as long as the benchmark is lower than the asset’s long-term mean compound return, the longer the investment horizon, the less likely the asset is to underperform the benchmark.
Putting these two ideas together and comparing two assets, one riskier than the other, it follows that, the longer the holding period, the more likely the riskier asset is to outperform the less-risky asset. For example, if we compare stocks and bonds, these arguments would suggest that, the longer the holding period, the more likely stocks are to outperform bonds. The evidence in Exhibit 12.2 seems to confirm that this is indeed the case.
These arguments are part of the hotly debated issue of time diversification, in which most practitioners believe and some (but certainly not all) academics don’t. Although this concept can be defined in many ways, all of them suggest that as the investment horizon increases, the probability that a riskier asset outperforms a less risky asset also increases. Note that this definition implies that the shortfall probability (the probability that the riskier asset delivers a return below that of the less risky asset) decreases as the holding period increases. It also implies that the longer the holding period, the more likely it becomes that above-average returns offset below-average returns (and that the asset delivers its long-term mean compound return).
Many theoretical arguments can be (and have been) made against time diversification. However, even those who don’t believe in this idea agree that, under mean reversion, time diversification indeed holds. What is, then, mean reversion? It is simply the tendency of an asset to revert to its long-term trend.
Flip a coin a few times, and the proportion of heads can be way off from the expected 50%. But keep flipping the coin, and the larger the number of flips, the more that the proportion of heads will approach 50%. Or spin a roulette a few times, and the proportion of 17s can be way off from its expected proportion of 1/37. But spin the roulette one million times and the proportion of 17s will be quite close to 1/37. Mean reversion is, in fact, as simple as that.
Whether or not there is mean reversion in returns is at the end of the day an empirical question, so let’s look at some evidence. Exhibit 12.3 shows the path followed by a $100 investment in the US stock market at the end of 1870 and compounded at the market’s annual real returns through the end of 2003. The straight line is simply a trend increasing constantly at the market’s long-term mean annual compound real return of 6.8%. As the graph clearly shows, periods of above-average returns tend to be followed by periods of below-average returns (and all along returns fluctuate rather closely around their long-term trend). This is exactly what mean reversion is all about.
Note that, during the years 1995 to 1999, the market delivered returns way above its long-term mean annual compound return. Mean reversion would not
$
1,000,000
100,000
10,000
1,000
100
1870 1889 1908 1927 1946 1965 1984 2003
Year EXHIBIT 12.3
Mean reversion
predict when, but it would predict that a correction was only a matter of time.
Sure enough, during the years 2000 to 2002, the market delivered returns way below its long-term mean annual compound return. Interestingly, note that by the end of 1999, after five consecutive years of far above-average returns, the market was way above its long-term trend. The three following years of far below-average returns took the market below its long-term trend.
Forecasting target returns
We’ll conclude our discussion of long-term risk and return with a simple tool designed to answer a question financial advisors face repeatedly. Given an asset, a target return, and a holding period, how likely is the asset to deliver at least the target return in the planned investment horizon? (In what follows we’ll use the concepts of simple and continuously compounded returns, as well as normal and lognormal distributions. If you’re not clear about the difference between these two types of returns and distributions, you may want to read Chapters 1, 28, and 29 before proceeding.)
Let’s start with a bit of notation. Let’s call AMand SDthe (arithmetic) mean and standard deviation of a series of continuously compounded returns (r), and let’s assume that these returns follow a normal distribution. This implies that simple returns (R) follow a lognormal distribution. Then, the probability of obtaining at least a mean annual compound return of R*over T years follows from a two-step procedure:
■ Calculate
(12.3)
■ Calculate
P(R≥R*) = P(z≥z*)
This procedure basically transforms the lognormal variable 1 + R into a standard normal variable z, with the purpose of bypassing the lognormal distribution and calculating probabilities out of the more widely used standard normal distribution. As the second step indicates, once we find the probability
z*= ln(1 +R*) – AM SD/√T