Fisher College of Business
Working Paper Series
Charles A. Dice Center for
Research in Financial Economics
Does Aggregate Riskiness Predict Future
Economic Downturns?
Turan G. Bali
McDonough School of Business, Georgetown University
Nusret Cakici
Graduate School of Business, Fordham University
Fousseni Chabi-Yo
Fisher College of Business, Ohio State University
Dice Center WP 2012-9
Fisher College of Business WP 2012-03-009
Original: May 2012
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fisher.osu.edu
Does Aggregate Riskiness Predict Future Economic Downturns?
Turan G. Balia ∗
a McDonough School of Business, Georgetown University, Washington, D.C.20057
Nusret Cakicib †
b Graduate School of Business, Fordham University, New York, NY 10023, USA
Fousseni Chabi-Yoc ‡
c Fisher College of Business, Ohio State University, Columbus, OH 43210-1144, USA
First draft: January 2011
This draft: March 2012
Abstract
Aumann and Serrano (2008) and Foster and Hart (2009) introduce riskiness measures based on the physical return distribution of gambles. This paper proposes model-free options’ implied measures of riskiness
based on the risk-neutral distribution of financial securities. In addition to introducing the forward-looking
measures of riskiness, the paper investigates the significance of aggregate riskiness in predicting future
economic downturns. The results indicate strong predictive power of aggregate riskiness even after controlling for the realized volatility of the U.S. equity market, the implied volatility of S&P 500 index options
(V IX) proxying for financial market uncertainty, as well as the TED spread proxying for interbank credit
risk and the perceived health of the banking system.
JEL C LASSIFICATION C ODES : G11, G12, G14, G33
KEY WORDS: Riskiness, economic index of riskiness, operational measure of riskiness, risk-neutral measures, economic downturns.
∗ Tel.:
+1-202-687-5388; fax: +1-202-687-4031. E-mail address: tgb27@georgetown.edu
636 6776; fax: +1-212-586-0575. E-mail address: cakici@fordham.edu
‡ Tel.:+1-614-292-8477; fax: +1-614-292-7062. E-mail address: chabi-yo 1@fisher.osu.edu
† Tel.:+1-212
Does Aggregate Riskiness Predict Future Economic Downturns?
Abstract
Aumann and Serrano (2008) and Foster and Hart (2009) introduce riskiness measures based on the physical return distribution of gambles. This paper proposes model-free options’ implied measures of riskiness
based on the risk-neutral distribution of financial securities. In addition to introducing the forward-looking
measures of riskiness, the paper investigates the significance of aggregate riskiness in predicting future
economic downturns. The results indicate strong predictive power of aggregate riskiness even after controlling for the realized volatility of the U.S. equity market, the implied volatility of S&P 500 index options
(V IX) proxying for financial market uncertainty, as well as the TED spread proxying for interbank credit
risk and the perceived health of the banking system.
JEL C LASSIFICATION C ODES : G11, G12, G14, G33
KEY WORDS: Riskiness, economic index of riskiness, operational measure of riskiness, risk-neutral measures, economic downturns.
1. Introduction
Aumann and Serrano (2008) introduce a measure of riskiness based on investors’ risk tolerance. They
define the riskiness of a gamble as the reciprocal of the constant absolute risk aversion of an individual,
implying that less-averse individuals accept riskier gambles. Foster and Hart (2009) also develop a riskiness measure that depends only on the gamble and not on the decision maker’s aversion to risk. This
alternative measure of riskiness determines the critical wealth level below which it becomes risky to accept
the gamble.1 The economic index measure of riskiness pioneered by Aumann and Serrano (2008) looks
for the critical utility regardless of wealth, whereas the operational measure of riskiness initiated by Foster
and Hart (2009) looks for the critical wealth regardless of utility.2 Bali, Cakici, and Chabi-Yo (2011) introduce a generalized measure of riskiness that nests the original measures proposed by Aumann and Serrano
(2008) and Foster and Hart (2009).
Risk is a central issue in optimal asset allocation, financial risk management, and derivative pricing.
The question is whether the riskiness measures of Aumann and Serrano (2008) and Foster and Hart (2009)
provide a better characterization of underlying risk than various measures already used by regulators and
finance professionals. First, the widely used measures of risk such as standard deviation, variance, and
mean absolute deviation determine only “dispersion”, taking little account of the gamble’s actual values.
For example, if g and h = g + c are gambles, where c is a positive constant, then any of these standard
risk measures rates h precisely as risky as g, in spite of its being sure to yield more than g. One important
drawback of these dispersion measures is that they are not monotonic with respect to first-order stochastic
dominance, i.e., a better gamble with higher gains and lower losses may well have a higher standard
deviation, variance, and mean absolute deviation and thus be wrongly viewed as having a higher riskiness.3
Second, the riskiness measures of Aumann and Serrano (2008) and Foster and Hart (2009) do not
depend on any ad hoc parameters that need to be specified. The downside risk measure used extensively by
commercial banks, investment banks, and insurance companies is Value at Risk (VaR) which depends on
a parameter called a confidence level.4 The problem is that an appropriate value of the confidence level is
1 Foster
and Hart (2009) show that for every gamble g there exists a unique critical wealth level R(g) such that accepting
gambles when the current wealth is below the corresponding R(g) leads to bad outcomes, such as decreasing wealth and even
bankruptcy in the long run.
2 As shown in Hart (2011), there is a similarity between the approaches to riskiness and standard decision and consumer theory.
Foster and Hart (2011) show that when investors have CRRA utilities, the Aumann and Serrano index can be interpreted as the
maximal riskiness, whereas the Foster and Hart measure can be viewed as the minimal riskiness measure.
3 See Hadar and Russell (1969), Hanoch and Levy (1969), Levy (2006, 2008), and Rothschild and Stiglitz (1970, 1971).
4 VaR measures market risk by determining how much the value of a portfolio (or a security) could decline over a given
probability as a result of changes in market prices or rates.
1
not clear. Also, VaR ignores the gain side of the gamble and even on the loss side, it concentrates only on
that loss which hits the confidence level. The losses beyond the VaR threshold are not taken into account
when computing the maximum likely loss of a portfolio.
Since these traditional measures of dispersion and downside risk do not satisfy the monotonicity and/or
duality conditions, the riskiness measures provide more accurate characterization of underlying true risk.
Therefore, we think that the recently proposed measures of riskiness deserve further investigation. Both
Aumann and Serrano (2008) and Foster and Hart (2009) introduce riskiness measures based on the physical
return distribution of gambles. This paper contributes to the literature by introducing a model-free options’
implied measure of riskiness based on the risk-neutral return distribution of financial securities.
We should note that it is difficult to obtain accurate estimates of riskiness under the physical measure
because one has to make a distributional assumption. Aumann and Serrano (2008) assume a Normal
distribution and Foster and Hart (2009) assume a Binomial distribution to illustrate the meaning of their
riskiness measures. However, there is ample evidence showing significant departures from normality, i.e.,
the empirical distribution of financial securities is typically skewed, is peaked around the mean (leptokurtic)
and has fat tails. Therefore, generating empirical measures of riskiness requires precise estimates of the
mean, standard deviation, and higher order moments of the return distribution. However, the literature
points out that computing the moments of the return distribution is a difficult task because one has to know
the exact return distribution under the physical measure. Since this is not possible, one needs to make
a distributional assumption, but then she needs a very long sample to generate reliable estimates of the
moments under the assumed distribution.
This paper makes an innovative contribution to the literature by providing a distribution-free riskiness
measure that can be obtained from actively traded options and does not rely on any particular assumptions
about the empirical return distribution. Suppose an investor needs to find a one-month ahead expected
riskiness of a financial security. Under the physical measure, riskiness can only be obtained from the past
historical data (e.g., daily returns over the past one year) and the investor has to use this historical measure
to proxy for future riskiness. However, this physical (or historical) measure does not reflect the market’s
expectation of future riskiness because the history does not generally repeat itself. Using options’ implied
measures of riskiness solves this problem by making future riskiness observable because option prices
incorporate the market’s expectation of future return distribution.
Starting with the global financial meltdown, there has been great interest among academics, market
professionals, and regulators on the causes, remedies and future of the current financial crisis. There is
2
now a long literature examining the recent economic downturns, high volatility in financial markets, and
transmission of tail risk across the world. An important contribution of our paper is to investigate the
significance of aggregate riskiness in predicting future economic downturns.
The late-2000s financial crisis (often called the Credit Crunch or the Global Financial Crisis) is considered by many economists to be the worst financial crisis since the Great Depression of the 1930s. It resulted
in the collapse of large financial institutions, the bailout of banks by national governments, and downturns
in stock markets around the world. In many areas, the housing market had also suffered, resulting in numerous evictions, foreclosures and prolonged vacancies. It contributed to the failure of key businesses,
declines in consumer wealth estimated in the trillions of U.S. dollars, and a significant decline in economic
activity, leading to a severe global economic recession in 2008. The financial crisis was triggered by a
liquidity shortfall in the United States banking system in 2008. The collapse of the U.S. housing bubble,
which peaked in 2007, caused the values of securities tied to U.S. real estate pricing to plummet, damaging financial institutions globally. Questions regarding bank solvency, declines in credit availability and
damaged investor confidence had an impact on global stock markets, where securities suffered large losses
during 2008 and early 2009. Economies worldwide slowed during this period, as credit tightened and international trade declined. Governments and central banks responded with unprecedented fiscal stimulus,
monetary policy expansion and institutional bailouts.
While many causes for the financial crisis have been suggested, with varying weight assigned by experts, the United States Senate Financial Crisis Report indicates that “the crisis was not a natural disaster,
but the result of high risk, complex financial products; undisclosed conflicts of interest; and the failure
of regulators, the credit rating agencies, and the market itself to rein in the excesses of Wall Street.”5
Critics argued that credit rating agencies and investors failed to accurately price the risk involved with
mortgage-related financial products, and that governments did not adjust their regulatory practices to address 21st-century financial markets. In response to the financial crisis, both market-based and regulatory
solutions have been implemented or they are under consideration.
After introducing the forward-looking options’ implied measures of riskiness, we examine whether
these newly proposed measures of riskiness predict future economic activity. We use individual equity
options with various strikes and maturities (1 month to 12 months) and estimate expected future riskiness
of individual stocks. Then, we define aggregate riskiness of the U.S. equity market as the value-weighted
and the equal-weighted average of individual stocks’ riskiness. The results show strong predictive power
5 Senate
Financial Crisis Report, 2011: http://hsgac.senate.gov/public/ files/Financial Crisis/FinancialCrisisReport.pdf.
3
of aggregate riskiness even after controlling for the options’ implied volatility (V IX) and the realized
volatility of the U.S. equity market as well as the TED spread which is an indicator of interbank credit
risk and the perceived health of the banking system. We also compare the options’ implied measures
of aggregate riskiness with the options’ implied volatility of the S&P 500 index (V IX) in terms of their
power to forecast future economic activity.6 The results indicate that aggregate riskiness provides much
more accurate predictions of future downturns than the V IX index, implying significant contribution of
higher-order moments and tails of the risk-neutral distribution in predicting declines in economic growth.
This paper is organized as follows. Section 2 provides the original, physical measures of riskiness
developed by earlier studies. Section 3 presents the newly proposed options’ implied measures of riskiness.
Section 4 describes the data and variables. Section 5 presents the empirical results. Section 6 concludes
the paper.
2. Physical Measures of Riskiness
This section describes and compares the original, physical measures of riskiness developed by Aumann
and Serrano (2008) and Foster and Hart (2009).
2.1. Aumann and Serrano’s Economic Index Measure of Riskiness
Aumann and Serrano (2008) assume a von Neumann-Morgenstern utility function for money which
is strictly monotonic, strictly concave, and twice continuously differentiable, and defined over the entire
real line. A gamble g is a random variable with real values - interpreted as dollar amounts - some of
which are negative, and that has positive expectation. That is, an individual with utility function u accepts
a gamble g at wealth w if E (u [w + g]) > u [w], where E stands for “expectation”. Aumann and Serrano
(2008)’s measure of riskiness is based on a “duality” axiom between riskiness and risk aversion and positive
homogeneity of degree one:
DUALITY: Roughly duality says that less risk-averse decision makers accept riskier gambles. Define
an index Q as a positive real-valued function on gambles and assume that gamble g is riskier than gamble
h, i.e., Q [g] > Q [h]. If an individual i is more risk-averse than individual j, then whenever i accepts g at
6V IX is the Chicago Board Options Exchange’s (CBOE) market volatility index and it is a popular measure of the implied
volatility of the aggregate stock market portfolio. Often referred to as the fear index or the fear gauge, it represents a measure of
the market’s expectation of stock market volatility over the next 30-day period.
4
some wealth w, and Q [g] > Q [h], then j accepts h at w.
POSITIVE HOMOGENEITY: Positive homogeneity represents the cardinal nature of riskiness, i.e.,
Q [ng] > nQ [g] for all positive numbers n. If g is a gamble, positive homogeneity implies that 2g is “twice
as” risky as g, not just “more” risky.
Aumann and Serrano (2008) show that for each gamble g, there is a unique positive number R [g] with
(
(
−g
E exp
R [g]
))
−1 = 0
(1)
The index of riskiness denoted in (1) satisfies duality and positive homogeneity. Aumann and Serrano
(2008) consider an agent with constant absolute risk aversion (CARA) coefficient α, who is indifferent
between accepting and rejecting g. Applying (1) to the CARA utility function, u [x] = − exp (−αx), gives
R [g] = α1 . That is, the riskiness of a gamble is the reciprocal of the CARA of an individual who is indifferent
between taking and not taking that gamble.
2.2. Foster and Hart’s Operational Measure of Riskiness
Foster and Hart (2009) shows that for every gamble g, there exists a unique real number R [g] > 0 such
that a simple strategy with critical-wealth function Q guarantees no-bankruptcy if and only if Q [g] ≥ R [g]
for every gamble g. R[g] is uniquely determined by the equation:
(
(
g
E log 1 +
R [g]
))
=0
(2)
The condition Q [g] ≥ R [g] says that the minimal wealth level Q [g] at which g is accepted must be R [g]
or higher, and so g is for sure rejected at all wealth levels below R [g], that is, at all w < R [g]. Thus, the
riskiness measure of Foster and Hart (2009) is the minimal wealth level at which g may be accepted. R [g] in
(2) may also be viewed as a sort of minimal “reserve” needed for g. Comparing the two physical measures
of riskiness yields the following distinctions:
(a) The riskiness measure of Aumann and Serrano (2008), RAS [g], is an index of riskiness based
on comparing the gambles in terms of their riskiness, whereas the riskiness measure of Foster
and Hart (2009), RFH [g] , is defined for each gamble separately.
(b) RAS [g] is based on risk-averse expected-utility decision makers, whereas RFH [g] does not require utility functions and risk aversion, and just compares two situations: bankruptcy versus
5
no-bankruptcy or loss versus no-loss.
(c) RAS [g] is based on the critical level of risk aversion, whereas RFH [g] is based on the critical
level of wealth. The comparison between decision makers in Aumann and Serrano (2008) being more or less risk-averse - must hold at all wealth levels. In other words, RAS [g] looks for
the critical risk aversion coefficient regardless of wealth, whereas RFH [g] looks for the critical
wealth regardless of risk aversion.
We further refer RFH to as the Foster and Hart (2009) measure of riskiness and RAS to as the Aumann and
Serrano (2008) measure of riskiness.
3. Risk-Neutral Options Implied Measures of Riskiness
This paper contributes to the existing literature by introducing the risk-neutral options’ implied measures of riskiness based on the Bakshi and Madan (2000) spanning formula. They show that any function
of the form H(S) with E[H(S)] < ∞ can be spanned by as a collection of call and put options:
[ ] (
) [ ] ∫
H [S] = H S + S − S Hs S +
S
∞
HSS [K] (S − K)+ dK +
∫ S
0
HSS [K] (K − S)+ dK
(3)
where HS (.) and HSS (.) represent the first and second derivative of H with respect to S. We denote
gt+τ =
Si (t, τ) − Si (t)
Si (t)
(4)
the return on the risky asset i with an investment horizon τ. Si (t, τ) is the price of the individual security at
time t + τ, and Si (t) is the price of the individual security at time t. In Propositions (1) and (2), we derive
a model-free measure of riskiness from option prices.7
Proposition 1 Let RAS
i,t be the riskiness measure of the risky asset with return-payoff gt+τ . Let C (Si (t) , K, τ)
be the price at time t of the call option with strike price K, maturity τ. Let P (Si (t) , K, τ) be the price at time
t of the put option with strike price K, maturity τ. Denote 1 + r f (t, τ) the return on the risk-free security.
7 Aumann and Serrano (2008) develop their physical measure of riskiness R relative to a gamble g interpreted as a random
dollar amount, whereas we introduce options implied measures of riskiness by applying R to returns on g, not to dollar amounts.
We should note that our newly proposed measure of riskiness retains the properties of R shown by Aumann and Serrano.
6
RAS
i,t is the fixed point solution to (5)
r f (t, τ) 1
=
1 + r f (t, τ) RAS
i,t
∫ ∞
Si (t)
[K]C (Si (t) , K, τ) dK +
fRAS
i,t
∫ Si (t)
0
where
fRAS
[K] = (
i,t
−
1
)2 e
RAS
S
(t)
i
i,t
[K] P (Si (t) , K, τ) dK
fRAS
i,t
(5)
(K−Si (t))
1
RAS
i,t
Si (t)
.
(6)
Proof: See Appendix A.
Proposition 2 Let RFH
i,t be the riskiness measure of the risky asset with return-payoff gt+τ . Let C (Si (t) , K, τ)
be the price at time t of the call option with strike price K, maturity τ. Let P (Si (t) , K, τ) be the price at time
t of the put option with strike price K, maturity τ. Denote 1 + r f (t, τ) the return on the risk-free security.
RFH
i,t is the fixed point solution to (7)
r f (t, τ)
1
=
FH
1 + r f (t, τ) Ri,t
∫ ∞
Si (t)
fRFH
[K]C (Si (t) , K, τ) dK +
i,t
∫ Si (t)
0
fRFH
[K] P (Si (t) , K, τ) dK
i,t
(7)
where
fRFH
[K] = (
i,t
1
1
)2 (
)2 .
1 K−Si (t)
FH
Ri,t Si (t)
1 + RFH Si (t)
(8)
i,t
Proof: See Appendix A.
The advantage of the option implied measures (5) and (7) is that they can be computed using option
prices at any time, and they are not model-dependent. We also provide in Appendix B, the Aumann and
Serrano (2008) and Foster and Hart (2009) option implied measures of riskiness when the return on the
underlying assets are defined in terms of log returns.8
4. Data and Variable Definitions
In this section we first describe the individual equity options data used to estimate the options’ implied
measures of riskiness for individual stocks. Second, we introduce generalized physical measures of riskiness accounting for higher-order moments of the empirical return distribution. Third, we present aggregate
8 The results indicate that there is no significant empirical difference between the riskiness measures when simple or log returns
are used.
7
riskiness of the U.S. equity market based on the physical and risk-neutral measures of individual stocks’
riskiness. Finally, we describe the TED spread proxying for default risk and the V IX index proxying for
financial market uncertainty.
4.1. Equity Options Data
The daily data on call and put option prices, and the corresponding strikes, maturities, and volatilities
are from OptionMetrics. The OptionMetrics Volatility Surface computes the interpolated implied volatility
surface separately for puts and calls using a kernel smoothing algorithm using options with various strikes
and maturities. The volatility surface data contain prices and implied volatilities for a list of standardized
options for constant maturities and deltas. A standardized option is only included if there exists enough
underlying option price data on that date to accurately compute an interpolated value. The interpolations
are done each day so that no forward-looking information is used in computing the volatility surface. One
advantage of using the Volatility Surface is that it avoids having to make potentially arbitrary decisions on
which strikes or maturities to include when computing options’ implied measures of riskiness based on the
Bakshi and Madan (2000) spanning formula for each stock. In our empirical analyses, we use out-of-themoney call and put option prices with expirations of 1, 3, 6, and 12 months to estimate options’ implied
measures of riskiness. In Volatility Surface, at-the-money call (put) options have a delta of 0.50 (-0.50).
Out-of-the-money call options have delta of 0.20 to 0.50 and out-of-the-money put options have delta of
-0.20 to -0.50. We use the longest sample available from January 1996 to October 2010.
4.2. Generalized Physical Measures of Riskiness
In Section 2, we present the original measures of riskiness developed by Aumann and Serrano (2008)
and Foster and Hart (2009). As discussed earlier, Aumann and Serrano (2008) assume a Normal distribution
and Foster and Hart (2009) assume a Binomial distribution to illustrate the meaning of their riskiness
measures. In this section, we use a Taylor series expansion to introduce more generalized physical measures
of riskiness that take into account higher-order moments of the empirical return distribution. Under the
physical measure, the Aumann and Serrano (2008) riskiness measure RAS,P
is solution to
i,t
(
(
Et exp −
))
gt+1
RAS,P
i,t
8
= 1.
(9)
)
(
t+1
− gAS,P
Ri,t
The Taylor expansion series of exp
)
(
exp −
(
gt+1
≃ exp −
RAS,P
i,t
1
+ (
2!
1
− (
3!
around the expected value produces
Et (gt+1 )
)
RAS,P
i,t
1
(
1
−
RAS,P
i,t
(gt+1 − Et (gt+1 )) exp −
(
2
RAS,P
i,t
1
)2 (gt+1 − Et (gt+1 )) exp −
(
3
RAS,P
i,t
)3 (gt+1 − Et (gt+1 )) exp −
Et (gt+1 )
)
Et (gt+1 )
)
(10)
RAS,P
i,t
RAS,P
i,t
Et (gt+1 )
RAS,P
i,t
)
.
We apply the expected value to (10) and deduce
(
1 − exp
Et (gt+1 )
RAS,P
i,t
)
(
+
1
2! RAS,P
i,t
)2 Var [gt+1 ] −
3
1
(
)3 (Var [gt+1 ]) 2 SKEW [gt+1 ] = 0,
AS,P
3! Ri,t
(11)
where Var [gt+1 ] and SKEW [gt+1 ] represent the variance and skewness of asset i return. The Aumann and
Serrano measure of riskiness RAS,P
is solution to (11).
i,t
Under the physical measure, the Foster and Hart (2009) riskiness measure RFH,P
is solution to (12):
i,t
(
EtP
(
log 1 +
)
gt+1
RFH,P
i,t
(
≃ log 1 +
))
log 1 +
gt+1
= 0.
RFH,P
i,t
(12)
)
Taylor expansion series of log 1 +
(
(
gt+1
RFH,P
i,t
Et (gt+1 )
RFH,P
i,t
around Et (gt+1 ) is
)
+ (gt+1 − Et (gt+1 ))
(
1
FH,P
Ri,t
(13)
t+1 )
1 + Et (g
FH,P
Ri,t
)2
(
)3
1
−
1
(gt+1 − Et (gt+1 ))2 (
2!
RFH,P
i,t
1+
9
Et (gt+1 )
RFH,P
i,t
1
)2 +
2
(gt+1 − Et (gt+1 ))3 (
3!
FH,P
Ri,t
1+
Et (gt+1 )
FH,P
Ri,t
)3 .
The expected value of (13) can be simplified as
(
log 1 +
Et (gt+1 )
(
)
RFH,P
i,t
1
− Var [gt+1 ] (
2!
)2
(
)3
1
1
FH,P
Ri,t
RFH,P
i,t
1+
3
2
)2 + (Var [gt+1 ]) 2 SKEW [gt+1 ] (
3!
Et (gt+1 )
RFH,P
i,t
1+
)3 = 0.
Et (gt+1 )
RFH,P
i,t
(14)
The Foster and Hart riskiness measure is solution to (14).
To generate the physical measures of riskiness for each month from January 1996 to October 2010,
we first compute the mean, standard deviation, and skewness of daily returns over the past 1, 3, 6, and 12
months and then numerically back out the physical measure of riskiness from equation (11) for Aumann
and Serrano (2008) and equation (14) for Foster and Hart (2009). The physical measures of riskiness are
estimated for each optionable stock in OptionMetrics. We should note that optionable stocks are generally liquid and big in terms of market capitalization, hence they do not carry significant size or liquidity
premium.
4.3. Aggregate Measures of Riskiness
Figures 1 and 2 show the options’ implied measures of aggregate riskiness (Foster-Hart and AumannSerrano), computed as the value-weighted and the equal-weighted average of firm-level options’ implied
measures of riskiness obtained from individual equity options with 1, 3, 6, and 12 months to maturity.
Figures 3 and 4 display the physical measures of aggregate riskiness (Foster-Hart and Aumann-Serrano),
computed as the value-weighted and the equal-weighted average of firm-level physical measures of riskiness obtained from daily stock returns over the past 1, 3, 6, and 12 months. The figures present significant
time-series variation in both the risk-neutral and the physical measures of aggregate riskiness. Although
there is no specific pattern in the physical measures of riskiness, the forward-looking measures of aggregate riskiness seem to track lower economic activity. Notably, the options’ implied measures of aggregate
riskiness exhibit extremely high values during large falls of the market corresponding to financial crisis
period (2007 - 2010).
4.4. TED Spread
The TED spread is the difference between the interest rates on interbank loans and short-term U.S.
government debt (T-bills). TED is an acronym formed from T-Bill and ED, the ticker symbol for the
10
Eurodollar futures contract.9 The size of the spread is usually denominated in basis points (bps). For
example, if the T-bill rate is 5.10% and ED trades at 5.50%, the TED spread is 40 bps. The TED spread
fluctuates over time but generally has remained within the range of 10 and 50 bps (0.1% and 0.5%) except
in times of financial crisis. A rising TED spread often presages a downturn in the U.S. stock market, as it
indicates that liquidity is being withdrawn. The TED spread is an indicator of perceived credit risk in the
general economy. This is because T-bills are considered risk-free while LIBOR reflects the credit risk of
lending to commercial banks. When the TED spread increases, that is a sign that lenders believe the risk
of default on interbank loans (also known as counterparty risk) is increasing. Interbank lenders therefore
demand a higher rate of interest, or accept lower returns on safe investments such as T-bills. When the risk
of bank defaults is considered to be decreasing, the TED spread decreases.
Figure 5 plots the TED spread for each month from January 1996 to October 2010. Excluding the
Long Term Capital Management (LTCM) and post-LTCM periods (1998-2000) and the recent financial
crisis period (2007-2010), the sample average of the TED spread is about 35 basis points with a maximum
of 73 bps. During 2007, the subprime mortgage crisis increased the TED spread to a region of 150-200
bps. For the crisis period in 2007 (July 2007-December 2007), the average TED spread is about 140 basis
points with a maximum of 202 bps. On September 17, 2008, the TED spread exceeded 300 bps, breaking
the previous record set after the Black Monday crash of 1987. Some higher readings for the spread were
due to inability to obtain accurate LIBOR rates in the absence of a liquid unsecured lending market. On
October 10, 2008, the TED spread reached another new high of 457 basis points (after the collapse of
Lehman Brothers). Figure 5 shows that during 2008 the monthly difference between the 3-month LIBOR
and T-bills is averaged at 160 bps with a maximum of 365 bps. In the first half of 2009, the monthly TED
spread remained high in the range of 73 to 89 bps with an average of 84 bps. In 2010, the TED spread has
returned slowly to its long-term average of 30 basis points, hitting a low of 11 basis points in March, as
confidence returned. But as the Greek debt crisis escalated into widespread fears about the health of the
Eurozone, the TED spread started to rise again, moving above 45 basis points by mid-June.
9 Initially, the TED spread was the difference between the interest rates for three-month U.S. Treasuries contracts and the
three-month Eurodollars contract as represented by the London Interbank Offered Rate (LIBOR). However, since the Chicago
Mercantile Exchange dropped T-bill futures, the TED spread is now calculated as the difference between the three-month T-bill
interest rate and three-month LIBOR.
11
4.5. VIX Index
V IX is the ticker symbol for the Chicago Board Options Exchange Market Volatility Index, a popular
measure of the implied volatility of S&P 500 index options. The V IX is the square-root of the risk neutral
expectation of the S&P 500 variance over the next 30 calendar days. The V IX is quoted as an annualized
standard deviation. Although the V IX is often called the “fear index”, a high V IX is not necessarily bearish
for stocks. Instead, the V IX is a measure of market perceived volatility in either direction, including to the
upside. In practical terms, when investors anticipate large upside volatility, they are unwilling to sell
upside call stock options unless they receive a large premium. Option buyers will be willing to pay such
high premiums only if similarly anticipating a large upside move. The resulting aggregate of increases
in upside stock option call prices raises the V IX just as does the aggregate growth in downside stock put
option premiums that occurs when option buyers and sellers anticipate a likely sharp move to the downside.
Hence high V IX readings mean investors see significant risk that the market will move sharply, whether
downward or upward. The highest V IX readings occur when investors anticipate that huge moves in either
direction are likely. Only when investors perceive neither significant downside risk nor significant upside
potential will the V IX be low.
Figure 6 plots the end-of-month V IX index from January 1996 to October 2010. Between 1996 and
2010, the average value of monthly V IX is about 22% per annum with a maximum of 60%. During 2007,
the subprime mortgage crisis increased the V IX to a region of 20% to 25%. In 2008, the V IX index is
averaged at 32% with a maximum of 60% in October 2008. In the first half of 2009, the monthly V IX
index remained high in the range of 20% to 45% with an average of 32% per annum. In 2010, similar to
the TED spread, the V IX index has returned slowly to 24% per annum hitting a low of 18% in March, as
confidence returned. But as the Greek debt crisis increased financial market uncertainty across the world,
the V IX index started to rise again, moving to 35% per annum in June 2010.10
5. Empirical Results
The recent subprime mortgage crisis is an ongoing real estate and financial crisis triggered by a dramatic
rise in mortgage delinquencies and foreclosures in the United States, with major adverse consequences for
banks and financial markets around the globe. Between June 2007 and November 2008, Americans lost
10 Over the sample period January 2, 1996 to October 29, 2010, the daily V IX index reached its highest value (80.86% per
annum) on November 28, 2008. The daily VIX is in the range of 60% to 80% in most of October and November of 2008.
12
more than a quarter of their net worth. By early November 2008, a broad U.S. stock index, the S&P 500,
was down 45% from its 2007 high. Housing prices had dropped 20% from their 2006 peak, with futures
markets signaling a 30-35% potential drop. Total home equity in the United States, which was valued at
$13 trillion at its peak in 2006, had dropped to $8.8 trillion by mid-2008 and was still falling in late 2008.
Total retirement assets, Americans’ second-largest household asset, dropped by 22%, from $10.3 trillion
in 2006 to $8 trillion in mid-2008. During the same period, savings and investment assets (apart from
retirement savings) lost $1.2 trillion and pension assets lost $1.3 trillion. Taken together, these losses total
a staggering $8.3 trillion.11 During 2008, three of the largest U.S. investment banks either went bankrupt
(Lehman Brothers) or were sold at fire sale prices to other banks (Bear Stearns and Merrill Lynch). These
failures augmented the instability in the global financial system. The significant declines and uncertainty
in financial markets, low economic growth, and high unemployment rate continued until mid-2009. Falling
prices also resulted in 23% of U.S. homes worth less than the mortgage loan by September 2010, providing
a financial incentive for borrowers to enter foreclosure. Although there have been aftershocks, the financial
crisis itself ended sometime between late-2008 and mid-2009.
A relevant question is whether the time-varying riskiness of financial markets contains significant information about future macroeconomic activity. In other words, can we use information in the stock and
options markets to predict future economic downturns and financial crisis? The newly proposed forwardlooking measures of riskiness well capture lower economic activity and the recent financial crisis. As
shown in Figures 1 and 2, between July 2007 and October 2010 (end of our sample), there is a significant
upward trend as well as jump in the options’ implied measures of aggregate riskiness that track instability
in the global financial system.
In this section, we investigate whether the aggregate measures of riskiness can predict real economic
activity. As discussed earlier, the riskiness measure of Foster and Hart (2009) is a combination of the
statistical moments such as the mean, volatility, skewness, kurtosis, and tails of a physical return distribution. The options’ implied riskiness measure of Foster and Hart introduced in the paper can be viewed as
a combination of the first, second and higher-order moments as well as the tails of a risk-neutral distribution. Since the options’ implied risk-neutral distributions incorporate the market’s forecast of future return
distributions and they take into account downturns in financial markets, we think that when measured at
the aggregate level, riskiness of the U.S. equity market may potentially predict future declines in economic
activity.
11 These
figures are taken from the 2009 speeches of the Federal Reserve Bank (FRB) Chairman Ben Bernanke:
http://www.federalreserve.gov/newsevents/speech/2009speech.htm.
13
Aumann and Serrano (2008) propose a measure of riskiness based on investors’ risk tolerance. Risk
tolerance is one of the most important factors influencing asset allocation because it takes into account
investors’ ability to take risks. A conservative or risk averse investor would favor investments in which
her capital is preserved, whereas an aggressive investor can risk losing her investment to generate higher
profits. According to Aumann and Serrano (2008), aggregate riskiness is related to aggregate risk aversion
of market investors. Since aggregate risk aversion affects investors’ investment and consumption decisions,
aggregate riskiness may potentially affect future economic activity. The options’ implied riskiness measure
of Aumann and Serrano introduced in the paper takes into account time-series variation in aggregate risk
aversion and may potentially be linked to business cycle fluctuations.
We determine increases and decreases in real economic activity by relying on the Chicago Fed National
Activity Index (CFNAI index), which is a monthly index designed to assess production, consumption,
employment, and related inflationary pressure. The CFNAI is a weighted average of 85 existing monthly
indicators of national economic activity. It is constructed to have an average value of zero and a standard
deviation of one.12 Since economic activity tends toward trend growth rate over time, a positive index
reading corresponds to growth above trend and a negative index reading corresponds to growth below
trend. Since the underlying monthly macroeconomic data series are volatile, the monthly CFNAI index
is also quite volatile. The Chicago Fed generates the three-month moving average of the CFNAI index
(CFNAI MA3 index) to reduce the month-to-month volatility. In our empirical analyses, we use both the
CFNAI and the CFNAI MA3 indices.
We also compare the options’ implied and physical measures of aggregate riskiness with the options’
implied volatility of the S&P 500 index. We use the Chicago Board Options Exchange (CBOE)’s V IX
implied volatility that provides the market’s forecast of aggregate volatility by using real-time S&P 500
index option prices. Since V IX is known as the fear index and captures financial market uncertainty, it
may potentially predict future economic downturns as well. We investigate the relative performance of
aggregate riskiness and aggregate implied volatility in predicting future economic activity.
Panel A of Table 1 reports the correlation matrix for the value-weighted average options’ implied
measures of riskiness, the V IX implied volatility, the CFNAI and CFNAI MA3 economic activity indices
for the sample period January 1996 - October 2010. As reported in Panel A, the correlations between the
12 The 85 economic indicators that are included in the CFNAI are drawn from four broad categories of data: production and
income; employment, unemployment, and hours; personal consumption and housing; and sales, orders, and inventories. Each of
these data series measures some aspect of overall macroeconomic activity. The derived index provides a single, summary measure
of a factor common to these national economic data.
14
V IX and the CFNAI and the CFNAI MA3 indices are about the same as the correlations between aggregate
riskiness measures and the CFNAI and the CFNAI MA3 indices. Hence we rely on multivariate regressions
to determine whether the V IX or aggregate riskiness measures have a stronger link with macroeconomic
activity index. A notable point in Panel A is that the correlations between the options’ implied measures of
aggregate volatility and aggregate riskiness are in the range of 0.46 to 0.53. This relatively low correlation
at the aggregate level indicates significant contribution of other moments (such as skewness, kurtosis, and
tails) to the aggregate measure of riskiness. Panel B of Table 1 reports the correlation matrix for the
equal-weighted average options’ implied measures of riskiness, the V IX implied volatility, the CFNAI and
CFNAI MA3 economic activity indices and the qualitative results are very similar to those reported in
Panel A. Panels C and D of Table 1 show lower association between the physical measures of aggregate
riskiness and the CFNAI and the CFNAI MA3 indices. At an earlier stage of the study, we investigate
the predictive power of aggregate measures of physical riskiness and find that the value-weighted and the
equal-weighted average measures of physical riskiness do not predict future economic downturns. The
regressions results with the physical measures and the control variables are available upon request.
We now examine the relative performance of aggregate riskiness and V IX in predicting future economic downturns after controlling for the T ED spread proxying for default risk of the financial sector.
Specifically, we estimate the time-series regressions of one-month ahead CFNAI MA3 index on the valueweighted average options’ implied measures of riskiness, the V IX index, and the T ED spread:
CFNAI MA3t+1 = λ0 + λ1 RFH,VW
+ λ2V IXt + λ3 T EDt + εt+1
i,t
(15)
CFNAI MA3t+1 = λ0 + λ1 RAS,VW
+ λ2V IXt + λ3 T EDt + εt+1
i,t
(16)
where CFNAI MA3t+1 is the 3-month moving average of the CFNAI index in month t + 1, RFH,VW
denotes
i,t
the value-weighted average options’ implied measure of Foster-Hart riskiness in month t, RAS,VW
denotes
i,t
the value-weighted average options’ implied measure of Aumann-Serrano riskiness in month t, V IXt is the
annualized implied volatility of the S&P 500 index options in month t, and T EDt is the T ED spread in
month t.
Table 2 presents the predictive regression results for the period January 1996 - October 2010. The
Newey and West (1987) t-statistics are reported in parentheses. The last column shows the adjusted R2
values. The first four regressions in the top panel of Table 2 report significantly negative slopes on RFH,VW
i,t
with the Newey-West t-statistics ranging from -2.38 to -3.42, indicating that higher levels of aggregate
riskiness predict lower macroeconomic activity. This result is robust across all measures of the value15
weighted average riskiness obtained from individual equity options with 1, 3, 6, and 12 months to maturity.
The options’ implied measure of riskiness, RFH,VW
, is a combination of the mean, volatility, and higheri,t
order moments as well as the tails of a risk-neutral distribution. Since RFH,VW
captures not only the average
i,t
fluctuations, but also the extreme fluctuations of the risk-neutral distribution, it accounts for downturns in
predicts future declines in economic growth.
financial markets. Hence, RFH,VW
i,t
The first four regressions in Table 2 also exhibit significantly negative slopes on the T ED spread with
the t-statistics ranging from -3.41 to -3.76, indicating that an increase in the spread between the 3-month
LIBOR and the 3-month T-bill rate predicts lower economic activity. This makes economic sense because
as the T ED spread increases, the default risk is considered to be increasing. Since, the risk of a bank
defaulting is slightly higher than that of the U.S. government defaulting, the T ED spread measures the
estimated risks that banks pose on each other. The higher the perceived risk that one or several banks
may have liquidity or solvency problems, the higher the rate you will ask from your loans to other banks
(LIBOR) compared to your loans to the government (T-bill rate). Consequently, the T ED spread is a great
indicator of interbank credit risk and the perceived health of the banking system. The significantly negative
slope on the T ED spread indicates that an increase in interbank credit risk (or decrease in the stability of
the financial system) predicts future economic downturns.13
The first four regressions in Table 2 provide no evidence for a robust, significant link between the
V IX index and the CFNAI MA3 macroeconomic activity index. Although the slopes on the V IX index
are estimated to be negative, they are not statistically significant at conventional levels. These results
suggest that aggregate riskiness provides much more accurate predictions of future economic downturns
than the V IX index, implying significant contribution of higher-order moments and tails of the risk-neutral
distribution in predicting declines in economic growth.
In addition to the risk-neutral measure of market volatility (V IX), we now control for the physical
measure of aggregate volatility (realized volatility) in predicting future economic downturns. Specifically,
we estimate the time-series regressions of one-month ahead CFNAI MA3 index on the options’ implied
aggregate riskiness, the V IX index, the T ED spread, and the realized volatility of the S&P 500 index:
CFNAI MA3t+1 = λ0 + λ1 RFH,VW
+ λ2V IXt + λ3 T EDt + λ4 RVt + εt+1
i,t
(17)
CFNAI MA3t+1 = λ0 + λ1 RAS,VW
+ λ2V IXt + λ3 T EDt + λ4 RVt + εt+1
i,t
(18)
13 When
there is a downturn in the economy, banks suspect that some banks may encounter problems. However, they do not
know which banks, so they restrict interbank lending, resulting in higher TED spreads and lower liquidity in the interbank market,
which ultimately produces lower credit availability for consumers and corporates.
16
where RVt is the annualized realized volatility of the U.S. equity market in month t.14 The last four regressions in Table 2 show that after controlling for the realized volatility of the stock market portfolio, the
negative link between RFH,VW
and CFNAI MA3 remains significant. The Newey-West t-statistics of the
i,t
slopes on RFH,VW
are in the range of -2.55 to -3.52. Similar to our earlier findings, this result is robust
i,t
across all measures of the value-weighted average riskiness.
After including the realized volatility of the S&P500 index, the economic and statistical significance
of the T ED spread reduced, but the slopes on TED are still negative and significant with the t-statistics
ranging from -2.11 to -2.35. The slopes on the realized volatility are estimated to be negative and significant
with the t-statistics between -2.18 and -2.60, implying that an increase in stock market volatility predicts
declines in future economic activity. A notable point in the last four regressions is that the slopes on the
V IX index are almost zero and they are statistically insignificant with the t-statistics ranging from 0.13 to
0.80. Overall, these results show that after controlling for the T ED spread and the realized volatility of
the U.S. equity market, the options’ implied measures of aggregate riskiness successfully predict future
economic activity, whereas the options’ implied measure of aggregate volatility (V IX) has no significant
association with future economic downturns.
As shown in the bottom panel of Table 2, the results from the value-weighted average options’ implied
measure of Aumann-Serrano riskiness, RAS,VW
, are very similar to those obtained from RFH,VW
. We find a
i,t
i,t
and the CFNAI MA3 index. The significantly negative
negative and highly significant link between RAS,VW
i,t
relation between the T ED spread, the realized volatility, and the CFNAI MA3 index remains intact.
Another notable in Table 2 is the large R2 values. The adjusted R2 ’s reported in the last column of Table
2 are between 48% and 56% without the realized volatility and between 51% and 59% with the realized
volatility of the market. To clarify the marginal contribution of RFH,VW
and RAS,VW
in forecasting future
i,t
i,t
economic downturns, we estimate the predictive regressions with the T ED spread and the V IX index only:
CFNAI MA3t+1 = λ0 + λ1V IXt + λ2 T EDt + εt+1
(19)
The adjusted R2 from this regression is about 39%. Adding RFH,VW
or RAS,VW
to equation (19) increases the
i,t
i,t
adjusted R2 to the range of 48%-56% when we use the options’ implied measures of riskiness. Improvement
in the adjusted R2 values provides clear evidence for the strong predictive power of riskiness.
14 Following a series of papers by Andersen, Bollerslev, and Diebold et al. (2001, 2003), daily realized volatility of the U.S.
equity market is computed as the sum of squared 5-minute returns on the S&P 500 index. We then annualize the daily realized
volatility assuming 252 trading days in a year.
17
To provide further evidence for the weak performance of the V IX index in forecasting future economic downturns, we run the following regression with the 1-month options’ implied measure of riskiness,
RFH,VW
, and the T ED spread:
i,t
CFNAI MA3t+1 = λ0 + λ1 RFH,VW
+ λ2 T EDt + εt+1
i,t
(20)
The adjusted R2 from equation (20) is 54.27%. As shown in the last column of Table 2, adding the V IX
index to equation (20) increases the adjusted R2 to 55.84%. This small improvement in the adjusted R2
(only 1.57%) provides another evidence for the weak predictive power of the V IX index.
Table 3 replicates the main predictive regressions using the equal-weighted average options’ implied
measures of riskiness (RFH,EW
and RFH,EW
), the V IX index, the T ED spread, and the realized volatility of
i,t
i,t
the U.S. equity market. Similar to our earlier results in Table 2, we find a strongly negative link between
aggregate riskiness and the CFNAI MA3 index. The significantly negative relation between the T ED
spread, the realized volatility, and the CFNAI MA3 index remains intact as well. However, the V IX index
has no forecasting power for future declines in economic activity.
Table 4 provides further robustness checks by presenting evidence for the CFNAI index. Panel A
(Panel B) shows that the value-weighted (equal-weighted) average options’ implied measures of riskiness
strongly predict the one-month ahead CFNAI index, whereas the V IX index remains a poor determinant of
future economic activity. At an earlier stage of the study, we replicate our main findings by excluding the
recent crisis period 2008-2010. Specifically, we estimate equations (17)-(18) for the sample period January
1996 - December 2007 and find that the predictive power of aggregate riskiness remains economically and
statistically significant.
6. Conclusion
Aumann and Serrano (2008) introduce an economic index measure of riskiness that looks for the critical utility regardless of wealth. Foster and Hart (2009) develop an operational measure of riskiness that
looks for the critical wealth regardless of utility. Both measures of riskiness are originated based on the
physical return distribution of risky assets. In this paper, we introduce generalized physical measures of
riskiness that take into account higher-order moments of the empirical return distribution. More importantly, we develop new measures of riskiness based on the risk-neutral return distribution of underlying
18
assets. Riskiness of an underlying financial security (e.g., equity) is derived from the prices of derivative
securities written on the underlying asset (i.e., prices of call and put options on equity). The newly proposed
forward-looking measures of riskiness condense options’ implied risk-neutral probability distribution to a
scalar and satisfy the monotonicity and duality conditions.
We also introduce the aggregate measures of riskiness for the U.S. equity market and investigate their
predictive power for future economic downturns. Although there is no specific pattern in the physical
measures of aggregate riskiness, the forward-looking options’ implied measures of aggregate riskiness
successfully track lower economic activity. Finally, we examine the relative performance of aggregate
riskiness and aggregate volatility (V IX) in forecasting declines in economic growth. The results indicate
that aggregate riskiness provides accurate predictions of future economic downturns, whereas the V IX
implied volatility does not add significant, marginal predictive power for real economic activity. The strong
predictive power of aggregate riskiness remains intact even after controlling for the realized volatility of the
U.S. equity market, the V IX index proxying for financial market uncertainty, and the T ED spread proxying
for interbank credit risk and the perceived health of the banking system.
Our findings suggest significant contribution of the higher-order moments and the tails of the riskneutral distribution in forecasting declines in economic growth. Hence, regulators can utilize readily available information in the options market to predict downturns in financial markets and macroeconomic activity. Our aggregate measure of riskiness can be viewed as a complement to bank systemic risk measures
(e.g., CoVaR and Marginal Expected Shortfall), and can be used to calibrate systemic risk premiums set by
bank regulators.
19
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20
Appendix A: Recovering the Aumann and Serrano (2008) and Foster and
Hart (2009) riskiness measures from option prices: The case of simple
returns
We use simple returns, and derive the Aumann and Serrano (2008) and Foster and Hart (2009) riskiness
measures from option prices.
Aumann and Serrano (2008)
Theorem A in Aumann and Serrano (2008) show that for each gamble gt+τ , there is a unique positive
number Rt [gt+τ ] such that
(
Et e
)
gt+τ
t [gt+τ ]
−R
− 1 = 0.
(A1)
UNDER THE RISK NEUTRAL MEASURE, (A1) can be expressed as
Et∗
(
gt+τ
t [gt+τ ]
−R
e
)
−1 = 0
(A2)
where
gt+τ =
Si (t, τ) − Si (t)
Si (t)
(A3)
represents the return on the risky asset i with an investment horizon τ. Notice that, under the risk neutral
measure
Et∗ (gt+τ ) = r f (t, τ) .
(A4)
)
(
g
− t+τ
where r f (t, τ) represents the risk-free rate for the time period [t,t + τ]. Since Et∗ e Rt [gt+τ ] − 1 is finite,
we can use the Bakshi and Madan (2000) spanning formula:
[ ] (
) [ ] ∫
H [S] = H S + S − S Hs S +
S
∞
+
HSS [K] (S − K) dK +
∫ S
0
HSS [K] (K − S)+ dK.
(A5)
We use the return’s definition (A3) and apply the Bakshi and Madan (2000) formula (A5) to
gt+τ
t [gt+τ ]
−R
H [S (t, τ)] = e
21
− 1.
(A6)
with S = Si (t). We obtain
e
gt+τ
t [gt+τ ]
−R
(
)
1
− 1 = (Si (t, τ) − Si (t)) −
Si (t) Rt [gt+τ ]
(
)
∫ ∞
(K−S (t))
− S (t)R ig
1
+
e i t [ t+τ ] (Si (t, τ) − K)+ dK
2
2
Si (t) Si (t) Rt [gt+τ ]
(
)
∫ Si (t)
(K−S (t))
− S (t)R ig
1
[
]
+
e i t t+τ (K − Si (t, τ))+ dK.
Si2 (t) Rt2 [gt+τ ]
0
(A7)
Now, we apply the expectation operator to (A7) and get
1
r f (t, τ)
Rt [gt+τ ]
)
(K−S (t))
− S (t)R ig
1
=
e i t [ t+τ ] Et∗ (Si (t, τ) − K)+ dK
2
2
Si (t) Si (t) Rt [gt+τ ]
(
)
∫ Si (t)
(K−S (t))
− S (t)R ig
1
+
e i t [ t+τ ] Et∗ (K − Si (t, τ))+ dK.
2
2
Si (t) Rt [gt+τ ]
0
∫ ∞
(
(A8)
Notice that the prices of the call and put options are:
1
E ∗ (Si (t, τ) − K)+ = C (Si (t) , K, τ) ,
(1 + r f (t, τ)) t
1
E ∗ (K − Si (t, τ))+ = P (Si (t) , K, τ) .
(1 + r f (t, τ)) t
Hence (A8) can be written as
r f (t, τ)
1
(1 + r f (t, τ)) Rt [gt+τ ]
(
)
(K−S (t))
− S (t)R ig
1
=
e i t [ t+τ ] C (Si (t) , K, τ) dK
2
2
Si (t) Si (t) Rt [gt+τ ]
(
)
∫ Si (t)
(K−S (t))
− S (t)R ig
1
+
e i t [ t+τ ] P (Si (t) , K, τ) dK.
Si2 (t) Rt2 [gt+τ ]
0
∫ ∞
(A9)
The riskiness measure Rt [gt+τ ] is, therefore, solution to (A9).
Foster and Hart (2009)
Theorem 1 in Foster and Hart (2009) show that there exists a critical wealth level Rt [gt+τ ] such as
(
Et
(
gt+τ
log 1 +
Rt [gt+τ ]
))
= 0.
UNDER THE RISK NEUTRAL MEASURE, (A10) can be expressed as
22
(A10)
Et∗
(
(
gt+τ
log 1 +
Rt [gt+τ ]
))
= 0.
(A11)
where gt+τ is defined in (A3). Notice that, under the risk neutral measure,
Et∗ (gt+τ ) = r f (t, τ) .
(A12)
( (
))
where r f (t, τ) represents the risk-free rate for the time period [t,t + τ]. Since Et∗ log 1 + Rtg[gt+τ
is
t+τ ]
finite, we can use the Bakshi and Madan (2000) spanning formula (A5). Now, we consider
(
gt+τ
H [Si (t, τ)] = log 1 +
Rt [gt+τ ]
)
(A13)
with S = Si (t) and apply (A5) to (A13) gives
(
gt+τ
log 1 +
Rt [gt+τ ]
)
(
= log (1) + (Si (t, τ) − Si (t))
−
−
∫ ∞

1
Rt [gt+τ ]Si (t)

Si (t)
∫ Si (t)
1+

1
Rt [gt+τ ]
K
Si (t)
1
Rt [gt+τ ]Si (t)

0
(
1+
1
Rt [gt+τ ]
(
1
)
Rt [gt+τ ] Si (t)
2
(A14)
)  (Si (t, τ) − K)+ dK
−1
2
K
Si (t)
−1
)  (K − Si (t, τ))+ dK.
Therefore, (A14) can be simplified to
(
gt+τ
log 1 +
Rt [gt+τ ]
)
)
Si (t, τ) − Si (t)
=
Si (t)
∫ ∞
1
1
+
−
(
)2 (Si (t, τ) − K) dK
2
Si (t) Si (t)
K
Rt [gt+τ ] + Si (t) − 1
1
Rt [gt+τ ]
−
1
2
Si (t)
(
∫ Si (t)
0
(A15)
1
+
(
)2 (K − Si (t, τ)) dK.
K
Rt [gt+τ ] + Si (t) − 1
We apply the expectation operator under the risk neutral measure to (A15):
1
r f (t, τ)
Rt [gt+τ ]
=
1
2
Si (t)
+
∫ ∞
1
+
∗
(
)2 Et (Si (t, τ) − K) dK
Si (t)
Rt [gt+τ ] + SiK(t) − 1
1
Rt [gt+τ ]
∫ Si (t)
0
1
1
(
Si2 (t)
Rt [gt+τ ] + SiK(t)
23
+
∗
)2 Et (K − Si (t, τ)) dK
−1
(A16)
We recall that the prices of the call and put options with strike K and maturity τ are given by (A17) and
(A18) respectively
1
Et∗ (Si (t, τ) − K)+ = C (Si (t) , K, τ)
(1 + r f (t, τ))
1
E ∗ (K − Si (t, τ))+ = P (Si (t) , K, τ)
(1 + r f (t, τ)) t
(A17)
(A18)
where (1 + r f (t, τ)) represents the risk-free return for the time period [t,t + τ]. Hence, (A16) reduces to
r f (t, τ)
1
1 + r f (t, τ) Rt [gt+τ ]
1
Si2 (t)
=
∫ ∞
1
(
)2 C (Si (t) , K, τ) dK
Si (t)
K
Rt [gt+τ ] + Si (t) − 1
∫ Si (t)
+
0
1
1
(
Si2 (t)
Rt [gt+τ ] + SiK(t)
(A19)
)2 P (Si (t) , K, τ) dK
−1
which simplifies to
r f (t, τ)
1
=
1 + r f (t, τ) Rt [gt+τ ]
∫ ∞
Si (t)
fR [K]C (Si (t) , K, τ) dK +
∫ Si (t)
0
fR [K] P (Si (t) , K, τ) dK
(A20)
with
fR [K] =
1
1
(
Si2 (t) Rt2 [gt+τ ]
1+
1
K
1
Rt [gt+τ ] ( Si (t)
)2
− 1)
(A21)
Equation (A20) can be numerically solved to deduce the Foster and Hart (2009) critical wealth level
Rt [gt+τ ] (solve for the fixed point f (x) = x). We further denote RtFH = Rt [gt+τ ].
24
Appendix B: Recovering the Aumann and Serrano (2008) and Foster and
Hart (2009) riskiness measures from option prices: The case of log returns
We use log returns, and derive the Aumann and Serrano (2008) and Foster and Hart (2009) riskiness
,
measures from option prices. We denote the log return: gt+τ = log SSi (t,τ)
i (t)
Aumann and Serrano (2008)
UNDER THE RISK NEUTRAL MEASURE, the riskiness measure in Theorem A of Aumann and Serrano
(2008) is
(
)
Et∗ e−gt+τ /Rt [gt+τ ] − 1 = 0.
(B1)
We apply Bakshi and Madan (2000) formula (A5) to
H [Si (t, τ)] = e−gt+τ /Rt [gt+τ ] − 1
(B2)
with S = Si (t). We obtain
e
−gt+τ /Rt [gt+τ ]
(
)
1
− 1 = (Si (t, τ) − Si (t)) −
(B3)
Rt [gt+τ ] Si (t)
)
((
)
∫ ∞
− R g1
log S K(t)
1
1
[
]
i
e t t+τ
+
+
(Si (t, τ) − K)+ dK
Rt [gt+τ ] K 2 [Rt [gt+τ ] K]2
Si (t)
((
)
)
∫ Si (t)
− R g1
log S K(t)
1
1
i
+
+
e t [ t+τ ]
(K − Si (t, τ))+ dK.
2
2
R
[g
]
K
0
t t+τ
[Rt [gt+τ ] K]
Now, we apply the expectation operator to (B3) and get
1
r f (t, τ)
Rt [gt+τ ]
∫ ∞
((
)
1
=
Si (t)
∫ Si (t)
Rt [gt+τ
((
+
] K2
+
1
Rt [gt+τ ] K 2
0
1
2
[Rt [gt+τ ] K]
+
−R
1
t [gt+τ ]
e
1
[Rt [gt+τ ] K]2
)
log S K(t)
Et∗ (Si (t, τ) − K)+ dK (B4)
i
−R
e
)
1
t [gt+τ ]
log S K(t)
)
i
Et∗ (K − Si (t, τ))+ dK.
Hence, (B4) can be written as
r f (t, τ)
1
(1 + r f (t, τ)) Rt [gt+τ ]
∫ ∞
=
((
)
1
1
−R
1
t [gt+τ ]
log S K(t)
)
i
C (Si (t) , K, τ) dK(B5)
+
e
Rt [gt+τ ] K 2 [Rt [gt+τ ] K]2
)
)
((
∫ Si (t)
− R g1
log S K(t)
1
1
i
e t [ t+τ ]
P (Si (t) , K, τ) dK.
+
+
Rt [gt+τ ] K 2 [Rt [gt+τ ] K]2
0
Si (t)
25
Therefore, Rt [gt+τ ] is the fixed-point solution to (B5).
Foster and Hart (2009)
UNDER THE RISK NEUTRAL MEASURE,
Et∗
(
(
gt+τ
log 1 +
Rt [gt+τ ]
))
=0
(B6)
We then apply the Bakshi and Madan (2000) formula (A5) to
(
gt+τ
H [Si (t, τ)] = log 1 +
Rt [gt+τ ]
)
(B7)
with S = Si (t). We obtain
(
log 1 +
gt+τ
Rt [gt+τ ]
)
(
)
1
= log (1) + (Si (t, τ) − Si (t))
Rt [gt+τ ] Si (t)


(
)
K
∫ ∞
1
 Rt [gt+τ ] + log Si (t) + 1 
+
− 2
(
(
))2  (Si (t, τ) − K) dK
K Si (t)
Rt [gt+τ ] + log SiK(t)


(
)
∫ Si (t) Rt [gt+τ ] + log K + 1
Si (t)
1


+
− 2
(
(
))2  (K − Si (t, τ)) dK
K 0
K
Rt [gt+τ ] + log Si (t)
(B8)
Applying the expectation operator under the risk neutral measure to (B8) gives:
r f (t, τ)
1
=
(1 + r f (t, τ)) Rt [gt+τ ]
∫ ∞
Si (t)
fR [K]C (Si (t) , K, τ) dK +
∫ Si (t)
0
fR [K] P (Si (t) , K, τ) dK

(
)
1
K
1
1
+
log
+
Rt [gt+τ ]
Si (t)
Rt [gt+τ ] 
1  1
fR [K] = 2 
(
(
))2  .
K
Rt [gt+τ ]
1 + Rt [g1t+τ ] log SiK(t)
(B9)

Equation (B9) can be solved numerically to recover Rt [gt+τ ] (solve for the fixed point f (x) = x).
26
(B10)
Table 1: Correlation Matrix
This table presents the correlation matrix for the options’ implied and the physical measures of aggregate
riskiness (Foster-Hart and Aumann-Serrano), the V IX implied volatility, the CFNAI and the CFNAI MA3
indices for the sample period January 1996 - October 2010. The aggregate options’ implied measures of
riskiness are computed as the value-weighted and the equal-weighted average of firm-level options’ implied
measures of riskiness obtained from individual equity options with 1, 3, 6, and 12 months to maturity.
The aggregate physical measures of riskiness are computed as the value-weighted and the equal-weighted
average of firm-level physical measures of riskiness obtained from daily stock returns over the past 1, 3, 6,
and 12 months. The V IX implied volatility is obtained from the Chicago Board Options Exchange (CBOE)
that provides the market’s forecast of aggregate volatility by using real-time S&P 500 index option prices.
The CFNAI and the CFNAI MA3 economic activity indices are obtained from the Federal Reserve Bank
of Chicago.
Panel A. Correlations with the Value-Weighted Average Options’ Implied Measures of Riskiness
Foster-Hart
Aumann-Serrano
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
V IX
CFNAI
RFH,Q
i,t
RFH,Q
i,t
RFH,Q
i,t
RFH,Q
i,t
RAS,Q
i,t
RAS,Q
i,t
RAS,Q
i,t
RAS,Q
i,t
-0.5301 -0.5453 -0.4136 -0.4249 -0.4554
-0.5453 -0.4138 -0.4251 -0.4558
CFNAI MA3 -0.5947 -0.6406 -0.4981 -0.5087 -0.5363
-0.6406 -0.4983 -0.5089 -0.5368
V IX
1.0000 0.5277 0.4595 0.4700
0.4978
0.5278 0.4596 0.4701
0.4981
Panel B. Correlations with the Equal-Weighted Average Options’ Implied Measures of Riskiness
Foster-Hart
Aumann-Serrano
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
V IX
CFNAI
RFH,Q
i,t
RFH,Q
i,t
RFH,Q
i,t
RFH,Q
i,t
RAS,Q
i,t
RAS,Q
i,t
RAS,Q
i,t
RAS,Q
i,t
-0.5301 -0.4777 -0.3512 -0.3644 -0.3945
-0.4777 -0.3514 -0.3643 -0.3943
CFNAI MA3 -0.5947 -0.5710 -0.4326 -0.4465 -0.4755
-0.5710 -0.4328 -0.4463 -0.4752
V IX
1.0000 0.4517 0.3790 0.3924
0.4251
27
0.4517 0.3793 0.3923
0.4250
Table 1 (continued)
Panel C. Correlations with the Value-Weighted Average Physical Measures of Riskiness
Foster-Hart
Aumann-Serrano
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
V IX
CFNAI
RFH,P
i,t
RFH,P
i,t
RFH,P
i,t
RFH,P
i,t
RAS,P
i,t
RAS,P
i,t
RAS,P
i,t
RAS,P
i,t
-0.5301 -0.0763 -0.3478 -0.3191 -0.3338
-0.0765 -0.3477 -0.3185 -0.3331
CFNAI MA3 -0.5947 -0.0444 -0.3186 -0.3228 -0.3754
-0.0444 -0.3186 -0.3222 -0.3746
V IX
1.0000 0.0394 0.4223 0.3117
0.4243
0.0393 0.4220 0.3113
0.4239
Panel D. Correlations with the Equal-Weighted Average Physical Measures of Riskiness
Foster-Hart
Aumann-Serrano
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
V IX
CFNAI
RFH,P
i,t
RFH,P
i,t
RFH,P
i,t
RFH,P
i,t
RAS,P
i,t
RAS,P
i,t
RAS,P
i,t
RAS,P
i,t
-0.5301 -0.2464 -0.2359 -0.2487 -0.2587
-0.2466 -0.2360 -0.2481 -0.2573
CFNAI MA3 -0.5947 -0.2637 -0.1892 -0.3033 -0.3001
-0.2639 -0.1894 -0.3028 -0.2987
V IX
1.0000 0.3259 0.3890 0.3830
0.2834
28
0.3258 0.3890 0.3827
0.2826
29
1-month
Intercept RFH,VW
i,t
0.7622 -0.0472
(4.82)
(-3.42)
0.9495
(4.15)
0.9485
(4.18)
0.9367
(4.36)
0.5977 -0.0460
(3.84)
(-3.52)
0.7814
(3.33)
0.7862
(3.36)
0.7827
(3.52)
0.7622
(4.82)
0.9495
(4.15)
0.9483
(4.18)
0.9364
(4.36)
0.5977
(3.84)
0.7813
(3.33)
0.7860
(3.36)
0.7824
(3.52)
-0.0530
(-2.55)
-0.0554
(-2.38)
3-month
RFH,VW
i,t
-0.0666
(-2.67)
-0.0700
(-2.52)
6-month
RFH,VW
i,t
-0.0975
(-2.69)
-0.1026
(-2.61)
12-month
RFH,VW
i,t
-0.0460
(-3.52)
-0.0472
(-3.42)
1-month
AS,VW
Ri,t
-0.0530
(-2.55)
-0.0554
(-2.38)
3-month
AS,VW
Ri,t
-0.0667
(-2.68)
-0.0701
(-2.52)
6-month
AS,VW
Ri,t
-0.0978
(-2.70)
-0.1028
(-2.62)
12-month
AS,VW
Ri,t
VIX
-0.0184
(-1.48)
-0.0286
(-1.96)
-0.0269
(-1.85)
-0.0232
(-1.62)
0.0126
(0.80)
0.0026
(0.13)
0.0032
(0.16)
0.0053
(0.29)
-0.0184
(-1.48)
-0.0286
(-1.96)
-0.0269
(-1.85)
-0.0231
(-1.62)
0.0126
(0.80)
0.0026
(0.13)
0.0032
(0.16)
0.0054
(0.29)
TED
-0.6266
(-3.41)
-0.6092
(-3.56)
-0.6256
(-3.66)
-0.6520
(-3.76)
-0.4443
(-2.20)
-0.4212
(-2.11)
-0.4407
(-2.21)
-0.4728
(-2.35)
-0.6266
(-3.41)
-0.6093
(-3.56)
-0.6258
(-3.66)
-0.6524
(-3.76)
-0.4444
(-2.20)
-0.4213
(-2.11)
-0.4409
(-2.21)
-0.4732
(-2.35)
-0.0393
(-2.57)
-0.0398
(-2.20)
-0.0386
(-2.18)
-0.0369
(-2.18)
-0.0393
(-2.60)
-0.0398
(-2.20)
-0.0386
(-2.18)
-0.0369
(-2.18)
RV
This table presents the parameter estimates from the predictive regressions of one-month ahead CFNAI MA3 index on aggregate riskiness (RFH,VW
,
i,t
AS,VW
Ri,t
), the annualized implied volatility of the S&P 500 index options (V IX), the T ED spread defined as the difference between the 3-month LIBOR
and the 3-month T-bill rate, and the annualized realized volatility (RV ) of the S&P 500 index. The aggregate measures of riskiness are computed as
the value-weighted average of firm-level options’ implied measures of riskiness obtained from individual equity options with 1, 3, 6, and 12 months
to maturity. The results are reported for the sample period January 1996-October 2010. Newey and West (1987) t-statistics are given in parentheses.
The last column presents the adjusted R2 values.
Table 2: Predicting CFNAI 3MA with the Value-Weighted Average Riskiness, Credit Risk, and Market Uncertainty
53.09%
51.59%
50.98%
58.60%
50.74%
48.98%
48.20%
55.84%
53.07%
51.57%
50.97%
58.60%
50.72%
48.97%
48.19%
55.84%
R2
30
Intercept
0.7102
(3.89)
0.8395
(3.30)
0.8454
(3.35)
0.8532
(3.50)
0.7102
(3.89)
0.8394
(3.30)
0.8455
(3.35)
0.8532
(3.50)
1-month
RFH,EW
i,t
-0.0173
(-3.00)
-0.0202
(-2.58)
3-month
RFH,EW
i,t
-0.0274
(-2.66)
6-month
RFH,EW
i,t
-0.0458
(-2.64)
12-month
RFH,EW
i,t
-0.0173
(-3.00)
1-month
RAS,EW
i,t
-0.0202
(-2.58)
3-month
RAS,EW
i,t
-0.0274
(-2.66)
6-month
RAS,EW
i,t
-0.0457
(-2.63)
12-month
RAS,EW
i,t
VIX
0.0065
(0.38)
0.0009
(0.04)
0.0001
(0.01)
0.0016
(0.08)
0.0065
(0.38)
0.0009
(0.04)
0.0001
(0.01)
0.0015
(0.08)
TED
-0.4353
(-2.17)
-0.3986
(-2.02)
-0.4219
(-2.13)
-0.4581
(-2.28)
-0.4353
(-2.17)
-0.3988
(-2.02)
-0.4219
(-2.13)
-0.4578
(-2.28)
RV
-0.0401
(-2.45)
-0.0413
(-2.24)
-0.0403
(-2.22)
-0.0385
(-2.19)
-0.0401
(-2.45)
-0.0413
(-2.24)
-0.0403
(-2.22)
-0.0385
(-2.19)
This table presents the parameter estimates from the predictive regressions of one-month ahead CFNAI MA3 index on aggregate riskiness (RFH,EW
,
i,t
AS,EW
), the annualized implied volatility of the S&P 500 index options (V IX), the T ED spread defined as the difference between the 3-month LIBOR
Ri,t
and the 3-month T-bill rate, and the annualized realized volatility (RV ) of the S&P 500 index. The aggregate measures of riskiness are computed as
the equal-weighted average of firm-level options’ implied measures of riskiness obtained from individual equity options with 1, 3, 6, and 12 months
to maturity. The results are reported for the sample period January 1996-October 2010. Newey and West (1987) t-statistics are given in parentheses.
The last column presents the adjusted R2 values.
Table 3: Predicting CFNAI 3MA with Equal-Weighted Average Riskiness, Credit Risk, and Market Uncertainty
51.37%
50.20%
49.57%
55.59%
51.38%
50.21%
49.57%
55.60%
R2
31
1-month
Intercept RFH,VW
i,t
0.6760 -0.0375
(4.11)
(-3.28)
0.8214
(3.79)
0.8241
(3.83)
0.8203
(4.00)
0.6759
(4.11)
0.8214
(3.79)
0.8241
(3.83)
0.8203
(4.00)
-0.0449
(-2.44)
3-month
RFH,VW
i,t
-0.0572
(-2.56)
6-month
RFH,VW
i,t
Panel A. Value-Weighted Average Riskiness
-0.0844
(-2.67)
12-month
RFH,VW
i,t
-0.0375
(-3.28)
1-month
AS,VW
Ri,t
-0.0449
(-2.44)
3-month
AS,VW
Ri,t
-0.0572
(-2.56)
6-month
AS,VW
Ri,t
-0.0845
(-2.67)
12-month
AS,VW
Ri,t
VIX
0.0131
(0.81)
0.0055
(0.29)
0.0062
(0.33)
0.0082
(0.47)
0.0131
(0.81)
0.0055
(0.29)
0.0062
(0.33)
0.0082
(0.47)
TED
-0.4850
(-2.35)
-0.4738
(-2.30)
-0.4933
(-2.38)
-0.5230
(-2.50)
-0.4850
(-2.35)
-0.4739
(-2.30)
-0.4933
(-2.38)
-0.5230
(-2.50)
RV
-0.0462
(-2.95)
-0.0464
(-2.60)
-0.0454
(-2.58)
-0.0439
(-2.58)
-0.0462
(-2.95)
-0.0464
(-2.60)
-0.0454
(-2.58)
-0.0439
(-2.58)
This table presents the parameter estimates from the predictive regressions of one-month ahead CFNAI index on aggregate riskiness, the annualized
implied volatility of the S&P 500 index options (V IX), the T ED spread defined as the difference between the 3-month LIBOR and the 3-month T-bill
rate, and the annualized realized volatility (RV ) of the S&P 500 index. The aggregate measures of riskiness are computed as the value-weighted
(Panel A) and the equal-weighted (Panel B) average of firm-level options’ implied measures of riskiness obtained from individual equity options with
1, 3, 6, and 12 months to maturity. The results are reported for the sample period January 1996-October 2010. Newey and West (1987) t-statistics
are given in parentheses. The last column presents the adjusted R2 values.
Table 4: Predicting CFNAI with Aggregate Riskiness, Credit Risk, and Market Uncertainty
47.10%
46.08%
45.59%
49.42%
47.10%
46.08%
45.58%
49.42%
R2
32
Intercept
0.7683
(4.22)
0.8714
(3.77)
0.8758
(3.82)
0.8822
(3.97)
0.7683
(4.22)
0.8713
(3.77)
0.8759
(3.82)
0.8823
(3.97)
1-month
RFH,EW
i,t
-0.0141
(-2.82)
-0.0169
(-2.44)
3-month
RFH,EW
i,t
-0.0232
(-2.53)
6-month
RFH,EW
i,t
Panel B. Equal-Weighted Average Riskiness
Table 4 (continued)
-0.0388
(-2.58)
12-month
RFH,EW
i,t
-0.0141
(-2.82)
1-month
RAS,EW
i,t
-0.0169
(-2.44)
3-month
RAS,EW
i,t
-0.0231
(-2.53)
6-month
RAS,EW
i,t
-0.0388
(-2.58)
12-month
RAS,EW
i,t
VIX
0.0088
(0.47)
0.0024
(0.12)
0.0031
(0.16)
0.0046
(0.25)
0.0080
(0.47)
0.0024
(0.12)
0.0031
(0.16)
0.0046
(0.25)
TED
-0.4767
(-2.31)
-0.4526
(-2.23)
-0.4741
(-2.32)
-0.5053
(-2.42)
-0.4766
(-2.31)
-0.4527
(-2.23)
-0.4740
(-2.32)
-0.5051
(-2.42)
RV
-0.0469
(-2.84)
-0.0478
(-2.62)
-0.0469
(-2.61)
-0.0454
(-2.58)
-0.0469
(-2.84)
-0.0478
(-2.62)
-0.0469
(-2.61)
-0.0454
(-2.58)
45.83%
45.11%
44.66%
47.71%
45.84%
45.11%
44.66%
47.71%
R2
Aggregate Options’ Implied Foster−Hart Riskiness Measure (Value−Weighted)
45
40
35
30
25
20
15
10
5
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Aggregate Options’ Implied Foster−Hart Riskiness Measure (Equal−Weighted)
90
80
70
60
50
40
30
20
10
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Figure 1. Options’ Implied Measures of Aggregate Riskiness (Foster-Hart)
This figure presents the aggregate options’ implied measures of riskiness (Foster-Hart), computed as the
value-weighted average (top panel) and the equal-weighted average (bottom panel) of firm-level options’
implied measures of riskiness obtained from individual equity options with 1, 3, 6, and 12 months to
maturity. The sample period is January 1996-October 2010.
33
Aggregate Options’ Implied Aumann−Serrano Riskiness Measure (Value−Weighted)
45
40
35
30
25
20
15
10
5
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Aggregate Options’ Implied Aumann−Serrano Riskiness Measure (Equal−Weighted)
90
80
70
60
50
40
30
20
10
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Figure 2. Options’ Implied Measures of Aggregate Riskiness (Aumann-Serrano)
This figure presents the aggregate options’ implied measures of riskiness (Aumann-Serrano), computed
as the value-weighted average (top panel) and the equal-weighted average (bottom panel) of firm-level
options’ implied measures of riskiness obtained from individual equity options with 1, 3, 6, and 12 months
to maturity. The sample period is January 1996-October 2010.
34
Foster and Hart Physical Measures of Aggregate Riskiness (Value−Weighted)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Foster and Hart Physical Measures of Aggregate Riskiness (Equal−Weighted)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Figure 3. Physical Measures of Aggregate Riskiness (Foster-Hart)
This figure presents the aggregate physical measures of riskiness (Foster-Hart), computed as the valueweighted average (top panel) and the equal-weighted average (bottom panel) of firm-level physical measures of riskiness obtained from daily returns over the past 1, 3, 6, and 12 months. The sample period is
January 1996-October 2010.
35
Aumann and Serrano Physical Measures of Aggregate Riskiness (Value−Weighted)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Aumann and Serrano Physical Measures of Aggregate Riskiness (Equal−Weighted)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Figure 4. Physical Measures of Aggregate Riskiness (Aumann-Serrano)
This figure presents the aggregate physical measures of riskiness (Aumann-Serrano), computed as the
value-weighted average (top panel) and the equal-weighted average (bottom panel) of firm-level physical
measures of riskiness obtained from daily returns over the past 1, 3, 6, and 12 months. The sample period
is January 1996-October 2010.
36
4
TED Spread
3.5
3
2.5
2
1.5
1
0.5
0
Jan 96
Aug−97
Apr−99
Dec−00
Aug−02
Apr−04
Dec−05
Aug−07
Apr−09
Dec−10
Figure 5. TED Spread
This figure plots the TED spread defined as the difference between the 3-month LIBOR (an average of
interest rates offered in the London interbank market for 3-month dollar-denominated loans) and the 3month Treasury bill rate. The size of this gap reflects interbank credit risk or liquidity premium. The
sample period is January 1996-October 2010.
37
0.6
VIX
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
Jan 96
Aug−97
Apr−99
Dec−00
Aug−02
Apr−04
Dec−05
Aug−07
Apr−09
Dec−10
Figure 6. VIX Index
This figure plots the end-of-month V IX index, a popular measure of the implied volatility of S&P 500
index options with 30 days to maturity. The V IX is the square-root of the risk neutral expectation of the
S&P 500 variance over the next 30 calendar days. The V IX is quoted as an annualized standard deviation.
The V IX is calculated and disseminated in real-time by the Chicago Board Options Exchange. The sample
period is January 1996-October 2010.
38