11
STATE STREET ASSOCIATES
The Divergence of High- and Low-Frequency Estimation:
Causes and Consequences
Will Kinlaw
Mark Kritzman
David Turkington
22
STATE STREET ASSOCIATES
Motivation
33
STATE STREET ASSOCIATES
The divergence of high- and low-frequency estimation
Emerging Markets Stock Return 9.30%
U.S. Stock Return 9.50%
Correlation of Monthy Returns 69%
January 1990 to December 2013
44
STATE STREET ASSOCIATES
The divergence of high- and low-frequency estimation
Emerging Markets Stock Return 9.30%
U.S. Stock Return 9.50%
Correlation of Monthy Returns 69%
Emerging Markets - US Stocks 121%
Emerging Markets - US Stocks -62%
January 1990 to December 2013
January 2005 to December 2007
January 2011 to December 2013
55
STATE STREET ASSOCIATES
U.S. and emerging markets stocks: monthly returns
Source: State Street Associates
-40%
-30%
-20%
-10%
0%
10%
20%
-20% -15% -10% -5% 0% 5% 10% 15%
Em
erg
ing
ma
rke
ts e
qu
ity
U.S. equity
Correlation = 0.69
66
STATE STREET ASSOCIATES
U.S. and emerging markets stocks: annual returns
-80%
-60%
-40%
-20%
0%
20%
40%
60%
80%
100%
-60% -40% -20% 0% 20% 40% 60%
Em
erg
ing
ma
rke
ts e
qu
ity
U.S. equity
Source: State Street Associates
Correlation = 0.44
77
STATE STREET ASSOCIATES
U.S. and emerging markets stocks: triennial returns
-100%
-50%
0%
50%
100%
150%
200%
250%
-50% 0% 50% 100% 150%
Em
erg
ing
ma
rke
ts e
qu
ity
U.S. equity
Source: State Street Associates
Correlation = 0.04
88
STATE STREET ASSOCIATES
The divergence of high- and low-frequency estimation
It is common practice to assume that correlations estimated from high-frequency returns are similar to correlations estimated from low-frequency returns.
It is also common practice to assume that volatility scales with the square root of time.
Neither of these assumptions is supported by evidence.
99
STATE STREET ASSOCIATES
Emerging markets and U.S. equities: Relative return distribution
-16
0.0
%
-14
0.0
%
-12
0.0
%
-10
0.0
%
-80
.0%
-60
.0%
-40
.0%
-20
.0%
0.0
%
20
.0%
40
.0%
60
.0%
80
.0%
10
0.0
%
12
0.0
%
14
0.0
%
16
0.0
%
18
0.0
%
20
0.0
%
Actual triennial returns Monthly TE tri-annualized
Excess Dispersion Excess
Dispersion
Source: State Street Associates
Excess dispersion refers to the fraction of a distribution that falls outside the one-standard deviation tails of the distribution implied by monthly returns.
ImpliedActual
1010
STATE STREET ASSOCIATES
Divergence is different from sampling error
1
2
3
4
5
6
Sampling error arises when we use parameters from one sample…
7
8
9
10
11
12
… to predict parameters in another sample.
1111
STATE STREET ASSOCIATES
Divergence is different from sampling error
1
2
3
4
5
6
7
8
9
10
11
12
Divergence arises when we extrapolate parameters derived from high-frequency observations…
2
1
3
4
… to estimate low-frequency parameters within the same sample.
1212
STATE STREET ASSOCIATES
Mathematics of divergence
1313
STATE STREET ASSOCIATES
The relation of high- and low- frequency volatility
1
1 ,1 )(2)(q
k xxxqtt kttkqqxx
The volatility of the cumulative continuous returns of x over q periods is given by:
1414
STATE STREET ASSOCIATES
The relation of high- and low- frequency volatility
The volatility of the cumulative continuous returns of x over q periods is given by:
1
1 ,1 )(2)(q
k xxxqtt kttkqqxx
This term reflects annualization in the absence of lagged effects
1515
STATE STREET ASSOCIATES
The relation of high- and low- frequency volatility
1
1 ,1 )(2)(q
k xxxqtt kttkqqxx
This term captures the impact of auto-correlation
The volatility of the cumulative continuous returns of x over q periods is given by:
1616
STATE STREET ASSOCIATES
The relation of high- and low- frequency volatility:Impact of first-order auto-correlation
-1.0 -0.5 0.0 0.5 1.0-40%
-20%
0%
20%
40%
Lagged first order x auto-correlation
Exc
ess
dis
pe
rsio
n
Source: State Street Associates
Notes: This example assumes x and y have a contemporaneous correlation of zero.
1717
STATE STREET ASSOCIATES
The relation of high- and low- frequency correlation
The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by:
1
1 ,
1
1 ,
1
1 ,,,
11
)(2)(2
))((
),(
q
k yy
q
k xx
q
k yxyxyx
qttqtt
kttktt
ktttkttt
kqqkqq
kqq
yyxx
1818
STATE STREET ASSOCIATES
The relation of high- and low- frequency correlation
1
1 ,
1
1 ,
1
1 ,,,
11
)(2)(2
))((
),(
q
k yy
q
k xx
q
k yxyxyx
qttqtt
kttktt
ktttkttt
kqqkqq
kqq
yyxx
This term captures the lagged cross-correlation between x and y
The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by:
1919
STATE STREET ASSOCIATES
The relation of high- and low- frequency correlation
1
1 ,
1
1 ,
1
1 ,,,
11
)(2)(2
))((
),(
q
k yy
q
k xx
q
k yxyxyx
qttqtt
kttktt
ktttkttt
kqqkqq
kqq
yyxx
This term captures the auto-correlation of x
The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by:
2020
STATE STREET ASSOCIATES
The relation of high- and low- frequency correlation
1
1 ,
1
1 ,
1
1 ,,,
11
)(2)(2
))((
),(
q
k yy
q
k xx
q
k yxyxyx
qttqtt
kttktt
ktttkttt
kqqkqq
kqq
yyxx
This term captures the auto-correlation of y
The correlation between the cumulative returns of x and the cumulative returns of y over q periods is given by:
2121
STATE STREET ASSOCIATES
The relation of high- and low- frequency correlation: Impact of first-order cross-correlation
-1.0 -0.5 0.0 0.5 1.0-40%
-20%
0%
20%
40%
Lagged first order cross-correlation
Exc
ess
dis
pe
rsio
n
Source: State Street Associates
Notes: This example assumes x and y have a contemporaneous correlation of zero.
2222
STATE STREET ASSOCIATES
The relation of high- and low- frequency: Shorter versus longer lags
Source: State Street Associates
Notes: This example assumes x and y have a contemporaneous correlation of zero.
-1.0 -0.5 0.0 0.5 1.0-3%
-2%
-1%
0%
1%
2%
3%
Lagged first order cross-correlation and opposite sign lagged t-2 cross-correlation
Exc
ess
dis
pe
rsio
n
2323
STATE STREET ASSOCIATES
Low-frequency tracking error
We can now use the long-horizon standard deviations of x and y, together with their correlation, to compute the standard deviation of their relative performance after q periods, which we define as the tracking error between x and y:
yxyxyxTE 2),( 22
2424
STATE STREET ASSOCIATES
Evidence and causes of divergence
2525
STATE STREET ASSOCIATES
Attribution of excess dispersion of triennial relative returns
Excess Dispersion
IIDNon-
normalityAuto-
correlation
Lagged Cross-
correlation
Auto and Cross-
correlation ActualU.S. Domestic Assets
Hedge Funds Private Equity 0.0%Real Estate Government Bonds 0.0%Large Cap Equity Government Bonds 0.0%Corporate Bonds Government Bonds 0.0%Energy Stocks Utilities Stocks 0.0%
International Assets
U.S. Equities Emerging Equities 0.0%Canadian Equities U.S. Equities 0.0%Global Portfolio in USD USD/AUD 0.0%USD/GBP USD/JPY 0.0%German Equities U.K. Equities 0.0%
All underlying data is monthly from Jan 1990 through Dec 2013, except for private equity and real estate which are quarterly from Q1 1996 through Q3 2013. Country equity data are MSCI total return indices in US dollars. Equity sector data are total returns for 10 sector portfolios from Ken French's website. Bond data are Barclays total return indices. Currency data are WM/Reuters spot rates.
Source: State Street Associates
2626
STATE STREET ASSOCIATES
Attribution of excess dispersion of triennial relative returns
Excess Dispersion
IIDNon-
normalityAuto-
correlation
Lagged Cross-
correlation
Auto and Cross-
correlation ActualU.S. Domestic Assets
Hedge Funds Private Equity 0.0% 0.7%Real Estate Government Bonds 0.0% -0.8%Large Cap Equity Government Bonds 0.0% -0.4%Corporate Bonds Government Bonds 0.0% -1.9%Energy Stocks Utilities Stocks 0.0% -0.5%
International Assets
U.S. Equities Emerging Equities 0.0% -0.9%Canadian Equities U.S. Equities 0.0% 0.3%Global Portfolio in USD USD/AUD 0.0% 0.2%USD/GBP USD/JPY 0.0% -0.2%German Equities U.K. Equities 0.0% 0.4%
All underlying data is monthly from Jan 1990 through Dec 2013, except for private equity and real estate which are quarterly from Q1 1996 through Q3 2013. Country equity data are MSCI total return indices in US dollars. Equity sector data are total returns for 10 sector portfolios from Ken French's website. Bond data are Barclays total return indices. Currency data are WM/Reuters spot rates.
Source: State Street Associates
2727
STATE STREET ASSOCIATES
Attribution of excess dispersion of triennial relative returns
Excess Dispersion
IIDNon-
normalityAuto-
correlation
Lagged Cross-
correlation
Auto and Cross-
correlation ActualU.S. Domestic Assets
Hedge Funds Private Equity 0.0% 0.7% 24.8%Real Estate Government Bonds 0.0% -0.8% 20.3%Large Cap Equity Government Bonds 0.0% -0.4% 9.7%Corporate Bonds Government Bonds 0.0% -1.9% -13.4%Energy Stocks Utilities Stocks 0.0% -0.5% -9.0%
International Assets
U.S. Equities Emerging Equities 0.0% -0.9% -2.7%Canadian Equities U.S. Equities 0.0% 0.3% -1.2%Global Portfolio in USD USD/AUD 0.0% 0.2% -3.0%USD/GBP USD/JPY 0.0% -0.2% -1.5%German Equities U.K. Equities 0.0% 0.4% -4.2%
All underlying data is monthly from Jan 1990 through Dec 2013, except for private equity and real estate which are quarterly from Q1 1996 through Q3 2013. Country equity data are MSCI total return indices in US dollars. Equity sector data are total returns for 10 sector portfolios from Ken French's website. Bond data are Barclays total return indices. Currency data are WM/Reuters spot rates.
Source: State Street Associates
2828
STATE STREET ASSOCIATES
Attribution of excess dispersion of triennial relative returns
Excess Dispersion
IIDNon-
normalityAuto-
correlation
Lagged Cross-
correlation
Auto and Cross-
correlation ActualU.S. Domestic Assets
Hedge Funds Private Equity 0.0% 0.7% 24.8% 5.3%Real Estate Government Bonds 0.0% -0.8% 20.3% 4.3%Large Cap Equity Government Bonds 0.0% -0.4% 9.7% 1.4%Corporate Bonds Government Bonds 0.0% -1.9% -13.4% 7.0%Energy Stocks Utilities Stocks 0.0% -0.5% -9.0% -18.7%
International Assets
U.S. Equities Emerging Equities 0.0% -0.9% -2.7% 20.6%Canadian Equities U.S. Equities 0.0% 0.3% -1.2% 14.8%Global Portfolio in USD USD/AUD 0.0% 0.2% -3.0% 19.7%USD/GBP USD/JPY 0.0% -0.2% -1.5% 13.2%German Equities U.K. Equities 0.0% 0.4% -4.2% -11.8%
All underlying data is monthly from Jan 1990 through Dec 2013, except for private equity and real estate which are quarterly from Q1 1996 through Q3 2013. Country equity data are MSCI total return indices in US dollars. Equity sector data are total returns for 10 sector portfolios from Ken French's website. Bond data are Barclays total return indices. Currency data are WM/Reuters spot rates.
Source: State Street Associates
2929
STATE STREET ASSOCIATES
Attribution of excess dispersion of triennial relative returns
Excess Dispersion
IIDNon-
normalityAuto-
correlation
Lagged Cross-
correlation
Auto and Cross-
correlation ActualU.S. Domestic Assets
Hedge Funds Private Equity 0.0% 0.7% 24.8% 5.3% 26.8%Real Estate Government Bonds 0.0% -0.8% 20.3% 4.3% 22.9%Large Cap Equity Government Bonds 0.0% -0.4% 9.7% 1.4% 10.8%Corporate Bonds Government Bonds 0.0% -1.9% -13.4% 7.0% -4.6%Energy Stocks Utilities Stocks 0.0% -0.5% -9.0% -18.7% -25.7%
International Assets
U.S. Equities Emerging Equities 0.0% -0.9% -2.7% 20.6% 21.7%Canadian Equities U.S. Equities 0.0% 0.3% -1.2% 14.8% 15.1%Global Portfolio in USD USD/AUD 0.0% 0.2% -3.0% 19.7% 17.3%USD/GBP USD/JPY 0.0% -0.2% -1.5% 13.2% 11.1%German Equities U.K. Equities 0.0% 0.4% -4.2% -11.8% -17.9%
All underlying data is monthly from Jan 1990 through Dec 2013, except for private equity and real estate which are quarterly from Q1 1996 through Q3 2013. Country equity data are MSCI total return indices in US dollars. Equity sector data are total returns for 10 sector portfolios from Ken French's website. Bond data are Barclays total return indices. Currency data are WM/Reuters spot rates.
Source: State Street Associates
3030
STATE STREET ASSOCIATES
Attribution of excess dispersion of triennial relative returns
Excess Dispersion
IIDNon-
normalityAuto-
correlation
Lagged Cross-
correlation
Auto and Cross-
correlation ActualU.S. Domestic Assets
Hedge Funds Private Equity 0.0% 0.7% 24.8% 5.3% 26.8% 26.2%Real Estate Government Bonds 0.0% -0.8% 20.3% 4.3% 22.9% 22.7%Large Cap Equity Government Bonds 0.0% -0.4% 9.7% 1.4% 10.8% 14.5%Corporate Bonds Government Bonds 0.0% -1.9% -13.4% 7.0% -4.6% -12.0%Energy Stocks Utilities Stocks 0.0% -0.5% -9.0% -18.7% -25.7% -26.6%
International Assets
U.S. Equities Emerging Equities 0.0% -0.9% -2.7% 20.6% 21.7% 27.6%Canadian Equities U.S. Equities 0.0% 0.3% -1.2% 14.8% 15.1% 24.4%Global Portfolio in USD USD/AUD 0.0% 0.2% -3.0% 19.7% 17.3% 22.4%USD/GBP USD/JPY 0.0% -0.2% -1.5% 13.2% 11.1% 11.0%German Equities U.K. Equities 0.0% 0.4% -4.2% -11.8% -17.9% -19.1%
All underlying data is monthly from Jan 1990 through Dec 2013, except for private equity and real estate which are quarterly from Q1 1996 through Q3 2013. Country equity data are MSCI total return indices in US dollars. Equity sector data are total returns for 10 sector portfolios from Ken French's website. Bond data are Barclays total return indices. Currency data are WM/Reuters spot rates.
Source: State Street Associates
3131
STATE STREET ASSOCIATES
Excess dispersion of triennial returns: Major asset classes
U.S. Large Cap
U.S. Small Cap
EAFE Equities
Emerging Equities
Global Sovereign
BondsU.S. Govt.
Bonds
U.S. Corporate
BondsCommod-
itiesHedge Funds
Real Estate*
Private Equity*
Upper Triangle of Matrix: Triennial Correlation minus Monthly Correlation
U.S. Large Cap -0.02 -0.06 -0.60 -0.29 -0.12 -0.11 -0.05 0.00 0.29 0.14
U.S. Small Cap 3.4% 0.09 -0.26 -0.12 -0.13 0.09 0.11 -0.04 0.32 -0.07
EAFE Equities 5.8% -15.9% -0.22 -0.49 -0.47 -0.39 -0.02 -0.20 0.44 0.07
Emerging Equities 27.6% 14.5% 1.1% 0.27 -0.02 -0.15 0.06 -0.32 0.21 -0.38
Global Sovereign Bonds 15.3% -12.8% 3.4% -0.1% 0.00 0.00 -0.03 0.23 -0.28 -0.32
U.S. Government Bonds 14.5% -10.4% -1.3% 3.8% -4.9% -0.20 0.02 0.44 -0.20 0.10
U.S. Corporate Bonds 14.1% -15.1% -0.1% 1.9% -9.6% -12.0% 0.06 -0.05 -0.13 -0.20
Commodities 3.1% -15.1% -4.5% -4.1% -1.7% -3.7% -5.2% 0.05 0.24 -0.29
Hedge Funds 14.5% -15.5% -2.1% 12.5% 5.4% 3.1% 4.2% -3.7% 0.56 -0.08
Real Estate* -7.2% -17.7% -12.4% -5.4% 29.7% 22.7% 17.4% -19.5% -1.9% 0.23
Private Equity* -5.4% 5.1% 1.6% 6.9% 31.4% 26.2% 27.9% 1.6% 26.2% -1.9%
Lower Triangle of Matrix: Actual Excess Dispersion of Triennial Returns
All data monthly from Jan 1990 - Dec 2013 except Real Estate and Private Equity, which are quarterly from Q1 1996 - Q3 2013.
Source: State Street Associates
3232
STATE STREET ASSOCIATES
Implications for portfolio construction
3333
STATE STREET ASSOCIATES
Benchmark and portfolio weights
Asset Class Benchmark Weight Portfolio Weight
U.S. Large Cap 24.33% 44.33%
U.S. Small Cap 2.70% 2.70%
EAFE Equities 19.64% 9.64%
Emerging Equities 5.80% 0.80%
Global Sovereigns 23.76% 18.76%
U.S. Government Bonds 11.88% 6.88%
U.S. Corporate Bonds 11.88% 16.88%
Source: State Street Associates
3434
STATE STREET ASSOCIATES
Standard deviations and correlations
MONTHLY Standard Deviation
Correlation
U.S. Large Cap 4.29% 1.00 U.S. Small Cap 5.59% 0.81 1.00 EAFE Equities 5.07% 0.75 0.66 1.00 Emerging Equities 6.81% 0.69 0.70 0.73 1.00 Global Sovereigns 1.93% 0.10 0.01 0.31 0.08 1.00 U.S. Government Bonds 1.23% -0.06 -0.16 -0.08 -0.16 0.61 1.00 U.S. Corporate Bonds 2.51% 0.28 0.20 0.28 0.24 0.49 0.68 1.00
TRIENNIAL Standard Deviation
Correlation
U.S. Large Cap 42.98% 1.00 U.S. Small Cap 33.49% 0.73 1.00 EAFE Equities 35.34% 0.57 0.71 1.00 Emerging Equities 62.30% 0.04 0.53 0.71 1.00 Global Sovereigns 12.14% -0.15 0.06 -0.06 0.26 1.00 U.S. Government Bonds 8.10% -0.04 -0.13 -0.51 -0.29 0.47 1.00 U.S. Corporate Bonds 15.20% 0.23 0.42 -0.07 0.04 0.42 0.42 1.00
Source: State Street Associates
3535
STATE STREET ASSOCIATES
Low-frequency underperformance risk
Tracking error of triennial returns Value at risk (3-year, 95%)
4.58%4.03%
8.27%
7.54%
Estimated from monthly Actual
Source: State Street Associates
3636
STATE STREET ASSOCIATES
Balancing high- and low-frequency optimality
If investors care only about performance over short horizons or within long horizons they could construct portfolios that reflect aversion to risk based on covariances of high-frequency returns.
Alternatively, if they are concerned only with performance at the conclusion of long horizons, they could estimate covariances of low-frequency returns.
Or, as is the most likely case, they care about performance over both short and long horizons, they could include separate estimates of risk; one based on covariances of high-frequency returns and one based on covariances of low-frequency returns.
22)( LLHHUE
2)( HHUE
2)( LLUE
3737
STATE STREET ASSOCIATES
Iso-expected return curve balancing high- and low-frequency tracking error
0.55% 0.60% 0.65% 0.70% 0.75% 0.80% 0.85% 0.90% 0.95% 1.00%5%
6%
7%
8%
9%
10%
Monthly tracking error
Tri
en
nia
l tra
ck
ing
err
or
Portfolio 2
Portfolio 1
Portfolio 3
Source: State Street Associates
3838
STATE STREET ASSOCIATES
Portfolio weights
37.54%
9.44%
0.45%
8.84%
12.26%
19.24%
12.22%
33.57%
14.66%
-2.46%
11.69%
8.88%
29.75%
3.92%U.S. Large Cap
U.S. Small Cap
EAFE Equities
Emerging Equities
Global Sovereigns
U.S. Govt Bonds
U.S. Corp Bonds
44.33%
2.70%
9.64%0.80%
18.76%
6.88%
16.88%
Portfolio 1:Monthly
Portfolio 2:Blended
Portfolio 3:Triennial
Source: State Street Associates
3939
STATE STREET ASSOCIATES
Implications for currency hedging
4040
STATE STREET ASSOCIATES
Illustrative portfolio – USD based
Source: State Street Associates
Note: We assume a 60/40 allocation into equities and bonds. The country equity allocation is based on the MSCI DM Equity Index as of March 2013. The country bond allocation is based on the Citigroup Global Bond Index as of March 2013.
Foreign Bond
US Bond
Foreign Equity
US Equity
28.04%
11.96%
26.70%
33.30%
Portfolio Structure (%) Currency Exposure (%)
Note: The chart shows the currency exposure as of March 2013. We use the Euro to proxy for the Danish kroner and the U.S. dollar to proxy for both the Singapore dollar and Hong Kong dollar, as these currencies trade within tight bands around the currencies to which they are pegged. We ignore the Israeli Shekel because it comprises a very small portion of the portfolio and is relatively more costly to hedge than the other developed market currencies.
EUR JPY GBP CAD AUD CHF SEK NOK NZD0.00
5.00
10.00
15.00
20.00
25.00
20.10
16.95
7.66
3.49 2.77 2.41
1.01 0.32 0.03
4141
STATE STREET ASSOCIATES
Excess dispersion of triennial returns
US Equity
Foreign Equity US Bond
Foreign Bond AUD CAD CHF EUR GBP JPY NOK NZD SEK
Upper Triangle of Matrix: Triennial Correlation minus Monthly Correlation
US Equity (0.10) (0.13) (0.38) (0.72) (0.71) (0.55) (0.57) 0.13 (0.42) (0.66) (0.60) (0.53)
Foreign Equity 19.0% (0.40) (0.37) (0.51) (0.26) (0.45) (0.36) 0.16 (0.32) (0.44) (0.46) (0.37)
US Bond 19.0% 2.2% (0.17) (0.14) (0.06) (0.25) (0.27) (0.20) (0.07) (0.01) (0.25) (0.22)
Foreign Bond 23.6% 9.7% -5.7% 0.31 0.26 (0.01) (0.02) (0.11) (0.11) 0.16 0.19 0.11
AUD 32.5% 30.6% 12.5% -20.1% 0.16 0.37 0.19 (0.03) 0.37 0.27 0.09 0.24
CAD 31.5% 5.9% 5.9% -15.9% -15.9% 0.24 0.20 0.03 0.37 0.19 0.13 0.13
CHF 24.5% 17.6% 4.1% -6.1% -24.3% -20.1% (0.02) (0.26) 0.16 0.17 0.27 0.15
EUR 23.6% 13.9% 13.9% 6.4% -18.2% -13.1% -1.5% (0.03) (0.00) 0.10 0.20 0.06
GBP 14.3% -2.9% 1.8% -7.5% 1.3% -8.5% 2.7% 4.1% (0.42) (0.03) 0.13 0.04
JPY 25.0% 9.7% 1.8% 18.0% -8.5% -8.0% -8.9% 6.4% 6.9% 0.19 0.05 0.06
NOK 24.1% 14.8% 4.1% -18.2% -19.2% -19.6% -21.5% -22.4% -8.5% -2.4% 0.31 0.10
NZD 22.7% 25.0% 12.0% -2.4% -14.1% -6.6% -15.5% -12.7% -0.6% 9.2% -19.6% 0.26
SEK 23.2% 17.1% 3.2% -11.3% -28.0% -15.9% -17.8% -14.1% -15.9% 5.0% -24.3% -18.2%
Lower Triangle of Matrix: Actual Excess Dispersion of Triennial Returns
Data covers Jan 1994 to Oct 2014.
Source: State Street Associates
4242
STATE STREET ASSOCIATES
Optimal hedging amount – USD based
Source: State Street Associates
Unhedged Monthly Annual TriennialOptimal Hedging Amount 48.04 35.37 14.92
Currency Hedging Position
AUD 2.77 (2.77) (2.77) (2.77)
CAD 3.49 (3.49) (3.49) (3.49)
CHF 2.41 (2.41) - -
EUR 20.10 (20.10) (20.10) -
GBP 7.66 (7.66) (7.66) (7.66)
JPY 16.95 (10.25) - -
NOK 0.32 (0.32) (0.32) -
NZD 0.03 (0.03) (0.03) -
SEK 1.01 (1.01) (1.01) (1.01)
Data covers Jan 1994 to Oct 2014.Orange cells highlight currencies that are not fully hedged.
4343
STATE STREET ASSOCIATES
Source: State Street Associates
Low-frequency underperformance risk
Estimated from monthly Actual0%
5%
10%
15%
20%
25%
14.46%
20.03%
Portfolio risk of triennial returns
4444
STATE STREET ASSOCIATES
Optimal hedging amount – JPY based
Source: State Street Associates
Unhedged Monthly Annual TriennialOptimal Hedging Amount 83.05 83.05 79.59
Currency Hedging Position
AUD 2.77 (2.77) (2.77) (2.77)
CAD 3.49 (3.49) (3.49) (3.49)
CHF 2.41 (2.41) (2.41) -
EUR 20.10 (20.10) (20.10) (20.10)
GBP 7.66 (7.66) (7.66) (7.66)
USD 45.26 (45.26) (45.26) (45.26)
NOK 0.32 (0.32) (0.32) (0.32)
NZD 0.03 (0.03) (0.03) -
SEK 1.01 (1.01) (1.01) -
Data covers Jan 1994 to Oct 2014.Orange cells highlight currencies that are not fully hedged.
4545
STATE STREET ASSOCIATES
Source: State Street Associates
Low-frequency underperformance risk
Estimated from monthly Actual0%
5%
10%
15%
20%
25%
14.54%
20.68%
Portfolio risk of triennial returns
4646
STATE STREET ASSOCIATES
Optimal hedging amount – AUD based
Source: State Street Associates
Unhedged Monthly Annual TriennialOptimal Hedging Amount 54.32 57.42 15.40
Currency Hedging Position
USD 45.26 (27.81) (45.26) (3.25)
CAD 3.49 (3.49) (3.49) (3.49)
CHF 2.41 - - -
EUR 20.10 (11.29) - -
GBP 7.66 (1.17) (7.66) (7.66)
JPY 16.95 (9.20) - -
NOK 0.32 (0.32) - -
NZD 0.03 (0.03) - -
SEK 1.01 (1.01) (1.01) (1.01)
Data covers Jan 1994 to Oct 2014.Orange cells highlight currencies that are not fully hedged.
4747
STATE STREET ASSOCIATES
Source: State Street Associates
Low-frequency underperformance risk
Estimated from monthly Actual0%
5%
10%
15%
20%
25%
12.12%
22.15%
Portfolio risk of triennial returns
4848
STATE STREET ASSOCIATES
Implications for performance measurement
4949
STATE STREET ASSOCIATES
The distortion of hedge fund performance
The conventional practice for evaluating the Sharpe ratios of hedge funds is to estimate their standard deviations from monthly returns and then to multiply these monthly standard deviations by the square root of 12.
This approach implicitly assumes that hedge fund returns are serially independent at all lags, which is not the case.
5050
STATE STREET ASSOCIATES
Percentage of hedge funds that change quantileMonthly versus triennial Sharpe ratio
Source: State Street Associates
Change Decile Change Quintile Change Quartile Cross Median
-10%
0%
10%
20%
30%
40%
50%
60%
31%
14%12%
4%
Percentage assuming serial independence
5151
STATE STREET ASSOCIATES
Percentage of hedge funds that change quantileMonthly versus triennial Sharpe ratio
Source: State Street Associates
Change Decile Change Quintile Change Quartile Cross Median0%
10%
20%
30%
40%
50%
60%
31%
14%12%
4%
50%
31%27%
11%
Percentage assuming serial independence
Actual percentage
5252
STATE STREET ASSOCIATESPercentage of hedge funds within categories that change quantileMonthly versus triennial Sharpe ratio
Percentage of Funds that:
Change Decile Change Quintile Change Quartile Cross Median
Hedge Fund StyleNumber of Funds Result p-value Result p-value Result p-value Result p-value
U.S. L/S Equity 110 51% 0% 30% 0% 35% 0% 16% 0%
International L/S Equity 57 47% 4% 33% 2% 23% 8% 14% 3%
Global Macro 32 63% 6% 47% 0% 47% 0% 9% 25%
Debt 27 70% 1% 52% 0% 22% 6% 7% 67%
Long-Only Equity 16 50% 38% 31% 6% 25% 36% 0% 100%
FoFs / Multi-strategy 209 46% 0% 26% 0% 23% 0% 11% 0%
Systematic Futures 73 45% 16% 22% 14% 25% 15% 5% 82%
Other 45 58% 1% 38% 0% 27% 0% 13% 0%
All Hedge Funds 569 50% 31% 27% 11%
Sample includes 569 hedge funds from the CISDM/Morningstar database that report monthly returns from January 2000 through December 2013. Sharpe ratios are computed from monthly and triennial return and standard deviation, as well as the monthly and triennial return of the JP Morgan U.S. Cash Index (0.20% and 7.39%, respectively) over the same period. To increase the sample size with each category, we consolidated the CISDM/Morningstar style categories as follows: U.S. Long/Short Equity includes Equity Market Neutral, U.S. Long/Short Equity, and U.S. Small Cap Long/Short Equity; International Long/Short Equity includes Asia/Pacific Long/Short Equity, Emerging Markets Long/Short Equity, Europe Long/Short Equity, and Global Long/Short Equity; Global Macro includes Global Macro and Currency; Debt includes Debt Arbitrage, Distressed Securityies, and Long/Short Debt; Long-Only Equity includes Long-Only Equity and Emerging Markets Long-Only Equity; Fund of Funds/Multistrategy includes Multistrategy funds as well as six Fund of Funds categories (Macro/Systematic, Debt, Equity, Event, Multistrategy, and Relative Value); Systematic Futures was not consolidated; Other includes Bear Market Equity, Convertible Arbitrage, Event Driven, Long-Only Other, and Merger Arbitrage. We employ a bootstrap simulation to estimate the p-values. Specifically, we compute the frequency with which each result would arise if monthly returns were independently distributed.
Source: State Street Associates
5353
STATE STREET ASSOCIATES
The distortion of mutual fund performance
Unlike hedge funds, mutual funds are usually evaluated by their performance relative to their given benchmarks.
They are compared to each other based on their information ratios, which equals a fund’s excess return relative to its benchmark divided by its excess risk relative to its benchmark.
The conversion of monthly information ratios to longer-interval information ratios is therefore subject to three sources of distortion:
• the auto-correlations of the fund’s returns, • the auto-correlations of the benchmark’s returns, and • the lagged cross-correlations between the fund’s returns and the benchmark’s
returns.
5454
STATE STREET ASSOCIATES
Percentage of mutual funds that change quantileMonthly versus triennial information ratio
Source: State Street Associates
Change Decile Change Quintile Change Quartile Cross Median
-10%
0%
10%
20%
30%
40%
50%
60%
34%
17%
13%
4%
Percentage assuming serial independence
5555
STATE STREET ASSOCIATES
Percentage of mutual funds that change quantileMonthly versus triennial information ratio
Source: State Street Associates
Change Decile Change Quintile Change Quartile Cross Median
-10%
0%
10%
20%
30%
40%
50%
60%
34%
17%
13%
4%
44%
25%
18%
5%
Percentage assuming serial independence
Actual percentage
5656
STATE STREET ASSOCIATES
Percentage of mutual funds within categories that change quantileMonthly versus triennial information ratio
Source: State Street Associates
Percentage of Funds that:
Change Decile Change Quintile Change Quartile Cross Median
Mutual Fund StyleNumber of Funds Result p-value Result p-value Result p-value Result p-value
U.S. L/S Equity 786 47% 2% 29% 0% 22% 0% 8% 0%
International L/S Equity 358 49% 3% 28% 2% 18% 12% 2% 99%
Global Macro 392 33% 15% 15% 33% 11% 43% 2% 99%
All Mutual Funds 1536 44% 25% 18% 5%
The U.S. Large Cap Blend sample includes 786 out of 1,428 mutual funds listed in the "Large Blend" category on Yahoo! Finance that meet the following two criteria: monthly returns are available from January 2008 through December 2013 and tracking error relative to the S&P 500 is greater than 1% per annum. We impose the latter filter to eliminate index funds. The U.S. Small Cap Blend sample includes 358 out of 640 funds listed in the "Small Blend" category that meet the same criteria. The Foreign Large Cap Blend sample includes 392 funds listed in the "Foreign Large Blend" category whose monthly returns are available for the same period and whose name does not include "Idx" or "Index" (unless it also includes "Enhanced"). Monthly total returns are from Bloomberg. Returns are net of fees and assume reinvestment of dividends. The benchmarks are S&P 500, Russell 2000, and MSCI All Country World ex USA, respectively. We employ a bootstrap simulation to estimate the p-values. Specifically, we compute the frequency with which each result would arise if monthly returns were independently distributed.
5757
STATE STREET ASSOCIATES
The fallacy of risk parity
5858
STATE STREET ASSOCIATES
The fallacy of risk parity
Risk parity over weights low-risk assets relative to high-risk assets.
It has been shown to outperform portfolios that are not biased toward low-risk assets.
Conceptually, risk parity delivers the highest Sharpe ratio if all assets have identical Sharpe ratios and correlations.
It is argued that its outperformance is enhanced because the security market line appears to have a slope that is significantly less than 1.0, implying that low-risk assets have higher risk-adjusted returns than high-risk assets.
These observations do not hold when we account for lagged auto- and cross-correlations.
5959
STATE STREET ASSOCIATES
Security market lineU.S. stocks and government bonds (1929 – 2010)
0% 2% 4% 6% 8%0%
2%
4%
6%
8%
Beta * Annualized excess market return
An
nu
aliz
ed
ex
ce
ss
re
turn
Source: State Street Associates
Monthly
Average annual excess returns are calculated as the annualized geometric mean of historical returns, minus the annualized geometric mean of historical 1-month treasury bill yields (or of 10-year treasury bond yields) over the same period. Standard deviations, betas and correlations for the 10-year holding period are calculated from historical rolling log returns in excess of the 10 year treasury yield, and splice the beginning of the data set onto the end to avoid unequal representation of returns in statistics. Returns for stocks and bonds are from Ibbotson. The market portfolio is a 60/40 stock/bond portfolio rebalanced monthly.
6060
STATE STREET ASSOCIATES
Security market lineU.S. stocks and government bonds (1929 – 2010)
0% 2% 4% 6% 8%0%
2%
4%
6%
8%
Monthly10-year
Beta * Annualized excess market return
An
nu
aliz
ed
ex
ce
ss
re
turn
Source: State Street Associates
Slope = 0.98
Slope = 0.59
Average annual excess returns are calculated as the annualized geometric mean of historical returns, minus the annualized geometric mean of historical 1-month treasury bill yields (or of 10-year treasury bond yields) over the same period. Standard deviations, betas and correlations for the 10-year holding period are calculated from historical rolling log returns in excess of the 10 year treasury yield, and splice the beginning of the data set onto the end to avoid unequal representation of returns in statistics. Returns for stocks and bonds are from Ibbotson. The market portfolio is a 60/40 stock/bond portfolio rebalanced monthly.
6161
STATE STREET ASSOCIATES
Annualized Sharpe ratiosU.S. stocks and government bonds (1929 – 2010)
Source: State Street Associates
Monthly 10 year
0.36
0.23
0.42
0.16
60/40
Risk parity
Risk parity weights each month are proportional to the inverse of the trailing 36 month asset volatilities, rescaled to sum to one so there is no leverage. Average annual excess returns are calculated as the annualized geometric mean of historical returns, minus the annualized geometric mean of historical 1-month treasury bill yields (or of 10-year treasury bond yields) over the same period. Standard deviations for the 10-year holding period are calculated from historical rolling log returns in excess of the 10 year treasury yield, and splice the beginning of the data set onto the end to avoid unequal representation of returns in statistics. Stock and bond data are from Ibbotson.
6262
STATE STREET ASSOCIATES
Attribution of divergenceU.S. stocks and government bonds (1929 – 2010)
Source: State Street Associates
Risk parity weights each month are proportional to the inverse of the trailing 36 month asset volatilities, rescaled to sum to one so there is no leverage. Average annual excess returns are calculated as the annualized geometric mean of historical returns, minus the annualized geometric mean of historical 1-month treasury bill yields (or of 10-year treasury bond yields) over the same period. Standard deviations for the 10-year holding period are calculated from historical rolling log returns in excess of the 10 year treasury yield, and splice the beginning of the data set onto the end to avoid unequal representation of returns in statistics. Stock and bond data are from Ibbotson.
6363
STATE STREET ASSOCIATES
The economic intuition of divergence
We hypothesize that high-frequency variability arises from changes in discount rates whereas low-frequency variability is caused by differences in cash flows.
Changes in discount rates occur relatively often because a constant flow of new
information causes investors to reassess the riskiness of a stream of cash flows, which therefore leads to high-frequency variability.
In addition, the value of a portfolio or strategy may gradually appreciate or erode over a long horizon because the drift of cash flows shifts upward or downward as fundamentals change. This process introduces to low-frequency variability.
6464
STATE STREET ASSOCIATES
The economic intuition of divergence
Source: State Street Associates
Monthly 3-year 10-year
0.47
0.17
0.23
0.08
0.32
0.44
Changes in discount rate
Changes in earnings
R-s
qu
are
d
We test this hypothesis by analyzing the returns of 10 U.S. sectors from December 1978 through December 2013. Each month, we regress the cross section of monthly, three-year, and 10-year sector returns on the cross section of changes in earnings during the same periods. We also regress the cross-section of monthly, three-year, and 10-year sector returns on the cross section of changes in beta during the same periods, where beta is computed with a three-year look back window relative to the S&P 500 index. We use beta as a proxy for sector discount rates with the intuition that investors demand higher returns from sectors that have greater systematic risk.
6565
STATE STREET ASSOCIATES
Summary
Financial analysts are often surprised by the extent to which assets that are thought to be strongly correlated diverge over time.
This result can occur within the same sample from which the parameters are derived. It occurs because one or both assets may have non-normal returns, or because they are auto-correlated or cross-correlated at one or more lags.
We find evidence of excess dispersion for a variety of asset pairs, and we attribute this excess dispersion to its sources. We find that most excess dispersion arises from non-zero auto-correlations and lagged cross-correlations.
We mathematically relate high frequency estimates of mean, standard deviation, and correlation to their low frequency values, and investigate comparative statics.
We introduce a portfolio construction framework that jointly accounts for aversion to high- and low-frequency risk.
We show that optimal currency hedge ratios based on monthly returns overstate the optimal amount to hedge for longer horizons.
We provide evidence that the risk-adjusted performance of hedge funds, mutual funds, and risk parity strategies is highly sensitive to the return interval used to estimate it.
We present evidence that high-frequency variability is due to changes in discount rates whereas low-frequency variability is due to differences in cash flows.
6666
STATE STREET ASSOCIATES
Disclaimers and Important Risk Information
The information provided herein is not intended to suggest or recommend any investment or investment strategy, does not constitute investment advice, does not constitute investment research and is not a solicitation to buy or sell securities. It does not take into account any investor's particular investment objectives, strategies or tax status. Clients should be aware of the risks trading foreign exchange, equities, fixed income or derivative instruments or in investments in non-liquid or emerging markets. Derivatives generally involve leverage and are therefore more volatile than their underlying cash investments. Past performance is no guarantee of future results. This communication is not intended for and must not be provided to retail investors. The products and services described in this communication may not be available in all jurisdictions. 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These opinions may not be suitable to your financial status and your risk and return preferences. Therefore, an investment decision based solely on the information herein may not be appropriate to your expectations. This communication is made available in United Arab Emirates by State Street Bank and Trust Company and its affiliates. This communication does not, and is not intended to, constitute an offer of securities anywhere in the United Arab Emirates and accordingly should not be construed as such. Nor does the addressing of this research publication to you constitute, or is intended to constitute, the carrying on or engagement in banking, financial and/or investment consultation business in the United Arab Emirates under the rules and regulations made by the Central Bank of the United Arab Emirates, the Emirates Securities and Commodities Authority or the United Arab Emirates Ministry of Economy. 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