Select up to 5 explanatory variables (but not necessarily 5) and estimate the co
esponding regression model to forecast volatility in June 2018. Briefly explain why these variables are selected. You need to ensure that the 3 regression model satisfies the underlying assumptions. You can use the full sample or a sub-sample to estimate your model. Please justify your sample selection and report the in-sample estimated coefficients in Table 1 below. Fill in the sample period and replace X’s with variable names.
Compare the forecasting performance of the best of DMA and Holt’s models against the linear regression in the hold-out period. Use the best model from above analyses to forecast volatility in June 2018. Based on daily returns in June, the team with the lowest forecast e
or gets 2 extra marks.
Investment Research
Predicting Volatility
Stephen Ma
a, CFA, Senior Vice President, Portfolio Manage
Analyst
Uncertainty is inherent in every financial model. It is driven by changing fundamentals, human psychology, and the
manner in which the markets discount potential future states of the macroeconomic environment. While defining uncer-
tainty in financial markets can quickly escalate into philosophical discussions, volatility is widely accepted as a practical
measure of risk. Most market variables remain largely unpredictable, but volatility has certain characteristics that can
increase the accuracy of its forecasted values. The statistical nature of volatility is one of the main catalysts behind the
emergence of volatility targeting and risk parity strategies.
Volatility forecasting has important implications for all investors focused on risk-adjusted returns, especially those that
employ asset allocation, risk parity, and volatility targeting strategies. An understanding of the different approaches used
to forecast volatility and the implications of their assumptions and dependencies provides a robust framework for the
process of risk budgeting.
In this paper, we will examine the art and science of volatility prediction, the characteristics which make it a fruitful
endeavor, and the effectiveness as well as the pros and cons of different methods of predicting volatility.
2
Introduction:
Statistical Properties of Volatility
“The starting point for every financial model is the uncertainty facing
investors, and the substance of every financial model involves the
impact of uncertainty on the behavior of investors and, ultimately, on
market prices. The very existence of financial economics as a discipline
is predicated on uncertainty.”1
A big part of risk management, asset allocation, and trading in finan-
cial markets is quantifying the potential loss of assets. In order to
measure these potential losses and make sound investment decisions,
investors must estimate risks.
Volatility is the purest measure of risk in financial markets and
consequently has become the expected price of uncertainty. The trade-
off between return and risk is critical for all investment decisions.
Inaccurate volatility estimates can leave financial institutions bereft of
capital for operations and investment. In addition, market volatility
and its impact on public confidence can have a significant effect on the
oader global economy.
Volatility targeting and risk parity are asset allocation methodolo-
gies that are directly impacted by volatility forecasting. Funds that
are managed for the insurance industry with a volatility band of
8%–12%—for example—use asset allocation aiming to control the
overall fund returns remaining within that range of volatility, as inves-
tors with different levels of risk tolerance and time horizons demand
differentiated levels of volatility. Maintaining this range of volatility
equires that a view be taken on the expected future volatility for the
asset classes in the fund. In addition, funds which use risk parity are
focused on the allocation of risk rather than the allocation of capital,
assuming that each asset class contributes the same degree of volatility
to the overall fund. As the volatility of each of these asset classes is not
constant, a forecast for the expected volatility for each is required to
maintain this type of investment approach.
It is well established that volatility is easier to predict than returns.
Volatility possesses a number of stylized facts which make it inherently
more forecastable. As such, volatility prediction is one of the most
important and, at the same time, more achievable goals for anyone
allocating risk and participating in financial markets.
The volatility of asset returns is a measure of how much the return
fluctuates around its mean. It can be measured in numerous ways but
the most straightforward is historical, observed volatility, which is
measured as the standard deviation of asset returns over a particular
period of time. When volatility is calculated by reverse-engineering
options market prices, it essentially becomes both a market price for
and an expectation of uncertainty.
The stochastic or random nature of asset prices and returns necessitates
the use of statistics and statistical theory to help describe and predict
these market fluctuations. The entire field of financial econometrics is
predicated on the integration of the theoretical foundations of economic
theory with finance, statistics, probability, and applied mathematics to
make inferences about the financial and market relationships critical
in the disciplines of asset allocation, risk management, securities regula-
tion, hedging strategies, and derivatives pricing. Volatility is forecastable
ecause of a number of persistent statistical properties.
Volatility Clustering
There is a delay for large or small changes in the absolute value of
financial returns to revert back to mean levels. In other words, the
magnitude of financial returns have latency—large changes in financial
eturns tend to be immediately followed by large changes and small
changes tend to be immediately followed by small changes. This can
lead to volatility clusters over time.
Many studies have found that volatility clustering is likely due to
investor inertia, caused by investors’ threshold to the incorporation
of new information. Except during times of extreme market turmoil,
only a fraction of market participants are actually trading in markets
at any given point in time. As such, it takes a period of time for these
investors to engage in the market and implement their changing views
as new information is revealed. The evidence for volatility clustering is
shown by the positive serial co
elation (co
elation of a return series
with itself lagged) in the absolute value of returns which eventually
decays over a period of observations (Exhibit 1). Volatility cluster-
ing can enhance the ability to forecast volatility. This clustering can
e shown by plotting a scatter chart of cu
ent month versus next
month’s volatility (Exhibit 2).
Exhibit 1
Autoco
elation of Global Equity Returns
Autoco
elation of Absolute Value of Returns
-0.1
0.0
0.1
0.2
0.3
0.4
MSCI EM
MSCI ACWI
S&P 500 Index
XXXXXXXXXX
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
Exhibit 2
Past Volatility May Be Indicative of Future Volatility…
0
40
80
120
XXXXXXXXXX
Following Monthly Volatility (%)
Monthly Volatility (%)
R squared: 0.35
For the period Fe
uary 1984 to September 2015, monthly returns
Data are based on the S&P 500 Index.
Source: Bloomberg
3
In contrast, if one plots the cu
ent month’s return versus the next
month there is no linear relationship and the serial co
elation
of actual returns—not the absolute value of returns—remains
insignificant (Exhibit 3).
Leverage Effect
The hypothesized leverage effect along with the volatility feedback
effect describes the negative and asymmetric relationship between
volatility and returns. The mathematical calculation of volatility
is indifferent to the direction of the market. However, volatility is
negatively co
elated to returns. At the same time, negative returns
esult in larger changes in volatility than positive returns. The beta of
the CBOE Volatility Index (VIX) to the S&P 500 Index on negative
eturn days is -3.9 with an r-squared of 0.36 whereas the beta of VIX
to the S&P 500 Index on positive return days is -2.8 with an r-squared
of 0.23 (Exhibit 4).
The volatility feedback effect suggests that as volatility rises and is
priced into the market, there is a commensurate rise in the required
eturn 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. As stock prices fall, companies
ecome more leveraged as the value of their debt rises relative to the
value of their equity. As a result, the stock price becomes more vola-
tile. This effect is more pronounced in well-developed markets that
have more analyst coverage.
Mean Reversion
Another stylized property of volatility is that it reverts to the mean
over time. The half-life of volatility is measured as the time it takes
volatility to move halfway towards its long-term average. Volatility
has a half-life of about 15–16 weeks—based on autoregressive models
which we will discuss later. With regards to implied volatility, the
degree of mean reversion is both asymmetric and accelerated (Exhibit 5).
The half-life of VIX mean reversion is about 11 weeks and is consider-
ably less than the half-life for equity returns, which is roughly 15 to 16
weeks (shown by the autoco
elation in Exhibit 1). In addition, VIX
mean reversion is far more pronounced when the VIX reaches higher
levels than when it dips below its long-term average. So, historically
the VIX has dropped with greater frequency and magnitude when ele-
vated than it has increased when at depressed levels. This suggests that
Exhibit 3
...But Not Indicative of Future Returns
Following Monthly Return (%)
Monthly Return (%)
-30
-20
-10
0
10
20
XXXXXXXXXX30
R squared: 0.00
Autoco
elation of Returns
-0.2
0.0
0.2
0.4
S&P 500 Index
MSCI EM
MSCI ACWI
XXXXXXXXXX
Lag (weeks)
Top chart: for the period Fe
uary 1984 to September 2015, monthly returns and data
are based on the S&P 500 Index.
Bottom chart: 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
Exhibit 4
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 S&P 500 Index (%)
-40
-20
0
20
40
60
XXXXXXXXXX0
Change in VIX (%)