Almost Exact Risk Budgeting with Return Forecasts for Portfolio Allocation Avinash Bhardwaj12 Manjesh K Hanawal1 Purushottam Parthasarathy1_2

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Almost Exact Risk Budgeting with Return Forecasts for

Portfolio Allocation

Avinash Bhardwaj1,2, Manjesh K Hanawal1, Purushottam Parthasarathy1,∗,

1IEOR, Indian Institute of Technology, Bombay

2Mechanical Engineering, Indian Institute of Technology, Bombay

∗Corresponding author: 194192001@iitb.ac.in

Abstract

In this paper, we revisit the portfolio allocation problem with designated risk-budget [Qian, 2005]. We

generalize the problem of arbitrary risk budgets with unequal correlations to one that includes return

forecasts and transaction costs while keeping the no-shorting (long-only positions) constraint. We oﬀer

a convex second order cone formulation that scales well with the number of assets and explore solutions

to the problem in diﬀerent settings. In particular, the problem is solved on a few practical cases - on

equity and bond asset allocation problems as well as formulating index constituents for the NASDAQ100

index, illustrating the beneﬁts of this approach.

Keywords:

risk-parity portfolio-optimization risk-budgeting mean-variance tactical-asset-allocation

portfolio-allocation

1 INTRODUCTION

In portfolio allocation problems, risk-budgeting and risk-

parity are two important criteria that are closely related

to each other. Although the investment management

community is quite familiar with risk-parity as a con-

cept, the term risk-budgeting has been less heard and

talked about, both in academic and practitioner circles.

In fact, it may come as a surprise that the term risk-

budgeting, mentioned in [

] actually predates the term

risk-parity that was coined around the same time as

[

], where the authors provided a clear deﬁnition of risk

contributions to a portfolio. Speciﬁcally, the problem of

risk budgeting the portfolio was deﬁned as the following

-given covariance information on a basket of assets, risk

budgeting seeks to form a portfolio whose partial risks

are weighted as per a pre-determined scheme.

Consider

assets indexed by set [

] =

{

, . . . , n}

Let

bi∈

1] to be the fractional risk budget of asset

i∈

[

]and

C∈Rn×Rn

their covariance matrix. Let

x∈

representing the portfolio’s fractional composition.

The partial risk of asset

under a variance risk measure

can be expressed as

(

)

. Then, the risk budgeting

portfolio for a speciﬁc risk measure, namely the variance,

satisﬁes :

xi(Cx)i=bix>Cx, i ∈[n].(1)

Additionally, since the portfolio is fully invested and

fractional risk budgets must add to 1 we have

i=1

xi= 1 and

i=1

bi= 1 (2)

Summarizing the above observations, we note that the

risk budget problem (P)can be formulated as:

minimize √x>Cx

(P)subject to xi(Cx)i=bix>Cx, i ∈[n]

1>x= 1

x≥0

It should be noted that the risk parity problem is a

speciﬁc case of the risk budgeting problem where all

fractional risk budgets are equal, i.e.

n∀i∈

[

]. In

this work we re-formulate the generalized risk-budgeting

problem with return forecasts and transaction costs

as a min-risk budgeting problem (a second order cone

program) and discuss the applications of this result to

systematic asset allocation.

2 PRIOR WORK AND CONTRIBUTION

2.1 Prior Work

As the concept of risk-parity took hold in the investment

community through the 2008 banking crisis, [

] made a

arXiv:2210.00969v1 [cs.CE] 3 Oct 2022

compelling case for risk-parity as an allocation strategy

by focusing on the large risk allocation that popular

60-40 portfolios gave to equity markets. [

] deﬁned the

risk budgeting problem as a general case of the risk

parity problem and presented theoretical results on the

variance of the resulting portfolio - that it is in between

the minimum variance and the corresponding weight

budgeting portfolio. The authors also analytically solved

the problem for the two-asset case and presented exis-

tence and uniqueness results for the general case. The

authors in [

] were able to extend the risk budgeting

approach to risk factors - as an illustration, they showed

that this approach can be used to allocate risk to the

Fama-French factors in a systematic way. In a work

that seeks to understand the fundamental workings of

risk-parity, [

] proposed leverage aversion as a plausible

reason as to why the average investor does not hold

the risk parity portfolio. The authors also pointed out

that not all investors have access to leverage, however,

some do. These more sophisticated investors can indeed

beneﬁt from the superior risk adjusted returns of the

levered risk parity portfolio. Focusing more on work

that makes computational advances towards calculating

the weights in the risk-parity portfolio, [

] reviewed

existing formulations of the risk-parity portfolio (ERC -

equal risk contribution portfolio), compared the empiri-

cal eﬃciency of solving this problem using a variety of

techniques and proposed an alternate formulation that

relied upon converting a hyperbolic constraint to a sec-

ond order cone constraint. Consequently, they showed

that the ERC portfolio with non-homogeneous corre-

lations across assets can be solved as a second order

cone program. [

] showed that the ERB (Equal Risk

Bounding) is a superior technique than ERC for portfo-

lio selection. In the case where short selling is allowed,

the ERB portfolio was shown to be the same as the

ERC portfolio. In [

], the 2014 ERC portfolio SOCP

formulation was extended to equal CVaR contributions.

[

] presented a formulation of the ERC portfolio that

was relaxed to deviate from the ERC allocations to in-

corporate asset forecasts. In more recent work, [

] solved

the ERC portfolio with a cardinality constraint. The

authors empirically demonstrate that these portfolios

show good out of sample performance.

In this work we extend the results of [

] to formu-

late an arbitrary risk budget portfolio with unequal

correlations, return forecasts, transaction costs, and po-

tentially also position constraints (the most generic case

in portfolio optimization). We present a second order

cone reformulation of the proposed problem and provide

a computational analysis to demonstrate the eﬃcacy

and eﬃciency of the reformulation for examples with a

large (≈100) number of assets.

2.2 Problem Setup And Contribution

We consider a set of

assets. We further deﬁne by

C∈Rn×n

and

r∈Rn

, the positive-deﬁnite covariance

matrix and the vector of expected returns for these assets,

respectively. We denote by

a vector of investment sizes

(denominated in $) of a long-only portfolio and

the

corresponding fractional holdings in the kth asset:

xk=1

Σisi

sk(3)

As shown by [

] and [

] in prior work, a solution

to the risk budget problem (

)can be computed by

solving an alternate problem (P∗)given as follows:

min

x∈Rn

2x>Cx −

i=1

bilog xi

As a brief explanation of why this works, observe that

the ﬁrst order optimality conditions for (P∗)are

(Cx)i−bi

= 0, i ∈[n],(4)

which are exactly the risk budgeting conditions in (

)if

we set the total variance of the portfolio in

(1)

, without

loss of generality, to 1. Note that the ﬁnal portfolio

satisfying the summation constraint

(2)

can be obtained

by re-scaling the weights so they sum to 1. It is worth-

while to note that this works as equation

(1)

is scale-

invariant - if a solution

x∗

satisﬁes

(1)

, so does

kx∗

Problem (

P∗

)is usually solved using some variant of

Newton’s method or block coordinate descent [18].

Further observe that any additions of more parame-

ter(s) to the objective function in the original problem

(adding asset return forecasts or accounting for trans-

actions cost) or adding additional constraints will alter

the ﬁrst order optimality conditions

(4)

. Consequently,

target risk budgets may not be achieved. In other words,

the convex formulation above only works in a very spe-

ciﬁc (almost impractical) case - with no return estimates,

no position constraints, or transaction costs. It is note-

worthy that [

] use a diﬀerent approach in formulating

the ERC portfolio, a special case of the risk budgeting

portfolio, as an SOCP program. Their approach could po-

tentially handle the extra constraints that are proposed

in this work, however the formulation is speciﬁcally for

computing an ERC portfolio.

Our work extends the work of [

] to show the gen-

eralized second order cone program formulation for an

arbitrary risk budgeting portfolio allocation with return

forecasts and transaction costs. We solve arbitrary risk

budgeting exactly and argue its merits as a portfolio

allocation process. Further, we provide examples that

show it’s beneﬁts on a few uses cases. Finally we ex-

plore variations of exact risk budgeting that relax the

risk budgeting equality constraints to provide long-term

economic value to the portfolio.

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AlmostExactRiskBudgetingwithReturnForecastsforPortfolioAllocationAvinashBhardwaj1;2,ManjeshKHanawal1,PurushottamParthasarathy1;,1IEOR,IndianInstituteofTechnology,Bombay2MechanicalEngineering,IndianInstituteofTechnology,BombayCorrespondingauthor:194192001@iitb.ac.inAbstractInthispaper,werevisitthep...

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Almost Exact Risk Budgeting with Return Forecasts for Portfolio Allocation Avinash Bhardwaj12 Manjesh K Hanawal1 Purushottam Parthasarathy1_2

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