Lazard Insights
The Art and Science of Volatility Prediction
Stephen Marra, CFA, Director, Portfolio Manager/Analyst
Summary
• Statistical properties of volatility make this variable forecastable to some degree. This is one of
the most profound findings in financial economics
with far-reaching implications for asset allocation.
• The relationship between risk (volatility) and
returns is not constant. If one assumes this
relationship holds through all environments,
this implies that only a static asset allocation is
sufficient. However, risk-return patterns can fluctuate dramatically, as such: volatility forecasting
models are often used to improve upon static
allocations through volatility targeting.
• There are almost infinite possibilities in terms of
designing a volatility forecasting model. In our
view, simpler, broader models are preferable
over more complex ones which can lead to overconfident predictions.
• Return streams across asset classes differ, so the
statistical properties of volatility differ as well. This
means that certain asset classes are better suited
than others for a volatility targeting approach.
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This paper is published in conjunction with a presentation
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via www.LazardNet.com/insights.
Introduction
Uncertainty is inherent in every financial model. While there are many
ways to quantify this uncertainty, volatility is the most widely accepted
barometer for this measure. The most practical and straightforward
way to measure volatility is as a statistical measure of the dispersion
of market returns over a particular period. As a result, volatility has
effectively become the expected price of uncertainty in markets. One
of the most profound findings in financial economics is that volatility can—to a large extent—be forecasted. Predictability follows from
some of the statistical properties of volatility.
The implications of volatility forecasts are evident in several ways.
Volatility reaches all corners of the economy. Inaccurate volatility
estimates can leave financial institutions without enough capital for
operations and investment. Market volatility and its impact on public
confidence can have a significant effect on the broader global economy. So, the trade-off between risk (volatility) and return is critical for
all investment decisions. In addition, we believe there are relationships
between volatility and asset returns that make it possible to systematically increase the risk-adjusted returns on investments (to varying
degrees) by applying volatility forecasts to asset allocation decisions.
2
Volatility Is Not Constant
It would not make much sense to produce forecasts for a variable that
does not change much—weather forecasters would not be needed
if it was 80 degrees and sunny 365 days a year. In financial markets,
volatility is not constant. Any strategic or tactical asset allocation decision is ultimately governed by the trade-off between risk and reward.
Conventional wisdom says that greater return is compensation for
greater risk.
Assuming a constant relationship between risk and reward leaves only
the investor’s risk tolerance as the determining factor for a fixed asset
allocation: maximize return at a particular level of risk; or minimize
risk at a particular level of target return. Managing portfolios in this
way assumes that a static relationship between risk and return exists in
all environments and implies a fixed asset allocation.
While the relationship between risk and return appears constant over
long periods, the relationship fluctuates dramatically when analyzing it
over shorter time frames. For example, over a 35+ year span and using
US equities and bonds data, one can see that the risk-reward profile of
equities and fixed income has been quite different through the decades
(Exhibit 1).
During the 1990s for example, equities added almost 10% annualized
return for a roughly 10% increase in volatility. Whereas, in the first
decade of the 2000s, equities actually detracted almost 8% annualized
for the 13% they added in volatility.
The lack of uniformity of this risk-return profile indicates to us that
opportunities existed during the period studied to add considerable
value—both from return enhancement as well as risk reduction—from
an asset allocation process that was more active. In our view, this
argues against having a fixed asset allocation and moving towards a
more active asset allocation approach. In addition, the inclusion of
asset classes beyond global equities and bonds—specific factor exposures for example—would make the relationship between risk and
return even more unstable and less consistent.
Forecasting Volatility Is Possible
because of Key Statistical Features
Latency (or volatility clustering) is the characteristic denoting that periods of both high and low volatility tend to persist. This follows from
the observation that large changes in asset returns tend to be immediately followed by large changes; the same is true for small changes. To
illustrate this, we calculated the autocorrelation (i.e., the correlation
of one asset’s return to itself) of the absolute value of returns for three
equity indices at different time lags (Exhibit 2). Using the absolute
value of the returns enabled us to focus on the change in magnitude of
the return rather than the direction of the return.
The magnitude of future returns is dependent on the magnitude of past
returns, with this dependence becoming weaker over time. As a result
of this relationship, past volatility actually has a reasonable amount of
explanatory or predictive power over future volatility. The latency of
volatility makes it significantly more forecastable than asset returns.
Exhibit 1
Risk-Reward Is Not Static
Efficient Frontiers for Sub-Periods
Return (%)
20
100% Equities
15
10
5
100% Bonds
0
-5
0
3
1980s
6
1990s
9
2000s
12
2010−2016
15
18
Volatility (%)
1980−2016
For the period January 1980 to May 2016
Equities = S&P 500 Index; Bonds = Barclays Capital US Aggregate Bond Index. The
performance quoted represents past performance. Past performance does not guarantee future results. The indices listed above are unmanaged and have no fees. It is
not possible to invest in an index. Index performance does not represent the performance of any product managed by Lazard.
Source: Bloomberg
Exhibit 2
Autocorrelation of Global Equity Returns: The Magnitude of
Returns Shows Some Dependence
Autocorrelation of Absolute Value of Returns
0.4
S&P 500 Index
MSCI EM
MSCI ACWI
0.3
0.2
0.1
0.0
-0.1
0
5
10
15
20
25
30
35
Lag (weeks)
For the period January 1999 to October 2015, weekly returns
The performance quoted represents past performance. Past performance is not a
guarantee of future results. This is not intended to represent any product or strategy
managed by Lazard. It is not possible to invest directly in an index.
Source: Bloomberg
Another notable characteristic of volatility is the negative and asymmetric relationship between returns and volatility. The calculation of
volatility is indifferent to the direction of the market. However, volatility generally rises when the market falls and volatility tends to fall when
the market rises (Exhibit 3, page 3). It has been theorized that this
relationship is fundamental in nature and is due to what is called the
leverage effect. The leverage effect posits that as stock prices fall, companies become more leveraged as the value of their debt rises relative to the
value of their equity. As a result, the stock price becomes more volatile.
3
Exhibit 3
There Is a Negative Relationship between Returns and Volatility
Change in VIX on Negative S&P 500 Index Days
Change in VIX on Positive S&P 500 Index Days
Change in VIX (%)
Change in VIX (%)
60
60
40
40
20
20
0
0
Beta: -3.9
R-squared: 0.36
-20
-40
-12
Beta: -2.8
R-squared: 0.23
-20
-40
-10
-8
-6
-4
-2
0
0
2
4
6
8
10
12
Change in S&P 500 Index (%)
Change in S&P 500 Index (%)
For the period January 1990 to October 2015, daily returns
The performance quoted represents past performance. Past performance is not a guarantee of future results. This is not intended to represent any product or strategy managed by Lazard.
It is not possible to invest directly in an index.
Source: Bloomberg
In addition to this negative correlation, the magnitude of the impact
on volatility is significantly larger for downward market moves than
it is when the market moves higher. Based on Exhibit 3, the beta of
the VIX Index to the S&P 500 Index on negative return days was -3.9
with an r-squared of 0.36 whereas on positive return days the beta was
-2.8 with an r-squared of 0.23. The volatility feedback effect suggests
that as volatility rises and is priced into the market, there is a commensurate rise in the required return on equity as investors place a higher
hurdle rate on returns to achieve their desired risk-adjusted upsides.
This leads to an instant decline in stock prices as the volatility immediately reduces the risk-adjusted attractiveness of equities.
The last property we will discuss is mean reversion—volatility tends to
revert to a long-term mean. Exhibit 4 shows the behavior of the VIX
Index when it is in the top and bottom decile of its historical distribution as it moves back to a long-term average. One can also see that
the degree and speed of mean reversion is more pronounced when the
level of volatility is high compared to when it is low.
The prevailing thesis behind the mean reverting nature of volatility is
related to investor psychology. In periods of low volatility, investors
reduce their expectations and thresholds for volatility and as a result,
become more sensitive to near-term news flow. This leads to a larger
reaction function to new information and higher volatility as a result.
Conversely, during periods of elevated volatility investors will increase
their expectations and thresholds for volatility and become less sensitive to new information. This should result in lower levels of volatility
in subsequent periods.
Models for Volatility Forecasting
The statistical properties of volatility discussed in the preceding section make volatility inherently predictable—to a sufficient degree to
be useful to practitioners. There are a number of different models that
can be used to forecast volatility, which incorporate different degrees
of these characteristics. Moreover, models have varying degrees of
Exhibit 4
Reversion to the Mean
Top/Bottom VIX Observations Converge to the Long-Term Average
VIX
40
From High
30
Long-Term Average
20
From Low
10
0
4
8
12
16
20
24
Weeks after observation in top/bottom 10%
For the period 5 January 1990 to 22 April 2016
Based on the top/bottom 10% of weekly observations, we tracked the pattern for 24
weeks and aggregated the top/bottom deciles by average.
Source: CBOE, Haver Analytics
success when dealing with different asset classes. In brief, some of the
most well-known models are:
HIS: Historical models are the most straightforward, using calculations
ranging from as simple as a historical mean to moving averages with different weighting schemes (to balance short- and long-term fluctuations).
Autoregressive Models: This includes a whole library of methods
involving the statistical technique of regressions with a lag of the original data series as predictors.
Implied Standard Deviation: These models take information from
option markets to calculate the implied volatility.
We discuss these methodologies in greater detail in our paper
Predicting Volatility.1 The takeaway here would be that there are end-
4
less possibilities to design a forecasting model with inherent trade-offs
as it relates to the volatility properties one wishes to capture.
In our view, a good forecasting model captures all of the key characteristics of volatility with a healthy dose of humility. After all, financial
markets are not akin to physical sciences which behave according to
some prescribed laws of nature. The standard statistical view assumes
that there is some constant and unknown underlying structure to
markets. This has led statisticians to build models that are extremely
specific and complex. These excessive levels of complexity and precision belie the random nature of future asset prices and engender
dangerous levels of overconfidence that these models can predict
future events with a high degree of certainty.
Consequently, simpler, broader models—which are able to capture
the more general features of volatility and financial returns—will likely
provide more robust and transparent predictive abilities over longer,
out-of-sample time horizons.
Applications of Volatility Forecasting:
Volatility Targeting
We have shown that the risk–reward trade-off is not constant—particularly over shorter time frames. This means that a static asset allocation
can be ineffective. With this in mind, volatility forecasting models
are often used to improve upon a fixed asset allocation framework by
allocating to a fixed level of volatility instead. This is known as volatility targeting and enables a portfolio to potentially take advantage of
volatility forecasts and make allocation decisions.
The return streams of different asset classes display differing degrees of
these characteristics (Exhibit 5). The ones which exhibit more of the aforementioned characteristics are better candidates for volatility targeting.
To illustrate the success of volatility targeting on different asset classes
we examined each asset class individually and then in combination
with cash picking a volatility target. The measure of success is the
Sharpe ratio, a measure of risk-adjusted return. The more an asset class
is favored by volatility forecasting characteristics, the more risk targeting improves the results (Exhibit 6).
Since US Treasuries exhibit none of the volatility forecasting features,
the impact from volatility targeting on the risk-adjusted return is
minimal. Equities, on the other hand, strongly exhibit all of the characteristics. Hence, a volatility targeting approach can have a significant
positive effect on the resulting Sharpe ratio. The impact on commodities and high yield fixed income is minimal; although the latter has
many characteristics that would favor volatility targeting, it does not
have a strong negative correlation between volatility and returns.
Because a volatility targeting strategy involves being under-exposed to
an asset class when its volatility is high and completely exposed when
its volatility is low, a strong negative relationship between volatility and
returns is a pre-requisite for volatility targeting. High yield fixed income
markets have historically experienced some of their strongest returns in
periods when volatility was very high—in the 15%–20% range.
Exhibit 5
Volatility Properties Differ by Asset Class
High Yield
Fixed
US
Income Treasuries
DM
Equities
US
Equities
EM
Equities
Commodities
Volatility
Clustering
Effect
Strong
Strong
Strong
Strong
Strong
Weak
Negative
Correlation
between
Volatility
and Returns
Strong
Strong
Strong
Weak
Weak
Weak
Fat Tails
(NonNormality)
Strong
Strong
Strong
Weak
Strong
Weak
Leverage
Effect
Strong
Strong
Strong
Weak
Strong
Weak
For the period May 2002 to May 2016
DM Equities = MSCI World Index; US Equities = S&P 500 Index; EM Equities =
MSCI Emerging Markets Index; Commodities = Bloomberg Commodity Index; High
Yield Fixed Income = Barclays Global High Yield Index; US Treasuries = Barclays US
Treasury Index.
Source: Bloomberg
Exhibit 6
Results by Asset Class
Risk-Adjusted Results Comparison, 2002–2016
Sharpe Ratio
1.6
Stand-Alone Sharpe Ratio
Volatility-Targeted Sharpe Ratio
1.2
0.8
0.4
0.0
-0.4
Developed
Markets
Equities
US
Equities
Emerging
Markets
Equities
Commodities
High Yield
US
Fixed
Treasuries
Income
For the period May 2002 to May 2016
Developed Markets Equities = MSCI World Index, volatility target 12%; US Equities
= S&P 500 Index, volatility target 12%; Emerging Markets Equities = MSCI Emerging
Markets Index, volatility target 12%; Commodities = Bloomberg Commodity Index,
volatility target 12%; High Yield Fixed Income = Barclays Global High Yield Index, volatility target 7%; US Treasuries = Barclays US Treasury Index, volatility target 4%.
The performance quoted represents past performance. Past performance is not a
guarantee of future results. This is not intended to represent any product or strategy
managed by Lazard. It is not possible to invest directly in an index.
Source: Bloomberg
5
Conclusion
Uncertainty is at the center of all financial models and volatility is the
practical measure of that uncertainty. We should note that volatility
and its relationship with returns is not constant, which makes volatility forecasting a worthwhile effort. Fortunately, several characteristics
of financial return series make volatility inherently predictable.
As a result, we believe forecasting volatility has important implications
for all investors that are focused on risk-adjusted returns. Forecasting
methodologies are quite diverse and vary in their degrees of complexity and accuracy. They incorporate varying degrees of these common
characteristics of financial return series, with varying degrees of specification to a particular data sample.
Volatility targeting is one method of utilizing a forecasting model to
leverage the dynamic relationship between returns and risk in an asset
allocation framework. Different asset classes possess differing degrees of
these statistical characteristics. This is the key driver of the impact that
volatility targeting can have on risk-adjusted returns. Equities in general
have the best combination of these volatility properties which leads to
favorable risk-adjusted results from a volatility targeting approach.
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This content represents the views of the author(s), and its conclusions may vary from those held elsewhere within Lazard Asset Management.
Lazard is committed to giving our investment professionals the autonomy to develop their own investment views, which are informed by a
robust exchange of ideas throughout the firm.
Notes
1 Paper available at: http://www.lazardnet.com/docs/sp0/22430/PredictingVolatility_LazardResearch.pdf
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