Momentum, simulation, and factor timing: Insights from ... · Simulated Method of Moments (series...

NA10161 10/2019

Momentum, simulation, and factor timing: Insights from heterogeneous agent modelsKenneth BlayHead of Thought Leadership – Investment SolutionsInvesco

Northfield 31st Annual Research ConferenceOctober 25, 2019

George WharfQuantitative Research AnalystNeuron Capital

Robert HillmanChief Executive OfficerNeuron Capital

For Northfield Annual Research Conference Attendees Only - NA10161

NA10161 10/2019

A brief history of agent-based models (ABM) Heterogeneity, extrapolative beliefs, and noise Myth busting Why the increasing interest in ABM today?

A simple model Overview Recent advances in estimation and calibration Contributions from different agent-types

Practical applications: Long-horizon simulation Return forecasting and scenario analysis Insights into factor timing

Overview

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Observations on trading floors suggested people extrapolate a lot. And surveys said that they did. What happens when we insert this reality into our models? - See e.g. Beja & Goldman (1986), and Frankel & Froot (1989)We know individuals are not alike, and without heterogeneity there would be no trade. But analytical models with heterogeneity are hard.- See e.g. Kenneth Arrow (2004)Even those considered ‘orthodox’ and ‘equilibrium’ have explored the concepts of the interplay of heterogeneous agents. Kim & Markowitz used simulation in 1988….- See e.g. Fischer Black (1986) and Kim & Markowitz (1988)

A brief history of agent-based modelsHeterogeneity

Source: Beja, A., Goldman, M.B. (1980) On the dynamic behaviour of prices in disequilibrium. The Journal of Finance. 35 (2). Pp 235-248. Jeffrey A. Frankel and Kenneth A. Froot. Chartists, Fundamentalists, and Trading in the Foreign Exchange Market Source: The American Economic Review, Vol. 80, No. 2, Papers and Proceedings of the Hundred and Second Annual Meeting of the American Economic Association (May, 1990),pp. 181-185Kenneth Arrow, In: D. Colander, R.P.F. Holt and J. Barkley Rosser (eds.), The Changing Face of Economics. Conversations with Cutting Edge Economists. The University of Michigan Press, Ann Arbor, 2004, p. 301.)Black, Fischer (1986) Noise. The Journal of Finance. 41(3), pp 528-543.G. Kim and H. M. Markowitz (1989). Investment rules, margin and market volatility. Journal of Portfolio Management, 16:45–52.3

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A brief history of agent-based modelsExtrapolative beliefs – Overshooting the overshoot

Source: Jeffrey A. Frankel and Kenneth A. Froot. Chartists, Fundamentalists, and Trading in the Foreign Exchange Market Source: The American Economic Review, Vol. 80, No. 2, Papers and Proceedings of the Hundred and Second Annual Meeting of the American Economic Association (May, 1990),pp. 181-185

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CPI-adjusted Dollar against G-10

countries(right scale)

Long-term Real Interest rate differential(left scale)

Index, March 1973=100(quarterly)

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Four myths about ABM:1. Great toys but not institutionally realistic

OFR, Berndt et al (2016)

2. They assume daft trading modelsKim & Markowitz (1988)Hillman / Neuron (2017)

3. You cannot estimate, calibrate, or validate themHommes (2006) – nonlinear estimationBraun-Munzinger (2016, Bank of England) – estimation + calibration + method of simulated momentsPlatt (2019); Barde (2017); Lamperti et al (2017); Guerini et al (2017) – Bayesian + model selection

4. You cannot forecastBoswijk et al (2007) - forecastingBaranova et al (2017) - backtesting

A brief history of agent-based modelsMyth busting

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There is a demand from investors and their advisors to consider how to generate future scenarios for markets, because: Recognition of the limits and dangers of back-testing Awareness of path-dependence and other characteristics of financial time series And that financial products themselves are the market – concerns over crowding and endogenous risk

particularly from algorithmic products

And since the crisis… Limitations of representative rational agent models were revealed Policy-makers and regulators have pushed for new approaches Heterogeneity, learning and market structures matter

A brief history of agent-based modelsWhy the increasing interest in ABM today?

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A simple model

Source: Invesco, Neuron Capital, Online Data - Robert Shiller

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Our simple model will explore some of these ideas in the context of an equity valuation model using the cyclically adjusted price-to-earnings ratio or CAPE Here CAPE(t) = P(t) / Smoothed Earnings(t-1)

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A simple modelBlack’s factor of “2”

Source: Black, Fischer (1986) Noise. The Journal of Finance. 41(3), pp 528-543.

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Over/under valuation ratio

Fischer Black posited that an efficient market might be one in which price is within a factor of 2 of value, i.e. more than half and less than twice the value.

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A simple model


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If we assume there is a long-term average value of CAPE (V) that valuationsfluctuate around

Then we can construct a “fair price” 𝐹𝐹𝐹𝐹 𝑡𝑡 = 𝑉𝑉𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶

∗ 𝐹𝐹(𝑡𝑡)

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Following Beja and Goldman (1980) a number of models were formalized with a general form:

𝑝𝑝𝑡𝑡+1 − 𝑝𝑝𝑡𝑡 = λ �𝑖𝑖=1

𝑁𝑁𝐷𝐷(𝑖𝑖, 𝑡𝑡) + 𝜖𝜖𝑡𝑡

𝜆𝜆 is similar to “Kyle’s Lamba” – market impact The sum is over agent’s demand and represents a net order imbalance 𝜖𝜖 is often interpreted as noise trader demand

A simple model

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Source: Beja, A., Goldman, M.B. (1980) On the dynamic behaviour of prices in disequilibrium. The Journal of Finance. 35 (2). Pp 235-248.

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Price: 𝑝𝑝𝑡𝑡+1 − 𝑝𝑝𝑡𝑡 = 𝜅𝜅 𝑣𝑣𝑡𝑡 − 𝑝𝑝𝑡𝑡 + 𝛽𝛽 tanh 𝛾𝛾 𝑚𝑚𝑡𝑡 + 𝜖𝜖𝑡𝑡

Value Traders: 𝜅𝜅 𝑣𝑣𝑡𝑡 − 𝑝𝑝𝑡𝑡Value: 𝑣𝑣𝑡𝑡+1 = 𝑣𝑣𝑡𝑡 +𝑔𝑔 + 𝜂𝜂𝑡𝑡+1

Momentum* (Extrapolators): 𝛽𝛽 tanh 𝛾𝛾 𝑚𝑚𝑡𝑡

EWMA: 𝑚𝑚𝑡𝑡 = 1 − 𝛼𝛼 𝑚𝑚𝑡𝑡−1+ 𝛼𝛼 𝑝𝑝𝑡𝑡− 𝑝𝑝𝑡𝑡−1

Noise traders: 𝜖𝜖𝑡𝑡

In 1992 Carl Chiarella formalised the approach suggested by Beja and Goldman (1980).The set-up above follows Majewski et al (2018).

A simple model

Source: Invesco, Neuron Capital, Online Data - Robert Shiller Chiarella, C. (1992) The dynamics of speculative behaviour. Annals of Operations Research. 37 (1), pp. 101-123.* Momentum in this context refers to time-series momentum not cross-sectional momentum.

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Insight that small (n-type) models are nonlinear econometric models We can draw on a large earlier literature: Tong, Terasvirta, Granger, etc. Today much larger models can now be estimated and/or calibrated: Calibration with non-price data (e.g. surveys, flows, Braun-Munziger BoE 2016) Simulated Method of Moments (series of papers by Cars Hommes & others) Bayesian (Majewski et al, CFM, 2018)

Still an art but…where deep-learning was 10 years ago

A simple modelRecent advances in estimation and calibration

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Source: Tong, H. (1993) A Non-linear Time Series: A Dynamical System Approach. Oxford Statistical Science Series. 6.Granger, C., Terasvirta, T. (1993) Modelling Nonlinear Economic Relationships. Oxford University Press.Hamilton, J. (1994) Time Series Analysis. Princeton University Press

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A simple modelValue and trend influence on prices

Source: Invesco, Neuron Capital

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Over/under valuation

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Value effects kick in around +/10% and increasingly so as price deviates further Trend influence grows as the price trends more but eventually saturates.

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A simple modelOptimal extrapolator lookback = 6 months


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This chart shows a partial cost function holding other parameters constant.The “best” alpha decay parameter is 0.833 which is a decay lag of 6 months.

Extrapolator lookback (months)

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A simple modelOptimal extrapolator response function


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The black line shows the weights as estimated by our empirical modelThe red circles show the weights from Greenwood & Shleifer (2014) where the average alpha across 7 surveys is 0.56 on quarterly data – confession – the 0.56 is not precisely estimated sd = 0.21 but still….

Quarters

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A simple modelMomentum and value influences


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A simple modelSimulation samples

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Value heavy simulation Momentum heavy simulation

Source: Invesco, Neuron Capital, Online Data - Robert Shiller. The fair value series is constructed using the log ‘Real Total Return Price Index’ and the ‘Cyclically Adjusted Total Return Price Earning’s Ratio’ from Shiller’s website. The Value Heavy and Momentum Heavy series are two runs from the simulation model.

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A simple modelSimulation samples

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Historical returns1972-2018 Simulated returns

Source: Invesco, Neuron Capital, Online Data - Robert Shiller. The fair value series is constructed using the log ‘Real Total Return Price Index’ and the ‘Cyclically Adjusted Total Return Price Earning’s Ratio’ from Shiller’s website. The Value Heavy and Momentum Heavy series are two runs from the simulation model. The actual series on the left chart is the log ‘Real Total Return Price Index’ from Shiller’s website.

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Investment outcomes Short- and long-horizon risk Investment consumption

Practical application:Long-horizon simulation

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Long-horizon simulationsCommon practice

Parametric simulation requires a model to be estimated on historical data, and then data is generated from that model. Bootstrapping simulation is a means of generating possible future price or return scenarios by resampling single returns from the historical data set. Block bootstrapping resamples “blocks” of returns from the historical data set.

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Portfolio and asset return simulations are used for a variety of purposes including: Risk management Portfolio construction Multi-period portfolio (target date) evaluation Financial planning

Common simulation methods

Simulation methodDistributionassumption

Incorporatesauto

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Parametric Lognormal(most common) No No

Bootstrapping (i.i.d) Empirical No No

Block bootstrapping Empirical Yes No

However, simulation methods often represent a trade-off between ease of implementation and realism in incorporating well-known asset dynamics. These trade-offs have implications for the practical application of these methods in providing effective decision support.

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Long-horizon simulations…and expectations for investment outcomes

Source: Invesco, Neuron Capital, Online Data - Robert Shiller Historical estimates of average monthly (mean) return and standard deviation are 0.22% and 5.40% based on S&P 500 real equity returns for the period Dec 1927 through August 2019; bootstrapped returns are drawn from the same historical period. The blue lines indicate (simulated) 10th , 50th and 90th percentile confidence bands.

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BootstrappingMonthly returns (i.i.d.)

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Long-horizon simulations…and expectations for investment outcomes

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We use the variance ratio to assess short- and long-horizon risk implications for different simulation methods.

The intuition is related to the common practice of scaling volatility by the square root of time (σ × 𝑇𝑇)

Are stocks less volatile in the long run?

An old question that has implications for equity allocations and rules for target-date funds, etc.

A variance ratio test is one way to explore this. The idea is to measure how ‘diffusive’ a time series is.

It is closely linked to the Hurst exponent and Mandlebrot’s rescaled range statistic

For a recent application see Pastor & Stambaugh (2012)

Long-horizon simulationsShort- and long-horizon risk

Source: Campbell, Lo, and McKinlay (1997), Pastor and Stambaugh (2012)

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𝑉𝑉𝑅𝑅 𝑘𝑘 = variance of k-period returnsk * variance of 1-period returns

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Long-horizon simulations…and expectations for investment risk

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Months Months

Parametric (lognormal)Using historical estimates

BootstrappingMonthly returns (i.i.d.)

Varia

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Source: Invesco, Neuron Capital, Online Data - Robert Shiller Historical estimates of average monthly (mean) return and standard deviation are 0.22% and 5.40% based on S&P 500 real equity returns for the period Dec 1927 through August 2019; bootstrapped returns are drawn from the same historical period. The blue lines indicate (simulated) 10th , 50th and 90th percentile confidence bands..

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Long-horizon simulations…and expectations for investment risk

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Months

Block bootstrapping12-month blocks

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Heterogenous Agent Model

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We use the coverage ratio to assess expectations provided by different simulation methods.

𝐶𝐶𝐶𝐶𝑣𝑣𝐶𝐶𝐶𝐶𝐶𝐶𝑔𝑔𝐶𝐶 𝐶𝐶𝐶𝐶𝑡𝑡𝑖𝑖𝐶𝐶 = 𝐶𝐶𝑡𝑡 = �𝑌𝑌𝑡𝑡 𝐿𝐿

𝑌𝑌 = the number of years of withdrawals sustained by a strategy, both during and after the retirement period

𝐿𝐿 = the length of the retirement period considered

A kinked utility function is used to express asymmetric investor preferences for achieving consumption objectives:

𝐶𝐶1−𝛾𝛾 − 11 − 𝛾𝛾

for 𝐶𝐶 ≥ 1

11−𝛾𝛾 − 11 − 𝛾𝛾

− 𝜆𝜆 1 − 𝐶𝐶 for 𝐶𝐶 < 1

𝛾𝛾 = linear penalty coefficient𝜆𝜆 = risk aversion coefficient

Long-horizon simulationsConsumption expectations

Source: Invesco, Estrada, Javier and Kritzman, Mark, Toward Determining the Optimal Investment Strategy for Retirement (December 14, 2018).

Coverage ratio kinked utility(𝛾𝛾 = 0.9999, 𝜆𝜆 = 10)

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As 𝛾𝛾 approaches 1, utility equals the natural logarithm of the coverage ratio; hence, a 𝛾𝛾 value of 0.9999 effectively implies the use of a log-wealth utility function for the coverage ratio above 1.

𝑈𝑈 𝐶𝐶 =

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Long-horizon simulations…and expectations for investment consumption

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Common Simulation Methods Heterogenous Agent Model

Source: Invesco, Neuron Capital, Online Data - Robert Shiller The distribution of coverage ratios are presented using 2,000 sets of simulated monthly returns over 30+ year periods; Historical estimates of average monthly (mean) return and standard deviation are 0.54% and 4.46% based on real total S&P 500 equity returns for the period Jan 1927 through August 2019; bootstrapped returns are drawn from the same historical period; starting capital is $1,000; the initial withdrawal rate is 4%.

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Source: Invesco, Neuron Capital, Online Data - Robert Shiller The distribution of coverage ratio utilities are presented using 2,000 sets of simulated monthly returns over 30+ year periods; Historical estimates of average monthly (mean) return and standard deviation are 0.54% and 4.46% based on real total S&P 500 equity returns for the period Jan 1927 through August 2019; bootstrapped returns are drawn from the same historical period; starting capital is $1,000; the initial withdrawal rate is 4%.

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% ofcoverage

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Simulation type Mean 1% 5% 25% 50% 75% 95% 99%Historical 3.7 5.0 0.9 1.1 1.8 3.3 5.2 7.8 9.1Lognormal 4.8 8.0 0.6 0.8 1.8 3.2 5.8 14.7 23.9Bootstrap (i.i.d.) 5.0 7.0 0.6 0.9 1.9 3.5 6.2 13.8 29.4Block bootstrap 5.8 14.0 0.4 0.6 1.5 3.1 6.5 20.5 37.0ABM 3.4 4.0 0.7 1.1 2.0 3.0 4.2 7.0 9.8Values with a greater than a 25% difference from historical values are highlighted in red.

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Common Simulation Methods Heterogenous Agent Model

Source: Invesco, Neuron Capital, Online Data - Robert Shiller The distribution of coverage ratio utilities are presented using 2,000 sets of simulated monthly returns over 30+ year periods; Historical estimates of average monthly (mean) return and standard deviation are 0.54% and 4.46% based on real total S&P 500 equity returns for the period Jan 1927 through August 2019; bootstrapped returns are drawn from the same historical period; starting capital is $1,000; the initial withdrawal rate is 4%.

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Conditional forecasting Scenario analysis

Practical application:Return forecasting and scenario analysis

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How can we use ABM in a return forecasting (CMA) process?

Consider the following return forecasting process:

Expected return = yield + growth + valuation Yield = historical yield Growth = a + b*inflation +c*risk free Valuation is often treated as a deterministic “value” path

However, in reality we are unsure of how valuations might develop. We can use ABM to factor in this uncertainty Value: 𝑣𝑣𝑡𝑡+1 = 𝑣𝑣𝑡𝑡 +𝑔𝑔 + 𝜂𝜂𝑡𝑡+1

Return forecastingConditional capital market assumptions (CMAs)

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Return forecastingReturn outcomes depend on initial market valuation


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Charts show reversion to fair value from different initial levels of P/E.

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Return forecastingDistribution of price/return outcomes for different initial valuation


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Forward return scenariosHistorical coverage ratios (1927-1988)

Source: Invesco, Neuron Capital, Online Data - Robert Shiller. The historical coverage ratio is constructed using the log ‘Real Total Return Price Index’ from Shiller’s website. Each data point shows the coverage ratio for a 30 year retirement period beginning in the December of each year. Starting capital is $1,000; the initial withdrawal rate is 4%.

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Forward return scenariosCAPE and coverage ratios (1927-1988)

Source: Invesco, Neuron Capital, Online Data - Robert Shiller. The chart shows simulated coverage ratios using the log ‘Real Total Return Price Index’, versus the ‘Cyclically Adjusted Total Return Price Earning’s Ratio’ from Shiller’s website. Each data point shows the coverage ratio for a 30 year retirement period beginning in the December of each year. Starting capital is $1,000; the initial withdrawal rate is 4%.

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Conditional return forecasts…and expectations for investment consumption

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Conditional coverage ratios Conditional coverage ratio utility

Source: Invesco, Neuron Capital, Online Data - Robert Shiller The distribution of coverage ratios and utilities are presented using 2,000 sets of simulated monthly returns over 30+ year periods; starting capital is $1,000; the initial withdrawal rate is 4%.

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Practical application:Insights into factor timing

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“The relation between price-earnings ratios and subsequent returns appears to be moderately strong, though there are questions about its statistical significance, since there are only about twelve non-overlapping ten-year intervals in the 119 years’ worth of data.” - Robert Shiller (2000)

The short history of data and weak theory makes testing very hard. - See the discussion by Cliff Asness et al. (2017)

Can our model shine any insights into this?

Insights into Factor TimingIrrational Exuberance

Source: Shiller, Robert (2000) Irrational Exuberance. Princeton University Press. New Jersey.Asness, C., Ilmanen, A., Maloney, T. (2017) Market Timing: Sin a Little. Resolving the Valuation Timing Puzzle. Journal of Investment Management, Vol 15, No. 3. pp. 23-40

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Insights into factor timingHistorical and simulated CAPE


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Historical1901-2009

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Momentum: 𝑆𝑆𝑖𝑖𝑔𝑔𝑆𝑆𝐶𝐶𝑆𝑆(𝑡𝑡) = 12𝑀𝑀 𝑅𝑅𝑅𝑅𝑡𝑡𝑅𝑅𝑅𝑅𝑅𝑅(𝑡𝑡) – 𝑀𝑀𝑅𝑅𝑀𝑀𝑖𝑖𝑀𝑀𝑅𝑅 12𝑀𝑀 𝑅𝑅𝑅𝑅𝑡𝑡𝑅𝑅𝑅𝑅𝑅𝑅𝑠𝑠𝑡𝑡𝑀𝑀𝑅𝑅𝑀𝑀𝑀𝑀𝑅𝑅𝑀𝑀 𝑀𝑀𝑅𝑅𝑑𝑑 (12𝑀𝑀 𝑅𝑅𝑅𝑅𝑡𝑡𝑅𝑅𝑅𝑅𝑅𝑅𝑠𝑠)

Value: 𝑆𝑆𝑖𝑖𝑔𝑔𝑆𝑆𝐶𝐶𝑆𝑆(𝑡𝑡) = − 𝐶𝐶/𝐶𝐶(𝑡𝑡) – 𝑀𝑀𝑅𝑅𝑀𝑀𝑖𝑖𝑀𝑀𝑅𝑅 𝐶𝐶/𝐶𝐶𝑠𝑠𝑡𝑡𝑀𝑀𝑅𝑅𝑀𝑀𝑀𝑀𝑅𝑅𝑀𝑀 𝑀𝑀𝑅𝑅𝑑𝑑 (𝐶𝐶/𝐶𝐶)

Strategy Return: 𝑅𝑅𝐶𝐶𝑡𝑡𝑅𝑅𝐶𝐶𝑆𝑆 𝑡𝑡 + 1 = 𝑆𝑆𝑖𝑖𝑔𝑔𝑆𝑆𝐶𝐶𝑆𝑆 𝑡𝑡 ∗ [𝐹𝐹 𝑡𝑡 + 1 − 𝐹𝐹(𝑡𝑡)]

We create signals that are z-scores of either momentum or value Use full sample estimates of median & standard deviation on the real data

and a rolling 60-year window for the simulated data Gross of any costs (e.g. transaction costs, fees, etc.)

Insights into factor timingExploring some simple strategies

Source: Asness, C., Ilmanen, A., Maloney, T. (2017) Market Timing: Sin a Little. Resolving the Valuation Timing Puzzle. Journal of Investment Management, Vol 15, No. 3. pp. 23-40

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Insights into factor timingExpanding the data set

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Strategy return-to-risk ratiosHistorical data (12/1927 – 9/2018)

Strategy return-to-risk ratiosSimulated data (30,000 months)

Source: Invesco, Neuron Capital, Online Data - Robert Shiller. The strategy simulation on the historical data (left table) uses the log ‘Real Total Return Price Index’ and the ‘Cyclically Adjusted Total Return Price Earning’s Ratio’ from Shiller’s website. The strategy simulation on the simulated data (right table) uses data generated from the simulation model.

QuartileMeanP/E

BuyHold Value Momentum

Q1 12.6 0.2 0.3 0.1

Q2 18.1 0.2 1.0 0.2

Q3 23.2 0.7 0.4 0.2

Q4 31.4 0.7 -0.4 0.3

All - 0.4 0.1 0.1

Correlation (Value, Trend): -0.55

QuartileMeanP/E

BuyHold Value Momentum

Q1 15.0 0.1 0.4 0.2

Q2 17.8 0.4 0.9 0.4

Q3 21.1 0.4 0.3 0.6

Q4 27.0 0.6 -0.3 0.6

All - 0.4 0.1 0.5

Correlation (Value, Trend): -0.26

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Insights into factor timingExpanding the data set

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Average positions for strategies per P/E quantile

Source: Invesco, Neuron Capital, Online Data - Robert Shiller. The strategy simulation on the historical data (left table) uses the log ‘Real Total Return Price Index’ and the ‘Cyclically Adjusted Total Return Price Earning’s Ratio’ from Shiller’s website. The strategy simulation on the simulated data (right table) uses data generated from the simulation model.

Historical Simulated

Quartile Value Momentum Value Momentum

Q1 13.1 -5.8 10.3 -4.9

Q2 4.4 -3.0 3.2 0.1

Q3 -3.5 0.7 -5.2 1.5

Q4 -16.3 2.5 -19.6 2.8

Strategies are both long (or short) simultaneously 40% of the time in the real data, 41% in the simulated.

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Can we add some information from slightly more structure rather than relying on empirical series alone?

Recent work by Polk et al (2018) on factor timing and tilts uses much more contemporaneous and forward-looking data on which to condition and looks promising

We believe there is value in trying to develop these ideas further by adding in more realism, cross market and a macro-finance components along the lines seen in policy-maker heterogeneous macro models, etc.

This is analogous to trying to estimate extreme rainfall levels using the last 100 years versus simulating a model that incorporates some physics

Insights into factor timingWhere does this leave us?

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The recognition of the limits of representative rational agent models stimulated new techniques and micro data and is creating a new set of tools for investors to simulate, explore, and evaluate investment opportunities and risks. We demonstrated how: A simple model with momentum and value agents allowed for simulating more realistic market dynamics How this information could be applied to address real-world investor challenges

Heterogenous agent modelling provides capabilities not easily accomplished through traditional methods. We presented several practical use cases: Long-horizon simulation with more realistic return and risk dynamics Return forecasting conditioned on market variables Scenario analysis

More detailed models may offer opportunities for greater insights as evidenced by a growing literature on understanding liquidity; redemptions; flows; structural issues (e.g.; passive/active debate; systemic risk; contagion/networks; algorithm execution simulators). We used a simple model to provide insights regarding factor timing.

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Conclusion

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Asness, C., Ilmanen, A., Maloney, T. (2017) Market Timing: Sin a Little. Resolving the Valuation Timing Puzzle. Journal of Investment Management, Vol 15, No. 3. pp. 23-40

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Barberis, N., Greenwood, R., Jin, L., Shleifer, A. (2018) Extrapolation and Bubbles. Journal of Financial Economics. 129. Pp. 203-227

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Berndt,D., Boogers, D., McCart, J. (2017) ‘Agent-based models of the corporate bond market’ https://idscblog.files.wordpress.com/2016/03/donald-j-berndt-paper.pdf

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Bookstaber, Richard (2017) ‘The End of Theory’, Princeton University Press

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Braun-Munziger, K, Zijun, L, and Turrell A (2016) “An agent-based model of dynamics in corporate bond trading” Bank of England Working Paper No. 592

https://www.bankofengland.co.uk/working-paper/2016/an-agent-based-model-of-dynamics-in-corporate-bond-trading

Brock, W.A., Hommes, C.H. (1998). Heterogeneous beliefs and routes to chaos in a simple asset pricing model. Journal of Economics Dynamics and Control 22 (8-9), pp. 1235-1274.

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References

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https://idscblog.files.wordpress.com/2016/03/donald-j-berndt-paper.pdf

https://ssrn.com/abstract=1269050

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Grazzini, J., and Richiardi, M., 2015. "Estimation of ergodic agent-based models by simulated minimum distance," Journal of Economic Dynamics and Control, Elsevier, vol. 51(C), pages 148-165.

Greenwood, R., Jin, L., and Shleifer, A., "Extrapolation and Bubbles", Journal of Financial Economics 129, 203-227, August 2018, also see more generally http://faculty.som.yale.edu/nicholasbarberis/

Greenwood, R.and Shleifer, A., 2014. “Expectations of Returns and Expected Returns.” Review of Financial Studies 27 (3): 714-746

Hommes, C. (2013) Behavioural Rationality and Heterogeneous Expectations in Complex Economic Systems. Cambridge

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Lamperti, F., Roventini, A., Sani, A., (2017), Agent-Based Model Calibration using Machine Learning Surrogateshttp://www.isigrowth.eu/wp-content/uploads/2017/04/working_paper_2017_05.pdf

Lux, Thomas and Zwinkels, Remco C. J., Empirical Validation of Agent-Based Models (March 2, 2017). Available at SSRN: https://ssrn.com/abstract=2926442 or http://dx.doi.org/10.2139/ssrn.2926442

Majewski, A., Ciliberti, S., Bouchaud, J-P. (2018) Co-existence of Trend and Value in Financial Markets: Estimating an Extended Chiarella Model. Capital Fund Management. https://arxiv.org/abs/1807.11751

Shiller, R. (2000) Irrational Exuberance. Princeton University Press. New Jersey.

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References

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http://faculty.som.yale.edu/nicholasbarberis/

http://www.isigrowth.eu/wp-content/uploads/2017/04/working_paper_2017_05.pdf

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For Northfield Annual Research Conference attendees only. This presentation is for informational and educational purposes only. All material presented iscompiled from sources believed to be reliable and current, but accuracy cannot be guaranteed. This is not to be construed as an offer to buy or sell any financialinstruments and should not be relied upon as the sole factor in an investment making decision. As with all investments there are associated inherent risks. Thisshould not be considered a recommendation to purchase any investment product. This does not constitute a recommendation of any investment strategy for aparticular investor. Investors should consult a financial professional before making any investment decisions if they are uncertain whether an investment is suitablefor them. Please obtain and review all financial material carefully before investing. The opinions expressed are those of the presenters, are based on currentmarket conditions and are subject to change without notice. These opinions may differ from those of other Invesco investment professionals. Any simulation shownin this presentation is hypothetical (not real) and is shown for informational purposes only. It may not be possible to replicate these results and there is noguarantee that any future portfolio would achieve the same results as the simulation. The simulation results do not reflect the deduction of management fees.

Invesco Advisers, Inc. is an investment adviser that provides investment advisory services and does not sell securities. Neuron Capital is investment and riskmanagement consultancy. Invesco Advisers, Inc. and Neuron Capital are not affiliated. Invesco worked with Neuron Capital in developing this content and iscontinuing to advance research in this area.

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