Empirical Modeling and Analytics @ Empirics.com
RISK BUDGETING
Yi Wang
Empirical Modeling and Analytics
@Empirics.com
June 2000
ABSTRACT
We show how well established principles of modern portfolio theory can
be applied to risk budgeting at the portfolio level, at the asset class level,
and at the total fund level. Specifically we describe in detail how to use a
factor model of returns to quantify risk at the three levels and illustrate
how quantification of risk can be used to:
 Monitor risk
 Project the impact of proposed transactions
 Optimize portfolio modifications
 Construct optimal portfolios
 Choose a optimal mix of asset classes
We also outline the algorithms for achieving these objectives.
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RISK BUDGETING
WHAT IS RISK BUDGETING
Risk budgeting is a relatively new innovation in the investment
world. Simply put, it is a process of carefully allocating a risk budget
among assets or asset classes, closely monitoring changes in the portfolio
or fund’s risk level, and timely adjusting the allocation to keep the
portfolio or fund’s actual risk consistent with the desired level. It builds
on the acknowledgement that market conditions can and will change and
on the fact that portfolio or fund managers today can be more precise in
the way they acquire the risk data and analytical tools to calculate and
control their risk exposure. While traditional asset allocation is based on
infrequent estimates of volatility and correlation as long term averages,
risk budgeting focuses on quantifying and reallocating risk in a timely
fashion. In addition to keeping portfolio or fund risk on track, research
shows that risk budgeting can significantly enhance portfolio or fund
performance, measured by some risk-reward ratio, such as the active ratio
(active return / active risk, as defined below).
DEFINING RISK
To allocate, monitor, and keep portfolio or fund risk on target, we
need to define and quantify risk. The classical definition of investment
risk is uncertainty of total returns as measured by their standard
deviation. Investments with greater risk are expected to earn greater
returns than less risky alternatives. Asset allocation models help investors
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choose the asset mix (e.g., a mix of stocks, bonds, and cash) with the
highest expected return given their risk constraints.
Once investors have selected a desired asset mix, they often enlist
specialized asset managers to implement their investment goals. The
performance of an asset manager is usually compared with a benchmark.
Accordingly the risk of the portfolio is defined by its deviation from the
benchmark rather than by the standard deviation of its total returns. The
risk thus defined is the active risk of the portfolio, since it stems from
active management of the portfolio, in contrast to passive replication of
the benchmark. This definition of risk makes it clear that the portfolio
manager bears the risk of deviating from the benchmark, while the
benchmark risk belongs to the plan sponsor. The active risk of a portfolio
is measured by its tracking error, the standard deviation of the return
difference between the portfolio and the benchmark.
MODELING PORTFOLIO RETURNS
Typically, to develop and apply a risk budgeting model, a
relationship is first established between individual security returns and a
set of risk factors that drive them. This relationship forms the bridge by
which market experience in the form of past returns can be applied to
characterize the expected distribution of future returns.
Assume that a portfolio manager’s investment universe consists of
a finite set of N securities. The performance of the entire universe over
the coming period can then be represented by an N  1 random vector r
of individual security total returns. A return model attempts to explain the
return ri on any security i in terms of broader market movements. A set of
M risk factors (M << N) is chosen to represent the primary sources of
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risk (and return) to which the portfolio may be exposed. The extent to
which security i is exposed to a particular risk factor j is modeled by a
fixed factor loading fij. The 1  M row vector fi thus characterizes the
exposure of security i to systematic risk.
The return of any security i can be expressed in terms of the M  1
random factor vector x by
M
(1)
ri   fij x j   i  fix+i,
j 1
where fi = {fij} is the vector of factor loadings that characterizes security
i, and i is the non-systematic random error. That is, i, also referred to as
residual error, is the portion of the return ri that is not explained by the
systematic risk model. This reflects the possibility of events specific to
security i.
If we let F be the N  M matrix containing one row for the factor
loading vector of each of the N securities in the investment universe, and
denote by  the N  1 vector of non-systematic random errors, we can
restate Equation (1) in matrix form,
(2)
r = Fx + .
We can represent a given portfolio p by a 1  N allocation vector qp,
which states the proportion of the market value of the portfolio allocated
to each of the N securities in the investment universe. The portfolio
return rp is then given by
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(3)
rp = qpr = qpFx + qp = fpx + qp
where fp = qpF is the portfolio factor loading vector that summarizes the
systematic risk exposure of a portfolio as a weighted sum of the
exposures of its constituent securities.
QUANTIFYING PORTFOLIO RISK
Of primary importance in assessing portfolio risk are the secondmoment statistics, that is, the return variances. The variances 2p and 2b
of the portfolio and benchmark returns rp and rb may be expressed as
(4)
2p = VAR(rp) = fpfpT + qpqpT
(5)
2b = VAR(rb) = fbfbT + qbqbT
where  is an M  M matrix, which contains the variances and
covariances of the common risk factors, with jk = COV(xj, xk) and jj =
VAR(xj), and  is an N  N matrix, which contains the variances and
covariances of the security residual returns, with ij = COV(i, j) and ii
= VAR(i).
The total volatility of the portfolio, measured by 2p, thus has two
components: the first component, measured by fpfpT, is due to its
exposure to the common risk factors, while the second component,
measured by qpqpT, is due to its exposure to security-specific risk.
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Accordingly, the total portfolio risk can be seen to be composed of
systematic risk and non-systematic risk.
There are no cross terms in Equations (4) and (5), due to the
commonly made assumption that the error vector  and the systematic
factor vector x are uncorrelated and that the error terms have a mean of
zero. An additional simplifying assumption is that residual terms are
uncorrelated to each other. In that case, ij = COV(i, j) = 0 and 
becomes an N  N diagonal matrix.
The tracking error squared of the portfolio may be expressed as
(6)
2TE = VAR(rp - rb) =( fp – fb)(fp – fb )T + (qp – qb)(qp – qb)T
From this expression, it is clear that tracking error arises from
mismatches of composition between the portfolio and the benchmark (qp
 qb, and so fp  fb as well). The closer the portfolio tracks the
benchmark, the smaller the value of TE. If the portfolio and the
benchmark are identically composed (qp = qb, and so fp = fb as well), then
rp will be identical to rb under all possible states of the world, and we will
have TE = 0. Indeed this is the only way that a zero tracking error may
be achieved.
From Equation (6), it is also clear that, like the total risk, the risk
of the portfolio relative to the benchmark has two components: one is
systematic and the other is non-systematic. While the non-systematic
tracking error arises because mismatches of composition lead to
differences in the exposures to security-specific risk (qp  qb), the
systematic tracking error arises because mismatches of composition
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lead to differences in the sensitivities to common, systematic risk
factors ( fp  fb ).
Traditional portfolio optimization techniques are based on
quantification of total risk, as in Equation (4). Their focus is on
maximizing return for a given level of total risk. Risk budgeting,
however, builds on quantification of active risk, as in Equation (6), and
the objective could be either minimization of active risk or maximization
of active return for a given level of active risk.
Quantification of tracking error allows us to translate the structural
differences of the portfolio and the benchmark into a forecast of tracking
error. The systematic component is projected based on the differences in
risk factor exposures, while the non-systematic component is projected
based on the differences in security-specific risk exposures.
Conversely, by tracing tracking error back to the structural
differences between the portfolio and the benchmark, we can give
quantitative answers to questions such as: How different are they in
exposures to market risk, to size-related risk, or to financial distress risk?
How different are they in exposures to certain sectors? How different are
they in exposures to security-specific risk? By how much will a given
transaction increase (or decrease) systematic (or non-systematic) risk?
What transactions will reduce the risk of the portfolio relative to the
benchmark to the desired level without significantly increasing portfolio
turnover? Quantitative answers to such questions allow us to keep the
risk of the portfolio on target in an efficient way.
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USING HISTORICAL RISK DATA
To develop and apply a risk budgeting model, we need to estimate
the factor loadings of the securities in our investment universe and the
variances and covariances of the risk factors and the residual returns on
the individual securities. In other words, we need to estimate F, , and .
Typically these parameters are estimated by analyzing historical data.
Therefore the assumption is that the variances and covariances of the
historical risk factor and security return realizations provide a reasonable
characterization of their distribution for the coming period.
Since risk factors themselves are unobservable, returns on factormimicking portfolios are usually used as proxies for risk factors. Thus,
for example, we may estimate the variances and covariances of the risk
factors directly from the monthly returns on the factor-mimicking
portfolios in the recent past. To estimate the factor loadings of individual
securities, we may run regressions of their monthly returns on the
corresponding returns on the factor-mimicking portfolios. Such
regressions produce as output not only the coefficients, which are our
estimates of the factor loadings, but also the residual terms, from which
we can calculate the variances and covariances of the residual returns on
individual securities.
The estimated variances and covariances of the risk factors and
factor loadings of the securities provide a basis on which we estimate and
analyze the systematic component of tracking error, while the estimated
variances and covariances of security residual returns provide a basis on
which we estimate and analyze its non-systematic component.
If we estimate F, , and  from monthly return data, we probably
want to update our estimates monthly. With monthly updated volatility
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and correlation information, we are able to monitor the portfolio risk and
keep it on track on a monthly basis.
MONITORING PORTFOLIO ACTIVE RISK
Traditionally, portfolio managers are measured by returns relative
to benchmarks or peers. Little attention is paid to contribution to portfolio
risk. Such measures of performance are inadequate.
Many firms today use risk-adjusted measures to evaluate portfolio
performance. Control of risk is the focus. A high return achieved by a
series of successful but risky plays may not please a risk-conscious
pension plan sponsor. A more modest return, achieved while maintaining
much lower or targeted risk, might be seen as a healthier approach over
the long run. This point of view can be implemented in two ways.
Traditionally, it is done by adjusting performance by the amount of risk
taken. Under risk budgeting, we specify in advance the acceptable level
of risk for the portfolio (i.e., targeted or budgeted risk). In any case, the
portfolio manager should be cognizant of the risk inherent in the active
management of the portfolio and weight it against the anticipated gain in
return.
A risk-budgeting model based on quantification of tracking error
can be used as a monitoring tool. Whatever position a portfolio manager
has taken relative to the benchmark, the model will quantify how much
systematic/non-systematic risk has been assumed relative to the
benchmark. This helps measure the risk of the exposures taken to express
a market view and may point out the potential unintended risk in the
portfolio.
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PROJECTING EFFECT OF PROPOSED TRANSACTIONS
Using a risk budgeting model based on quantification of tracking
error, we can easily quantify the effect of proposed trades on the portfolio
risk relative to the benchmark.
Consider a one-for-one substitution, that is, selling one security in
the portfolio and using the proceeds to buy another. To see how we can
quantify its effect on the portfolio tracking error, let us rewrite the
formula for tracking error (Equation (6)) directly in terms of the security
allocation vectors qp and qb for the portfolio and the benchmark,
respectively:
(7)
2TE = ( qp – qb)FFT(qp – qb )T + (qp – qb)(qp – qb)T.
The gradient G of 2TE with respect to qp is given by
(8)
G = 2( qp – qb)FFT + 2(qp – qb).
The ith element of the gradient vector G, Gi, represents the sensitivity
of the portfolio tracking error to change in portfolio position in
security i.
Now let us investigate the effect on tracking error of a small
change  in portfolio weights. If we denote the new portfolio allocation
vector by qp’ = qp + , we can see that the tracking error of the new
portfolio is:
(9)
’2TE = ( qp’ – qb)FFT(qp’ – qb )T + (qp’ – qb)(qp’ – qb)T
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= 2TE + 2( qp – qb)FFTT + 2(qp – qb)T
+ FFTT + T.
We can express the resulting change of tracking error in terms of the
change  in portfolio weights and the gradient G as
(10) 2TE = ’2TE - 2TE = GT + (FFT + )T.
The transaction to sell security m and purchase an equivalent
amount of security l can be represented by a vector  whose elements are
given by
(11)
i  l, m
0
i =  x
 x

il
im
where x is the size of the trade, which is the percentage market value of
the purchase or the sale relative to the total portfolio market value. For
this special type of transaction, with only two non-zero entries in the
change vector , the expression for the change in tracking error squared
can be simplified dramatically. For instance, the first term of Equation
(10), which gives the first-order effect of the change in portfolio
composition, can be expressed as
(11) GT =
N
G 
i 1
i
i
 x(Gl  Gm ) ,
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where Gi is the ith element of the gradient vector G, which quantifies the
sensitivity of the portfolio tracking error to change in portfolio position in
security i.
The second term of Equation (10) incorporates the second-order
effects of the transaction, as well as the cross-effects of changing
positions in more than one security at a time. This term, which considers
both systematic and non-systematic risk, can be simplified as well, and
the change in tracking error squared can be expressed as
(12) 2TE = x(Gl – Gm) + x2[(fl – fm)( fl – fm)T + ( 2   2 ) ]
l
m
where fi and  2 are the factor loading vector and residual variance of
i
security i, respectively.
As discussed above, we can estimate security factor loadings and
residual variances and risk factor variances and covariances by analyzing
historical return data. Once we have estimated those risk parameters we
can use Equation (8) to calculate gradient Gi for each security. With the
risk parameter and gradient information in store, we can quickly quantify
the impact of any proposed transaction on the portfolio tracking error
using Equation (12).
OPTIMIZING PORTFOLIO MODIFICATIONS
Suppose we want to find a one-for-one transaction that will reduce
the portfolio risk relative to the benchmark the most. We need to
determine both the optimal securities and size to trade. A procedure can
be developed in a risk budgeting model that will allow us to do this
quickly.
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For a one-for-one transaction, Equation (12) gives the change in
tracking error as a quadratic function of the transaction size x,
(12) 2TE = bx + ax2
where
(13) b = (Gl – Gm)
(14) a = [(fl – fm)( fl – fm)T + ( 2   2 ) ].
l
m
To find the optimal market value to trade, we set d2TE / dx = 0, and
find that 2TE has a minimum at
(15)
x
b
.
2a
(This is clearly a minimum because  is a variance-covariance matrix,
which is positive semi-definite, guaranteeing that a  0.) The resulting
change in tracking error (as long as a  0) is
(16) 2TEmin = 
b2
.
4a
(Note that since 2TEmin is negative, it represents the maximum reduction
in tracking error by the transaction of the selected pair of securities.)
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Thus, for any selected pair of securities, it is straightforward to
calculate the optimal transaction size as well as the resulting tracking
error. In principle, the best such transaction could be found by a brute
force search across all pairs of securities in our investment universe,
selecting the one that gives the largest reduction in tracking error.
However, for N securities, there are (N–1)N/2 possible pairs of securities,
making this a time-consuming operation. Additionally, we may have
other reasons to favor one security over another.
A procedure can be designed that will make the search easily and
quickly. From the above equations, it is easy to see that the more
negative Gl is, the lower 2TE is; and the more positive Gm is, the lower
2TE is. Thus, the most reduction in tracking error would be achieved by
selling the security with the most positive gradient and using the
proceeds to buy the security with the most negative gradient.
Accordingly, the first step of the procedure would be to compute and sort
the gradients of all the securities in our investment universe. Thus, if we
wish to first pick the security to buy, the procedure presents the securities
in ascending order of their gradients; if we prefer to first select the
security to sell, it presents the securities in descending order of their
gradients. Once we have chosen a security, the procedure computes the
amount to buy (or sell) and the resulting tracking error for each of the
possible substitutions between this security and all other N-1 securities.
The results are then presented with the lowest tracking error first.
Our discussion in this section has focused on finding a one-for-one
transaction that will reduce the portfolio tracking error the most. Various
procedures can be developed depending on our objectives. For example,
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a procedure can be designed that will help us quickly find a transaction
that will increase the active ratio of the portfolio the most.
CONSTRUCTING OPTIMAL PORTFOLIOS
The procedures used to find an optimal transaction can be extended
into models for constructing optimal portfolios.
As in any portfolio optimization procedure, the first step is to
choose the set of securities that may be purchased. The composition of
this investable universe is critical. It should be large enough to provide
flexibility in matching benchmark risk exposures, yet contain only those
securities that are acceptable candidates for purchase.
Once the investable universe has been determined, a portfolio
optimization model begins an iterative process, searching for one-for-one
transactions that will achieve our objective. In the simplest case, the
objective is to minimize the risk of the portfolio relative to the
benchmark. In that case, the securities are ranked in terms of reduction in
tracking error per unit of each security purchased. The model indicates
which security, if purchased, will lead to the steepest decline in tracking
error. Once a security has been selected for purchase, the model offers a
list of possible one-for-one transactions of this security with various
securities in the portfolio (with the optimal transaction size for each
transaction), sorted in order of possible reduction in tracking error. Thus,
at each step of the iterative process, the model guides us to select the
transaction that will reduce the portfolio tracking error the most. The
procedure is repeated until the portfolio has reached the desired level of
risk relative to the benchmark.
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Minimization of tracking error is the most basic application of risk
budgeting. It is ideal for passive investors who want their portfolios to
track the benchmark as closely as possible. Risk budgeting also can be
used by those investors who hope to outperform the benchmark mainly
on the basis of security selection, without expressing views on markets or
sectors. For example, given a carefully selected universe of securities, a
procedure can be designed to help build security picks into a portfolio
with no significant systematic exposures relative to the benchmark.
For more active portfolios, the objective is no longer minimization
of tracking error. When minimizing tracking error, we may find that the
most efficient way is to reduce the largest differences in composition
between the portfolio and the benchmark. But what if, for example, the
portfolio is meant to be overweighted in a particular sector to express a
view. These views certainly should not be “optimized” away. However,
unintended exposures need to be minimized, while keeping the
intentional ones. In cases like this, a procedure can be designed that will
allow the investor to keep the desired sector exposure and optimize to
reduce the components of tracking error due to all other risk exposures.
More complicated procedures also can be developed to maximize
various portfolio risk-reward ratios, such as the active ratio.
QUANTIFYING ACTIVE RISK AT THE ASSET CLASS LEVEL
So far, our discussion of risk budgeting has focused on individual
portfolio manager level. After choosing a mix of asset classes, a plan
sponsor typically enlists a number of portfolio managers within each
asset class. For such an investor, risk budgeting ultimately must be at the
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total fund level. So we need to quantify the total fund tracking error and
identify its sources.
The total fund tracking error depends on the tracking error for each
asset class, which in turn depends on the tracking error taken by each of
the active portfolio managers. Thus, to quantify the total fund tracking
error we must first use information on the tracking error for each
manager to arrive at an asset class tacking error.
Table 1 illustrates how to compute the tracking error within an
asset class. The example uses a hypothetical set of U.S. Large Cap
managers, each of which is measured relative to the S&P 500 index.
Information on the tracking error and weight of each manager is shown in
the table.
Table 1:
U.S. Large Cap Tracking Error
=================================
Manager
Tracking Error (bp)
Weight
-------------------------------------------------------1
400
60%
2
600
40%
-------------------------------------------------------Total
380*
100%
=================================
* See derivation below
It is important to keep in mind that the asset class tracking error
depends on the correlation of active returns between managers within the
asset class. As the correlation between managers increases, the
investment in the asset class is taking on more exposures to the same
underlying source of active risk. Consequently, the tracking error for the
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asset class also increases, since the sources of active risk in the asset class
are becoming less diversified. Thus, one practical implication of this
analysis is that investors should look for investment styles that are
complementary (i.e., a correlation of active returns close to zero). By so
doing, the sources of tracking error can be better diversified and the asset
class tracking error reduced.
For the example in Table 1, we assume that two managers have
been enlisted within the U.S. Large Cap and that the correlation of their
active returns is 0.25. In that case, the formula for portfolio variance
applies:
(Tracking Error U.S. Large Cap)2 = (0.6*400)2 + (0.4*600)2
+ 2*0.25*(0.6*400)*(0.4*600)
= 144000
and
Tracking Error U.S. Large Cap = 380
Thus, the tracking error for the asset class is 380 basis points, which is
less than the tracking error of either of the two managers.
QUANTIFYING ACTIVE RISK AT THE TOTAL FUND LEVEL
Once we have calculated the tracking error for each asset class in
the asset mix, we can determine the tracking error for the total fund given
its asset allocation. Table 2 provides an example.
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Table 2:
Total Fund Tracking Error
====================================================
Asset Class
Benchmark
Tracking Error (bp) Weight
---------------------------------------------------------------------------------------U.S. Large Cap
S&P 500
380
45%
U.S. Small Cap
RUSSELL 2000
500
25%
U.S. Fixed Income
Lehman Aggregate
125
20%
International Equity
EAFE
400
10%
---------------------------------------------------------------------------------------Total Fund
217
100%
====================================================
Just as an assumption about the correlation of active returns across
managers within an asset class is required to determine the tracking error
of an asset class, so, too, is an assumption required about the correlation
of active returns across asset classes. This assumption can be interpreted
as measuring the degree to which active managers across asset classes are
exposed to the same active risk factors. The total fund tracking error in
Table 2 is calculated under the assumption that active returns are
uncorrelated across asset classes.
The correlation of active returns across asset classes has an
important impact on the total fund tracking error. Table 3 illustrates how
sensitive it is to the correlation assumption. It assumes that the fund
invests 60 percent of its capital in U.S. Large Cap and 40 percent in U.S.
Small Cap. As the correlation between active returns of the two asset
classes increases from  = 0 (uncorrelated) to  = 1.0 (perfectly
correlated), the tracking error of the total fund increases from 303 basis
points to 428 basis points. This type of sensitivity analysis is important,
as it provides a sense as to the potential ranges for the predicted total
fund tracking error.
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Table 3:
Total Fund Tracking Error and Asset Class Correlation
====================================================
Asset Class
Weight
TE (bp)
TE (bp)
TE (bp)
( = 0)
( = 0.5)
( = 1.0)
---------------------------------------------------------------------------------------U.S. Large Cap
60%
380
380
380
U.S. Small Cap
40%
500
500
500
---------------------------------------------------------------------------------------Total Fund
100%
303
371
428
====================================================
The total fund tracking error is a figure of interest in and of itself,
as it provides information about the risk of deviating from a purely
passive strategy at the total fund level. However, from the perspective of
adjusting risk allocation, additional information is required. In particular,
we need to identify the sources of the total fund tracking error.
Return to the example in Table 2. In that case, the bulk of the total
fund active risk is being taken in U.S. Equity. Only a small portion is
attributable to U.S. Fixed Income. Suppose we assume that U.S. Fixed
Income has a positive active return. Under this assumption, the fund
could possibly better diversify its total active risk away from U.S. Equity
and into U.S. Fixed Income. Because U.S. Fixed Income has a tracking
error (125 bps) less than one third of the tracking error of either U.S.
Large Cap (380 bps) or U.S. Small Cap (500 bps), shifting funds from
U.S. Equity to U.S. Fixed Income can largely reduce the total fund
tracking error.
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RISK BUDGETING AT THE TOTAL FUND LEVEL
Reiterating a point made earlier, risk budgeting ultimately must be
at the total fund level. At the total fund level, risk budgeting starts with a
strategic asset allocation.
The process of asset allocation usually begins by determining the
desired characteristics of the total risk of the fund. After the targeted total
fund risk has been set, a set of relevant asset classes is selected. At this
point, an optimizer based on modern portfolio theory is generally used to
develop an efficient frontier that shows the return-maximizing asset
allocations at various risk levels. From this efficient frontier, an asset
allocation is selected that best meets the overall desired risk
characteristics of the total fund and is taken as the strategic asset
allocation.
In principle, the investor has the option of passively investing in
the underlying indices of the asset classes selected. For example, the
investor can choose to invest in the S&P 500 and Russell 2000 index
funds for the capital allocated in the asset classes of U.S. Large Cap and
U.S. Small Cap. The strategic asset allocation in the underlying indices
for the asset classes chosen involves no active risk and constitutes the
strategic benchmark for the fund.
Most institutional investors take active risk relative to their
strategic benchmarks. Indeed, the trend is towards blending passive and
active investments. Active portfolio managers are enlisted because of
their presumed ability to add value at the total fund level. But, in which
asset classes should active risk be taken? What percentage of the assets
should be actively versus passively managed? Resolutions to such issues
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depend on what the targeted active risk is and what the objective function
is at the total fund level.
Suppose that, for a given target of total fund active risk, the
objective is to maximize the active ratio, i.e., the ratio of active return to
active risk, at the total fund level. In such a case, a computer model can
be used to determine the optimal allocation of active risk across the asset
classes and across the portfolio managers within each asset class. The
procedure typically starts with some relatively arbitrary allocation of
active risk and then recursively adjust it until the active ratio at the total
fund level is maximized. The resulting allocation of active risk is optimal
and subsequent deviations from it can be monitored by computing and
analyzing the active risk at the total fund level.
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