1

Higher-Frequency Analysis of Low-Frequency Data

Mark Szigety

Mickaël Mallinger-DoganHarvard Management Company

szigetym@hmc.harvard.edu

doganm@hmc.harvard.edu

November 2013

DRAFT/PRELIMINARY RESULTS

2

Problem

• Institutions invest heavily in asset classes characterized by illiquidity

• Illiquid asset classes are priced at a lower frequency than public market investments

• Risk measures, rebalancing, and asset allocation often rely on historical time series

• How do we incorporate all asset classes in a common analytical framework?

3

Highest Common Frequency

• Convert all series to common frequency, typically quarterly

• Advantages:– Easy

• Disadvantages:– Discards information, exacerbated by short evaluation window– Derived results may not be relevant at decision-making frequency

4

Other Possibilities

• Public markets proxies/Factor assignments– Bottoms-up analysis of illiquid holdings– NAV-blend appropriate higher-frequency series– Advantages: Thorough replication of current risks– Disadvantages: Complicated, especially for big portfolios; subjective;

needs to be constantly refreshed; not linked to realized returns (unsmoothed?)

• Replication of current valuation model(s)– Build and calibrate valuation model(s) in-house– Construct time series (or if possible search for analytical solutions)– Advantages: Thorough replication of current risks– Disadvantages: Essentially impossible; model error

5

Interpolation Approaches

• Can we make use of some econometric approaches to the problem?

• Economists and policy makers face a similar problem when measuring GDP

• Work on the “distribution” problem began in the 1950s and continues today

6

Contribution and Findings

• Apply econometric interpolation methodologies to the context of illiquid investments and institutional risk

• Rigorously compare through simulation the two primarily relevant interpolation methodologies

• Primary findings:– The more current methodology tends to excel when the serial

correlation is high and/or the explanatory power of the proxy is high– The proxy used can affect the risk attribution by changing the

relationship that the interpolated series has to factors– Whether to interpolate and which to use depends on context

7

Best Linear Unbiased Estimator (BLUE)

The estimator is

where

Assuming

we get

8

Let for all t

Assume a Gaussian VAR(p) model for {y*t}

A state-space representation of the VAR model is

Let for all t

we get

As {y+t} has no missing observations, the Kalman filter and smoother apply directly.

Mixed-Frequency VAR (MFV)

9

Comparison

BLUE: Chow and Lin (CL) MFV: Mariano and Murasawa (MM)

Year 1971 2003, 2010

Description 1) Applies low-frequency betas to higher-frequency observations of the proxy(ies)

2) Distributes estimated residuals

1) VAR model for partially latent high-frequency series

2) State-space model for the observable mixed-frequency series

Pros 1) Intuitive2) Fast3) Number of parameters to estimate grows

slowly when more proxies are added

1) Complex residuals structure built-in2) Can estimate multiple low-frequency

series at once

Cons 1) Very difficult to implement complex residuals structure (i.e., more than AR(1))

2) Only estimates one low-frequency series at a time

3) Biased(?)

1) Harder to implement2) Slow(er)3) Number of parameters to estimate grows

dramatically when a new series is added and/or the order of the VAR model is increased

10

Graphical Example: Real Estate

Source: Real Estate Index data from Cambridge Associates, LLC. Copyright 2013. All rights reserved. The Cambridge Associates, LLC data may not be copied, used, or distributed without Cambridge’s prior written approval. The data is provided “as is” without any express or implied warranties. Past performance is no guarantee of future results.

11

Simulation Framework• Three return series are simulated (P, X, F); P is converted to

lower frequency in 3-to-1 ratio

whereejt ~ N(0, Σ)

• Each methodology is then used to convert P back to higher frequency, and their relative success is assessed

12

Dimensions of Comparison

• RMSE: measure of the overall accuracy of each candidate interpolation methodology

• Volatility: useful in a variety of analytical measures of portfolio risk

• AR(1): impacts proper volatility scaling and could affect the usefulness of various downstream unsmoothing approaches

• Beta to Factor: useful in preserving relationships heavily relied upon for factor contribution to risk, correlations, and stress testing

13

Context of Comparison

• Low/High autocorrelation: does performance vary based on the level of autocorrelation?

• Low to High explanatory power of proxy: does performance vary based on the “quality” of the proxy?

• Low to High beta of portfolio to factor: does performance vary based on the true high-frequency beta of the portfolio to the factor?

• Minimum/Maximum possible beta of factor to proxy: does performance vary based on the relationship of the factor to the proxy?

14

Results: Low AR; Minimum Beta F to X

CL MM CL MM CL MMRMSE 0.00% 2.31% 2.30% 1.83% 1.74% 1.16% 0.87%

Volatility 15.0% 14.5% 15.7% 14.0% 15.0% 14.3% 15.0%AR(1) 0.20 0.36 0.48 0.16 0.33 0.06 0.22

Beta P to F 0.10 0.03 0.03 0.03 0.03 0.04 0.03Beta F to X 0.0 0.0 0.0 0.0 0.0 0.0

RMSE 0.00% 2.31% 2.30% 1.83% 1.74% 1.16% 0.87%Volatility 15.0% 14.5% 15.6% 14.0% 15.0% 14.3% 14.9%AR(1) 0.20 0.36 0.47 0.17 0.32 0.06 0.22

Beta P to F 0.27 0.08 0.07 0.09 0.08 0.16 0.14Beta F to X 0.0 0.0 0.0 0.0 0.1 0.1

RMSE 0.00% 2.31% 2.30% 1.82% 1.74% 1.16% 0.87%Volatility 15.0% 14.5% 15.7% 14.1% 15.0% 14.3% 14.9%AR(1) 0.20 0.36 0.47 0.17 0.33 0.06 0.22

Beta P to F 0.45 0.13 0.12 0.16 0.14 0.34 0.30Beta F to X 0.0 0.0 0.0 0.0 0.3 0.3

RMSE 0.00% 2.30% 2.30% 1.82% 1.74% 1.15% 0.87%Volatility 15.0% 14.4% 15.5% 14.1% 15.2% 14.3% 14.9%AR(1) 0.20 0.35 0.46 0.17 0.35 0.06 0.23

Beta P to F 0.62 0.20 0.18 0.32 0.28 0.59 0.52Beta F to X 0.0 0.0 0.2 0.2 0.6 0.6

RMSE 0.00% 2.31% 2.29% 1.82% 1.73% 1.16% 0.87%Volatility 15.0% 14.5% 15.7% 14.1% 15.1% 14.3% 15.0%AR(1) 0.20 0.35 0.47 0.17 0.34 0.06 0.23

Beta P to F 0.79 0.41 0.36 0.63 0.55 0.83 0.73Beta F to X 0.4 0.4 0.7 0.7 0.9 0.9

% of Remaining P Variance Explained by XTrue

ValueDimension30 60 90

15

Results: Low AR; Maximum Beta F to X

CL MM CL MM CL MMRMSE 0.00% 2.30% 2.29% 1.82% 1.73% 1.15% 0.87%

Volatility 15.0% 14.3% 15.6% 14.1% 15.0% 14.3% 15.0%AR(1) 0.20 0.35 0.48 0.16 0.33 0.06 0.23

Beta P to F 0.10 0.34 0.29 0.37 0.31 0.28 0.24Beta F to X 0.8 0.8 0.7 0.7 0.4 0.4

RMSE 0.00% 2.30% 2.29% 1.83% 1.74% 1.15% 0.87%Volatility 15.0% 14.4% 15.6% 14.1% 15.1% 14.3% 14.9%AR(1) 0.20 0.35 0.47 0.16 0.33 0.06 0.23

Beta P to F 0.27 0.43 0.37 0.49 0.42 0.46 0.40Beta F to X 0.9 0.9 0.8 0.8 0.6 0.6

RMSE 0.00% 2.31% 2.30% 1.82% 1.73% 1.15% 0.87%Volatility 15.0% 14.4% 15.6% 14.1% 15.1% 14.3% 15.0%AR(1) 0.20 0.35 0.47 0.17 0.34 0.06 0.23

Beta P to F 0.45 0.48 0.42 0.61 0.52 0.58 0.51Beta F to X 0.9 0.9 0.9 0.9 0.7 0.7

RMSE 0.00% 2.31% 2.29% 1.83% 1.74% 1.16% 0.87%Volatility 15.0% 14.4% 15.6% 14.1% 15.0% 14.3% 15.0%AR(1) 0.20 0.35 0.47 0.16 0.32 0.06 0.23

Beta P to F 0.62 0.53 0.46 0.67 0.59 0.77 0.67Beta F to X 0.9 0.9 0.9 0.9 0.9 0.9

RMSE 0.00% 2.30% 2.29% 1.82% 1.73% 1.16% 0.87%Volatility 15.0% 14.5% 15.7% 14.1% 15.0% 14.3% 14.9%AR(1) 0.20 0.36 0.48 0.17 0.32 0.06 0.22

Beta P to F 0.79 0.52 0.45 0.69 0.62 0.83 0.73Beta F to X 0.7 0.7 0.8 0.8 0.9 0.9

% of Remaining P Variance Explained by XTrue

ValueDimension30 60 90

16

Results: High AR; Minimum Beta F to X

CL MM CL MM CL MMRMSE 0.00% 1.06% 0.79% 1.16% 0.60% 1.26% 0.30%

Volatility 15.0% 13.4% 17.0% 12.0% 16.1% 11.6% 15.2%AR(1) 0.70 0.58 0.83 0.39 0.77 0.26 0.72

Beta P to F 0.10 0.04 0.03 0.04 0.03 0.05 0.03Beta F to X 0.0 0.0 0.0 0.0 0.0 0.0

RMSE 0.00% 1.08% 0.79% 1.17% 0.60% 1.26% 0.30%Volatility 15.0% 13.3% 17.0% 12.1% 16.0% 11.5% 15.2%AR(1) 0.70 0.57 0.83 0.39 0.77 0.26 0.72

Beta P to F 0.16 0.06 0.05 0.07 0.05 0.15 0.09Beta F to X 0.0 0.0 0.0 0.0 0.2 0.2

RMSE 0.00% 1.06% 0.79% 1.17% 0.60% 1.28% 0.30%Volatility 15.0% 13.4% 17.1% 12.0% 16.1% 11.6% 15.2%AR(1) 0.70 0.59 0.84 0.39 0.78 0.26 0.72

Beta P to F 0.21 0.08 0.06 0.09 0.07 0.26 0.16Beta F to X 0.0 0.0 0.0 0.0 0.4 0.4

RMSE 0.00% 1.06% 0.79% 1.18% 0.60% 1.28% 0.30%Volatility 15.0% 13.3% 17.0% 12.1% 16.1% 11.5% 15.2%AR(1) 0.70 0.58 0.83 0.39 0.78 0.25 0.72

Beta P to F 0.27 0.10 0.08 0.22 0.14 0.36 0.22Beta F to X 0.0 0.0 0.3 0.3 0.6 0.6

RMSE 0.00% 1.07% 0.79% 1.18% 0.60% 1.26% 0.30%Volatility 15.0% 13.3% 17.1% 12.1% 16.0% 11.6% 15.2%AR(1) 0.70 0.58 0.83 0.39 0.77 0.26 0.72

Beta P to F 0.32 0.20 0.14 0.34 0.21 0.46 0.28Beta F to X 0.3 0.3 0.6 0.6 0.8 0.8

% of Remaining P Variance Explained by XTrue

ValueDimension30 60 90

17

Results: High AR; Maximum Beta F to X

CL MM CL MM CL MMRMSE 0.00% 1.08% 0.79% 1.17% 0.60% 1.25% 0.30%

Volatility 15.0% 13.3% 17.0% 12.0% 15.9% 11.5% 15.2%AR(1) 0.70 0.58 0.83 0.39 0.77 0.26 0.72

Beta P to F 0.10 0.27 0.15 0.31 0.18 0.24 0.14Beta F to X 0.9 0.9 0.8 0.8 0.5 0.5

RMSE 0.00% 1.06% 0.79% 1.17% 0.60% 1.27% 0.30%Volatility 15.0% 13.3% 17.0% 12.1% 16.0% 11.6% 15.2%AR(1) 0.70 0.58 0.83 0.39 0.77 0.26 0.72

Beta P to F 0.16 0.28 0.17 0.37 0.22 0.35 0.21Beta F to X 0.9 0.9 0.9 0.9 0.7 0.7

RMSE 0.00% 1.06% 0.79% 1.18% 0.60% 1.26% 0.30%Volatility 15.0% 13.3% 16.9% 12.1% 16.0% 11.6% 15.2%AR(1) 0.70 0.58 0.83 0.38 0.77 0.26 0.72

Beta P to F 0.21 0.30 0.18 0.39 0.23 0.41 0.24Beta F to X 0.9 0.9 0.9 0.9 0.8 0.8

RMSE 0.00% 1.07% 0.79% 1.16% 0.60% 1.26% 0.30%Volatility 15.0% 13.3% 17.0% 12.0% 15.9% 11.6% 15.1%AR(1) 0.70 0.58 0.83 0.40 0.77 0.26 0.71

Beta P to F 0.27 0.32 0.20 0.41 0.25 0.48 0.29Beta F to X 0.9 0.9 0.9 0.9 0.9 0.9

RMSE 0.00% 1.06% 0.79% 1.18% 0.60% 1.27% 0.30%Volatility 15.0% 13.4% 17.2% 12.0% 16.0% 11.6% 15.2%AR(1) 0.70 0.58 0.84 0.38 0.77 0.26 0.71

Beta P to F 0.32 0.30 0.19 0.44 0.26 0.50 0.30Beta F to X 0.7 0.7 0.9 0.9 0.9 0.9

% of Remaining P Variance Explained by XTrue

ValueDimension30 60 90

18

Summary Findings

• The mixed-frequency VAR/MM model performs better than the best linear unbiased estimator/CL, ceteris paribus, when:– the AR coefficient increases– the explanatory ability of the proxy improves

• The relationship between the factor and the proxy dictates how accurately the beta of the portfolio to the factor will be recovered ceteris paribus, and which interpolator is best– In the case of high AR: the higher the beta of F to X, the more accurate the MFV

beta of P to F will be (CL tends to overestimate); low beta of F to X tends to favor CL

– In the case of low AR: both MM and CL perform similarly

• In some cases, it might not make sense to interpolate at all!– Low AR, low beta of F to X, low explanatory power…both perform terribly if you

care about factor decomposition and recovering AR…

19

Practical Example: HCF (Quarterly)

Stocks BondsPrivate Equity

Real Estate

Portfolio

Stocks 1.00 -0.81 0.84 0.56 0.99Bonds -0.81 1.00 -0.80 -0.58 -0.79

Private Equity 0.84 -0.80 1.00 0.85 0.89Real Estate 0.56 -0.58 0.85 1.00 0.63

Portfolio 0.99 -0.79 0.89 0.63 1.00

2010-2012 Stocks BondsPrivate Equity

Real Estate

Portfolio (45/25/20/10)

Volatility 18.60% 5.07% 7.55% 4.70% 7.63%VaR 95% 30.69% 8.36% 12.45% 7.76% 12.59%TE to 60/40 1.38%Beta to Stocks 1.00 -0.22 0.34 0.10 0.47Beta to Bonds -2.96 1.00 -1.20 -0.44 -1.30FPCTR

Stocks 100.00% 0.00% 47.42% 13.49% 103.95%Bonds 0.00% 100.00% 27.69% 22.41% -5.28%

StressesStocks -50.00% 11.11% -17.21% -5.68% -23.73%Bonds 44.81% -15.00% 18.03% 6.54% 20.67%

8/11 - 9/11Max Drawdown 15.30% 2.79% 5.40% 0.18% 6.35%

Source: Real Estate and Private Equity Index data from Cambridge Associates, LLC. Copyright 2013. All rights reserved. The Cambridge Associates, LLC data may not be copied, used, or distributed without Cambridge’s prior written approval. The data is provided “as is” without any express or implied warranties. Past performance is no guarantee of future results. Stocks and Bonds are represented by Russell 3000 and Barclays US Agg.

20

Practical Example: Interpolation (Monthly)

Stocks BondsPrivate Equity

Real Estate

Portfolio

Stocks 1.00 -0.73 0.90 0.45 0.99Bonds -0.73 1.00 -0.71 -0.19 -0.67

Private Equity 0.90 -0.71 1.00 0.61 0.93Real Estate 0.45 -0.19 0.61 1.00 0.54

Portfolio 0.99 -0.67 0.93 0.54 1.00

2010-2012 Stocks BondsPrivate Equity

Real Estate

Portfolio (45/25/20/10)

Volatility 15.95% 3.68% 6.73% 3.34% 7.19%VaR 95% 26.32% 6.07% 11.11% 5.50% 11.86%TE to 60/40 1.38%Beta to Stocks 1.00 -0.17 0.34 0.13 0.49Beta to Bonds -3.16 1.00 -1.18 -0.33 -1.43FPCTR

Stocks 100.00% 0.00% 81.35% 22.06% 108.27%Bonds 0.00% 100.00% -3.06% -3.95% -9.02%

StressesStocks -50.00% 8.63% -16.93% -5.87% -24.46%Bonds 43.28% -15.00% 14.04% 3.23% 18.96%

8/11 - 9/11 -13.29% 4.58% -5.58% -0.94% -6.19%Max Drawdown 17.75% 2.79% 5.58% 3.24% 7.23%

Source: Real Estate and Private Equity Index data from Cambridge Associates, LLC. Copyright 2013. All rights reserved. The Cambridge Associates, LLC data may not be copied, used, or distributed without Cambridge’s prior written approval. The data is provided “as is” without any express or implied warranties. Past performance is no guarantee of future results. Stocks and Bonds are represented by Russell 3000 and Barclays US Agg.

21

Open Questions

• When do we decide to use the interpolators, and which one?– Depends what we care about…

• How do we choose the appropriate proxies?

• Future methodological questions/issues:– How to incorporate higher order AR?– How are constraints introduced into the MFV?– Are there any benefits to interpolating multiple low-frequency series?– How effective forecasters are these methodologies?

22

Thank You

Questions?