Visualizing the Time Series Behavior
of Volatility, Serial Correlation and
Investment Horizon
Presented by:
Ralph Goldsticker, CFA
???? ??, 2014
Summary
The standard assumptions we use to model risk don’t hold.
• Standard deviations don’t grow with the square root of time.
• Cross-asset class correlations are sampling frequency dependent.
• Observed volatilities and correlations varied through time.
Recommendations:
1. Investors should calculate risk using return periods that are consistent
with their investment horizon. Correlations calculated using highfrequency returns can mask underlying relationships.
2. Rather than relying on summary statistics, investors should develop a
covariance matrix that represents their future expectations. This
requires examining risk and correlation behavior through time.
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Agenda
Using cumulative contribution charts to understand:
I.
Historical relationships between volatility and holding period
II. Historical relationships between correlation and holding period
III. Discussion
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I. Cumulative contribution to volatility charts provide insights
into the behavior of volatility over time.
1.
2.
3.
4.
5.
6.
7.
Chart of cumulative contribution to volatility
Cumulative contribution to volatility versus rolling volatility.
Cumulative contribution to volatility versus length of return period
Variation of volatility though time depends on length of return period
Volatility estimated using daily versus monthly returns
Volatility estimated 1-month versus 3- and 12-month returns
Volatility estimated 12-month versus 24- and 36-month returns
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Cumulative contribution to volatility charts provide insights into
the behavior of volatility through time.
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Rolling volatility doesn’t provide the same insights.
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Cumulative contribution to volatility charts provide shows the
effects of serial correlations and changing volatilities.
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Evolution of volatilities at different sampling horizons show
similar but not identical patterns.
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The volatility of daily returns has been higher than monthly
returns since the mid 1990s.
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Higher volatility of 3- and 12-month returns shows positive serial
correlation of monthly returns.
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Data shows time diversification at multi-year horizons.
Pattern was reversed during and after the Tech Bubble.
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Volatility Summary
Volatility varied through time
Full period behavior:
• Negative serial correlation of daily returns
• Positive serial correlation of monthly returns
• Negative serial correlation of annual returns
But:
• Volatility of monthly returns was higher during the 1970s and 1980s.
• Positive serial correlation of annual returns during and after the Tech
Bubble
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II. Cumulative contribution to correlation charts provide insights
into the behavior of correlations over time.
1. Charting time series contribution to correlation provides insights that are
obscured by rolling windows.
2. Stock versus bond correlation increased with the holding period.
3. Correlation between stocks and Treasuries was positive until the Tech
Bubble, and negative since.
4. Correlation between stocks and Treasuries was regime dependent.
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Cumulative contribution to correlation provides insights
unavailable using full period and rolling window correlations.
• Cumulative contribution shows
when stock versus bond
correlation changed from
positive to negative.
• Cumulative correlation is not
sensitive to data points dropping
out of the sample (e.g. Oct.
1987)
• Slope of cumulative correlation
line shows the evolution of
correlations through time.
Correlation
1972 - 2013
0.10
12/1972 – 9/2000
0.31
10/2000 – 12/2013
-0.37
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Stock versus Bond correlation increased with holding period.
• Investors should use a return
period that’s consistent with
their investment horizon.
• The correlation calculated using
3-year returns was larger than
correlations calculated using
returns for shorter periods.
• Higher degree of correlation
shows that there are regimes
and cycles that are not
captured in short-term returns.
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Cumulative contribution to correlation allows us to identify
turning points.
Holding Period for Return Calculation
Daily
Monthly
Quarterly
Annual
3-Year
Full Period: 1972 - 2013
0.00
0.10
0.07
0.21
0.35
While Positive
0.32
0.31
0.38
0.53
0.62
While Negative
-0.34
-0.37
-0.52
-0.65
-0.71
10/21/1997
Sep-00
Mar-98
Apr-99
Apr-02
Date Changed from
Positive to Negative
• The correlations calculated using annual returns are roughly 1½ times the
correlations calculated using monthly returns.
• The correlations calculated using 3-year returns are roughly two times as large.
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Stock versus Bond correlation is regime dependent.
Macro Environment
Correlation
1970s
Changing inflation expectations
1980s
Falling interest rates
1990s
Capital flows into both stock and bond markets
+
+
+
2000s
Responses to bursting of the Tech Stock Bubble and
the Financial Crisis
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Changes in the slope of lines highlights regimes
• The steep 3-year line between
1975 and 1981 shows that the
environment of highly volatile
inflation expectations
translated into high stock-bond
correlations for longer-term
investors.
• There wasn’t much variability in
correlations between 1972 and
2000 using daily and monthly
returns.
• The correlations were more
negative after the Tech Bubble
and Financial Crisis than during
the period between.
Financial
Crisis
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Correlation Summary
1.
2.
Correlation between US stocks versus Treasuries varied through time
• Positive through the Tech Stock Bubble
• Negative since
Correlation between US stocks versus Treasuries increased with holding
period.
• More positive when positive
• More negative when negative
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Discussion
• We use charts of cumulative returns to understand market performance,
why shouldn’t we put the same effort into understanding risk?
• While the presentation focused on the visualizing asset class risks and
correlations, similar cumulative contribution charts could be used to
visualize other types of risks such as tracking error and information ratios.
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Appendix:
Formula for cumulative contribution to standard deviation
Standard Deviation =
∑ Cumulative Contribution to Standard Deviation =
∑ StockReturnt − AvgStkRet
numberofperiods − 1
2
Note: All returns are log excess returns.
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Appendix:
Formula used to calculate cumulative contribution to correlation
Correlation(Stocks, Bonds) =
Covariance(Stocks, Bonds)
StdDev Stocks × StdDev(Bonds)
Covariance(Stocks, Bonds ) =
∑((StkRett − AvgStkRet) × (BndRett − AvgBndRet))
numberofperiods
Cumulative Contribution to Correlation(Stock, Bond) =
∑((StockReturnt − AvgStkRet) × (BondReturnt − AvgBndRet))
numberofperiods × StdDev Stocks × StdDev(Bonds)
Note: All returns are log excess returns.
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