Meucci_FactorsOnDemand

8/4/2019 Meucci_FactorsOnDemand

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FACTORS ON DEMAND

Optimized Flexible Factors for Risk

Estimation and Attribution

Attilio Meucci

http://ssrn.com/abstract=1565134


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EXECUTIVE SUMMARY

TRADITIONAL MULTI-PURPOSE FACTOR MODELS

FACTORS ON DEMAND THEORY

FACTORS ON DEMAND APPLICATIONS

REFERENCES


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EXECUTIVE SUMMARY




REFERENCES


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Attilio MeucciFACTORS ON DEMAND Executive Summary

Risk Estimation vs. Risk Attribution

Identify Risk Factorsto impose structure on estimate of largemultivariate market distribution

Compute overall portfolio risk (StandardDeviation and Tail Risk) from market

distribution Goal: maximize predictive power

Define Attribution Factors

Allocate overall portfolio risk obtained fromRisk Estimation to Attribution Factors

Goal: maximize interpretability andpracticality for hedging/trading

Risk Estimation Risk Attribution


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Attilio MeucciFACTORS ON DEMAND

Traditional Factor Models: same or similar factors for Risk Estimation and Attribution








Executive Summary

Tradit ional Mult i-Purpose Factor Models

Suboptimal choice of systematic factors- Suboptimal statistical properties for risk estimation- Risk attribution factors are not most practical for hedging/interpretation

- Not portfolio-specific estimation/attribution Inflexible choice of loadings (betas)

- Rigid bottom-up aggregation (beta of portfolio is sum of beta of securities)- Rigid maximization target (R-square)

- Rigid unconstrained maximization (CAPM beta) Incorrect modeling of non-linear products/derivatives

FACTORS ON DEMAND E ti S


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Factors On Demand: different factors for Risk Estimation and Risk Attribution








Executive Summary

Factors On Demand

Flexible choice of factors: dominant, instead of systematic- Ideal statistical properties for risk estimation- Ideal hedging/interpretation properties for risk attribution

- Portfolio-specific estimation/attribution Flexible choice of loadings (betas)

- Flexible top-down aggregation

- Flexible maximization target (R-square, CVaR, etc.)- Flexible constrained maximization (best pool, long-only, etc.)

Consistent across non-linear products/derivatives (full conditional distribution of )


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EXECUTIVE SUMMARY




REFERENCES

FACTORS ON DEMAND

Tradit ional Mult i Purpose Factor Models


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5. Potential attribution factors

6. Security-level attribution

7. Portfolio risk attribution: bottom up


3. Aggregation

4. Portfolio risk estimation

Risk AttributionRisk Estimation

= security return

= idiosyncratic shock

= loading= systematic factor

1. Stocks return estimation

Normal assumption


Estimation

Traditional Risk Estimation Techniques

- Regression analysis

Risk Estimation Rationales

- Estimate the joint distribution of security returns, imposing structure with factor model

FACTORS ON DEMAND



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3. Aggregation



= security return




Normal assumption


Estimation

Traditional Risk Estimation Techniques

- Regression analysis

- Dimension reduction- Parametric assumptions

Risk Estimation Rationales

- Estimate the joint distribution of security returns, imposing structure with factor model

- Use the portfolio positions wto determine aggregated portfolio return distribution- Define and compute risk: standard deviation, Value at Risk (tail risk), etc.

FACTORS ON DEMAND



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3. Aggregation



= security return




Normal assumption

: govt curve changes : log-return of underlying

: log-return of implied vol.: spread changes

: discount formula : Black-Scholes formulaExample: bond


Estimation

Traditional modeling of non-linear securities

- For non-equity securities such as bonds and derivatives, the returns Rare not invariants, i.e.

they do not behave identically and independently across time

Example: option

Pricing

FACTORS ON DEMAND



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3. Aggregation



= security return




Normal assumption





Estimation



they do not behave identically and independently across time- Therefore, estimation cannot be performed on returns, but rather on risk drivers X, which areinvariants

Example: option

1. Risk drivers estimation

= risk driver

= idiosyncratic shoc


2. Pricing

FACTORS ON DEMAND



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1. Risk drivers estimation 5. Attribution factors



= risk driver




3. Aggregation

2. Pricing



Normal assumption


Estimation







Example: option

FACTORS ON DEMAND



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= risk driver




3. Aggregation

2. Pricing



Normal assumption


Estimation




- Then, risk drivers Xare transformed into returns Rby delta or duration coefficients

: govt curve changes : log-return of underlying: log-return of implied vol.: spread changes

: spread duration : vegaExample: bond Example: option: curve duration : delta

FACTORS ON DEMAND



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= risk driver




3. Aggregation

2. Pricing



Normal assumption


Estimation




- Then, risk drivers Xare transformed into returns Rby delta or duration coefficients - The risk computations follow

FACTORS ON DEMAND



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= risk driver




3. Aggregation

2. Pricing



Normal assumption

Risk Attribution Rationales

- After obtaining aggregate portfolio risk (Sdev, VaR, CVaR, etc.), attribute it to individual factors- Purpose: see how factors contributed to portfolio risk and make hedging decision

Traditional Risk Attribution Techniques

- Use same factors for attribution as for estimation- Perform linear operations to define security-level risk attribution

p

Attribution

FACTORS ON DEMAND



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= risk driver



Attilio Meucci

3. Aggregation

2. Pricing



Normal assumption

Risk Attribution Rationales

- After obtaining aggregate portfolio risk (Sdev, VaR, CVaR, etc.), attribute it to individual factors- Purpose: see how factors contributed to portfolio risk and make hedging decision

Traditional Risk Attribution Techniques

- Use same factors for attribution as for estimation- Perform linear operations to define security-level risk attribution

- Perform bottom-up aggregation for portfolio-level risk attribution

p

Attribution

FACTORS ON DEMAND



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= risk driver



Attilio Meucci

3. Aggregation

2. Pricing



Normal assumption

p

Pitfalls

Pitfalls

- Same factors used for both estimation and attribution: choice neither optimizes the estimation

power nor the interpretability or practicality for hedging

- As an estimation model, band Fmaximize r-square- As an attribution model, band Fmaximize r-square (CAPM)- delta assumption can be inappropriate

- Bottom-up aggregation not flexible: small exposures better in residual

FACTORS ON DEMAND



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= risk driver





Attilio Meucci

3. Aggregation

2. Pricing



Normal assumption

Enhanced Attribution

Pitfalls

- Similar factors used for both estimation and attribution: choice neither optimizes the estimation

power nor the interpretability or practicality for hedging- Factors restricted by the systematic + idiosyncratic assumption- As an estimation model, band Fmaximize r-square- As an attribution model, band Fmaximize r-square (CAPM)- delta assumption can be inappropriate

- Bottom-up aggregation not flexible: small exposures better in residual

Enhanced attribution factors


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EXECUTIVE SUMMARY




REFERENCES

FACTORS ON DEMAND

FOD Theory


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3. Aggregation

2. Pricing



5. Attribution factors

= portfolio return

= residual

= attribution loadin

= attribution factor

= risk driver= loading

= residual= dominant factor

Attilio Meucci


exact

exact

6. Portfolio risk attribution: top down

FOD Theory

1 jointscenario

E.g. PCA facts

Attili M iFACTORS ON DEMAND

FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact


FOD Theory

1 jointscenario

E.g. PCA facts PCA res stocks log.rets

Attili M iFACTORS ON DEMAND

FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact


FOD Theory

1 jointscenario

E.g. PCA facts PCA res stocks log.rets

Attilio Me cciFACTORS ON DEMAND

FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact


y

1 jointscenario

E.g. PCA facts PCA res stocks log.rets stocks lin. rets


FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact


y

1 jointscenario

E.g. PCA facts PCA res stocks log.rets stocks lin. rets


FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact


y

1 jointscenario

E.g. PCA facts PCA res stocks log.rets stocks lin. rets port ret


FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact


1 jointscenario

E.g. PCA facts PCA res stocks log.rets stocks lin. rets port ret


FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact


1 jointscenario

conditional scenarios given X

E.g. PCA facts PCA res stocks log.rets stocks lin. rets port ret attribution GICS sectors


FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact



1 jointscenario



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





Attilio Meucci


exact

exact



1 jointscenario



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual





tt o eucc


exact

exact



1 jointscenario



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



1 jointscenario

? ?



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



1 jointscenario



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



1 jointscenario



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



1 jointscenario



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



1 jointscenario



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



1 jointscenario

~

~~

~ ~ ~

~ ~

~~ ~ ~E.g. PCA facts PCA res stocks log.rets stocks lin. rets port ret attribution hedges


FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact


Factors on Demand - Features1. Estimation factors Fand loadings bare chosen to optimize the explanation power


FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact


Factors on Demand - Features1. Estimation factors Fand loadings bare chosen to optimize the explanation power2. Exact risk numbers through exact pricing


Ri k A ib iRi k E i i

FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact


Factors on Demand - Features1. Estimation factors Fand loadings bare chosen to optimize the explanation power2. Exact risk numbers through exact pricing3. Attribution factors Zare chosen to be interpretable and practical for hedging


Ri k Att ib tiRi k E ti ti

FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact


Factors on Demand - Features1. Estimation factors Fand loadings bare chosen to optimize the explanation power2. Exact risk numbers through exact pricing3. Attribution factors Zare chosen to be interpretable and practical for hedging3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc4. Constraints allow for long-only, best-few-out-of-many, etc



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc4. Constraints allow for long-only, best-few-out-of-many, etc5. Exact Linear interpretation/hedge of non-linear securities



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual






exact

exact



3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc4. Constraints allow for long-only, best-few-out-of-many, etc5. Exact Linear interpretation/hedge of non-linear securities6. No linear relationship between Zand F: connection created by conditional distribution



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading



exact

exact



3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc4. Constraints allow for long-only, best-few-out-of-many, etc5. Exact Linear interpretation/hedge of non-linear securities6. No linear relationship between Zand F: connection created by conditional distribution7. Conditional distribution -> one estimation method, several possible interpretations/hedges

~

~~

~ ~ ~

~ ~

~



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading



exact

exact



3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc4. Constraints allow for long-only, best-few-out-of-many, etc5. Exact Linear interpretation/hedge of non-linear securities6. No linear relationship between Z and F: connection created by conditional distribution7. Conditional distribution -> one estimation method, several possible interpretations/hedges8. Systematic + idiosyncratic -> dominant + residual



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading



exact

exact



3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc4. Constraints allow for long-only, best-few-out-of-many, etc5. Exact Linear interpretation/hedge of non-linear securities6. No linear relationship between Z and F: connection created by conditional distribution7. Conditional distribution -> one estimation method, several possible interpretations/hedges8. Systematic + idiosyncratic -> dominant + residual9. Top-down attribution provides portfolio-specific best model



FOD Theory


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


s tt but os st at o

exact

exact



3. Attribution loadings dare chosen to optimize r-square, CVaR, downside risk, etc4. Constraints allow for long-only, best-few-out-of-many, etc5. Exact Linear interpretation/hedge of non-linear securities6. No linear relationship between Zand F: connection created by conditional distribution7. Conditional distribution -> one estimation method, several possible interpretations/hedges

8. Systematic + idiosyncratic -> dominant + residual9. Top-down attribution provides portfolio-specific best model



FOD Frequently Asked Questions


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


Factors on Demand Frequently Asked Questions

Q: Why not run a regression of portfolio returns Rvs. attribution factors Z?A: Rand Zare not necessarily invariants

Q: Why abandon systematic + idiosyncratic model?A: Uis where managers look for alpha factors ->A: otherwise we cannot merge irrelevant systematic factors with idiosyncratic residual toobtain more efficient attribution/hedgingA: in powerful estimation approaches (PCA,RMT) residual Uis never idiosyncraticA: that model is not a consequence of APT/CAPM





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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


80 90 100 110 120 130

80 90 100 110 120 1300

5

10

15

20

25

30

exact

order 2 approx.

50 100 150 200

50 100 150 2000

20

40

60

80

00

20

exact

order 2 approx.


1-day horizon call Multi-day horizon call

Q: Why should we not use

delta approximation?A: Risk of derivatives or nonlinear instruments atmulti-day horizon isdistorted





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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact



Q: Do we have to generate conditional scenarios for Z?A: Not always: if using historical scenarios, use historical (drivers for) Z

Q: Does FOD recommend specific estimation/attribution factors/techniques?A: No, FOD proposes a flexible, modular methodology that hosts all techniques

Q: Does FOD dismiss traditional multi-purpose factor modelsA: No, all traditional model are special cases of FOD


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EXECUTIVE SUMMARY




REFERENCES



FOD Appl ication #1Optimize Factor Choice for Risk and Portfolio Mgmt


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


b, F : high statistical power

Principal Component Analysis and RandomMatrix Theory can be applied

Factors and loadings are determined tominimize estimation error although they might bedifficult to interpret.

Z : high interpretability/tradability

Attribution factors examples

GICS Sectors: Material, Technology, Financials

Macro: S&P500, 10 year yield, Gold price, MSCIEM Index, Russell 2000



FOD Appl ication #1Optimize Factor Choice for Risk and Portfolio Mgmt


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


volatility decomposition per industry (RMT)

b, F : high statistical power Z : high interpretability/tradability

volatility decomposition per factor (GICS)



FOD Appl ication #2Custom attribution factors on the fly


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


X : historical

No factor modes for X, pure historicalrealization of risk drivers

Ris not the time series of the returns

Explicitly no idiosyncratic term

Z : g(X)

Attribution factors are deterministic functions ofrisk drivers

For instance, Zcan be user-supplied definitionsof value/momentum factors

FOD then allows to compare in real time theattribution to different, user-supplied factor models

Zand Z All models share the samerisk statistics

~



FOD Appl ication #3Integrated Global and Regional Risk Models


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


e.g. US Model: US sector factors

e.g. UK Model: UK financial, UK utilities,)

Global factors Zare deterministic, linearfunctions (aggregations) of the regional factors

b, F: regional equity factor model Z : global equity factors

e.g. global financial, global utilities,

Regional factors Fconstructed by cross-sectional regression on given loadings b

1 Ri k d i ti ti



FOD Appl ication #4New attribution target: minimize CVAR for hedging


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


Z : returns of hedging instruments; d: attribution target as CVaR

For hedging, the attribution factors must be the linear returns Z=P(t+1)/P(t)-1 of tradables

Linearattribution (6) is important for hedging: only portfolios, i.e. linear combinations, are traded

Profits and losses of hedged p&l play a non-symmetricalrole: non-linear pricing (2) properlyinduces asymmetries on R; downside target CVaR in (6) accounts for asymmetries in

Thus FOD hedging (full-pricing/CVaR) and Black-Scholes hedging (delta/r-square) are different

Example: units of underlyingto hedge call options

1 Ri k d i ti ti A ib i f



FOD Appl ication #5Best Pool on Demand / flexible constraints


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


0 5 10 15 20 25 30 35 40 45 500

5

10

15

20

25

30

35

40

45

50

num players out of total 50

naive

rec. rejectionrec. acceptance

d: constraint few relevant out of many in top-down attribution

rec. rejection rec. acceptancenave sorting

For hedging, traders prefer to put on fewerhedges. Therefore the selection of the best few

trades should be optimized

For factor modeling, it does not make sense toinclude minimally represented factors in analysis.Better to add them to residual

Other constraints can be added (e.g. long only,sum-to-one, etc.)

Num attribution factors

At

tributiontarge

t

1 Risk drivers estimation 5 Att ib ti f t



FOD Appl ication #6Turnover-trading persistence


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


Z : returns of sub-portfolios; portfolios: past holdings

The attribution of the current holdings to the past holdings allows the portfoliomanager to evaluate the turnover (half-life) of their positions

1 Risk drivers estimation 5 Att ib ti f t



FOD Appl ication #6Turnover-trading persistence


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


Z : returns of sub-portfolios; portfolios: past holdings

The attribution of the current holdings to the past holdings allows portfoliomanagers to evaluate the turnover (half-life) of their positions

If the attribution target in (6) is set as the r-square and the attributionoptimization is unconstrained we obtain the analytical solution in Grinold (2006)

FOD allows portfolio managers to customize their analysis, with arbitrarytargets and constraints

1 Risk drivers estimation 5 Attribution factors



FOD Applications #7Point in Time Style Analysis


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3. Aggregation

2. Pricing




= portfolio return

= residual



= risk driver

= loading


exact

exact


Z : style factors; constraints: long-only, sum-to-one

Traditional style analysis a-la-Sharpe runs a constrained regression of portfolio returns Rp(t) onstyle factors Z(t)

In traditional style analysis the past returns are affected by the past allocation decisions Rp(t-k)=w(t-k) x R(t-k) includes a component due to rebalancing w(t-k)

FOD allows to perform point-in-time style analysis based only the current exposures w(t)

EXECUTIVE SUMMARY


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EXECUTIVE SUMMARY




REFERENCES

Attilio MeucciFACTORS ON DEMAND References


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Article

Attilio Meucci - Factors on DemandRisk, July 2010, p 84-89available at http://ssrn.com/abstract=1565134

MATLAB examples

MATLAB Central Files Exchange (see above article)

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