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The Black-Litterman Model: Active Risk

Targeting and the Parameter Tau
Presented to: Presented by:
Northfield 29th Annual Research Randy O’Toole, CFA
Conference Federated Investors
March 21, 2017 Portfolio Manager/Senior Quantitative Analyst
212-905-4828
rotoole@federatedinv.com

For Institutional/Investment Professional Use Only. Not for Distribution to the Public. Tracking number: 12-44015
Overview

 In the Black-Litterman model, the parameter tau (τ) determines the overall
weight given to active versus passive investment views
 Tau originates from the seminal Bayesian derivation of the model; despite its
importance, tau has proved to be a very confusing aspect of Black-Litterman in
terms of interpretation, estimation, and implementation
 We offer a simple interpretation of tau, show that it is directly related to the level
of active risk implicit in Black-Litterman expected returns, and is easily calibrated
so that Black-Litterman expected returns produce a portfolio with a targeted level
of active risk
 We introduce an alternative derivation of Black-Litterman that affords a more
direct way of targeting active risk that doesn’t require a specific value for tau
 The derivation clearly shows that portfolio construction using the Black-Litterman
model is equivalent to creating a mean-variance optimal portfolio of active
strategies that is then overlaid onto a benchmark portfolio
 By targeting active risk directly, users of the Black-Litterman model can effectively
ignore tau

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1
Introduction

 Introduced 25 years ago, The Black-Litterman (BL) model was developed to

mitigate problems related to mean-variance optimization (MVO)
 Unconstrained MVO portfolios are typically (1) very sensitive to expected return
inputs, (2) concentrated in just a few assets, and (3) not reflective of underlying
active investment views
 BL incorporates passive “equilibrium” expected returns and active “views”
 Equilibrium expected returns proxy for consensus expected returns associated
with a benchmark
 Views are portfolio-level expected returns associated with active investment
strategies
 BL converts views into asset-level active expected returns that are blended
with equilibrium expected returns
 Unconstrained MVO holdings for assets not included in any active views will be
equal to benchmark weights
 Active weights will differ from the benchmark in proportion to the view portfolios
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2
Black-Litterman is apparently well-known and widely used...

 Google Scholar search yields 5,370 results for BL; 1,374 citations for
“Global Portfolio Optimization” (Black and Litterman, 1992)
 Asset managers: Barclays Wealth and Investment Management, Goldman-
Sachs, and UBS
 Investment advisors: Standard and Poor’s Investment Advisory Services
(SPIAS) and Betterment
 Investment software providers: offered as a portfolio construction tool from
Morningstar Direct and Zephyr Associates
 Sell-side: promoted in research from Deutsche Bank Securities and J.P.
Morgan
 “...many trillions of institutional and hedge fund dollars are invested in Black-
Litterman optimized portfolios.” (Michaud, 2012)

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3
...but is it well understood?

 BL literature includes publications such as:

 “The Intuition Behind Black-Litterman Model Portfolios” (He and Litterman, 1999)
 “A Demystification of the Black-Litterman Model” (Satchell and Scowcroft, 2000)
 “A Step-by-Step Guide to the Black-Litterman Model” (Izadorek, 2004)
 “The Black-Litterman Model Explained” (Cheung, 2010)
 “Deconstructing Black-Litterman” (Michaud, Esch, and Michaud, 2013)
 “Reconstructing the Black-Litterman Model” (Walters, 2014)
 BL is typically analyzed in the context of Bayesian statistical methods
 Emphasis on its Bayesian statistical underpinnings likely obfuscates the practical
workings of BL for many prospective (and even current) users of the model

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4
Black-Litterman expected returns

 The BL “master equation”:

rBL = (τΣ ) + P′Ω P [ −1 −1

] [(τΣ)
−1 −1
Π + P′Ω −1Q ]
rBL is the N x 1 vector of expected returns;

τ is a parameter that reflects the level of confidence in the equilibrium expected returns;

Σ is the N x N matrix of asset return covariances;

P is the K x N view matrix of asset portfolio weights for K active investment strategies;

Ω is the K x K diagonal matrix containing measures of confidence in each strategy;

Π is the N x 1 vector of equilibrium expected returns; and

Q is the K x 1 vector of views, which are the expected returns for the active strategies.

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5
Deriving Black-Litterman (two approaches out of many)

 Theil’s mixed estimator (generalized least squares derivation):

y = xrBL + e, e ~ N (0, V )
Π I  u  τΣ 0 
y =  , x =  , e =  , V =  
Q   P v   0 Ω
rBL = x′V x ( −1
) −1
′ −1
xV y
 Simultaneous least squares derivation:

1 ′ −1 ′ −1
min rBL ( rBL − Π ) (τΣ ) (rBL − Π ) + (Q − PrBL ) Ω (Q − PrBL )
2 
 Both are optimization problems, but do they convey any real investment
intuition...?
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6
Alternative derivation of Black-Litterman

 The BL master equation can also be written

rBL = Π + τΣP′(τPΣP′ + Ω ) [ −1
](Q − PΠ )
= Π + Λα S
= Π + α BL
BL expected returns are equal to the sum of passive and active expected returns
δ
 Unconstrained MVO: max h hP′ rBL − hP′ ΣhP
P
2
→ hP = δ −1Σ −1rBL
= δ −1Σ −1Π + δ −1Σ −1α BL
= hB + hA
BL portfolio holdings are equal to benchmark weights plus a vector of active weights
 Note: δ is a risk aversion parameter
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7
Equilibrium expected returns

 Equilibrium expected returns, also referred to as implied returns, are

calculated using reverse optimization:

hB = δ −1Σ −1Π ⇒ Π = δ Σ hB
Equilibrium expected returns are engineered to produce benchmark weights when
used in unconstrained MVO
 In Bayesian terminology, equilibrium returns are the “prior” estimates of
expected returns
 Equilibrium returns act as an anchor for active investment views and obviate
the need to calculate expected returns for all assets individually
 Original BL derivation uses market capitalization weights that correspond to
the equilibrium assumptions of the CAPM; in practice, any benchmark can
be used

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8
Active expected returns

 O’Toole (2013) uses reverse optimization to derive BL alphas:

 Strategy covariance matrix: Σ S = PΣP′
 Confidence-adjusted strategy alphas: α S = τΣ S (τΣ S + Ω )
~ −1
αS [ ]
δ ~
 Unconstrained MVO: max w w′α~S −
2
w′Σ S w → w = δ −1Σ −S1α S

 Calculate asset-level active holdings by multiplying each view portfolio by its

corresponding MVO weight and sum the scaled holding across strategies:
 p11   p12   p1K 
     
′
hA = P w =    w1 +    w2 +  +    wK
p  p  p 
 N1   N2   NK 
 Apply reverse optimization to the active holdings:

[
α BL = δΣh A = τΣP ′(τPΣP ′ + Ω )−1 (Q − PΠ ) ]
BL alphas are designed to produce the same unconstrained MVO active holdings
as those generated from MVO at the strategy level
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9
Practitioner interpretation of Black-Litterman

 Portfolio construction using the Black-Litterman model is equivalent to

creating a mean-variance optimal portfolio of active strategies that is then
overlaid onto a benchmark portfolio
 Unconstrained MVO holdings for assets not included in any active views will be
equal to benchmark weights
 Active weights will differ from the benchmark in proportion to the view portfolios
 Mean-variance mechanics are embedded in Black-Litterman expected
returns
 The active portion of Black-Litterman expected returns is associated with a
specific MVO active portfolio
 Consistent use of the risk model (covariance matrix) promotes stable results
vis-à-vis traditional mean-variance optimization

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10
Example: He and Litterman two market views case

Equilibrium
Benchmark Annualized Correlation Matrix Expected Returns,
Weights, hB (%) Volatilities (%) Australia Canada France Germany Japan UK USA Π = δΣhB (%)
Australia 1.6 16.0 1 3.94
Canada 2.2 20.3 0.488 1 6.92
France 5.2 24.8 0.478 0.664 1 8.36
Germany 5.5 27.1 0.515 0.655 0.861 1 9.03
Japan 11.6 21.0 0.439 0.310 0.355 0.354 1 4.30
UK 12.4 20.0 0.512 0.608 0.783 0.777 0.405 1 6.77
USA 61.5 18.7 0.491 0.779 0.668 0.653 0.306 0.652 1 7.56

View Matrix*, P' (%) Strategy Covariance Matrix, Strategy Equilibrium

North ΣS = PΣP' (%) Views, Q (%) Views, PΠ (%)
Europe/UK America Europe/UK North America Europe/UK 5.00 1.79
Australia 0 0 Europe/UK 213.0 20.1 North America 3.00 -0.64
Canada 0 100.0 North America 20.1 170.3
France -29.5 0 Strategy Adjusted
Germany 100.0 0 Strategy Confidence,Ω (%) Alphas, Alphas, α~S =
Japan 0 0 Europe/UK North America αS = Q-PΠ (%) τΣS(τΣS+Ω)-1αS (%)
UK -70.5 0 Europe/UK 10.6 0 Europe/UK 3.21 1.71
USA 0 -100.0 North America 0 8.5 North America 3.64 1.89
*Note that this is the transpose of the view matrix.

Strategy MVO Weights, Active Portfolio BL Expected BL Alphas,

w = δ ΣS α S (%) αBL = δΣhA
-1 -1 ~
Holdings, Holdings, Returns,
Europe/UK 28.2 hA = P'w (%) hP = hB+hA (%) rBL = δΣhP (%) = rBL - Π (%)
North America 41.1 Australia 0 1.6 4.42 0.48
Canada 41.1 43.3 8.73 1.81
Risk Aversion Parameter, δ = 2.5 France -8.3 -3.1 9.48 1.12
Strategy Correlation, ρ = 10.5% Germany 28.2 33.7 11.21 2.18
Ann. Active Risk, σA = 7.1% Japan 0 11.6 4.62 0.31
Ann. Portfolio Risk, σP = 18.8% UK -19.9 -7.5 6.97 0.20
Tau, τ = 0.05 USA -41.1 20.4 7.48 -0.08

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11
The parameter tau (τ)

 τ has proved to be a particularly confusing aspect of the BL model

 τ is a non-negative scalar that reflects an overall level of confidence in the active
views versus the equilibrium expected returns
 In Bayesian terms, τ measures the subjective degree of uncertainty as to how
precisely the equilibrium returns (the expected return priors) have been estimated
 τ is a critical component of BL as it effectively determines the overall weight
placed on the active views relative to the equilibrium expected returns
 The literature is replete with conflicting guidance as to how to interpret and
quantify τ, and includes some harsh criticism of BL related to τ:
 “The adjustment of τ to attain investability is an intervention which contaminates
the rigor of the analysis and must be viewed as an ad hoc correction of a flawed
procedure, and a major departure from rigorous statistical analysis.” (Michaud,
Esch, and Michaud, 2013)
 “...Black-Litterman is a failed scientific theory with vacuous investment value that
should be ignored by the investment community as any serious solution to the
instability and ambiguity of Markowitz optimization in practice.” (Michaud, 2012)
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12
Practitioner interpretation of τ

 A practical interpretation of τ is that it reflects an investor’s degree of belief

in efficient markets
 Smaller values indicate less subjective uncertainty regarding market efficiency
and shrink the weight on the active views toward zero
 In the limiting case of τ = 0 the weight on all of the active views is zero, and the
investor will passively hold the “market” as represented by the benchmark
portfolio
 Larger values indicate belief in exploitable market inefficiencies, with higher
values corresponding to greater confidence in the active views and more
willingness to take on active risk
 Active managers typically have explicit targets for active risk based on the
opportunity set, investment objectives, policy mandates, etc.
τ provides a natural mechanism for controlling active risk, and having a specific
target for active risk provides a sensible context for determining a value for τ

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τ and active risk

 BL portfolio holdings can be separated into benchmark and active weights:

hBL = hB + hA = δ Σ Π + δ τP′(τPΣP′ + Ω ) (Q − PΠ )−1 −1 −1 −1

Note that τ only enters the active component

 Active risk is equal to

σ A = h′AΣhA
τ
= (Q − PΠ )′ (τPΣP′ + Ω )−1 PΣP′(τPΣP′ + Ω )−1 (Q − PΠ )
δ
 Active risk is an increasing non-linear function of τ
Active risk increases at a decreasing rate with respect to τ, asymptotically
approaching a well-defined upper bound:

1
σ AMAX = limτ →∞ σ A =   (Q − PΠ )′ (PΣP ′)−1 (Q − PΠ )
δ 
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14
τ and active risk: two market views case

Active Risk as a Function of τ Active Risk and τ

14 Active Portfolio Holdings (%)

12 τ = 0.00786 τ = 0.05 τ = 0.353 τ→∞
10 Australia 0 0 0 0
Active Risk

8 Canada 11.5 41.1 70.1 79.4

6
France -2.4 -8.3 -13.9 -15.6
Germany 8.0 28.2 47.0 52.8
4
Japan 0 0 0 0
2
UK -5.7 -19.9 -33.1 -37.2
0
USA -11.5 -41.1 -70.1 -79.4
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
τ
Ann. Risk (%) 2.0 7.1 12.0 13.5

 He and Litterman set τ = 0.05, which produces σA = 7.1%

 For an investor with 100% confidence in the active views, σAMAX = 13.5%

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15
Targeting active risk directly

 There is a more direct way to target active risk that doesn’t require a specific
value for τ and for which there is no upper bound on active risk
 Recall that the active strategy MVO weights were solved for using δ, the
same risk aversion parameter used to derive the equilibrium expected
returns associated with the benchmark:
δ 1~
max w w′α~S − w′Σ S w → w = δ −1Σ − S αS 2
 O’Toole (2013, 2017) incorporates an active risk aversion parameter, φ, that
corresponds to a targeted level of active risk, σA,T :
φ
(
max wˆ wˆ ′α~S − σ A2 ,T − wˆ ′Σ S wˆ
2
) → w ~ ,
ˆ = φ −1Σ −S1α S

α~S′ Σ −S1α~S  τ 
φ = 
=   (Q − PΠ )′ (τPΣP′ + Ω )−1 PΣP′(τPΣP′ + Ω )−1 (Q − PΠ )
σ A,T 
 σ A,T 
The active risk aversion parameter, φ, will adjust for different values of τ

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Risk-targeted Black-Litterman alphas

 Asset-level active holdings using risk-targeted MVO weights:

hˆ = P′w ˆ = φ −1τP′(τPΣP′ + Ω )−1 (Q − PΠ )
A

 Use reverse optimization to calculate risk-targeted BL alphas:

δ 
[ δ 
αˆ BL = δΣhˆA =   τΣP′(τPΣP′ + Ω )−1 (Q − PΠ ) =  α BL ]
φ  φ 
The active component of BL expected returns can simply be scaled by the ratio of
benchmark risk aversion to active risk aversion in order to target a particular level of
active risk

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17
Targeting active risk directly: two market views case

Active Risk as a Function of φ (τ = 0.05) Active Risk and φ

25 Active Portfolio Holdings (%)

τ = 0.05 φ = 0.0888 φ = 0.0250 φ = 0.0148 φ = 0.0131 φ = 0.0104
20 τ=1 φ = 0.1620 φ = 0.0456 φ = 0.0270 φ = 0.0240 φ = 0.0191
τ=5 φ = 0.1678 φ = 0.0473 φ = 0.0280 φ = 0.0249 φ = 0.0197
Active Risk

15 τ = 20 φ = 0.1690 φ = 0.0476 φ = 0.0282 φ = 0.0250 φ = 0.0199

Australia 0 0 0 0 0
10 Canada 11.5 41.1 70.1 79.4 98.5
France -2.4 -8.3 -13.9 -15.6 -20.0
5 Germany 8.0 28.2 47.0 52.8 67.5
Japan 0 0 0 0 0
0 UK -5.7 -19.9 -33.1 -37.2 -47.6
0.01
0.02
0.04
0.06
0.07
0.09
0.10
0.12
0.14
0.15
0.17
0.18
0.20
0.22
0.23
0.25
0.26
0.28
0.30
0.31
0.33
0.34
0.36
0.38
0.39
0.41
USA -11.5 -41.1 -70.1 -79.4 -98.5
φ
Ann. Risk (%) 2.0 7.1 12.0 13.5 17.0

 Higher levels of active risk correspond to lower levels of active risk aversion

 τ is subsumed by the active risk aversion parameter: φ adjusts for τ to

generate portfolios with specific active risk targets, and there is no upper
bound on active risk
Targeting active risk directly eliminates the need to choose a specific value for τ

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18
Conclusion

 There is apparent persistent confusion over certain aspects of Black-

Litterman expected returns, with a number of publications offering various
explanations and clarifications as to how the model works in practice
 The parameter tau has proved to be especially perplexing and contentious,
with many authors offering widely varying suggestions as to how this
important component of the model should be interpreted and quantified
 We show there is a direct relationship between tau and active risk, and tau
can be calibrated to produce mean-variance optimal Black-Litterman
portfolios with targeted levels of active risk, but only up to a maximum
amount
 Our alternative derivation of Black-Litterman clearly reveals the mean-
variance mechanics of the model while affording a more direct way to target
active risk, with no need to set a specific value for tau and no upper bound
on active risk
Investors who target active risk directly using the Black-Litterman model need
not concern themselves with tau at all
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19
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