Behaviouralizing Black Litterman

Sebastien Lleo1

Joint work with Mark Davis

QUANT 12 WORKSHOPEM Lyon

November 27, 2015

1Finance Department, NEOMA Business School, Reims Campus, Email:sebastien.lleo@NEOMA-bs.fr

Portfolio Selection is a Two-Stage Process...

Figure : Does Harry Markowitz need anyintroduction?

At the very beginning of his seminal paper,Portfolio Selection, Markowitz (1952) states:

The process of selecting a portfoliomay be divided into two stages. Thefirst stage starts with observation andexperience and ends with beliefsabout the future performances ofavailable securities. The second stagestarts with the relevant beliefs aboutfuture performances and ends withthe choice of portfolio. This paper isconcerned with the second stage.

Today’s talk is concerned with the first stage...

... and we will also show how our results relateto the second stage!

Our objective is to present an extension of our earlier work on Black-Littermanin continuous time (Davis and Lleo, 2013, 2014).

The talk is based on two papers:

I M. Davis and SL. A Simple Procedure For Merging Expert Opinions ToAchieve Superior Portfolio Performance. Journal of Portfolio Management(To Appear).

I M. Davis and SL. Behaviouralizing Black-Litterman: Expert Opinions andBehavioural Biases in a Diffusion Setting. Working Paper,http://ssrn.com/abstract=2663650.

The ideas presented here also relate to:

I The recent papers by Frey, Gabih, Sass and Wunderlich (Frey et al., 2012);

I Savage’s pioneering work on personal probabilities and personal utility(Savage, 1954, 1971);

I Advances in mathematical psychology and behavioural finance;

I Lessons from stochastic programming models...

I ... and of course Black and Litterman (1991, 1992)!

Introduction: YOUR Opinion Matters

From analyst reports to CNBC pundits, internetblogs and Google trends, we have access alarger numbers of opinions, views and data onfinancial markets and the economy.

Why not use them to formulate ‘beliefs aboutthe future performances of available securities’?

Figure : James Cramer, Trader turned CNBCPundit

Financial Universe: m securi)es

Investment universe (m securi)es)

k views

Total number of observa)ons = m + k

Financial Universe: m securi)es

Investment universe (m securi)es)

k views

Total number of observa)ons = m + k

X(t)

Financial Universe: m securi)es

Investment universe (m securi)es)

k views

Total number of observa)ons = m + k

X(t)

⇒

X(t)

Overview

Our approach follows 6 steps:

Step 1 • Parametrize the financial market

Step 2 • Collect expert opinions and views

Step 3 • Address the impact of behavioral biases

Step 4 • Blend data with opinions and filter to esBmate the current level of factors

Step 5 • OpBmize the porFolio

Step 6 • Monitor

To illustrate the process, we consider the case of Irene, an investor whomanages a portfolio of m = 11 U.S. Exchange Traded Funds (ETFs).

The Probability Space

Let (Ω,F ,P) be the underlying probability space and define an Rd -valued(Ft)-Brownian motion W (t) with components Wk(t), k = 1, . . . , d andd = n + m + k with n ≥ 0, m > 0 and k ≥ 0.

Step 1: Parametrize The Financial Market

Consider an asset market comprised of m ≥ 1 risky securities Si , i = 1, . . .mand a money market account process S0.

The growth rates of the assets depend on n unobservable factorsX1(t), . . . ,Xn(t) which follow the affine dynamics given in the equation (3)below.

The dynamics of the money market asset S0 is given by

dS0(t)

S0(t)= r(t)dt, S0(0) = s0, (1)

where the Ft-adapted process r(t) ∈ R+, t ∈ R+ represents the money marketrate.

Denote by Si (t) = Si (t)S0(t)

the discounted price of asset i = 1, . . . ,m. The

dynamics of Si (t) is given by the diffusion SDE

dSi (t)

Si (t)= (a(t) + A(t)X (t))idt +

d∑j=1

Σik(t)dWk(t),

Si (0) = si , i = 1, . . . ,m, (2)

The point here is that the optimal investment policy ultimately depends on therisk premia, not on the nominal rates of return.

The drift of the discounted asset prices is subject to the evolution of an-dimensional vector of unobservable factors X (t) modelled using an affinedynamics:

dX (t) = (b(t) + B(t)X (t))dt + Λ(t)dW (t), X (0) = x . (3)

We derive an estimate X (t) for the factor process X (t) using filtering, and usethis estimate in subsequent sections to solve a range of optimal investmentproblems.

Implementation

We consider the case of Irene, an investor who manages a portfolio of m = 11U.S. Exchange Traded Funds (ETFs).

Asset Sector ETF Ticker Weight in the S&P 500

S1 S&P500 Technology XLK 21.77%S2 S&P500 Financials XLF 16.44%S3 S&P500 Health Care XLV 13.85%S4 S&P500 Consumer Discretionary XLY 11.75%S5 S&P500 Industrial XLI 10.34%S6 S&P500 Consumer Staples XLP 9.89%S7 S&P500 Energy XLE 9.34%S8 S&P500 Materials XLB 3.39%S9 S&P500 Utilities XLU 3.23%S10 Russell 2000 IWM -S11 U.S. Real Estate Investment Trusts (REITs) IYR -

The n = 3 factors follow the Fama-French model (see Fama and French, 1993):

I X1 is the market risk premium;

I X2 captures the difference in risk premia between small marketcapitalisation stocks and high market capitalisation stocks;

I X3 represents the difference in risk premia between stocks with a highbook-to-market ratio and stocks with a low book-to-market ratio.

The weekly data come from Kenneth French’s online database.

The rate of return of the money market instrument is the weekly rate of a1-month Treasury Bill as computed by Kenneth French based on rates providedby Ibbotson and Associates.

I We need this rate to discount the asset prices.

Model Estimation

I We use the definition of the quadratic variation of s(t) = ln S(t), that is:

〈s, s〉t = lim∆tk→0

∑tk≤t

(s(tk+1)− s(tk)) (s(tk+1)− s(tk))′ = ΣΣ′t, (4)

to estimate the historical diffusion matrix ΣΣ′.

I Similarly for the diffusion matrix ΛΛ′ and for the cross variation term ΣΛ′.

I In our case,

ΣΣ′ ≈

0.12 0.08 0.05 0.08 0.08 0.03 0.06 0.07 0.03 0.08 0.060.08 0.19 0.07 0.11 0.11 0.05 0.09 0.10 0.05 0.11 0.120.05 0.07 0.06 0.05 0.05 0.03 0.05 0.05 0.03 0.05 0.050.08 0.11 0.05 0.11 0.08 0.04 0.07 0.09 0.04 0.09 0.080.08 0.11 0.05 0.08 0.09 0.04 0.08 0.09 0.04 0.09 0.080.03 0.05 0.03 0.04 0.04 0.04 0.04 0.04 0.03 0.04 0.040.06 0.09 0.05 0.07 0.08 0.04 0.14 0.10 0.05 0.09 0.070.07 0.10 0.05 0.09 0.09 0.04 0.10 0.12 0.04 0.09 0.080.03 0.05 0.03 0.04 0.04 0.03 0.05 0.04 0.06 0.04 0.040.08 0.11 0.05 0.09 0.09 0.04 0.09 0.09 0.04 0.11 0.090.06 0.12 0.05 0.08 0.08 0.04 0.07 0.08 0.04 0.09 0.13

,

ΣΛ′ ≈

0.06 0.01 0.010.09 0.00 0.030.04 −0.00 0.000.07 0.01 0.010.07 0.01 0.010.04 −0.00 0.000.07 0.00 0.010.07 0.01 0.010.04 −0.00 0.010.07 0.02 0.010.07 0.01 0.02

, ΛΛ′ ≈

0.0726 0.0062 0.00470.0062 0.0147 −0.00240.0047 −0.0024 0.0189

.

based on K = 739 discounted weekly log returns from July 26, 2000 to August31, 2014.

I To estimate the drift parameters b and B, we start by discretising the SDEfor X (t), defined at (3), as:

Xt+1 = (b∆t + (B∆t + Im)Xt) + ΛZXt

√∆t,

where Im is the m ×m identity matrix and ZXt is a d-dimensional standard

normal random variable for every t.

I Because ∆t is fixed in our dataset, we can express the dynamics of Xt as afirst order vector autoregressive VAR(1) process:

Xt+1 = b1 + B1Xt + ΛZt

√∆t, (5)

where b1 := b∆t and B1 := Im + B∆t.

I Reversing the definition of b1 and B1, we obtain the following MaximumLikelihood estimates for b and B:

b =

−7.77E−04

6.79E−05

−5.03E−05

, B =

−56.00 9.84 3.32−0.40 −53.32 3.40−2.16 2.94 −50.64

. (6)

We can obtain the drift parameters, a and A, via an OLS regression.

I Start by discretizing the process s(t) given at (2) as:

∆sit =

(ai −

1

2ΣΣ′ii

)∆t +

(AXt

)i∆t + (ΣZ s

t )i√

∆t, (7)

where ∆st = st+1 − st , and Z st is a d-dimensional standard normal variable

for every t.

I Notice that Xt should be independent from the time discretisation scheme.However, the values of Xt in the Fama-French database are weeklyreturns, and as such are a function of the discretization scheme.

I To address this inconsistency, we consider the following linear modelinstead:

yit = αi + βxt + εit , (8)

where yit = ∆st∆t

, αi = ai − 12ΣΣ′ii , βi = A[i , ·]′∆t is the ith row of matrix

A transposed, xt = Xt∆t

and εit is an error term.

Parameter Estimate Standard Error t-Statistics p-value

XLKIntercept -0.06241 0.04530 -1.378 0.1687X1 0.92154 0.03479 26.486 < 2e−16 ***X2 0.16212 0.07642 2.121 0.0342 *X3 -0.20861 0.06451 -3.234 0.0013 **Adjusted R2 0.5112Std. Error Resid. 1.224F-statistic 258.3 < 2.2e−16 ***XLFIntercept -0.12349 0.03647 -3.386 0.0007 ***X1 1.20397 0.02801 42.983 < 2e−16 ***X2 -0.09319 0.06153 -1.515 0.1303X3 1.09226 0.05193 21.032 < 2e−16 ***Adjusted R2 0.7791Std. Error Resid. 0.9857F-statistic 868.7 < 2.2e−16 ***XLVIntercept 0.02707 0.03260 0.830 0.4070X1 0.67802 0.02504 27.077 < 2e−16 ***X2 -0.25550 0.05500 -4.645 4.02e−06 ***X3 -0.02012 0.04643 -0.433 0.6650Adjusted R2 0.5013Std. Error Resid. 0.8812F-statistic 248.3 < 2.2e−16 ***

Parameter Estimate Standard Error t-Statistics p-value

XLYIntercept -0.01003 0.03399 -0.295 0.7680X1 0.92990 0.02610 35.626 < 2e−16 ***X2 0.28379 0.05733 4.950 9.21e−07 ***X3 0.34332 0.04839 7.094 3.07e−12 ***Adjusted R2 0.6795Std. Error Resid. 0.9185F-statistic 522.5 < 2.2e−16 ***XLIIntercept -0.02413 0.02959 -0.815 0.4151X1 0.95666 0.02272 42.104 < 2e−16 ***X2 0.13644 0.04991 2.734 0.00641 **X3 0.33969 0.04213 8.063 3.01e−15 ***Adjusted R2 0.7393Std. Error Resid. 0.7996F-statistic 698.5 < 2e−16 ***XLPIntercept 0.02728 0.02600 1.049 0.2940X1 0.51915 0.01997 25.998 < 2e−16 ***X2 -0.26492 0.04386 -6.040 2.45e−09 ***X3 0.05549 0.03702 1.499 0.1340Adjusted R2 0.4857Std. Error Resid. 0.7027F-statistic 233.3 < 2.2e−16 ***

Parameter Estimate Standard Error t-Statistics p-value

XLEIntercept 0.006911 0.047031 0.147 0.8830X1 0.959066 0.036119 26.553 < 2e−16 ***X2 -0.017043 0.079336 -0.215 0.8300X3 0.418873 0.066969 6.255 6.75e−10 ***Adjusted R2 0.5989Std. Error Resid. 1.271F-statistic 365.8 < 2.2e−16 ***XLBIntercept -0.009194 0.038474 -0.239 0.8110X1 0.973241 0.029548 32.938 < 2e−16 ***X2 0.325417 0.064902 5.014 6.69e−07 ***X3 0.372338 0.054784 6.796 2.22e−11 ***Adjusted R2 0.5279Std. Error Resid. 1.146F-statistic 276.1 < 2.2e−16 ***XLUIntercept 0.01177 0.03742 0.314 0.7530X1 0.56233 0.02874 19.568 < 2e−16 ***X2 -0.34069 0.06312 -5.397 9.13e−08 ***X3 0.27622 0.05328 5.184 2.81e−07 ***Adjusted R2 0.647Std. Error Resid. 1.04F-statistic 451.8 < 2.2e−16 ***

Parameter Estimate Standard Error t-Statistics p-value

IWMIntercept -0.04679 0.02418 -1.935 0.0533 †X1 0.93277 0.01857 50.235 < 2e−16 ***X2 0.90894 0.04078 22.286 < 2e−16 ***X3 0.44481 0.03443 12.921 < 2e−16 ***Adjusted R2 0.375Std. Error Resid. 1.011F-statistic 148.6 < 2.2e−16 ***IYRIntercept -0.02801 0.04241 -0.661 0.5090X1 0.84445 0.03257 25.929 < 2e−16 ***X2 0.49090 0.07154 6.862 1.44e−11 ***X3 0.81080 0.06038 13.427 < 2e−16 ***Adjusted R2 0.5989Std. Error Resid. 1.146F-statistic 365.8 < 2.2e−16 ***

Table : Parameter estimated for the regression (8) and degree of significance. Thistable reports the key statistics of regression (8) performed on all 11 ETFs. The levelsof significance indicated in the table are as follows: *** indicates a significance levelnear 0, ** indicates a significance at the 0.001 level, * indicates a significance at the0.01 level, † indicates a significance at the 0.05% level. The F -statistic is tested on 3and 735 degrees of freedom.

Step 2: Collect Expert Opinions And Views

The second ingredient in our model consists in k ≥ 0 views formulated byanalysts.

The analysts express today their views about the evolution of factors up to theend of the time horizon. When the factors represent risk premia, typical analyststatements would be:

I ‘My research leads me to believe that the U.S. equity risk premium willslowly increase to 4% over the next two years. I am 90% confident thatthe risk premium will not go below 2% or go above 6% over the nexttwo years.’, or;

I ‘My research leads me to believe that the spread between 10-yearTreasury Notes and 3-month Treasury Bills will remain low over thenext year before gradually widening over the next 2 years to 200 basispoints in response to improving macroeconomic conditions. I am 90%confident that the spread will remain in a 50 basis point to 300 basispoint range.’

Mathematically, we can translate the system ofviews Z(t) into a system of ordinary differentialequations:

Z(t) = (aZ (t) + AZ (t)X (t))dt, Z(0) = z(9)

We introduce a white noise term to construct adynamic confidence interval around the views:

Z(t) = (aZ (t)+AZ (t)X (t))dt+ψ(t)Wdt, Z(0) = z ,(10)

where W is a k-dimensional Gaussian whitenoise process and ψ is a k × k matrix.

Dynamic View

Confidence Interval

Finally, we express (10) as a stochastic differential equation driven by theFt-Brownian motion W :

dZ(t) = (aZ (t) + AZ (t)X (t))dt + ΨZ (t)dW (t),

Z(0) = z , (11)

Enter Adam and Beth

The next task is to collect and model analyst views. Investor Irene worksclosely with two equity analysts: Adam and Beth.

Both of them follow closely the equity risk premium and will provide a view onthe future evolution of X (t).

I Adam is a new analyst;

I Beth already has an extensive experience.

To make the computations simpler, we assume that

Assumption

The noise generated by the asset price process and the factor process areindependent of the dynamic confidence interval around the views, implying thatΛΨ′Z = 0 and ΣΨ′Z = 0.

Adam’s View

Adam expresses the following view:

Over the next five years, I see the equity risk premium gradually goingback to 6%, which represents its long-term average in my opinion.Half of the move will take place in the next 12 month. I am 90%confident that the risk premium will not go below 5% or exceed 7%at the end of the five year horizon.

We model Adam’s view using an Ornstein-Uhlenbeck process:

dZ1(t) = (η1 − κ1Z1(t)) dt + ψ1W`+1(t), Z1(0) = z1, (12)

where we have set ` := n + m. The solution to this SDE is

Z1(t) = z1e−κ1t +η1

κ1(1− e−bt) + ψ1

∫ t

0

e−κ1(t−s)dW`+1(s).

The mean and variance of Z1(t) are respectively:

E [Z1(t)] = z1e−κ1t +η1

κ1

(1− e−κ1t

)Var [Z1(t)] =

ψ21

2κ1

(1− e−2κ1t

)We can readily translate Adam’s statement into a set of parameters for (13):

I κ1 = ln 2 because half of the move should take place over the next yearand the half life of the Ornstein-Ulhenbek process is ln 2

κ1;

I η1 = 0.06 ln 2 because the long term mean of the Ornstein-Uhlenbeck, η1κ1

,

should equal 6%;

I ψ1 = 0.011.645

√2κ1

1−e−2κ1T= 0.716% because Adam’s 90% confidence interval

is at -/+ 1% from the view.

Beth’s View

Beth’s opinion is the following:

The equity risk premium will increase to 10% over the coming year,before declining back to 7% at the end of the end of the five yearhorizon, which represents its long term-average. I am 90% confidentthat the risk premium will not go below 6% or exceed 8% at the endof the five year horizon.

We model this view with a generalized Ornstein-Uhlenbeck process (see Hulland White, 1994):

dZ2(t) = (η2(t)− κ2Z2(t)) dt + ψ2W`+2(t) (13)

The solution to this SDE is

Z2(t) = z2e−κ2t +

∫ t

0

e−κ2(t−u)η2(s)ds + ψ2

∫ t

0

e−κ2(t−s)ψ2dW`+2(s).

E [Z2(t)] = z2e−κ2t +

∫ t

0

e−κ2(t−s)η2(s)ds

Var [Z2(t)] =ψ2

2

2κ2

(1− e−2κ2t

)

Figure 31 suggests that the fifth order polynomial function

P(t) = 3× 10−5t5 − 0.001t4 + 0.0112t3 − 0.0572t2 + 0.1169t + 0.03 (14)

provides an adequate fit for Beth’s views.

y = 3E-­‐05x5 -­‐ 0.001x4 + 0.0112x3 -­‐ 0.0572x2 + 0.1169x + 0.03 R² = 0.99843

0.00%

2.00%

4.00%

6.00%

8.00%

10.00%

12.00%

0 1 2 3 4 5 6 7 8 9 10

Annu

al Risk Prem

ium

Time

Calibra2on Of The Func2on η2(t)

Figure : Polynomial Calibration Function For η2(t)

To fit Beth’s view of the overall evolution of the risk premium, we chose apolynomial function of order 4:

η2(t) = α0 + α1t + α2t2 + α3t3 + α4t4 (15)

Then,

E [Z2(t)] =e−κ2t

κ52

[−24α4 + κ2

(6α3 + κ2

(−2α2 + α1κ2 − α0κ

22

))]+α4

κ52

[24 + κ2t (−24 + κ2t (12 + κ2t (−4 + κ2t)))]

+1

κ42

[α3 (−6 + κ2t (6 + κ2t (−3 + κ2t)))]

+1

κ32

[α2(2 + κ2t(−2 + κ2t)) + α0κ2 + α1(−1 + κ2t))]

+ze−κ2t (16)

A Taylor expansion of this expression around t = 0 yields

E [Z2(t)] = z + (α0 − κ2z) t +1

2

(α1 − α0κ2 + κ2

2z)

t2

+1

6

(2α2 − α1κ2 + α0κ

22 − κ3

2z)

t3

+1

24

(6α3 − 2α2κ2 + α1κ

22 − α0κ

32 + κ4

2z)

t4

+1

120

(24α4 − 6α3κ2 + 2α2κ

22 − α1κ

32 + α0κ

42 − κ5

2z)

+ O(t6).

(17)

Selecting κ2 = 2 ln 2 = 1.3862, which implies a half life of 6 months,guarantees a rapid convergence back to Beth’s view, and matching the terms inexpression (17) with those in (14), we get:

I z = 0.03;

I α0 = 0.158488831;

I α1 = 0.047657811;

I α2 = −0.045696037;

I α3 = 0.011526497;

I α4 = −0.001236294.

Finally, Beth’s confidence interval implies that ψ2 = 0.011.645

√2κ2

1−e−2κ2T= 2.480%.

To summarize, we model the views that Adam and Beth expressed, as astochastic differential equation:

dZ(t) = (aZ (t) + AZ (t)X (t))dt + ΨZdW (t), Z(0) = z

with

aZ (t) =

(η1

η2(t)

), AZ (t) =

(−κ1 0 0−κ2 0 0

), ΨZ =

(0` ψ1 00′` 0 ψ2

).

where 0` is a `-element zero vector.

Step 3: Address The Impact Of Behavioral Biases

Warning! Behavioural finance has evidenced thatpsychological biases have an impact on thedecision-making process of individuals andorganisations:

I Hirschleifer (2001) classifies 22 differentpsychological biases into four categories:self-deception, heuristic simplification,emotion/affect and social;

I Shefrin (2005) identifies 12 main psychologicalpitfalls.

These biases will also impact the analyst views andopinions.

I How can we address these biases in amathematical model?

In this paper we consider four main psychological biases, examine theirpotential impact on the formulation and collection of analyst views, andpropose general modelling principles to correct their impact on the model:

(i) Overconfidence;

(ii) Excessive optimism;

(iii) Conservatism;

(iv) ‘Groupthink’.

In our follow-up paper (Davis and Lleo, 2015) we consider four additionalbiases, which are best addressed with jump-diffusion processes.

Overconfidence:

I Definition: tendency for individuals to be tooconfident in their beliefs.

I Impact on views: overconfidence may leadanalysts to overestimate the accuracy of theirviews, resulting in confidence bounds that are toonarrow.

I How to address: increase the magnitude of thediffusion term ΨZ used in (11) to widen theconfidence interval.

In our case,

I Adam has recently filled a survey designed to gauge his degree ofoverconfidence, such as the survey in Klayman et al. (1999). The surveyreveals that Adam’s 90% confidence interval corresponds in reality to a50% confidence interval. We adjust the value of the diffusion parameter

ψ1 accordingly. The updated value is ψ1 = 0.010.675

√2κ1

1−e−2κ1T= 1.746%,

where 0.675 is the parameter for a 50% confidence on a standard Normaldistribution.

I Looking at Beth’s extensive track record of views, the investor observesthat actual realisations of the market risk premium fall 25% of the timeoutside of Beth’ 90% confidence bounds. This suggests that the analystexhibits a small degree of overconfidence, which we can account for byadjusting the parameter ψ2 from 2.480% to 3.546%.

Excessive optimism:

I Definition: tendency for individuals to see theworld through ‘rose-colored glasses.’

I Impact on the views: Excessively optimisticanalysts will overestimate the probability ofscenarios that they perceive as positive.

I How to address:I Increase the magnitude of the diffusion term ΨZ ;I Not a perfect solution because Gaussian

confidence bounds are symmetric;I Better solution: use jump-diffusion processes to

model the asymmetry.

In our case, Beth and Adam may exhibit some degree of excessive optimismwhen they forecast a 6% to 7% risk premium.

We could widen the standard deviation of the confidence interval by adjustingthe parameters ψ1 and ψ2. However, this solution is not ideal because itincreases the confidence bounds on both sides of the expert’s view. Thisprovides a strong motivation for including non-Gaussian confidence intervals inour model.

After correcting for overconfidence, the standard deviation around Beth’s viewis 0.869% implying a 39% probability that the risk premium will end up below6% and a 29% probability that the risk premium will end up below 5%. Both ofthis probabilities are large enough to account for some degree of excessiveoptimism and investor Irene decides not to increase ψ2.

After correcting for overconfidence, the standard deviation around Adam’s viewis 1.483% implying a 28% probability that the risk premium will end up below5%, comparable to Beth’s probability. Irene decides not to increase ψ1.

Conservatism a.k.a anchoring-and-adjustment:

I Definition: tendency to overweight priorinformation relative to newly released information,often resulting in a failure to update one’s beliefsin a Bayesian manner.

I Impact on views: affects the point estimate givenby analysts as well as the confidence interval. It isa serious concern especially for multiperiod orcontinuous time models.

I How to address: our model does not require theanalyst to update their views: analysts formulatetheir views at the initial stage when the model isparametrised and they are not asked to updatethem after. Hence, any effect of the conservatismbias remains confined to the initial set of views.After that, the treatment of the views asobservations in the Kalman filter is Bayesian, soobservations will be incorporated accurately intothe model.

Groupthink:

I Definition: groupthink leads people in groups toact as if they value conformity over quality whenmaking decisions (Shefrin, 2010).

I Impact on views: finance professionals are at aparticular risk of exhibiting groupthink or offollowing the herd, for fear of falling behind therest of their colleagues.

I How to address: to reduce the effects ofgroupthink,

1. Add a correlation structure between the views viaΨZ ;

2. Seek dissenting analysts whose views differmarkedly from the majority;

3. Introduce historical and stress test scenarios.

Stress Test Scenarios

Adding stress test scenarios to the views is consistent with the guidingprinciple that it is more important to avoid large drawdowns in difficulttimes than to capture the highest returns in bullish markets.

Stress test scenarios may also be helpful to mitigate the effect of behaviouralbiases such as narrow framing, excessive optimism or groupthink.

Stress test scenarios should have a wide, markedly skewed confidence interval,because the realised value of X (t) is unlikely to be as extreme as suggested bythe stress test scenario.

It is however possible that the realised value of X (t) could be worse than thestress test scenario. For this reason, it is important to establish a two-sidedconfidence interval around the stress test scenario.

In our case, we model groupthink by adding both a correlation structure to theconfidence interval around the views, and a historical scenario.

The diffusion matrix ΨZ , subject to a correlation ρ, becomes:

ΨZ =

(0` ψ1 0

0` ρψ2

√1− ρ2ψ2

), so that ΨZΨ′Z =

(ψ2

1 ρψ1ψ2

ρψ1ψ2 ψ22

). (18)

The difficulty here is in estimating ρ.

I A possibility would be to use the historical correlation correlation of theforecasting error, defined as the difference between the forecast and theobservation, across analysts.

I Investor Irene considers that groupthink is not prevalent among heranalysts and picks ρ = 0.5.

As an additional measure, Irene decides to include a stress test scenario basedon data from the 2008 financial crisis.

She models this scenario as a generalized Ornstein-Uhlenbeck process:

dZ3(t) = (η3(t)− κ3Z3(t)) dt + ψ3W`+3(t) (19)

Figure 47 suggests a fifth order polynomial function

Q(t) = 0.0362t5 − 0.437t4 + 1.7867t3 − 2.7691t2 + 1.2301t − 0.0137

to model the stress test scenario, as higher order polynomial do not improvethe fit significantly.

y = 0.0362x5 -­‐ 0.437x4 + 1.7867x3 -­‐ 2.7691x2 + 1.2301x -­‐ 0.0137 R² = 0.32765

-­‐80.00%

-­‐60.00%

-­‐40.00%

-­‐20.00%

0.00%

20.00%

40.00%

60.00%

80.00%

100.00%

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Annu

al Risk Prem

ium

Time

Calibra2on Of The Func2on η3(t)

Figure : Polynomial Calibration Function For η3(t)

To fit this polynomial function, we express the function η3 as a fourth orderpolynomial:

η3(t) = β0 + β1t + β2t2 + β3t3 + β4t4 (20)

This yields an expectation and a Taylor expansion of the same form as(16)-(17). Selecting κ3 = 1

4ln 2 = 0.138629436, implies a half life of 5 years.

Matching the terms in expression (17) with those in (14), we get:

I z = −0.0137;

I β0 = 1.228200777;

I β1 = −5.367671931;

I β2 = 4.976221228;

I β3 = −1.500310786;

I β4 = 0.120418936.

The confidence interval around the stress test scenario needs to be wideenough to include the analyst views.

Setting Ψ3 = 117% guarantees that the 90% confidence interval around Adamand Beth’s views is also in the confidence interval around the stress testscenario.

Finalising The Views

We model the views expressed by Adam and Beth and the stress test scenariousing the stochastic differential equation

dZ(t) = (aZ (t) + AZ (t)X (t))dt + ΨZdW (t), Z(0) = z

with

aZ (t) =

η1

η2(t)η3(t)

, AZ (t) =

−κ1 0 0−κ2 0 0−κ3 0 0

,

ΨZ =

0` ψ1 0 0

0′` ρψ2

√1− ρ2ψ2 0

0′` 0 0 ψ3

,

where 0` is a `-element zero vector and ` = n + m = 14.

In fully expanded form,

aZ (t) =

0.06 ln 20.3664330 + 0.8579469t − 0.4421762t2 + 0.0891590t3 − 0.0081678t4

2.9272107 + 24.105017t + 57.000278t2 + 50.246147t3 + 19.511842t4 + 3.1183905t5

,

AZ (t) =

− ln 2 0 0−12 ln 2 0 0−12 ln 2 0 0

,

ΨZ =

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.017465 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0.005120 0.008868 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1.17

.

We have finished collecting the views and debasing them.

Step 4: Blend Data With Opinions, And Filter

We start by constructing the observation vector combining

1. m investable risky assets S1(t), . . . , Sm(t);

2. k analyst views Z1(t), . . . ,Zk(t), and;

Asset prices are not directly suitable for use in a Kalman filter because of theirgeometric dynamics. On the other hand, excess returns, defined as

si (t) = ln(Si (t)), i = 1, . . . ,m

are linear processes that can be used directly in the Kalman filter.

The dynamics of the excess returns is given by

dsi (t) =

[(a(t) + A(t)X (t))i −

1

2ΣΣii (t)′

]dt +

d∑k=1

Σik(t)dWk(t)

si (0) = ln si , i = 1, . . . ,M. (21)

The pair of processes (X (t),Y (t)), where

Yi (t) =

si (t) = ln Si (t)

S0(t), i = 1, . . . ,m,

Zi−m(t), i = m + 1, . . . ,m + k(22)

takes the form of the ‘signal’ and ‘observation’ processes in a Kalman filtersystem.

Consequently the conditional distribution of X (t) is normal N(X (t),P(t))where X (t) = E[X (t)|FY

t ] satisfies the Kalman filter equation and P(t) is adeterministic matrix-valued function.

We express the dynamics of Y (t) succinctly as

dY (t) = (aY (t) + AY (t)X (t))dt + Γ(t)dW (t), Y (0) = y0, (23)

where

aY =

(a

aZ

), AY =

(A

AZ

), Γ =

(Σ

ΨZ

).

Next, we define the filtration FYt = σS(s),Z(s),U(s), 0 ≤ s ≤ t generated

by the observation process Y (t) alone.

Proposition

Define processes Y 1(t), Y 2(t) ∈ Rm as follows.

Y 1(t) = AY (t)X (t)dt + Γ(t)dW (t), Y 1(0) = 0 (24)

Y 2(t) = aY (t)dt, Y 2(0) = y0 (25)

so that Y (t) = Y 1(t) + Y 2(t). Also, defineYit = σY i (u), 0 ≤ u ≤ t, i = 1, 2. Then(i) The processes Y 1,Y 2 are each adapted to the filtration FS

t .(ii) For any bounded measurable function f and t ≥ 0,

E[f (X (t))|FYt ] = E[f (X (t))|Y1t ].

In the present case we need to assume that X0 is a normal random vectorN(µ0,P0) with known mean µ0 and covariance P0, and that X0 is independentof the Brownian motion W .

I The estimation of µ0 and P0 depends closely dependent on the choice offactors: we will discuss estimation procedures for these parameters inrelation to the applications considered in our paper.

The processes (X (t),Y 1(t)) satisfying (3) and (24) and the filtering equations,which are standard, are stated in the following proposition.

Proposition (Kalman Filter:)

The conditional distribution of X (t) given FYt is N(X (t),P(t)), calculated as

follows.(i) The innovations process U(t) ∈ Rm+k defined by

dU(t) =(ΓΓ′(t)

)−1/2(dY (t)− AY (t)X (t)dt), U(0) = 0 (26)

is a vector Brownian motion.(ii) X (t) is the unique solution of the SDE

dX (t) = (b(t) + B(t)X (t))dt + Λ(t)dU(t), X (0) = µ0, (27)

whereΛ(t) =

(ΛΓ′(t) + P(t)AY (t)′

) (ΓΓ′(t)

)−1/2.

(iii) P(t) is the unique non-negative definite symmetric solution of the matrixRiccati equation

P(t) = ΛΥ⊥(s⊥)′Λ′(t)− P(t)AY (t)′(ΓΓ′(t)

)−1AY (t)P(t)

+(

B(t)− Λ(t)Γ(t)′(ΓΓ′(t)

)−1AY (t)

)P(t)

+P(t)(

B(t)′ − AY (t)′(ΓΓ′(t)

)−1Γ(t)Λ′(t)

), P(0) = P0,

where Υ⊥(t) := I − Γ(t)′ (Γ′(t)Γ(t))−1

Γ(t).

Now the Kalman filter has replaced our initial state process X (t) by anestimate X (t) with dynamics given in (27).

To recover the observation process, we use (26)

dY (t) = dY1(t) + dY2(t)

= (aY (t) + AY (t)X (t))dt + (ΓΓ′(t))1/2dU(t),

Y (0) = y0. (28)

The next step is to extract the dynamics of s(t), Z(t), H(t) and S(t) from theobservation vector Y (t). First observe that

ΓΓ′ :=

ΣΣ′ ΣΨ′Z ΣΨ′HΨZΣ′ ΨZΨ′Z ΨZΨ′HΨUΣ′ ΨHΨ′Z ΨHΨ′H

,

and define the (m + k)× (m + k) matrix (ΓΓ′)1/2 as

(ΓΓ′)1/2 :=

Σ

ΨZ

ΨH

This implies that

ΓΓ′ :=

ΣΣ′ ΣΨ′Z ΣΨ′HΨZΣ′ ΨZΨ′Z ΨZΨ′HΨUΣ′ ΨHΨ′Z ΨHΨ′H

=

ΣΣ′ ΣΨ′Z ΣΨ′HΨZ Σ′ ΨZ Ψ′Z ΨZ Ψ′HΨUΣ′ ΨHΨ′Z ΨHΨ′H

. (29)

As a result, the SDEs for s(t), Z(t) and S(t) are given by

dsi (t) =

[(a(t) + A(t)X (t))i −

1

2ΣΣii (t)′

]dt +

m+k∑k=1

Σik(t)dUk(t),

si (0) = ln si ,

dZ(t) = (aZ (t) + AZ (t)X (t))dt + ΨZ (t)dU(t), Z(0) = z ,

dSi (t)

Si (t)=

(a(t) + A(t)X (t)

)idt +

m+k∑k=1

Σik(t)dUk(t), Si (0) = si . (30)

Derive The Prior Expected Value of the Factor Process µ0

Continuous time Fund Separation results identify the Kelly portfolio as thecornerstone of all investment strategies.

If we knew the composition of the Kelly portfolio ex ante, we could use areverse optimization argument to derive the equilibrium risk premium.

A significant advantage of this approach is that it is preference-free: we do notneed to know anything about the level of risk-sensitivity of an investor or groupof investors.

The Kelly portfolio’s asset allocation at time 0 is:

hK (0) =(ΣΣ′

)−1(

a + AX (0))

The factor level X (0) is not observable but we could back an estimate µ0 outof the allocation of the Kelly portfolio, provided we know hK and provided A′Ais invertible:

µ0 = (A′A)−1A′(

ΣΣ′hK (0)− a)

(31)

Typically A′A will be invertible because n < m. The accuracy of the estimateµ0 depends on the accuracy of the estimates for ΣΣ′, a and A. It also dependscrucially on the allocation hK (0), which may not be directly observable.

We could construct a portfolio that approximates the Kelly strategy usingCover’s ‘universal portfolio’ (Cover, 1991; Cover and Thomas, 2006, Chapter16) which is shown to converge asymptotically to the Kelly strategy.

The major advantage of this method is that it does not make any assumptionon the shape of the return distribution and does not imply any view aboutfuture performance.

Step 5: Optimize The Portfolio

Using the idea developed in Step 1, we express and solve a stochastic controlproblem in which X (t) is replaced by X (t) and the dynamic equation (3) bythe Kalman filter. This very old idea in stochastic control goes back at leastto Wonham (1968).

Concretely, we can ‘just’ plug the estimations into our favorite stochasticinvestment management model, for us a risk-sensitive asset managementmodel maximising the criterion

Jθ,T (v , x ; h) :=

(−1

θ

)lnEe−θ ln V (T ;h) (32)

where V (t) is the value of the portfolio and θ ∈ (−1, 0) ∪ (0,∞) is the risksensitivity.

We like the risk-sensitive criterion because it has two complementaryinterpretations:

1. Dynamic “mean-variance” optimisation. To see this, perform a Taylorexpansion of the risk sensitive criterion J around θ = 0:

J(x , t, h; θ) = E [ln V (t, h)]− θ

2var [ln V (t, h)] + O(θ2) (33)

2. Utility Maximization:

E[e−θ ln V (t)

]=: E [U(V (t))]

defines the expected utility, under a Hyperbolic Absolute Risk Aversion(HARA) utility function, derived from the relative position of the investor’sportfolio with respect to its benchmark at time t.

In our case, it is easy to show that all the results derived in Kuroda and Nagai(2002) (see also Davis and Lleo, 2014) still hold with only minor modifications.In particular,

I The value function Φ is the C 1,2 solution to associated HJB PDE. It hasthe form Φ(t, x) = e−θΦ(t,x), where

Φ(t, x) =1

2x ′Q(t)x + x ′q(t) + k(t), (34)

I There is a unique Borel measurable maximiser h(t, x , p) for(t, x , p) ∈ [0,T ]× Rn × Rn, given by:

h(t, x , p) =1

1 + θ

(ΣΣ′

)−1 [a + AX (t)− θΣΛ′(t)p)

]=

1

1 + θ

(ΣΣ′

)−1[a + AX (t)− θΣΛ′(t)p)

].

I The maximiser is optimal, meaning h∗(t, X (t)) = h(t, X (t),DΦ).

Kelly Strategies

Take the limit as θ → 0 to recover the Kelly portfolio :

h =(

ΣΣ′)−1 (

a(t) + A(t)X (t)). (35)

... and we even have a Fractional Kelly Strategy:

I invest a fraction 1θ+1

in the Kelly portfolio;

I invest a fraction θθ+1

in an ‘intertemporal hedging’ portfolio

Note that we do not have a Fund Separation theorem in the classical sensebecause the ‘intertemporal hedging’ portfolio is not preference-free.

Optimising Investor Irene’s Portfolio

Investor Irene has a ten-year horizon and is willing to invest one third of herwealth in the Kelly portfolio.

The Fractional Kelly strategy implies that the investor’s optimal investmentstrategy at time 0, h∗(0), is an allocation of 1/3 of initial wealth in the Kellyportfolio hK (0) and 2/3 of initial wealth in an intertemporal hedging portfoliohI (0), which means that Irene’s risk sensitivity is θ = 2.

The Kelly allocation at time 0 given an initial state X (0) = µ0 is:

hK (0) = (ΣΣ′)−1(

a + AX (0))

=

(−50.87%,−140.39%, 117.42%, 116.25%,−52.31%, 58.45%, 74.29%, 22.9%, 7.58%,−72.86%, 83.19%

)′.

The Kelly portfolio has net leverage funded by a the short position in themoney market instrument equal to 63.64% of the investor’s wealth.

Irene’s intertemporal hedging portfolio is

hI (0) = (ΣΣ′)−1ΣΛ′(0)(

q(0) + Q(0)X (0))

=

(5.73%, 19.07%,−7.07%, 19.25%,−15.31%, 23.61%, 14.71%, 11.93%,−6.78%,−37.69%,−7.87%

)′.

The ETF positions may seem large when looked at individually. However, oncethese positions are netted, we see that the intertemporal hedging portfolio isstill invested at 80.41% in the money market instrument.

As a result, the overall portfolio allocation is

h∗(0) =1

3hK (0) +

2

3hI (0) =

(−13.13%,−34.11%, 34.43%, 51.58%,−27.65%, 35.22%, 34.56%, 15.58%,−2.00%,−49.41%, 22.48%

)with 32.45% allocated to the money market instrument.

The next table presents Irene’s asset allocation, including the Kelly portfolioand intertemporal hedging portfolio (IHP), for six models:

1. the Universal Portfolio due to Cover (1991) and already used to get theprior vector of factors X (0);

2. the Risk-Sensitive Asset Management Model proposed by Bielecki andPliska (1999) and Kuroda and Nagai (2002), with full observation.

3. the Risk-Sensitive Asset Management Model with partial observation dueto Nagai and Peng (2002). Here, the asset prices supply the sole source ofobservations;

4. Black-Litterman in Continuous Time: the optimal investment modelproposed in Davis and Lleo (2013). It includes Adam and Beth’s views,but ignores behavioural biases and the stress test scenario;

5. the Behavioural Black-Litterman model addressing the biases in Adam andBeth’s views, but without the stress test scenario;

6. the Behavioural Black-Litterman, derived in this paper, including thestress test scenario.

Asset Class Universal Kelly Portfolio Risk-Sensitive Asset Management Risk-Sensitive Asset ManagementPortfolio with full observation with partial observation

IHP Optimal Portfolio IHP Optimal PortfolioXLK 8.76% -50.87% -1.37% -17.87% 2.65% -15.19%XLF 8.91% -140.45% 3.41% -44.55% 17.39% -35.23%XLV 9.11% 117.43% -1.03% 38.46% -1.20% 38.34%XLY 9.16% 116.25% 0.12% 38.83% 17.66% 50.52%XLI 9.04% -52.34% -0.24% -17.61% -2.25% -18.94%XLP 9.09% 58.43% 0.33% 19.70% 7.02% 24.16%XLE 9.29% 74.27% 1.26% 25.60% 16.45% 35.72%XLB 9.18% 22.89% 1.18% 8.42% 12.66% 16.07%XLU 9.11% 7.54% -0.21% 2.37% -1.25% 1.68%IWM 9.12% -72.85% 6.72% -19.81% -39.64% -50.71%IYR 9.23% 83.18% 1.17% 28.51% -12.36% 19.48%Total risky allocation 100.00% 163.47% 11.34% 62.05% 17.14% 65.92%Money Market 0.00% -63.47% 88.66% 37.95% 82.86% 34.08%

Asset Class Black-Litterman in Continuous Behavioural Black-Litterman Behavioural Black-LittermanTime (with stress test scenario)

IHP Optimal Portfolio IHP Optimal Portfolio IHP Optimal PortfolioXLK 5.64% -13.20% 5.73% -13.13% 5.73% -13.13%XLF 29.65% -27.06% 19.07% -34.11% 19.07% -34.11%XLV -0.20% 39.01% -7.07% 34.43% -7.07% 34.43%XLY 29.06% 58.12% 19.25% 51.58% 19.25% 51.58%XLI -3.70% -19.91% -15.31% -27.65% -15.31% -27.65%XLP 13.93% 28.77% 23.61% 35.22% 23.61% 35.22%XLE 28.68% 43.88% 14.71% 34.56% 14.71% 34.56%XLB 20.47% 21.28% 11.93% 15.58% 11.93% 15.58%XLU -1.54% 1.49% -6.78% -2.00% -6.78% -2.00%IWM -70.93% -71.56% -37.69% -49.41% -37.69% -49.41%IYR -19.80% 14.53% -7.87% 22.48% -7.87% 22.48%Total risky allocation 31.27% 75.34% 19.59% 67.55% 19.59% 67.55%Money Market 68.73% 24.66% 80.41% 32.45% 80.41% 32.45%

Step 6: Monitor The Portfolio

Monitoring begins once the initial portfolio allocation is set and implemented.

Typically, long-term investors conduct a full asset allocation review once a yearor once every few years. A full review involves starting the portfolio selectionprocess over: identifying the objectives and constraints, listing the investableassets, gathering financial market data and experts, before embarking on thefull portfolio selection process.

In between full reviews, investors set up an ongoing monitoring of the portfolioagainst its target asset allocation. The filtering step in our procedurecontributes to this ongoing monitoring. We can use the filter to incorporatenew stock price data, measure the accuracy of the expert opinions and comparethe performance of the portfolio against today’s ‘optimal’ portfolio in real time.

Wrap-Up

I The formulation of beliefs about futureperformance of financial markets is an integralpart of the portfolio selection process.

I In addition to historical price data, we could useexpert opinions and even Big Data analytics toformulate these views... as long as we canformulate the problem in terms of linear filtering!

I Psychological biases have an impact on opinionsand views: we can address them in our model;

I The filtering step is separable from the portfolioestimation problem: just plug your estimate inyour favourite model!

I Jump-diffusion processes are required to make themodel more realistic.

Model Spe-­‐cifica.on: n, m, k

Data

Collec.on

Kalman Filter es.ma.on:

Model Input:

discounted price

process

Sta.s.cal es.ma.on: •  b, B, Λ•  ã,Ã,Σ

Model Input: views

Stochas.c Op.miza.on

Inputs: .me horizon T,

risk aversion θ

Stage 1 in Markowitz (1952) Stage 2 in Markowitz (1952)

Model output: asset

alloca.on

Monitoring

Full Asset Alloca.on review

No

Yes

View collec.on: aZ,AZ,ΨZ

X(t)

Z(t)

h*(t)!S t( )

Debiasing:

aZ,AZ,ΨZ,ζZ

Thank you!

2

Any question?

2Copyright: W. Krawcewicz, University of Alberta

T.R. Bielecki and S.R. Pliska. Risk-sensitive dynamic asset management.Applied Mathematics and Optimization, 39:337–360, 1999.

F. Black and R. Litterman. Global portfolio optimization. Journal of FixedIncome, 1991.

F. Black and R. Litterman. Global portfolio optimization. Financial AnalystsJournal, 48(5):28–43, Sep/Oct 1992.

T.M. Cover. Universal portfolios. Mathematical Finance, 1(1):1–29, 1991.

T.M. Cover and J.A. Thomas. Elements of Information Theory. Wiley Series inTelecommunications and Signal Processing. Wiley-Interscience, 2 edition,2006.

M.H.A. Davis and S. Lleo. Black-Litterman in continuous time: The case forfiltering. Quantitative Finance Letters, 1(1), 2013.

M.H.A. Davis and S. Lleo. Risk-Sensitive Investment Management, volume 19of Advanced Series on Statistical Science and Applied Probability. WorldScientific Publishing, 2014.

M.H.A. Davis and S. Lleo. Behaviouralizing Black-Litterman part II: Thejump-diffusion model. 2015.

E. Fama and K. R. French. Common risk factors in the returns on stocks andbonds. Journal of Financial Economics, 33(1):3–56, 1993.

R. Frey, A. Gabih, and R. Wunderlich. Portfolio optimization under partialinformation with expert opinions. International Journal of Theoretical andApplied Finance, 15(1):1250009, 2012.

D. Hirschleifer. Investor psychology and asset pricing. Journal of Finance, 56(4):1533–1597, 2001.

J. Hull and A. White. Branhcing out. Risk, 7:34–37, 1994.

J. Klayman, J.B. Solland C. Gonzalez-Vallejo, and S. Barlas. Overconfidence:It depends on how, what and whom you ask. Organizational Behavior andHuman Decision Processes, 79:216–247, 1999.

K. Kuroda and H. Nagai. Risk-sensitive portfolio optimization on infinite timehorizon. Stochastics and Stochastics Reports, 73:309–331, 2002.

H. Markowitz. Portfolio selection. The Journal of Finance, 7(1):77–91, March1952.

H. Nagai and S. Peng. Risk-sensitive dynamic portfolio optimization withpartial information on infinite time horizon. The Annals of AppliedProbability, 12(1):173–195, 2002.

L.J. Savage. The Foundations of Statistics. Dover, 1 edition, 1954.

L.J. Savage. The Foundations of Statistics. Dover, 2 edition, 1971.

H. Shefrin. Behavioral Corporate Finance. McGraw-Hill, 2005.

H. Shefrin. How psychological pitfalls generated the global financial crisis. InL. Siegel, editor, Voices of Wisdom: Understanding the Global FinancialCrisis. Research Foundation of CFA Institute, 2010.

W.M. Wonham. On the separation theorem of stochastic control. SIAMJournal on Control, 6(2):312–326, 1968.