total return total risk optimizer

7/29/2019 Total Return Total Risk Optimizer

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Total Return Utility Functions Suitable for Optimization

Alpha, Beta, Market Timing, and the Liability, All in One

M. Barton Waring

[email protected]

360 941-3566

Working Paper

May, 2007


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Most efforts to set up optimization routines simply try to find the desired weights

of asset classes. Since these optimizations are typically fed by expected return

assumptions for fully diversified asset classeswhich are justfactor beta positions

they are somewhat limited. What would an optimizer look like if it were a total return

optimizer, dealing correctly with beta and alpha -- and better yet, with both types of

alpha, those fromsecurity selection efforts and those from beta timingormarket timing

efforts?

What would this mean to the investor? It would mean a great deal. It would

provide the ability to simultaneously optimize for long term strategy, and to also

appropriately incorporate short term tactical views. What a plus, especially in

conjunction with the ability to incorporate views about the active managers that the

investor is considering hiring. One can imagine an optimization that simultaneously

works across all the strategic and tactical dimensions of investment strategy.

Further, what would the optimizer look like if it were also capable of dealing with

the liability in a comfortable and complete way? Again, it would mean a great deal.

Controlling liability-relative risk should be a central task of most investors.

Here we present such a talented optimizer, and we do it while staying comfortably

within the realm of single- and multiple-index models of return and risk. Nothing esoteric

or made up; the finance we will use should be comfortable to all, well within the heart

of those very basic single- and multiple index and market models so familiar to students

of finance. It is the assemblage that is novel, not most of the pieces.

The math will be presented first in scalar algebra for intuition, and then in vector-

matrix algebra for implementation. The former is useful to anyone wanting a better

understanding of how one incorporates these different inputs, but the latter will be of

greater interest to that hard core of serious practitioners that actually get their

fingernails dirty in the gardens of asset allocation.

Of necessity, this latter section is mathematically intense. But it is the authors

observation that the basic technology of surplus optimization hasnt been used much,

perhaps simply because the math for some of the ancillary parts of a full process is just

complicated enough that it is difficult to work out. So this, as a reference source, may aid

the process of dissemination of this technology.


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With that background, well build the fully generalized vector-matrix form of

these utility functions, again first starting in an asset-only form and then adding the

additional complexities of the liability.

Finally, well discuss how to make variations for optimization tasks that may not

require all the capabilities set up in the general version, adapting this mathematical

structure to different types of problems.

A.AN INTRODUCTION TO PARSED UTILITY FUNCTIONS:ASSET-ONLY

1. An asset-only model of total returns:

a. The Grinold and Kahn benchmark-relative approach to

modeling total returns

Theory tells us that the return of a portfolio of assets,A, can always be expressed

in terms of its market, or beta terms, and its residual terms. Grinold & Kahn [2000], in

the appendix to chapter 4 of their wonderful book, Active Portfolio Management, parse a

generalized form of the CAPMmore accurately, a single-index model against a

specified benchmarkas follows:1

A F A bR R f (1)

(This is read as the total return of a portfolio of assets is equal to the risk-free

rate, plus the beta of the assets relative to the benchmark portfolio times the expected

excess return over the risk-free rate of the benchmark portfolio, plus an alpha.) This

reference to afixedbenchmark was convenient to their purpose, saving some complexity

not needed to meet their objectives. They were focused, appropriately to the task of their

book, on dealing with active management issues across portfolios ofsecurities, where a

benchmark is typically pre-specified. They werent focused on selecting asset classes, onstrategic asset allocation (SAA) issues. By specifying the model in terms of a pre-set

benchmark it made some things easierfor example, the ideal beta in that case is always

1 While the story is brief in this reference, it is rich, and the student of this topic is encouraged to

start with this source.


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unity, or matched with the beta of the benchmark, and that made if appropriately

parsimonious for their purpose.

But the ambitions of this article, set forth above, require more generality. Since

were going to include strategic asset allocation solutions among our ambitions, we cant

settle for using a pre-determined benchmarkthe policy portfolio needs to be optimized

in pari pasuwith the active decisions. The optimal beta solution has to be able to float, to

be more conservative or more aggressive than the reference portfolio depending on the

investors lambda, or risk aversion parameter. To determine strategic asset allocation

policy is tosetan investment benchmark, a complex benchmark to be sure, but a

benchmark nonetheless. We cant be limited by the confines of having the benchmark

pre-specified.

b. A more generalized market model approach

We can improve and generalize their approach so that it is broadly usable for asset

allocation purposes by specifying the reference portfolio for beta calculations as the

consensus or equilibrium portfolio, Portfolio Q. To students of consensus expected

returns, Portfolio Q is recognizable as the market portfolio and so it is no surprise that

we work with it in this manner. At the heart of every efficient frontier, acknowledged or

not, is such a reference portfolio, a consensus portfolio representing the beliefs of the

collective market place. In our work we choose to select it intentionally and explicitly. As

a result, we in effect move from the single index model of equation (1) to a more fully

generalized market model:

A F A QR R f (2)

(Now, this reads that the total return of the asset portfolio is equal to the risk-free

rate, plus the beta of the assets relative to the marketportfolio, times the expected excess

return of the marketportfolio, plus an alpha.) Well think of this Portfolio Q as a

portfolio that closely approximates the world wealth portfolio described by Roll2 (or at

least the investable portion of it). However, it can be any reference portfolio thought by

the analyst to represent the consensus of the market.

2Roll, Richard. A Critique of the Asset Pricing Theorys Tests. Journal of Financial

Economics, March 1977 (volume 4, number 2, pp. 129-176).


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For notation in scalar algebra (well be operationalizing these equations in a later

section in vector-matrix algebra) well use uppercase letters to denote a return inclusive

of the risk free rate, and lowercase letters to denote a return in excess of the risk free rate,

FR . So Qf and bf are the excess return forecast for the consensus portfolio in

equation(2) (and for the benchmark portfolio in equation (1)). Each consists of the sum of

the equilibrium or consensus forecast plus the exceptional market return forecast, so that

Q Q Qf f . In effect, the forecast return for the market, Q , is an expected risk

premium, and the exceptional market return forecast termQ

f is a market timing input.3

c. Why reference Portfolio Q?

For ordinary asset allocation work across a set of asset classes, it isnt necessary

to include any reference to the consensus, or market, portfolio. Why here? There are a

couple of reasons. One is that it simply reflects the authors bias that when modeling

betas in the course of asset allocation work one should be modeling them consistently

with capital markets theory, either CAPM or some version of APT, in which the market

portfolio is the low risk portfolio and the reference point for most decision making.

But secondly, it facilitates a clear understanding of the differences between beta

and alpha, which will be important in our overall construct. In current practice, it is

aggravatingly normal to see expected return inputs to asset allocation studies that mix

equilibrium and nearer-term expectations (tactical expectations) without differentiation.

Since they are rewarded differently, the one being rewarded unconditionally and the other

being rewarded only on the joint condition of inefficiency and skill in the analyst, they

should be differentiated. And we will support that goal.

d. Including active asset allocation inputs: Beta timing

The exceptional market forecast, Qf , is a non-equilibrium expectation for excess

return such as one might hold if one had a view of what the consensus portfolio (or any

other particular benchmark) might return in the forecast period above or below the

3 For portfolio Q, expected alpha is zero, by construction. For other portfolios where expected

alpha might be non-zero, it could be included in our term f, such that Q Q Qf f .


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neutral expectation provided by the consensus. This is an input that would drive tactical

asset allocation, or beta timing positions within the optimal asset allocation policy: For

example, if one had a positive expectation in the next year for market returns, one would

be justified in holding a portfolio with a higher beta than would hold with only

equilibrium expectations.

As an active expectation, Qf has a horizon, a beginning and an end (only

equilibrium views are horizonless). It is useful only for that specific horizon, after which

time it must be reviewed, renewed, changed, deleted, or whateverbecause it expires.

When it is revised or changed, the optimization must be redone, and new active beta

positions established (discussed below). If the exceptional market forecasts are skillful,

they will add return, and over many cycles this will be visible in a regression of realized

returns as a positive regression alpha. If they are not skillful, they will randomize the

portfolio, and will only add positive alpha returns by chance. At the time of this writing,

in the early summer of 2004, many sponsors are expecting interest rates to rise and thus

that bonds will have a negative expected return.4

This type of non-equilibrium view

should be expressed, if strongly held, as an exceptional market forecast, not as a strategic

or equilibrium component, and it will cause an underweight to bonds only so long as the

view is held and incorporated in that way in the optimization.

e. Incorporating alpha from active managers

The residual or alpha, , is the uncorrelated or idiosyncratic component of

returns at difference from the market-related component of returns, and is appropriately

used in the context of both expectations and of realizations, but for optimization as here

we are using expectations for this value. It will be non-zero in expectation only in the

presence of the two conditions of Waring, et al[2000] and Waring and Siegel [2003]5,

i.e., that there is some inefficiency in the market, and that there is above-average skill

available with which to capture it. The first is pretty easy to accept; the second is much

4 And sometimes this view is expressed without regard to the likelihood thatit has already been

priced into the forward rate curve.5 The reader is presumed to have some familiarity with these two articles, as they lay a

groundwork for understanding how active positions are incorporated into the portfolio.


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harder to verify. Yet everyday most investors do behave consistent with a belief that they

satisfy both conditions.

f. Parsing the return model

Dividing the components of equation (2) between the consensus and the

exceptional market forecasts, and between the SAA beta and the portfolio active beta,

and multiplying through, we get the following 4-part parsed form of total return. It is

generally consistent with the Grinold and Kahn parsing, referred to above, yet different in

some details that will be important later. The differences are related to my plan to enable

the benchmark level of beta (the SAA policy itself) to be evaluated in the optimization,

rather than assuming it as a given):

A F SAA PA Q QR R f (3)

Part 4Part 1 Part 2 Part 3

A F SAA Q PA Q SAA Q PA QR R f f (4)

These parts, and this ordering of those parts, will be valuable as we proceed.

2. The asset-only model of total risk:

Similarly, portfolio variance can be expressed in terms of the same market and

residual risks that we just dealt with on the return side, and that it can likewise be parsed

into four component terms. We need only to separate beta into its SAA component and its

portfolio active component, and square the separated pieces. Again, we have changed

from the G&K single-index reference portfolio, to a market model based on Portfolio Q:

2 2 2 2

2 2 2

A A Q

SAA PA Q

(5)

2 2 2 2 2 2 2


2A SAA Q SAA PA Q PA Q (6)

The only new term not defined above is 2 , the residual variance. Again, these

pieces and this ordering of them will prove to be valuable.


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3. A total return utility function using these return and risk models

Here is the payoff: a total return utility function that properly accounts for beta, or

policy inputs, for market timing across beta components, and for manager selection

inputs (forecast alphas and active risks for managers). By simply combining these two

return and risk functions in the parts and in the order used above, a parsed mean-variance

utility function is developed that operates correctly in total return space:

fU R (7)

2 2

SAA Q SAA SAA Q (Part 1)

2cov2SAA Q PA Q SAA PA Qf (Part 2)

2 2

PA Q PA PA Qf (Part 3)

2

(Part 4)

Generally speaking, we can see that part 1 accomplishes the strategic asset

allocation task. Depending on the lambda used to determine policy, the optimal SAA

portfolio (the portfolio having market risk levelSAA ) would be on the capital market line

(CML), to the right or to the left of Portfolio Q. This is the form of mean-variance utility

that most of us are accustomed to seeing.

But there is more, as this form is capable of simultaneously evaluating active

decisions, making this a true total return utility function.

Part two is a covariance term relating exceptional market forecast utility and SAA

utility. If one studies the first order conditions of this utility function for SAA , one can

see that this is a zero term where several conditions are satisfied: there must be no

exceptional market forecast, all return/risk combinations must lie on the security market

line, and the lambdas for all parts of this utility function must be equal. But since we

want to optimize even when those difficult conditions are not met, well retain the term

rather than dropping it (as suggested in Grinold and Kahn [2000], chapter 4; they could

drop it only because they were assuming the benchmark to have already been set and the

lambda to be equal to the SAA lambda, presumptions that we will seldom be able to

repeat in practice. However, in circumstances where the SAA policy were in fact already

in place, and if we believed that the lambda for this covariance term is the SAA lambda,


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then again we could show that Part 2 becomes in that special and limited context

irrelevant.)

Part 3 is driven entirely by the exceptional market forecast, a TAA or active

beta form of active management. A little later when we move to multi-factor models

well use this input variable to support various beta timing activities not just in a single

beta context, but also across multiple betas, which organizes disciplines such as tactical

asset allocation, market timing, country rotation, and style rotation. Where in any period

there is a non-zero exceptional market return, the optimization will cause there to be a

non-zero portfolio active beta in that period.

Part 4, dealing with residual utility, has two parts. The first simply deals with the

mathematics of asset class mixes that are off of the CML, creating residuals as a result.

Well deal with the need to account for these residuals in later sections. This is

uninteresting in the sense that these residuals are uninformed by insights, but yet they

must be accounted for if one desires a complete utility solution. These residuals are a

natural result that happens when there are limited number of managers availableno

optimal combination of the managers is likely to perfectly represent the ideal aggregate

beta, but will have some degree of misfit risk.

The second, more interesting, part is driven by expected alphas sourced in active

management of the residual holdings within the overall portfolio or any of its sub-

portfolios (asset classes or styles). This part represents the utility function for manager

selection, or if we want to develop it as such, for security selection. Well focus on its

role in manager selection, but any economist that is comfortable with the rest of the

mathematics of this chapter (the vector-matrix forms) could comfortably adapt it to allow

for security selection within one or more of the investors asset classes.

Note that we have a potential to have as many as five lambdas, one for the basic

SAA policy decision, one for the covariance relationship between SAA and beta timing,

one for beta timing decisions, one for active manager decisions, and one for uninformed

residuals. These dont have to be set differently, and some will argue with some

persuasiveness that theyshouldntbe, but we are providing the flexibility that they can

be.


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So this utility function, so far, accomplishes all of our total return optimization

goals other than the inclusion of the liability: It develops our SAA policy. It allows for

tactical positioning. And it allows for manager structure decisions regarding the search

for alpha through active management.

4. The interaction of equilibrium and exceptional market forecasts

If we look at the first order conditions of equation (7), we see that there appears to

be an effect on the strategic asset allocation beta in the presence of exceptional

benchmark forecasts:

2

2

cov

2

2

Q SAA SAA Q

SAA

Q PA Q

U

f

(8)

and so the solution is:

2

cov*

2 2

2

2 2

Q Q PA Q

SAA

SAA Q SAA Q

f

. (9)

The left hand term in the solution is quite ordinary and expected. The right hand

term is the one of interest here. Im still trying to work out the implications of its

presence.

One can posit circumstances under which this term might be a zero term, and in

fact there is some intuition that this writer hasnt yet proven to himself that it might

always be a zero term, unconditionally. But until and unless this is proven, we will have

to assume an interaction. (not proven as of 10/1/2006)

I think it may mean that the SAA policy wouldnt be same if calculated without

exceptional benchmark forecasts as it would be if calculated when there were such

forecasts. If so, we have to decide how to handle it:

1. Do a first pass optimization without any exceptional market forecastinputs, to determine the static SAA beta policy. This preserves our notion

of the policy as a static benchmark. Then, in the second pass, using this

benchmark, we would separately determine the portfolio active beta

resulting from the addition of exceptional forecasts. This is probably the


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preferred alternative if we want the SAA result to feel similar to how it has

felt in past practice.

2. Or we just do a single-pass optimization, acepting a new and non-staticnotion of the policy portfolio, a portfolio having some interaction with the

exceptional market forecasts. This feels like the impact on the SAA beta

is partially active in that it is affected by the exceptional forecasts. This

isnt an appealing result, and my understanding of it is not yet complete,

but it may be exactly correct.

We will have to understand this better before finalizing this paper or locking

down the code for the optimizer.

B.A PARSED SURPLUS TOTAL RETURN UTILITY FUNCTION

1. A model ofsurplus total return

Lets now make the model fully generalized, by including the liability. Weve

been using subscriptAto designate the asset portfolio; now well add subscriptL to

designate the liabilities. The return of the surplus has been stated as follows, expressed in

scalar form:6

0

0

S A L

AR R R

L

(10)

Note: If rewriting this I would probably divide through by assets rather than by

liabilities, getting the alternative form of surplus return; possibly I would divide through

by surplus, using the native form and suffering the zero divide problem; minor, really.

Restating this in a form acknowledging that we can decompose both asset and liability

returns into their beta and alpha components; i.e., by populating it with our asset and

liability return models as hinted at by the notes in brackets below the equations:

00

A L

S F A Q A F L Q L

R R

AR R f R fL

(11)

6Waring, M. Barton. Liability Relative Investing II: Alpha, Beta, and etc.Journal of Portfolio

Management, Fall 2004.


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0 0 0

0 0 0

Residual surplus returnBeta-related surplus return

1S F A L Q A L

A A AR R f

L L L

(12)

This is useful, but we can further parse the beta terms, in the same way we did in

equation (4), above. We simply substitute that parsed form for the summary notation of

the return of the assets and the return of the liabilities. (Note that the portfolio active beta

for the liability is definitionally always zero as the liabilitys beta doesnt change

depending on the exceptional market forecast.) So the parsed return form for the liability

only includes part 1 and part 4, leaving parts 2 and 3 as zero terms. Restating equation

(11) in that four-part parsed form developed earlier, we have:

0

0

0 0

A

L

F SAA Q SAA Q PA Q PA Q A

R

F L Q L Q L

R

AR f f

L

R f

(13)

Combining like terms and setting surplus return into its parts:

0

0

1S FA

R RL

Risk-free rate (14)

0

0

SAA L Q

A

L

Part 1

0 0

0 0

SAA L Q PA Q

A Af

L L

Part 2

0

0

PA Q

Af

L Part 3

0

0

A L

A

L

Part 4

2. Surplus total risk

Next, lets develop surplus risk, based on the risks found in the return equation,

equation (12):

2

2 2 20

0

S A L Q S

A

L

(15)


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This is expressed in terms of the optimal asset portfolio beta, A . But presuming

that this beta may include a portfolio active beta, lets separate that portion out from the

strategic beta. Remembering that portfolio active beta can only be on the asset side, not

the liability side:

2

2 2 20 0

0 0

S SAA L PA Q S

A A

L L

(16)

This setup is now equivalent to that we used to arrive at equation (6). Following

through in parallel form, we multiply out the squared term (and dealing separately with

the residual risk term 2S in expanded format), we get a usable and intuitive parsing of

surplus risk in scalar form:

2

2 20

0

S SAA L Q

A

L

Part 1 (17)

20 0

0 0

2 PA SAA L QA A

L L

Part 2

22 20

2

0

PA Q

A

L Part 3

2

2 20 0,2

00

2A A L LA A

LL

Part 4

By inspection, one can see that the minimum variance portfolio for surplus (with

respect to beta) will be where the parenthetical strategic surplus beta term

0

0

SAA L

A

L

equals zero, i.e., where * 0

0

SAA L

L

A

. This itself is an important result,

as it suggests that an underfunded plan must invest aggressively simply in order to keep

up with the liability, if controlling surplus volatility is part of the objective function. At

this point, the portfolio would be solely a hedging portfolio, with no companion risky

asset portfolio.

3. Surplus Utility

Now we can set up the total return/total risk form of surplus utility, again in these

same four parts:


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0

0

1FS

AR

LMax U

(18)

2

20 0

0 0

SAA L Q SAA SAA L Q

A A

L L

Part 1

20 0 0 0cov

0 0 0 0

2SAA L Q PA Q PA SAA L QA A A A

fL L L L

Part 2

22 20 0

2

0 0

PA Q PA PA Q

A Af

L L Part 3

2

2 20 0 0,

0 0 0

2A L A A L LA A A

L L L

Part 4

This is a valuable and general form of utility function (even though we have yet to

deal with residuals sourced in constraints on the underlying optimization problem). Lets

proceed to implement these utility functions in optimizable, vector-matrix form.

C.VECTOR-MATRIX VERSIONS: MULTIPLE ASSET CLASSESThe development so far has been expressed in a single beta form. To be truly

useful in actual practice we need a multi-factor model, spanning the useful asset classes,

sub-asset classes, and styles that we are likely to be using.

The market-related component of any asset allocation policy, any manager, any

asset class, and any liability can be described in the form of some list, or more formally

some vectorof factor weights. All of these vectors are in reality vectors offactor betas

across asset classes and styles, and can be thought of not only in that regression sense but

as a sort of vector of mini-CAPM betas in a very real, market-related risk sense of the

term, as each style or factor weight is in fact a representative of some fully-diversified

component of market related risk.7

Moreover, these multiple factors can be readily

7 There are other factors that might be usable for asset allocation work besides asset classes and

styles, and that might be used one day for next generation asset allocation and strategy decisions. For

example, one can imagine a time in the future when we might use BARRAs factors, or some newer factor

set appropriate to the task of global strategic asset allocation and much more granular than todays asset

class factors.


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combined into a single factor representing the actual CAPM beta, ideally a world market

portfolio beta.

The notation we will develop will have as many consistencies as is possible with

the scalar algebra notation just used. In general, we have followed the standard

convention that scalars are italic, and not bolded, and now well follow the additional

convention that vectors are in lower case bold non-italic letters and that matrices are in

upper case non-italic, bold letters. Beta vectors will be row vectors, all others will be

column vectors. Transposes will be indicated with a bold superscripted T. There will be

various subscripts, hopefully intuitively connected to the scalar algebra subscripts, to

differentiate types of variables.

The beta vectors that we use for the assets,q

(the weights or betas of the asset

classes in Portfolio Q),A

(the current policy asset allocation weights, inclusive of both

SAA and PA effects),SAA

(the strategic component of asset allocation weights),PA

(the

portfolio active portion of the asset allocation weights), and for the liability, L , are

(1 x q) row vectors of asset class factor betas across the q Portfolio Q asset class and style

factors.

1. Portfolio Active Beta in Multi-Asset Class Form

A bit of refreshing on the concept of portfolio active beta may be in order in

preparation for developing the concept in vector-matrix form. It is driven from a zero

position, if at all, by a special type of active return forecast. In the first section we

referred to single factor beta timing inputs, or exceptional market forecasts with respect

to the equilibrium return of Portfolio Q itself, but now were going to address exceptional

market forecasts on each component beta, the element of Portfolio Q. After parsing the

scalar version of the return function equation (2), above, we separated out from the total

return forecast Qf , the consensus expected return Q , and the exceptional market

forecast,Q

f , defining the relationship between them as Q Q Qf f . In vector form to

capture the components of Q, we can write this return forecastqf for the (q x 1) asset

class and style return column vector as the sum of the consensus expected return vector

qr and a vector of exceptional market forecasts qf :


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q q qf = r + f . (19)

Well need additional notation,qV being the (q x q) variance-covariance matrix

for the asset classes or factors, andnV being the (n x n) variance-covariance matrix of

the nmanagers alphas.8We summarize the managers residual returns as alphas in a

single (n x 1) vectora across those n managers.

D.VECTOR-MATRIX FORM: ASSET-ONLY1. Return model

With this notation, we can write directly in vector-matrix form, and thus make

usable, the four-part parsed return of equation (4). Well take advantage of the fact that

we can add the managers beta vectors together, getting a weighted average combined

beta vector, by multiplying the managers holding percentage vector (also the

optimization change variable) nh by the (n x q)matrix of the managers individual beta

vectorsn,q

B (the (n x q) matrix of the nmanagers qbeta exposures; i.e., their normal

portfolios) , getting TA n n,q h B . (In this case, the forward-looking estimates of the beta

vectors of the n individual managers are taken as givens, estimated before the

optimization process.) Thus the holdings of the managers can act as the output vector for

the optimizationby controlling manager weights one indirectly also controls the asset

allocation policy, within the confines of the betas available to the portfolio through the

candidate managers.

Beyond that point, Ill skip the algebraic progression of the parsing process, as

this represents a very straightforward mimicry of the parsing process used to obtain

equation (4), written in a form that assembles the returns from the individual manager

building blocks of this portfolio, an improved level of construction detail relative to

more typical asset class building block approaches. Here is the complete version of the

8 The latter is often thought of as a diagonal matrix, but need not be and strictly speaking, could

notbe. Think of the extreme case: if there were only two managers in the world, and between them they

held all the securities, their alphas would have to be perfectly negatively correlated. In the real world,

diversification tends to drive managers from this bias to negative correlations towards a zero average, and

in practice zero is a good null hypothesis for manager correlations where there is no special contradictory

information.


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vector-matrix return modelthe labels below the vector-matrix terms translate each into

its scalar algebra equivalents:

Part 1

Part 2

Q Q Q

SAA A SAA SAA

PA

A F

f

R R

T T T T T

SAA q q n n,q q q SAA q q SAA q q

q q q q q qT T T T

q q q q q q q q q q q q

V h B V V V r - r f V V V V

Part 4

Part 3

Q

A SAA

PA

f

T T T

n n,q q q SAA q q T

q q n nT T

q q q q q q

h B V V - f h a

V V

(20)

a. Beta-related residuals

Well, I used the word complete. That was a bit of an overstatement. Equation

(20) is almostcomplete, but not fully.

Because we are going to use this in an actual optimization in a later step, we want

to make sure that we are properly representing beta-related residual returns, the alphas

from asset class misfit as opposed to the alphas from the active managers. This is a real

world issue that we didnt concern ourselves about when developing the scalar algebra

version earlier; we glossed over it. These alphas are uninformed by insights, simply being

a result of some deviations from market perfection. If we are in a happy world where

are beta expected returns lie neatly on the security market line, these alphas are all zeros,

but they will then be non-zero in practice for many consultants and advisors who ignore

such niceties. And the risk associated with these alphas will be positive in either event.

Heres the deal: Weve shown the pure alpha from the managers, but we havent

shown any residuals from the possibility (indeed the likelihood) that the beta vectors of

our asset mix and of our liability mix might be interior to, or off, the CML. This isnt

an unusual or exceptional issue, but a usual occurrence under common asset allocation


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situations, most obviously related to the practical constraints on beta imposed by the

limited beta factors usually available in the manager mix.

There might be other reasons for this on the asset side of the problem, perhaps for

as obvious a reason as a lack of available borrowing or lending capabilities at the risk-

free rate. This creates the usual curved efficient frontier just tangent to, but in whole or in

part below, the CML; this difference from the CML associated with residuals. Commonly

used constraints all cause a variation on this same problemin the presence of

constraints, the frontier is going to be interior to the CML at most or all points.

The liability beta vector is almost certainly not efficient, and thus it is also interior

to the CML. If the liabilities are exactly hedged, we eliminate it from the possibility of

contributing to a residual, but that isnt always going to be the case. The bottom lines is

that we need to differentiate and account for the difference between these actual factor

beta vectors and their CML counterparts, and that requires us to deal with the somewhat

messy math of the residuals.

A direct way to start thinking about these beta-related residuals (inspired by

Appendix to Chapter 4 of G&K) is to solve for them, for example by manipulation of

equation (2) for asset-related residuals, getting:

A F A Q

A A Q

R R f

r f

(21)

This simple manipulation confirms my assertion that there is a beta element to the

residuals, they arent solely the manager alphas as we are accustomed to expressing them.

To the extent that the beta vectors are interior to the CML, there will be a beta-related

residual component of returns, and also of risks.

b. Capital market line beta vectors and factor beta vectors

To write this in matrix form, well need to have the tools to differentiate between

the factor beta vector, loosely equivalent to an asset class holdings vector and which may

include residuals, and the pure or CML beta vector, which has no residuals as it is

composed of beta values that place it somewhere on the CML. This latter vector is the

classic two-fund portfolio, part Portfolio Q,q

, the world market portfolio, and part the


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riskless asset. In this section well set up these tools, including the CAPM single factor

beta which controls the leverage ofq

in this two-fund format.

The CAPM beta is the ratioof a covariance of a portfolio with the market, to the

variance of the market. For example, if the factor beta vector for the SAA policy

9

is SAA ,

and if for the asset classes as invested through managers it is the sum of those managers

asset class exposures:

TA n n,q

h B (22)

then the scalar asset and SAA betas would be formed by dividing the covariance (with q)

of our factor beta vectors, by the variance ofq:

A T T T

A q q n n,q q q

T T

q q q q q q

V h B V

V V

(23)

SAA T

SAA q q

T

q q q

V

V (24)

Lets use these scalar betas to find the residual terms in vector-matrix form.

Expressing equation (21) in this form:

T T

n n,q q qT T

n n,q q n n q qT

q q q

h B V h B f h a f

V

T Tn n,q q qT T

n n n n,q q qT

q q q

h B V h a h B f

V (25)

But we can simplify this with the scalar beta abbreviations so that we have an

easier equation to read, and it makes it easier to note that were just comparing the actual

factor beta vector of the assets to the CML beta vector that represents those same assets:

A Tn n A q qh a f , (26)

or, if the beta-related residual vector (the portion in the parenthesis) is indicated as a beta

vector with subscript R, then we can abbreviate this even further, to:

Tn n R qh a f . (27)

9 If there are any constraints, the SAA policy will not be on the CML but will be on some interior

efficient frontier.


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The first term of equation (26), summing the managers alphas, is already familiar

weve been using it to represent the managers alphas already (Waring, et al [2000]). It

isnt a new component of the residual for our optimization.

The second term isnt really new, either. It is the misfit return of the same paper,

weve simply generalized it as a total portfolio measure. Perhaps it better renamed from

misfit return to beta-related residual return.

c. Total return equation, with beta-related residuals

The addition of this residual component allows us to state a generalized, usable

total portfolio return term, to truly complete equation (20). Expressed fully including

the new material in Part 4, it is:

A fR R (28)

T

SAA q q

q qT

q q q

V r

V Part 1

Q Q

SAAPA

f

T T T T

n n,q q q SAA q q SAA q q

q q q qT T T

q q q q q q q q q

h B V V V - r f

V V V Part 2

PA

T T T

n n,q q q SAA q q

q qT T

q q q q q q

h B V V - f

V V

Part 3

R

T T

n n,q q qT T

n n n n,q q qT

q q q

h B V h a h B f

V Part 4

Or equivalently, we can state this quite compactly, using more abbreviated

notation:

Part 1 Part 2 Part 3

Part 4

A f SAA A SAA SAA A SAAR R

q q q q q q q q

T

n n R q

r - r f - f

h a f(29)


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2. Risk model:

The scalar algebra version of the riskmodel in equation (6), above, can also be

expressed in vector-matrix form, again across the same n managers and q asset classes or

styles (or other market factors).

Well start with equation (5) and follow a similar progression to that we used for

parsing the scalar algebra version. To keep it simple, well use the scalar values of beta

developed in equation (23), et seq:

2 2 2 2

A A Q (5)

2

AT T

q q q n n n V + h V h

Separating the beta of the assets into its strategic and portfolio active components:

2

2A SAA A SAA

T Tq q q n n n V h V h

22 2


2A SAA SAA A SAA A SAA T T T T

q q q q q q q q q n n n V V V h V h (30)

Well, again, almost. In a world with no beta misfit residuals, we would be done.

But alas, it still isnt such a world. We need the beta misfit residuals to parallel the

additional term we added to the return term, in equation (28). Well do this by solving

equation (5) for residual risk, and rewriting it in vector-matrix form:

2 2 2 2

A A Q (5)

2 2 2 2

A A Q

If we write 2A directly, it will be in terms of the vector of asset factor betas, not

in terms of the scalar betas:

2 22

2

A QA

A

T T TA q A n n n q q q V h V h V

2

A T T T

n n n A q A q q qh V h V V (31)

Were already familiar with the term on the left hand side of equation (31), the

residual risk introduced by the search for alpha engaged in by the managers. The new

term, in parentheses on the right, is the beta-relatedresidual risk term. With this more


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complete residual risk term in hand, we can restate total portfolio risk, equation (30), in a

more complete and general form that will fit the real world:

2 2

A SAA T

q q q V Part 1 (32)

2 SAA A SAA T

q q q V Part 2

2

A SAA T

q q q V Part 3

2A T T Tn n n A q A q q qh V h V V Part 4

These return and risk equations admittedly look a bit hairy, particularly when

remembering that we have used abbreviated scalar forms for many beta terms, but this

parsing is valuable. And look at the bright side: this is what computers are for! Once

programmed, a long equation is as easy as a short equationbut you better understand it

at least once!

3. Utility

Collecting return and risk terms developed above, we get the following fully

generalized version of the utility function (a reminder: were still in asset-only space).

This uses the abbreviated versions of the beta terms, but of course would have to

implemented with the full versions:

Max p fU Rnh (33)

2

SAA SAA SAA T

q q q q q r V (Part 1)

cov2A SAA SAA A SAA SAA T

q q q q q q q- r f - V (Part 2)

2

A SAA PA A SAA T

q q q q q- f V (Part 3)

2A A T T T T

A q q n n A q A q q q n n n f h a V V h V h (Part 4)

The optimization change variable is simply nh , the manager weights, the basic

unit of trade for a sponsor portfolio, and the key to summing up both the asset beta and

the alpha terms. Because this term is hidden through our abbreviations, the operation of

this utility function is a bit obscured. But this is a key to using this as a total return

optimization function. Through the manager weights, which tie in the managers betas,

the optimizer allows for both active beta and alpha to be optimized relative to the SAA


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policy, and simultaneously for the SAA policy to be established. This is important, and

handy. What this means to the investor is that the output of the optimizer tells her what

managers to hold and in what weights, and this is controlling all the important underlying

variables, including strategic asset allocation policy in Part 1, tactical asset allocation

policy in Part 3, and manager optimization policy in Part 4. Thats a lot!

E.VECTOR-MATRIX FORM: SURPLUS UTILITY1. The surplus residual terms

This time, well develop the residual terms before building the return and risk

models. Well solve for the total surplus residual return (the unsystematic residuals plus

the beta-related residuals) by manipulating equation (12), repeated here solely for

convenience:

0 0 0

0 0 0

1S F A L Q A LA A A

R R fL L L

(12)

Rearranging to solve for the residual return with respect to beta, and with respect

to both the assets and the liability:

0 0 0

0 0 0

1A L S F A L QA A A

R R fL L L

Because the term 0

0

1S F

AR R

L

is simply the excess return of the surplus, Sr ,

this simplifies further. We subtract the portfolio surplus beta, and rearrange, to get a

complete expression of the entire residual:

0 0

0 0total residual

A L S A L Q

A Ar f

L L

0 0 0

0 0 0

unsystematic residuals

S

L A L

r

A A A

L L L

T

A L q n n q q f h a f


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0 0 0

0 0 0

L A L

A A A

L L L

T

n n A L q qh a f (34)

Notice that we have used the same type of compact notation that we developed

above to describe the CML beta of the liability, L

T

L q qT

q q q

V V

.

This can be furtherabbreviated, and although well lose the details we may see

the intuition more clearly:

0 0

0 0

A L L S

A A

L L

T

n n S q qh a f (35)

This reads that the total surplus residual return, on the left hand side, equals the

sum of the residuals from manager alphas uncorrelated with the asset class factors, and

the beta-related residuals from the combined liability model and asset class vectors where

they are off of the CML.

While were dealing with the residuals, lets figure out a full definition of surplus

residual risk, expecting to find a risk term parallel to the surplus residual return, equation

(34):

2

2 2 20

0

S A L Q S

A

L

(15)

2

2 2 20

0

0

0

0

0

2

2

20 0

0 0

2

2 0 0

0 0

beta-related surplus residual r

S S A L Q

A L

A L

S

S

S

A

L

A

L

A

L

A A

L L

A A

L L

T

T

A L q A L q q q

T

T

A L q A L q q q

V V

V V

isk


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0

0

2

2 20 0,

0 0

Unsystematic surplus residual risk

2

0 0

0 0

beta-related surplus residual risk

2

A L

S A L L

A

L

A A

L L

A A

L L

T

n n n

T

T

A L q A L q q q

h V h

V V

(36)

We lose the detail, but this can be abbreviated and expressed as:

2

2 2 20 0,

0 0beta-related surplus residual risk

Unsystematic surplus residual risk

2S A L L S

A A

L L

T T T

n n n S q S q q qh V h V V (37)

Again, there is no point in using vector-matrix algebra to represent the liabilitys

alpha-related residual risks nor its covariance with the asset residual risks, so we will

continue to show these terms in scalar form.

While looking at the unsystematic portion of residual risk, there are some

observations that we can make: The third of these terms, 2L is a constant, and could be

dropped. And one would expect the middle term, the correlation of the asset and liability

unsystematic residuals, to tend towards zero, so this risk term shouldnt be especially

important to the optimization process However, the total risk may more accurately berepresented if an estimate of these risks are made and included.

10

2. Surplus return

With this result, we can write our surplus return equation (14) in matrix form,

including the more complete surplus residual of equation (35):

0

0

1S f

AR R

L

(38)

0

0

SAA L

A

L

q q

r Part 1

10 These liability residual terms can become very real and important, where as in some cash

balance plans the liability itself is credited with a rate that includes an explicit manager alpha with its

attendant residual risk, or where it includes company stock with all the residual risk of a single stock

portfolio.


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0 00 0

SAA L A SAA

A A

L L

q q q q f r Part 2

00

A SAA

A

L

q q f Part 3

0

0

L S

A

L

T

n n S q qh a f Part 4

Note that the alpha, or residual return, for the liability, L , i.e., the portion of the

liability return not related to market risk factors, is simply a scalar alpha, not one in

vector-matrix form. With todays data, its probably impossible to accurately estimate

this numberbut thats ok, as it is a constant and the largest error wont affect the

optimization result. While we wont be precise, we can probably back into a number that

is reasonable. One probably improves the clarity of the risk-return view by making a dull

axe estimate and including it simply so that it shows up in the output, but I would

understand that some analysts might exclude it and simply note that the resulting return

number is understated for lack of its inclusion, its true magnitude being unknown. The

same, of course, will be true on the risk side.

I should be clearthis liability alpha is trivial to the optimization, but it may not

be trivial to the sponsor that actually experiences this very real element of return and risk.

There are risks in DB plans that cannot be hedged away in the markets under any

circumstances. These can be moderatedby being less ambitious in increasing benefits,

by pricing benefits correctly,by using mortality tables with more improvement built in,

etc. But not all risk can be eliminated through the asset portfolio. This is the alpha of the

liability, or the residual risk of the liability. It is risk not related to the markets.

3. Surplus risk

2

2 0

0

S SAA LAL

Tq q q V Part 1 (39)

0 00 0

2 A SAA SAA LA A

L L

T

q q q V Part 2


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2

20

0

A SAA

A

L

T

q q q V Part 3

2

2 20 0,

0 0

2 A L L SA A

L L

T T T

n n n S q S q q qh V h V V Part 4

4. Surplus utility

Now we assemble the risk and return terms into a utility function:

0

0

Max 1S fA

U RL

(40)

2

0 0

0 0

SAA L SAA SAA L

A A

L L

T

q q q q q r V (Part 1)

0 0

0 0

0 0cov

0 0

2

SAA L A SAA

A SAA SAA L

A A

L L

A A

L L

q q q q

T

q q q

f r

V

(Part 2)

2

20 0

0 0

A SAA PA A SAA

A A

L L

T

q q q q q f V (Part 3)

2

20 0 0| ,

0 0 0

2

|

2

L A L L

S S

A A A

L L L

T T

n n n n n

T T

S q q S q S q q q

h a h V h

f V V

(Part 4)

Ive abbreviated this through my notation a great deal, so that some of the

operations cant be seen. The most important one that cant be seen with this abbreviation

is the optimization change vector, the manager holdings vectornh . It is found inside of

everyA

through the required term found in its numerator, TA n n,q

h B . So apologies

there is much accounting to do in properly handling total returns, and even more so in a

surplus context.

Note that part four is really two partsthe informed residuals and the beta-related

residuals.


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This is a very complete total return utility function, capable of informing a

number of the common strategy decisions that every plan sponsor and other serious

investor routinely faces:

Basic strategic asset allocation across asset classes, on either or a liability-relativeor on an asset-only basis.

Making the manager selection decision and the manager structure decision. Making tactical decisions to re-weight asset classes and styles based on

exceptional market forecasts (views).

Beta-alpha separation decisions.F.TACTICAL BETS, BETA TIMING, AND THE LIKE

1. A Bit More on Beta Timing

There are two reasons why an optimization might result in a beta different than

the strategic asset allocation beta,SAA , i.e., why it might have a non-zero PA . Weve

just discussed the beta-related residuals.

The otherof these reasons is the active management reason discussed earlier,

the forecasting of non-zero exceptional market forecasts. Recall, this results in an active

beta, the beta that is different than the beta of the SAA policy in single beta space, and

different from the CML SAA policy beta vector in multiple beta space, i.e.,

PA p SAA , and the more generally useful PA p SAA = - . We discussed beta timing

briefly above, in section A.1.d, but now that we have built the vector-matrix version we

can make some additional observations.

This is beta timing, or market timing, or active beta, which in different

applications is known as tactical asset allocation, style rotation, country rotation, etc. A

sponsor or other investor could hold intentional mis-weights of the beta factors in the

total portfolio benchmark (such as styles, asset classes, or other factors) in an effort to

add alpha to the portfolio through active positioning of the asset classes. The exceptional

market forecast component of expected return,Q

f or in vector form,q

f , is the input

that would drive any such intentional beta mis-weights.


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This of course, is as active as active gets, by any definition. Each time the

optimization is run, the forecast exceptional market return might well be different, so the

resulting active beta position will also be different during that period.If these forecasts

are skillful, value will be added over a time series of such forecasts. That value will be

experienced in realization as a regression alpha over and above the policy beta return.

In todays practice, this particular active management discipline is growing to be

unusual, often disparaged even, though in the hands of the skillful it really is an

appropriate investment discipline. The search for alpha today is more often through hired

money managers expected to add alpha through their carefully selected residual holdings

ofsecurities(and occasionally of styles and factors) relative to the managers underlying

benchmark.

Ive organized our work here with the goal of supporting the sponsors beta

timing activity, processing the sponsors exceptional market forecasts appropriately into

the current asset allocation policy. While many scoff, nonetheless sponsors often want

their portfolio to reflect a view that value will beat growth over the next period, or

bonds will beat stocks, or domestic equities will beat international equities, or whatever.

The exceptional returns vector allows a sponsor to readily incorporate such views.

This has the quite important side benefit of taking some of the forecasting angst

out of the basic SAA decision. Using consensus expected returns for the asset classes that

are the elements of Portfolio Q is pretty easyreverse optimization is quickly and readily

available and can usually be applied sensibly. Modifications to that set of forecasts can be

understood for what they are, exceptional forecasts, and processed in a manner and with a

risk aversion term appropriate to their active nature.

G.ADAPTING THE OPTIMIZER TO DIFFERENT SOLUTIONNEEDS: A SIDE NOTE FOR INTERNAL DESIGN USE

The biggest problem facing anyone implementing an optimizer with these talents

may not be the development of the mathematics, now set forth above. The biggest thing

may be the design of the user interface that allows a user to control such an optimizer in

actual use. Seldom will a strategist use all the talents of this optimizer in a single grand


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optimization, although that might happen. Can we identify the tasks that it might be used

for, in an effort to support the design of the user interface?

There are four major inclusion/exclusion questions that the strategist must answer

before setting up this very generalized optimizer to solve a particular problem:

1. Are we solving for the underlying SAA policy, or has it already beendetermined so that we can take it as a given benchmark?

a. And if we are solving for the SAA policy, are we doing so on anasset-only basis, or on a surplus basis relative to a liability?

2. Will there be exceptional market forecasts, for beta timing (portfolioactive beta)?

3. Will there be manager structure (alpha) decisions, or are we simplyoptimizing for SAA and portfolio active beta (beta decisions) across the

asset classes?

Depending on which of these decisions are to be included in the optimization,

there are implications for the inputs that will be needed, the outputs that will be

interesting, and for which ones of the five terms of the surplus utility function will be

used. It is conceivable that one might do all of these tasks together in well-coordinated

total return optimization, or that we take some one or more of them one at a time, as we

have done for years in optimizing manager structure. There are many different

combinations, and this presents many design challenges for the user interface, for the

schedules of input parameters, and for the type of output graphs and data tables that will

be useful. Well only figure this out over time.

1. Question 1: Is the strategic asset allocation policy to be

determined, or not?

If, in answer to the first question, the policy asset allocation, or SAA policy, is to

be determined by the optimization, then PA A SAA = , i.e., the SAA beta vector is a

variable and will be determined in the process; it is the optimization change vector (either

directly, or through the manager betas and the manager holdings, see below). If, on the

other hand, the SAA policy has already been set, it is not going to be up for change, then


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it can be expressed as a fixed benchmark beta vector, and PA A b = . The vector is

frozen.

2. Question 1A: Is a liability to be considered?

If the SAA policy is to be determined, then it will be determined either with, or

without, the inclusion of a liability (the present value of the future consumption for which

the assets are being held). If the liability is to be considered, then the utility function in

equation (40) is OK, as is. If not, then the values for the vectorL and the scalars ,A L ,

and L are all set to zeroes, and the value for the A/L ratio 0 0A L is set to 1, except

where it modifies the risk free rate term, where it is also set to zero. With this, equation

(40) is identical to the asset-only utility function, equation (33).

If the answer to the first question was that the SAA was not to be determined,

then the answer to question 2 is also determined, since liability considerations would only

seldom be valuable where the SAA policy were not at issue. The exception that disproves

the rule is for cash balance plans: in the rare occasion where the liability is set by

reference to a third-party managed active fund, there is an alpha that is hedgeable, since

the plan can actually invest in that fund if it chooses to do so. In that case, we want the

liability terms even if not doing SAA policy and just doing manager structure.

3. Question 3: Is beta timing included?

If the optimizer is going to be used to decide whether to over- or under-weight

some of the asset classes in the beta vectors, then it would be done by putting non-zero

entries in one or more of the elements of the exceptional market forecast vector,Q

f .

This will have the effect of altering the PA vector to reflect those biases. This is an easy

one to handle: the vector is by default a zero vector.

If the Qf vector is made to have any non-zero elements, then we may want to

display output data for thePA

vector, and the risk and return impacts of this portfolio

active holding, perhaps separately for Parts 1 and 2 of the utility function.


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4. Question 4: Are managers being allocated, is this in whole or in

part a manager structure optimization process?

If managers are included in the optimization, we often refer to the active

return/active risk part as a manager structure optimization process (MSO). In this case,

the optimization change vector is nh , as it is determinative of A through the handy

mechanismA

T

n n,q q

T

q q q

h B V

V , which sums the managers raw beta vectors and calculates the

vector of CML betas that the manager holding vector thus implies.

Without managers being included in the optimization, we are simply determining

the asset allocation solution and not the simultaneous manager structure solution.A is

the optimization change vector.So depending on whether the problem involves managers, the optimization

change vector changes its level of aggregation. A simple A when they are not included,

and the function TA n n,q h B translates the manager holdings vector into the total assets

beta vector when they are.

H.MISCELANEOUS1. Manager structure optimization:

If managers are included, even if the exceptional market forecast is a zero vector,

then in addition to the Part 4 treatment of their alphas and true active risks, the beta-

related residuals from Part 4 and the (sometimes zero) risk from Part 2 will be needed.

If we take out the liability terms (which might be very valuable to leave in for a

cash balance plan that has alphas in its crediting rates, for example), and if we combine

the beta-related residual risk from Part 4 with the portfolio active risk from part 3

(requiring us to set the lambda equal to each other), then this becomes a very

straightforward manager structure optimization utility function.

I . ALPHA-BETA SEPARATIONWe are in the habit of presuming that alpha always has to come with beta in a pre-

packaged form. But the reality is that real alpha is definitionally uncorrelated to beta, and


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that observation frees the mind to consider unleashing alpha and beta from each other.

The new class of market-neutral long-short funds are good evidence of this reality

being pure alpha (plus the risk free rate) they can readily be ported onto any readily

available source of beta, from index funds to futures contracts, etc. The long-standing

practice of taking alpha from a fixed income portfolio by shorting Lehman Aggregate

and going long S&P 500 is another example.

But true alpha (regression alpha), being independent of beta, need not be ported

from one asset class to another. In the ideal world, one would build a portfolio of the best

alpha sources one could identify, regardless of their beta exposures or lack thereof. And

when done, one would fix the betas so as to make them look like the intended SAA

policy. We call this portable beta, and we see it as the fully generalized version of

portable alpha, full separation of alpha and beta.

There are costs and other frictions on this ideal process, so in reality it is not two

steps, but must be done as one, with costs on beta transfer and availability of beta

transfers being taken into account.

To do this with this optimizer, we only need to set up a system of constraints that

allows this to happen. We might (or might notit is potentially a strategists decision

variable) start with a budget constraint on the dollars of alpha exposure. And usually,

alphas sources are long-only with respect to the investor, because it is unfortunately

difficult toshorta manager (although the alpha source itself may be short, long, or

mixed). These two constraints operate on the holdings of the managers. What about

betas?

The beta constraints would be separately stated from the alpha constraints. It has

been the practice to have both a budget constraint and a long-only constraint on the total

beta holdings ofallasset classes and an implied constraint on the combined beta-alpha

exposure of each manager. Of course, to return the CML, cash must be available for use

in leveraging Portfolio Q, the reference portfolio, which implies that one must be able to

short cash. In practice, most strategists dont allow cash to be shorted, as one cant in the

real world borrow at the risk free rate. But this isnt enough. An example will illustrate

why.


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Thus unconstrained with respect to manager sources of alpha, we want the

optimizer to put the weight of the portfolio into the best combination of the best

managers without regard whatsoever to the beta of those alpha sources. One can imagine

the entire portfolio being invested in one small cap growth tech sector specialistif that

is where ones assumptions indicated that the best alpha was to be obtained. Of course,

then we would have toshortmost of that managers beta in order to go long the other

asset class betas to get back to the optimal beta portfolio.

The point is, we have to allow individual positions in beta that are short; it is only

the netbeta positions, across short and long individual positions, that must be compliant

with the budget and long only constraints. So if we allow particular managers to go short,

most easily managers that are really just beta (indexes, futures, etc.), so long as the net of

all asset classes meets the budget and long-only constraint, the optimizer can short beta

where it needs to in order to favor one alpha source over another.

Costs for beta shorting and longing need to be estimated in the form of rates of

return, and incorporated into the short beta opportunity set.

The approach to actually conducting such an optimization is deceptively simple

merely include any available short beta sources (managers, futures, swaps, trusts,

whatever) as additional managers in the optimization, with their cost estimates. If the

costs of shorting and holding these short beta sources is included as a negative alpha

(take care to get the signs rightits a short position, so the negative alpha will end up

being positive), then the optimizer will do the best it can within the constraints of costs

and availability to maximize utility across alpha and beta separately.

J . CONCLUSIONThe idea of conducting a total return, total risk portfolio optimization across all

important variables is an appealing one. To date, those that have said they are doing this

have done so without really parsing the utility functions correctly. While the math is

tedious to work out, once set up it is no big deal; the computer takes care of it.

There is still an open question in my mind about whether we can do a one-step

total return optimization and get sensible results, or whether we need to first determine

the SAA benchmark (with the liability, if desired) and then move on to the other


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decisions. I suspect that we have something yet to learn about the interactions of strategic

and tactical inputs.

We also have a lot to learn about interface design and output design.


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To do list:

Capture the two fund (three fund) theorem page from my Q-Group presentation to

simplify some of the concepts here; citing Waring, M. Barton and Duane Whitney. 2009.

An AssetLiability Version of the Capital Asset Pricing Model with a Multi-Period

Two-Fund Theorem.Journal of Portfolio Management, Vol. 35 No. 4, Summer 2009,

http://www.iijournals.com/doi/pdfplus/10.3905/JPM.2009.35.4.111.
http://www.iijournals.com/doi/pdfplus/10.3905/JPM.2009.35.4.111http://www.iijournals.com/doi/pdfplus/10.3905/JPM.2009.35.4.111http://www.iijournals.com/doi/pdfplus/10.3905/JPM.2009.35.4.111

total return total risk optimizer

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