## FINC6009 – Portfolio Theory and its Applications

## Task description

Betting against beta (BAB) is an investment strategy developed by AQR

capital which has also gained attention from academics with the creators

of the strategy publishing an article in the Journal of Financial Economics

(see Frazzini and Pedersen; 2014). The BAB strategy exploits an anomaly

in financial markets where investors aren’t compensated sufficiently for the

systematic risk (CAPM β) they take on in their investments.1 To employ

the BAB strategy, we aim to make a portfolio that has a β of zero. This

is achieved by taking a long position in a portfolio of assets with low betas

which is then combined with the risk-free asset (levered up) to make the

portfolio have a beta of 1. A similar portfolio is constructed from high beta

assets and levered down to have a beta of 1. The BAB portfolio is then a

long position in the levered low beta portfolio and a short position in the

levered high beta portfolio.

To provide clarity, recall that the CAPM suggests that returns follow

Ri,t = αi + βiRM,t + i,t

where Ri/M are the excess returns on asset i and the market respectively.

The betting against beta portfolios is a self financed portfolio which consists

of a long leg and a short leg, both of which involve a 100% investment. To

create the short leg, we identify the set of assets i = 1, 2, . . . , n which have

the lowest betas β

i

. This has excess return given by

RL,t =

1

n

Xn

i=1

Ri,t (1)

1This is one of several documented low volatility anomalies.

1

and will have βL =

1

n

Pn

i=1 β

i,t. Since the risk free rate has a beta of 0, we

can lever up the beta by borrowing at the risk free rate. To do this we invest

1

βL

in the risky assets and fund it at the risk free rate. The excess returns

on this portfolio is given by

Rˆ

L,t =

1

βL

RL,t (2)

A similar approach can be taken with the highest beta assets to produce

a portfolio RH,t which can then be levered down to produce the portfolio

RˆH,t. The returns to the betting against beta portfolio are given by

RBAB,t = Rˆ

L,t − RˆH,t (3)

The success of the BAB strategy suggests that this portfolio has a positive

expected excess return.

The BAB portfolio can be expressed as a view within the Black-Litterman

model and your task is to implement the Black-Litterman model where views

are produced using the BAB portfolio. To create the BAB portfolio for your

analysis, you are to create the long leg from the assets with the 5 smallest

betas and the short leg from the assets with the 5 highest betas. To implement this view within the Black-Litterman framework, you may use some

simplifying conditions.

First, you may use the 30 industry portfolios as your investment assets.

These may be found in the Ken French data library

https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_

library.html

You are to use the average value weighted returns with monthly frequency.

These are provided in an excel spreadsheet called BL BAB Data.xlsx together with the market capitalisations of each industry and some preliminary

calculations to get you started.2

To construct your views (and other statistical parameters) at date t,

you are to use the previous 60 months of returns. Your view will be that

the BAB portfolios expected excess return will be the same as the average

estimated from the previous 60 months, and the uncertainty in your view

2Note that the market capitalisations are approximate as there are some data errors

between the data I used for the market capitalisations and those used by Ken French.

Nonetheless, these discrepancies are small. When asked for the market portfolio returns

you should use the returns provided by Ken French. The market cap weights are required

to implement the Black-Litterman model.

2

will be the variance of this portfolio computed over the same time period.

You may assume that the covariance of all error terms in CAPM regressions

is zero. Note that since there is only one BAB portfolio on any date there

will only be one view per date. The covariance matrix you use should in

your analysis be the sample covariance matrix.

Using a risk aversion parameter A = 3 and weight-on-views τ = 0.5,

compute the returns on the portfolio which uses the BAB strategy (via

views) if it is rebalanced each month (portfolio weights are recomputed for

each month) beginning in January 2010 and finishing in December 2019.

You are to use teh standard CAPM formula (x = A−1Σ−1

(r − rf1)) to

produce your new allocation vector even if this requires additional leverage.

You are to write a short report which addresses the following:

• Provides a brief overview of the betting against beta strategy and how

it is implemented (including maths).

• Provides a brief explanation of the Black-Litterman model and how

the BAB views are implemented.

• Provides a back test of an implementation of the Black-Litterman

model with views generated from the BAB strategy.

• Provides the results from a Fama-French three factor regression

RBL,t = α + βMRM,t + βSMBSMB + βHMLHML + t

where RBL,t is the excess return on your Black-Litterman portfolio, to

identify which factors this strategy s exposed to.

• Examines the performance of the Black-Litterman portfolio with comparison to the market portfolio as a benchmark.

• Provides a recommendation to your employer about using your tested

strategy in the markets, making sure to discuss any potential shortcomings of your research that you think may need to be addressed.

When writing your report, you should take the view of a research analyst

who works for a large active fund manager. Your explanations should be

sufficiently clear so that someone who is experienced in finance, but not

necessarily the Black-Litterman model or the BAB strategy, can understand

and could potentially replicate your study. Your report cannot be more

than 5 pages, including references/appendices and should be typeset in a

standard setup (12 point font, 1 inch margins and single spacing). Make

3

sure you think carefully about how to best present your results. You should

use tables and figures to present data so that it can be easily understood by

the reader.

The marking scheme will be as follows:

Component Marks

BAB discussion 15

Black-Litterman discussion 15

Back test accuracy 20

Regression analysis 10

Performance measure 10

Recommendation 15

Presentation/discretionary 15