Practical PricingM.J. Brennan

Institute of Financial Studies

SWUFE, Chengdu

July 2017

Practical Pricing

• A Simple Asset Pricing Model – the Stochastic Liquidation Model – with Yuzhao Zhang

• Asset Pricing for Small Economies – a data driven approach - with Alex Taylor

Graham and Harvey (2001) report that 73.5% of firms in their survey `always or almost always use' the CAPM, and its use was much more common among large firms.

Asset Pricing:

• Originally a normative exercise – DCF growing out of work of Fisher – but what is the discount rate when cash flows are risky?

• CAPM, 1964 provides the answer answer

Does the CAPM work (in the US)?

Yes, if we allow for the changing and uncertain horizon of investors •

Asset Pricing with Stochastic Liquidation (Brennan and Zhang, 2016)

A. (Annualized) mean returns vary with the horizon (return interval)

Betas also vary with the horizonB.

So, fit of CAPM varies with the horizonC.

Actual and Excess Returns Predicted by the CAPM using monthly returns and annual returns FF25 1926-2013

Not enough spread in monthly betas

Asset Pricing with Stochastic Liquidation (Brennan and Zhang, 2016)

Representative Agent:

FoC:

𝜋𝜏 probability of liquidation after τ periods

𝑀𝑎𝑥 𝐸

𝜏=1

𝜏∗

𝜋𝜏𝑈(𝑊𝜏)

𝐸

𝜏=1

𝜏∗

𝜋𝜏(1 − 𝑏𝑅𝑚𝜏 )(𝑅𝑗

𝜏 − 𝑅𝐹𝜏) = 0, 𝑗 = 1,……𝑁

Approximate 𝜋𝜏 with a 2 parameter Gamma function and estimate risk aversion parameter, b, by GMM

Initial sample: FF25 portfolios 1926-2013

quadratic utility

Agent does not know liquidation date:

0

0.2

0.4

0.6

0.8

1 4 8 12 16 20 24

Horizon (months)

Probabilities of Liquidation Date for Different Sample Periods

1926-47 1947-68 1968-88 1989-2013

Average horizon of investors has been falling

1926-2013 1926-62 1963-2013 1926-47 1947-68 1968-88 1988-2013

b 1.67 1.69 2.48 0.94 3.38 1.86 3.06(6.43) (8.94) (11.27) (4.27) (10.24) (3.71) (2.39)

J 28.82 30.48 29.62 13.72 18.42 15.58 10.06p-value [0.17] [0.11] [0.12] [0.88] [0.62] [0.79] [0.98]

Parameter estimates for Stochastic Liquidation Model 1926 – 2013, 25 FF portfolios

Predicted and actual excess returns 1923-2013

CAPM FF3 SL Model

γ1 6.11

0.67

γ2 5.29

0.43

brmrf 2.1 1.17 1.67

(4.44) (1.96) (6.43)

bsmb 0.62

(0.74)

bhml 4.59

(4.97)

J 43.09 43.58 28.82

p-value [0.01] [0.00] [0.17]

MAE 0.022 0.018 0.015

Comparison with classical asset pricing models – FF25 1926- 2013

FF25 FF25 + 38 Industries

γ1 6.11 11.000.67 0.91

γ2 5.29 9.330.43 0.87

Mean horizon months 12.3 12.9

brmrf 1.67 1.92(6.43) (6.87)

J 28.82 36.21[0.17] [0.99]

MAE 0.015 0.013

Extending the sample to include 38 Industry portfolios 1926 - 2013

The key parameters are stable as we extend the sample

FF25 MEOP MEINV OPINV

b 2.48 2.35 2.57 2.11(11.27) (6.66) (4.56) (5.21)

Mean horizon 7.4 5.0 4.0 5.3

J 29.62 21.16 22.8 23.6p-value [0.12] [0.45] [0.35] [0.31]

MAE 0.017 0.017 0.019 0.02

MEOP MEINV OPINV

SLM CAPM FF3 FF5 SLM CAPM FF3 FF5 SLM CAPM FF3 FF5

J 21.16 29.5 32.15 17.89 22.8 37.61 26.74 19.99 23.6 35.38 33.9 18.07

p-value [0.45] [0.20] [0.07] [0.59] [0.35] [0.04] [0.22] [0.46] [0.31] [0.06] [0.05] [0.58]

MAE 0.017 0.019 0.019 0.017 0.019 0.022 0.024 0.01 0.02 0.023 0.023 0.012

Alternative Portfolios 1963 - 2013

Parameter Stability

Comparison with Classical Models

More recent work

Allow the two parameters of Gamma distribution to be time varying, depending on NYSE turnover so then probability distribution of liquidation dates tied to turnover

Estimates highly significant:

𝜽𝟏 = 11.79 – 4.94 NYSETOt

𝜽𝟐 = -0.46 + 20.00 NYSETOt

Imply mean horizon decreasing in turnovermean horizon decreasing over time

Time series correlation between NYSETOt and model implied turnover is 0.97

Summary

The Stochastic Liquidation Model

• Is parsimonious – 3 parameters

• Robust across samples• FF3, FF5 have very different prices of risk across samples

• Simple theoretical basis• Average horizon declining over time consistent with increased

turnover rates.

Asset Pricing in a Small Economy

• Canadian motivation

• A Data Driven approach

• Theory• Empirical Evidence

• US• Chile

Asset Pricing in Canada

What is the market portfolio? US, Canada, (US + Canada) – or something else?

• Asset Pricing in a Small Economy: A Test of the Omitted Assets Model, Brennan and Schwartz, 1986

• Integration vs Segmentation in the Canadian Stock Market, Jorion and Schwartz, 1986

• Test for Integration: H0: γ2 = 0 in ത𝑅𝑗 − 𝑅𝐹 = 𝛾0 + 𝛾1 𝛽𝑗𝐺𝑙𝑜𝑏𝑎𝑙 + 𝛾2 𝛽𝑗

𝐶𝑎𝑛𝑎𝑑𝑎

• Test for Total Segmentation: H0: γ2 = 0 in ത𝑅𝑗 − 𝑅𝐹 = 𝛾0 + 𝛾1 𝛽𝑗𝐶𝑎𝑛𝑎𝑑𝑎 + 𝛾2 𝛽𝑗

𝑈𝑆

• Find that Canadian and US markets not integrated and not totally segmented

Asset Pricing in Chile - the domestic CAPM• CAPM – what is market portfolio?? Chilean market, world market, Latin America?

-5%

0%

5%

10%

15%

20%

25%

0 0.2 0.4 0.6 0.8 1 1.2 1.4

Beta

Annualized Average Returns on 18 Industry portfolios and CAPM Betas: Chile 1992 - 2016

The Chilean CAPM does not seem to work too well

The alternative

Rely less on theory • – the market portfolio

Rely instead on • data to estimate the pricing kernelIn CAPM the pricing kernel is the market portfolio•

Seek a • robust estimate of pricing kernel that does not rely on exact estimation of variance covariance matrix

Analogue with robust portfolio selection • – Uppal et al.

Estimating the Pricing Kernel

Basic theory:

Pricing kernel definition: 𝐸 𝑚𝑡𝑅𝑗𝑡 = 1 ∀ 𝑗, 𝑡

implies 𝐸 𝑚𝑡(𝑅𝑗𝑡 − 𝑅𝐹𝑡 ≡ 𝐸 𝑚𝑡𝑟𝑗𝑡 = 0 ∀ 𝑗, 𝑡

Write (scaled) pk as: 𝑚𝑡 = 1 + 𝑧𝑡, where 𝐸 𝑧𝑡 = 0

𝐸 𝑚𝑡𝑟𝑗𝑡 = 𝐸 𝑚𝑡 𝐸 𝑟𝑗𝑡 + 𝑐𝑜𝑣(𝑚𝑡, 𝑟𝑗𝑡) = 0

𝜇𝑗 = −𝑐𝑜𝑣(𝑚𝑡, 𝑟𝑗𝑡)

𝜇𝑗 = −𝑐𝑜𝑣(𝑚𝑡, 𝑟𝑗𝑡)

Consider the regression of excess returns on the pricing kernel innovation, 𝑧𝑡:

𝑟𝑗𝑡 = 𝑎𝑗 + 𝑏𝑗𝑧𝑡 + 휀𝑗𝑡

Then 𝑏𝑗 ≜𝑐𝑜𝑣(𝑟𝑗𝑡,𝑧𝑡)

𝜎𝑧2 =

−𝜇𝑗

𝜎𝑧2 , 𝑎𝑗 = 𝜇𝑗

So,

𝑟𝑗𝑡 − 𝜇𝑗 =−𝜇𝑗

𝜎𝑧2𝑧𝑡 + 휀𝑗𝑡

𝑟𝑗𝑡 − 𝜇𝑗 =−𝜇𝑗

𝜎𝑧2 𝑧𝑡 + 휀𝑗𝑡

Consider the constrained cross-section regression of 𝑟′𝑗𝑡 ≜ 𝑟𝑗𝑡 − 𝜇𝑗 on −𝜇𝑗.

The regression coefficient is proportional to Ƹ𝑧𝑡, an estimate of the pricing kernel innovation.

We can convert this into the return on a portfolio whose return is perfectly correlated with the pk innovation.

Now we have a standard (CAPM like) pricing equation:𝑅𝑗𝑡 − 𝑅𝐹𝑡 = 𝛽𝑗𝑧 𝑅𝑧𝑡 − 𝑅𝐹𝑡 + 𝑒𝑗𝑡

The cross section regression to identify the pk innovation, 𝑧𝑡:

𝑟𝑗𝑡 − 𝜇𝑗 =−𝜇𝑗

𝜎𝑧2 𝑧𝑡 + 휀𝑗𝑡

Problems

1. The independent variable, mean return, is measured with error – biases down Ƹ𝑧𝑡bias constant each period

2. Ignoring measurement error, the estimator can be written as

Ƹ𝑧𝑡 =−𝝁′𝜴−𝟏𝒓′𝒕

𝝁′𝜴−𝟏𝝁=

𝑧𝑡

𝜎𝑧2 +

𝝁′𝜴−𝟏𝝐′𝒕

𝝁′𝜴−𝟏𝝁= 𝑧𝑡 + 𝜂𝑡

3. How important is the estimation error term?

Some analytical results

If • 𝑬 𝝐𝒋𝜺𝒌 = 𝟎 (diagonal assumption)

−𝒄𝒐𝒗 𝒓𝒋, ො𝒛 = 𝝀𝝁𝒋

where λ is coefficient from regression of true mean returns, 𝝁𝒌 ,on estimated means, mk

And −𝒄𝒐𝒗 𝒓ො𝒛, ො𝒛 = 𝝀𝝁𝒓ො𝒛 where 𝒓ො𝒛 is return on estimated `pricing kernel portfolio’

This implies:

𝝁𝒋 = 𝜷𝒋ො𝒛 𝝁𝒓ො𝒛 `CAPM’ type relation wrt estimated pricing kernel

• If 𝑬 𝝐𝒋𝜺𝒌 ≠ 𝟎

Then GLS estimator of kernel innovation, ො𝒛𝒈𝒍𝒔

And we still get

𝝁𝒋 = 𝜷𝒋ො𝒛 𝝁𝒓ො𝒛 `CAPM’ type relation wrt gls estimate of pricing kernel

And this is true if we estimate kernel using noisy estimates, m, of true mean returns, μ

• Suppose 𝑬 𝝐𝒋𝜺𝒌 ≠ 𝟎 and we estimate zt by OLS

𝒓𝒋𝒕 − 𝝁𝒋 = − 𝝁𝒋𝒛𝒕𝝂𝒛𝟐+ 𝜸𝒋𝒚𝒕 + 𝜺𝒋𝒕

where yt is a common factor orthogonal to kernel.

Then, if 𝜸𝒋is a linear function of 𝝁𝒋 (plus noise), we have a zero beta

version of CAPM (plus noise):

𝝁𝒋 = 𝜷𝒋ො𝒛 𝝁𝒓ො𝒛 (plus noise)

Ƹ𝑧𝑡 =𝝁′𝜴−𝟏𝒓′𝒕

𝝁′𝜴−𝟏𝝁

Ƹ𝑧𝑡 is the innovation in the return on a portfolio whose weights are proportional to the vector 𝝁′𝜴−𝟏. The excess return on this kernel portfolio is

𝑟𝑧Ω =𝝁′𝜴−𝟏𝒓𝒕

𝝁′𝜴−𝟏𝒋

Estimation error:1. mitigate by using large number of portfolios

2. different sets of portfolios3. different weighting matrix, Ω.4. evaluate using known pricing kernels (CAPM, FF3)

Data:• 25 size and btm p’tfolios + 30 Industry p’tfolios + 10 Dividend yield p’tfolios = 65 portfolios• Sample: July 1927 to December 2015

4 estimators of zt corresponding to different covariance matrices, Ω.

1. OLS – identity matrix2. diagonal covariance matrix - inversely proportional to variance of estimate of μj

3. residual covariance matrix (GLS) – covariance matrix of residuals from market modelassuming FF3

𝑟𝑗𝑡 = 𝛼𝑗 + 𝛽𝑗𝑟𝑀𝑡 + 𝑒𝑗𝑡𝑒𝑗𝑡 = 𝛾𝑗𝐻𝑀𝐿𝑡 + 𝛿𝑗𝑆𝑀𝐵𝑡 + 𝑢𝑗𝑡

4. full (factor) covariance matrix – covariance matrix of excess returns, assuming FF3

𝑟𝑗𝑡 = 𝛼𝑗 + 𝛽𝑗𝑟𝑀𝑡 + 𝛾𝑗𝐻𝑀𝐿𝑡 + 𝛿𝑗𝑆𝑀𝐵𝑡 + 𝑢𝑗𝑡

Experiment 1: CAPM

Returns mean adjusted so that CAPM holds ( alpha = 0) – market is (true) kernel portfolio

How well can we recover the market portfolio ?

How well can we price the 65 portfolios with the estimated kernel ?

How well do we recover market portfolio?

How closely correlated are the different estimators of zt?

TGLS is slightly better at recovering market portfolio

Correlations of portfolio weights of different pricing kernel portfolios

Portfolios are quite different even though returns highly correlated

zols zdiag zgls ztgls xrm

zols 1.00 1.00 1.00 1.00 0.97

zdiag 1.00 1.00 1.00 0.97

zgls 1.00 1.00 0.97

ztgls 1.00 0.98

xrm 1.00

wols wdiag wgls wtgls

wols 1.00 0.16 0.43 0.10

wdiag 1.00 0.34 0.69

wgls 1.00 0.52

wtgls 1.00

When portfolio returns satisfy CAPM

The estimated kernel portfolios are only slightly inefficient

We will see how well they do in pricing

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Mea

n E

xces

s R

etu

rn

SIGMA

Mean and Standard deviation of kernel portfolios and market 65 portfolios with expected returns given by CAPM

Kernel Portfolios

Market Portfolio

0

0.002

0.004

0.006

0.008

0.01

0.012

0 0.002 0.004 0.006 0.008 0.01 0.012

Act

ual

(C

AP

M)

Ret

urn

GLS Kernel Predicted Return

Simulated CAPM return versus prediction of GLS Kernel estimate 1927-2015 65 portfolios

The pricing errors from our estimation are small when the true kernel is the market portfolio

Experiment 2: FF3

Returns mean adjusted so that FF3 holds ( alpha = 0) – market is no longer (true) kernel portfolio

𝑟𝑗𝑡 = 𝛽𝑗𝑟𝑀𝑡 + 𝛾𝑗𝐻𝑀𝐿𝑡 + 𝛿𝑗𝑆𝑀𝐵𝑡 + 𝑢𝑗𝑡

What does our estimated kernel look like relative to the market?

How well can we price the 65 portfolios with the estimated kernel?

Now returns mean adjusted so that FF3 holds ( alpha = 0) – market is no longer (true) kernel portfolio- Market is inefficient portfolio

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

Me

an E

xces

s R

etu

rn

Sigma

Mean and Standard deviation of kernel portfolios and market formed from 65 portfolios under FF3

Market portfolio

Kernel Portfolios

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Act

ual

(FF

3)

Ret

urn

GLS Kernel Predicted Return

Simulated FF3 return versus prediction of GLS Kernel estimate 1927-2015 65 portfolios

Pricing continues to be good when we assume that the true kernel is FF3

So far we have assumed that realized return satisfies CAPM (FF3) exactly

Now lets see what happens when we have a sample from a set of portfolios that satisfy CAPM (FF3) ex-ante

Take 65 portfolio returns 1927-65 ( + market return)

Estimate betas and alphas•

Subtract alphas to get set of returns which exactly satisfy CAPM•

Randomly sample from the return months to generate a random •

sample with same covariance structure and same population means (CAPM)

Estimate kernel on the random sample and report•

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Mea

n R

etu

rn

PK Predicted Return

Mean Return and Pricing Kernel Prediction for 65 portfolio returns simulated under CAPM 1927-2015

𝑹𝒑𝒕 = 𝟎. 𝟎𝟎𝟏𝟓 + 𝟎. 𝟎𝟎𝟔𝟗 𝜷𝒛 , 𝑹𝟐 = 𝟎. 𝟔𝟓

(2.32) (10.80)

Correlation (zgls, RM ) = 0.98

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Me

an R

etu

rn

CAPM Predicted Return

Mean Return and CAPM Prediction for 65 portfolio returns simulated under CAPM 1927-2015

How well does the CAPM prediction do when CAPM holds? Same sample

𝑹𝒑𝒕 = 𝟎. 𝟎𝟎𝟎𝟐 + 𝟎. 𝟎𝟎𝟕𝟖 𝜷𝑴 , 𝑹𝟐 = 𝟎. 𝟔𝟓

(0.32) (11.93)

Estimated kernel does as well as true (ex-ante) kernel i.e. market portfolio for CAPM – same R2

Now, how well does the pricing kernel do in pricing actual returns?

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Me

an R

etu

rn

GLS PK Predicted Return

Actual Returns vs GLS pk prediction - 65 portfolios 1927-2015

Small Lo BM2

Other

Services

Small Hi BM

𝑹𝒑𝒕 = 𝟎. 𝟎𝟎𝟒𝟓 + 𝟎. 𝟎𝟎𝟑𝟓 𝜷𝒛 , 𝑹𝟐 = 𝟎.22

(5.67) (4.23)

Small Lo BM1Steel

Small2 LoBM1

Coal

Smoke

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

Ave

rage

Ret

urn

CAPM Predicted Return

Actual Returns vs CAPM prediction 65 portfolios 1927-2015

Small Lo BM1

Small Hi BM

𝑅𝑝𝑡 = 0.0042 + 0.0033 𝛽𝑀 , 𝑅2 = 0.14

(3.70) (3.19)

Smoke

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

Pricing Kernel Portfolio Weights 1927-2016

wols wdiag wgls

----------- Size and Book to market ------- xxxxxxxxxxx Industry xxxxxxxxxxxxxxxxxxxxxxxx ooo D/P oooo

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

SLo S2 S3 S4 Shi

2Lo 2

2

23

24

2H

i

3Lo 3

2

33

34

3H

i

4Lo 4

2

43

44

4H

i

Blo B2

B3

B4

Bh

i

Foo

d

Be

er

Smo

ke

Gam

es

Bo

oks

Hsh

ld

Clt

hs

Hlt

h

Ch

ems

Txtl

s

Cn

str

Stee

l

Fab

Pr

ElcE

q

Au

tos

Car

ry

Min

es

Co

al Oil

Uti

l

Telc

m

Serv

s

Bu

sEq

Pap

er

Tran

s

Wh

lsl

Rta

il

Me

als

Fin

Oth

er

LoD

P 2 3 4 5 6 7 8 9

HiD

P

Kernel Estimates of Required Monthly Excess Returns for different Omegas 1927-2015

OLS DIAG GLS

0.0%

2.0%

4.0%

6.0%

8.0%

10.0%

12.0%

14.0%

Industry costs of capital (in excess of risk free rate) from Kernel Estimates and CAPM

GLS CAPM

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0 0.002 0.004 0.006 0.008 0.01 0.012

Mea

n R

etu

rn

Pricing Kernel Prediction

MEBM

Ind

DP

INV

OP

Actual and Predicted Returns for 85 PortfoliosJuly 1963 to December 2015

25 SIZE AND BOOK-TO-MARKET PORTS plus 30 INDUSTRY + 10 DP + 10 INV + 10 OP PORTFOLIOS

Corr(z_65, z_85) = 0.998 - the INV and OP portfolios do not change the estimated pricing kernel

Application to Chilean industry portfolios

• January 1992 – May 2016

• 35 industries – screen out < 210 observations

• Left with 18 industries

• MSCI market index

• No risk free rate

CAPM in Chile

-5%

0%

5%

10%

15%

20%

25%

0 0.2 0.4 0.6 0.8 1 1.2 1.4

CAPM β

Annualized Average Returns on 18 Industry portfolios and CAPM Betas: Chile 1992 - 2016

The Chilean CAPM does not seem to work too well

Correlation(Return, CAPM beta) = -0.23

Pricing without a risk free asset

• Approach I

The Zero Beta Approach – estimate the zero beta rate

• Approach II

Relative pricing – price assets relative to the market return

The Pricing Kernel Approach for Chile

1: The Zero Beta Approach

Equilibrium:

𝐸 𝑅𝑖 = 𝑅𝐹 −𝑐𝑜𝑣 𝑚,𝑅𝑖

𝐸 𝑚= 𝑅𝐹 − 𝛽𝑖𝑧

𝜎𝑧2

𝐸 𝑚≝ 𝑅𝐹 − 𝜆−1𝛽𝑖𝑧

So, 𝛽𝑖𝑧 =- (𝐸 𝑅𝑖 − 𝑅𝐹) λ

But realized returns satisfy 𝑅𝑖𝑡 − 𝐸[𝑅𝑖] = 𝛽𝑖𝑧𝑧𝑡 + 휀𝑖𝑡 = −(𝐸 𝑅𝑖 − 𝑅𝐹) λ 𝑧𝑡 + 휀𝑖𝑡

𝑅𝑖𝑡 − 𝐸[𝑅𝑖] = 𝑎𝑡 + 𝑏 𝜇𝑖 𝑧𝑡 + 휀𝑖𝑡

Where 𝑎𝑡 = 𝑅𝐹𝑡 λ 𝑧𝑡, 𝜇𝑖= 𝐸 𝑅𝑖 , 𝑏 = −𝜆

Step 1: Estimate zt by unconstrained cross section regressions:

𝑅𝑖𝑡 − 𝜇𝑖 = 𝑎𝑡 + 𝜇𝑖 𝑧𝑡 + 휀𝑖𝑡

Step 2: Estimate pricing kernel betas by time series regression:

𝑅𝑖𝑡= 𝛼𝑖 + 𝛽𝑖𝑧𝑧𝑡 + 𝑢𝑖𝑡

Step 3: Cross section regression:

ത𝑅𝑖𝑡 = 𝑘0 + 𝑘1 መ𝛽𝑖𝑧

Predicted Return = 𝑘0 + 𝑘1 𝛽𝑖𝑧

The Zero Beta Approach

-5%

0%

5%

10%

15%

20%

25%

-5% 0% 5% 10% 15% 20% 25%

Correlation 0.91

Annualized Average Returns and GLS pk Predicted Returns 18 Industries Chile 1992 - 2016

-5%

0%

5%

10%

15%

20%

25%

Pricing Kernel Required ReturnsChile 1992 - 2016

ols diag gls

2. Relative pricing

Replace 𝐸 𝑚𝑡(𝑅𝑗𝑡 − 𝑅𝐹𝑡) ≡ 𝐸 𝑚𝑡𝑟𝑗𝑡 = 0 ∀ 𝑗, 𝑡

with 𝐸 𝑚𝑡(𝑅𝑗𝑡 − 𝑅𝑀𝑡 ≡ 𝐸 𝑚𝑡𝑟𝑗𝑡Δ = 0 ∀ 𝑗, 𝑡

where 𝑟𝑗𝑡Δ = 𝑅𝑗𝑡 − 𝑅𝑀𝑡, returns measured relative to market return

The CAPM for relative returns

𝐸 𝑟𝑗𝑡Δ = 𝐸 𝑅𝑗𝑡 − 𝑅𝑀𝑡 = 𝛽𝑗 − 1 𝐸 𝑅𝑀 − 𝑅𝑧 = 𝜆𝛽𝑗

Δ

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

0.008

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

Mean return relative to market and (β_j -1) for 18 Chilean industries1992-2016

Corr( 𝑟𝑗𝑡Δ , 𝛽𝑗

Δ) = −0.20

𝛽𝑗Δ

𝑟𝑗𝑡Δ

The Pricing Kernel Approach for Market Relative Returns

Estimate cross section regression each month:1.

𝑟𝑗𝑡Δ − 𝜇𝑗

Δ =−𝜇𝑗

Δ

𝜎𝑧2 𝑧𝑡 + 휀𝑗𝑡 by OLS, WLS, GLS etc

2. Calculate returns on portfolio perfectly correlated with zt - kernel portfolio returns, Rzt

3. Regress market relative return on kernel portfolio returns to calculate 𝛽𝑗Δ

𝑟𝑗𝑡Δ = 𝛼𝑗

Δ + 𝛽𝑗Δ 𝑅𝑧𝑡 + 𝜖𝑗𝑡

Δ

Then pricing kernel predicted market relative return is 4. 𝛽𝑗Δ ത𝑅𝑧

and we expect 𝛼𝑗Δ = 0

-1.3%

-1.1%

-0.9%

-0.7%

-0.5%

-0.3%

-0.1%

0.1%

0.3%

0.5%

-1.5% -1.0% -0.5% 0.0% 0.5% 1.0% 1.5%

Mea

n R

elat

ive

Ret

urn

GLS Pricing Kernel Expected Return

Expected return relative to market return and GLS pkpredicted relative return - 18 industries Chile 1992 - 2016

Ind goods

Correlation(Mean relative return, pk predicted return) = 0.85

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

CAPM and GLS pk market relative return alphas18 industries Chile 1992 - 2016

alpha_M alpz

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

Pricing kernel portfolio weights -kernel for pricing market relative returns

Chile 1992 - 2016

wols wdiag wgls

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

banks mats resourc chem conmat consume conserv finserv finance health indgood ind oilgas realest retail tech telec util

Annualized alphas from pk model for absolute pricing and for relative pricing

ols diag gls gld rel price

Correlations of GLS Relative Pricing alpha with Absolute Pricing Alphas

OLS 0.49

DIAG 0.54

GLS 0.49

Standard deviations of alphas

Absolute pricing 2.50%Relative pricing 6.91%

Summary

Asset Pricing Models can help us to estimate costs of capital for investment•

The CAPM in US works reasonably well when all for different horizons•

In recent period horizon approaches • 1 month

For other countries the market portfolio is not too compelling as a •

candidate for the pricing kernel

Suggest discard (CAPM) theory or ad hoc empiricism of factor models •

and estimate kernel directly