11 STATE STREET ASSOCIATES The Divergence of High- and Low-Frequency Estimation: Causes and...

11

STATE STREET ASSOCIATES

The Divergence of High- and Low-Frequency Estimation:

Causes and Consequences

Will Kinlaw

Mark Kritzman

David Turkington

22


Motivation

33


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



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%




55


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


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


Correlation = 0.44

77


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


Correlation = 0.04

88



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


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


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


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


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


Mathematics of divergence

1313


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




1

1 ,1 )(2)(q

k xxxqtt kttkqqxx

This term reflects annualization in the absence of lagged effects

1515



1

1 ,1 )(2)(q

k xxxqtt kttkqqxx

This term captures the impact of auto-correlation


1616


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


Notes: This example assumes x and y have a contemporaneous correlation of zero.

1717


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



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


1919



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


2020



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


2121


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



2222


The relation of high- and low- frequency: Shorter versus longer lags



-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


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


Evidence and causes of divergence

2525


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.


2626



Excess Dispersion

IIDNon-

normalityAuto-

correlation

Lagged Cross-

correlation

Auto and Cross-


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%


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%



2727



Excess Dispersion

IIDNon-

normalityAuto-

correlation

Lagged Cross-

correlation

Auto and Cross-


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%


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%



2828



Excess Dispersion

IIDNon-

normalityAuto-

correlation

Lagged Cross-

correlation

Auto and Cross-


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%


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%



2929



Excess Dispersion

IIDNon-

normalityAuto-

correlation

Lagged Cross-

correlation

Auto and Cross-


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%


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%



3030



Excess Dispersion

IIDNon-

normalityAuto-

correlation

Lagged Cross-

correlation

Auto and Cross-


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%


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%



3131


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.


3232


Implications for portfolio construction

3333


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%


3434


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


3535


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


3636


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


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


3838


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


3939


Implications for currency hedging

4040


Illustrative portfolio – USD based


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


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.


4242


Optimal hedging amount – USD based


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




Estimated from monthly Actual0%

5%

10%

15%

20%

25%

14.46%

20.03%

Portfolio risk of triennial returns

4444


Optimal hedging amount – JPY based




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) -


4545





5%

10%

15%

20%

25%

14.54%

20.68%


4646


Optimal hedging amount – AUD based




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)


4747





5%

10%

15%

20%

25%

12.12%

22.15%


4848


Implications for performance measurement

4949


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


Percentage of hedge funds that change quantileMonthly versus triennial Sharpe ratio


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


Percentage of hedge funds that change quantileMonthly versus triennial Sharpe ratio


Change Decile Change Quintile Change Quartile Cross Median0%

10%

20%

30%

40%

50%

60%

31%

14%12%

4%

50%

31%27%

11%


Actual percentage

5252

STATE STREET ASSOCIATESPercentage of hedge funds within categories that change quantileMonthly versus triennial Sharpe ratio

Percentage of Funds that:


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.


5353


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


Percentage of mutual funds that change quantileMonthly versus triennial information ratio



-10%

0%

10%

20%

30%

40%

50%

60%

34%

17%

13%

4%


5555


Percentage of mutual funds that change quantileMonthly versus triennial information ratio



-10%

0%

10%

20%

30%

40%

50%

60%

34%

17%

13%

4%

44%

25%

18%

5%


Actual percentage

5656


Percentage of mutual funds within categories that change quantileMonthly versus triennial information ratio


Percentage of Funds that:


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


The fallacy of risk parity

5858


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


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


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


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


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


Annualized Sharpe ratiosU.S. stocks and government bonds (1929 – 2010)


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


Attribution of divergenceU.S. stocks and government bonds (1929 – 2010)


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


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


The economic intuition of divergence


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


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


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11 STATE STREET ASSOCIATES The Divergence of High- and Low-Frequency Estimation: Causes and...

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