William F. Sharpe*

STANCO 25 Professor of Finance, Emeritus, Stanford University

Chairman, Financial Engines Inc.

September, 2001

This paper describes a set of mean/variance procedures for setting targets for the risk characteristics of components of a pension fund portfolio and for monitoring the portfolio over time to detect significant deviations from those targets. Due to the correlations of the returns provided by the managers of a typical defined benefit pension fund it is not possible to simply characterize the risk of the portfolio as the sum of the risks of the individual components. However, expected returns can be so characterized. We show that the relationship between marginal risks and implied expected excess returns provides the economic rationale for the risk budgeting and monitoring systems being implemented by a number of pension funds. Next, we show how a fund's liabilities can be taken into account to make the analysis consistent with goals assumed in asset/liability studies. We also discuss the use of factor models as well as aggregation and disaggregation procedures. The paper concludes with a short discussion of practical issues that should be addressed when implementing a pension fund risk budgeting and monitoring system.

Investment portfolios are composed of individual investment vehicles. A personal portfolio might be made up of equity and fixed income securities. An institutional portfolio might be made up of investments run by individual managers, with each such investment made up of equity and/or fixed income securities. Traditionally, each of the components of a portfolio is an asset, the future value of which cannot fall below zero. In this environment the total monetary value of the portfolio is typically considered an overall budget, to be allocated among investments. In a formal portfolio model, the decision variables are the proportions of total portfolio value allocated to the available investments. For example, in a portfolio optimization problem, the "budget constraint" is usually written as:

S

_{i}X_{i}= 1

where X_{i }is the proportion of total value allocated to investment i.

This approach does not work as well for portfolios that include investments that combine equal amounts of long and short positions. For example, a trading desk may choose to take a long position of $100 million in one set of securities and a short position of $100 million in another. The net investment is zero But this would also be true of a strategy involving long positions of $200 million and short positions of $200 million. For this type of portfolio, some other budgeting approach may be more desirable. One solution is to include a required margin, to be invested in a riskless security, and state gains and losses as percentages of that margin. This may suffice for a fund which uses few such investments, but is less than satisfactory for funds and institutions that utilize large short and long positions.

In recent years hedge funds and financial institutions with multiple trading desks have
developed and applied a different approach to this problem. Instead of (or in addition to)
a dollar budget, they employ a *risk budget*. The motivation is
straightforward. The goal of the organization is to achieve the most desirable
risk/return combination. To do this it must take on risk in order to obtain expected
return. One may think of the optimal set of investments as maximizing expected return for
a given level of overall portfolio risk. The latter provides the risk budget, and
the goal is to allocate this budget across investments in an optimal manner. Once a risk
budget is in place, the portfolio components can be monitored to assure that risk
positions do not diverge from those stated in the risk budget by more than pre-specified
amounts.

Recently, managers of defined benefit pension funds have taken an interest in employing the techniques of risk budgeting and monitoring.. To some extent this has been motivated by a desire to better analyze positions in derivatives, hedge funds and other potentially zero-investment vehicles. But even a fund with traditional investment vehicles can achieve a greater understanding of its portfolio by analyzing the risk attributes of each of the components.

Much of the practice of risk management in financial institutions is concerned with short-term variations in values. For example, a firm with a number of trading desks may be concerned with the effect of each desk on the overall risk of the firm. Risk management systems for such firms are designed to control the risk of each of the trading desks and to monitor each to attempt to identify practices that may be "out of control" -- adding more risk to the portfolio than was intended. Often such systems employ valuations made daily or even more frequently. Moreover, the horizon over which risk is calculated is typically measured in days rather than weeks, months or years.

Pension funds differ from such institutions in a number of respects. The components in a pension fund portfolio are typically accounts managed by other investment firms or by groups within the organization. Such accounts may or may not be valued daily, but major performance reports are typically produced monthly, using end-of-month valuations. Horizons for risk and return projects are often measured in years if not decades. Finally, to identify and control an external manager who is taking excessive risk can be very difficult.

Another key attribute of a defined benefit pension fund is its obligation to pay benefits in the future. This gives rise to a liability, so that its investment practices should, in principle, be viewed in terms of their impact on the difference between asset and liability values.

It is possible for a pension fund to gather data on the individual securities held by its managers and to establish a risk measurement and monitoring system using daily data on a security-by-security level, and some funds have implemented systems designed to do this, thereby replicating the types of risk management tools used by financial institutions. However, such systems are complex, require a great deal of data, and are costly. For this reason, a pension fund manager may choose a less ambitious approach, relying instead on data about the values of manager accounts provided on a less frequent basis. This paper focuses on procedures that can be employed in such a system, with investment managers evaluated on the basis of returns on invested capital, as in standard performance reporting and analysis.

Central to any risk budgeting and monitoring system is a set of estimates of the impacts of future events that can affect the value of the portfolio. In some systems, actual historic changes in values are used as estimates of possible future scenarios. In others a portfolio is subjected to a "stress test", for example, by assuming that future events might combine the worst experiences of the past at the same time even though they actually occurred at different times.

While the direct use of historic data provides useful information, for longer-horizon projections it is more common to consider a broader range of possible future scenarios based on models of the return-generating process. A standard approach utilizes estimates of risks and correlations, with assumptions about the shapes of possible probability distributions. Often a factor model is employed in order to focus on the key sources of correlated risks. In this paper we assume that such an approach is utilized. Moreover, we assume that only a single set of estimates for such a model is needed, although the procedures described here can be adapted relatively easily for use with alternative estimates as part of stress testing.

Most large pension funds take a two-stage approach when allocating funds among investments. The top stage involves a detailed study of the fund's allocation among major asset classes, usually (but not always) taking into account the liabilities associated with the fund's obligation to pay future pensions. For example, such an asset allocation or asset/liability study might be performed every 3 years. Its result is a set of asset allocation targets and a set of allowed ranges for investment in each asset class. Based on the asset allocation analysis, funds are allocated among investment managers, subject to the constraints that each asset exposure fall within the specified range.

In many cases, the fund asset allocation analysis uses a standard one-period mean/variance approach, often coupled with Monte Carlo simulation to make long-term projections of the effects of each possible asset allocation on the fund's future funded status, required contributions, etc.. For most large funds asset allocation typically accounts for well over 90% of total risk, justifying the attention given it by management and the fund's investment board.

The world of investments is very complex, with hundreds of thousands of possible
investment vehicles. This makes it virtually impossible to estimate risks and
correlations on a security-by-security basis. For this reason, almost all risk
estimation systems employ some sort of *factor model*. A relatively small
number of factors are identified and their risks and correlations estimated. It is
assumed that the return on any individual investment component can be expressed as a
function of one or more of these factors plus a residual return that is independent of the
factors and of the residual returns on all other investments. This reduces the estimation
problem to one requiring estimates of the factor risks and correlations plus, for each
investment component, the estimation of the parameters in the function relating its return
to the factors and the risk associated with its residual return. In some systems,
factors are assumed to exhibit serial correlation over time, but a component's residual
returns are generally assumed to be independent from period to period. One-period returns
are generally assumed to be normally or log-normally distributed, with multi-period
returns distributions determined by the characteristics of the underlying process.

In practice, the factors used in a risk budgeting and monitoring system are likely to differ from the asset classes used for the fund's asset allocation studies. Ideally there should be a clear correspondence between the two. We do not consider such issues here. Instead we allow for the possibility that the factor model and risk and correlation estimates used in the risk budgeting system may differ from the model and estimates used in the asset/liability analyses.

Following common practice in the pension fund industry, we assume that factor models used in the overall process are linear. More precisely, we assume that the return on any investment component can be expressed as a linear function of the returns on the factors (or asset classes) plus an independent residual.

We are concerned with a *pension fund *that employs a number of *investment
managers. *The fund has a fixed number of dollars in assets, *A *and a
liability *L*, representing the present value of its accrued obligations. Ideally,
both assets and liabilities should be measured in terms of market value. In
practice, however, liabilities are often obtained as a by-product of actuarial
analyses designed to determine appropriate current fund contributions. Such
liabilities are typically not (nor were they intended to be) equal to the market value of
any particular definition of accrued liabilities. Such actuarial liability values tend to
respond relatively slowly to changes in the values of assets, interest rates, etc..
For this reason, analyses that compare market values of assets with actuarial values of
liabilities give results that are affected less by the inclusion of liabilities than would
be the case if the market values of liabilities were used.

Ultimately, the decision variables for the fund's overall allocation are the amounts
given to each of the investment managers (henceforth, *managers*). We assume that
it is possible to determine each manager's factor exposures and residual risk. This
could be performed using a top-down approach such as returns-based style analysis or a
bottom-up procedure based on the manager's security holdings. The best approach
depends on the horizon over which risk is to be estimated, the costs and accuracies of
alternative models, and other considerations. Here we simply assume that such
measures have been obtained.

To fix ideas, we start with an extremely simple example. A fund has identified three asset classes -- cash, bonds and stocks. It plans to allocate its money ($100 million) among three managers, in accordance with the results of an asset allocation study. Each manager runs an index fund that tracks one of the classes exactly. Thus the manager selection problem and the asset allocation problem are one and the same. For now we ignore the fund's liabilities, concentrating solely on the risk and return of the fund's assets.

To simplify the analysis, all calculations deal with *excess returns*, where*
*the excess return on an asset is defined as its return minus the return on cash. Thus
we are concerned with expected excess return (EER), standard deviation of excess return
(SDER) and the correlations of excess returns. The assumptions used for the asset
allocation study are these:

Expected Excess Returns and Risks

EER SDER Cash 0 0 Bonds 2 10 Stocks 6 18

Correlations

Bonds Stocks Bonds 1.0 1.0 Stocks 0.4 0.4

For computational ease, the standard deviations and correlations can be combined to
produce a *covariance matrix*, which includes all the risk estimates. The
covariance between two asset classes is simply their correlation times the product of
their standard deviations. Including cash, we have:

Covariances

Cash Bonds Stocks Cash 0 0 0 Bonds 0 100 72 Stocks 0 72 324

Given the estimated expected excess returns and covariances, it is possible to find a Markowitz-efficient portfolio for any given level of risk. By definition, such a portfolio provides the greatest possible expected excess return for the level of risk. For our example we assume that after considering the long-term effects of each of several such portfolios using Monte Carlo simulation, the fund selected the following:

Optimal Asset Allocation

Proportion Cash .0741 Bonds .2976 Stocks .6283

This portfolio's expected excess return and standard deviation of excess return are:

Portfolio Expected Excess Return and Risk

EER SDER Portfolio 4.3651 12.7942

In this case, each manager is given the specified amount for its asset class and each
manager's return equals that of its asset class. Given this, we can compute the *dollar
expected excess return* ($EER) for each manager and its proportion of the portfolio's
dollar expected excess return.. For example, the bond manager has 29.76% (.2976) of the
total asset value of $100 million, or $29.76 million. The expected excess return on
this part of the portfolio is 2% per year (.02), so that the manager is expected to add
$595.2 thousand or $0.5952 million to the portfolio, over and above the amount that
could have been earned by putting the money in cash. Similar computations show that
the stock manager is expected to add $3.7698 million over and above the amount that could
have been earned in cash. Since the cash manager cannot contribute any expected
excess return, the total expected added value is $ 4.3651 million, which is consistent
with the portfolio expected excess return obtained in the asset allocation study.

Dollar Expected Excess Returns

Percent Dollars

($ million)EER $EER Proportion of $EER Cash Manager .0741 7.41 0 0 0.00 Bond Manager .2976 29.76 0.02 0.5952 0.1364 Stock Manager .6283 62.83 0.06 3.7698 0.8636 PORTFOLIO 1.00 100.00 4.3651 1.0000

The final column in this table shows the proportion of total dollar expected excess return contributed by each manager, obtained by dividing its $EER by the portfolio's total $EER. In this case the bond manager is expected to contribute approximately 13.6% of the expected excess return and the stock manager to contribute 86.4%. Note that this differs significantly from the amount of assets allocated to each (approximately 29.8% and 62.8%, respectively).

It would be straightforward to compute the risk of each manager's position, in percentage or in dollar terms. We could also compute a value-at-risk (VAR) amount for each manager, indicating, for example, an amount that the manager's value could fall below with 1 chance out of 20. But such measures are not sufficient to determine the effect of a manager's investments on the risk (or VAR) of the portfolio as a whole, since they do not take into account the correlations among manager returns.

It would seem that the entire concept of risk budgeting is doomed to failure in cases
involving risks that are correlated, since risks are not additive when they are
correlated. Fortunately there is a measure of *manager risk contribution* that can
serve as the foundation of a budgeting procedure, as we now show.

Define the *marginal risk* of a manager as the change in total portfolio risk
per unit change in the amount allocated to the manager when the amount is small.
More precisely, it is the derivative of portfolio risk, expressed as *variance*
(standard deviation squared) with respect to the amount allocated to the manager. The
marginal risk of a manager ( MR_{i} ) will equal twice its covariance with the
portfolio as a whole, which is in turn a weighted average of the covariances of the
manager's returns with those of the other managers, using the current portfolio
proportions as weights.

MR_{i} = dVp/dX_{i} = 2 C_{ip}
= 2 S_{j} X_{j} C_{ij}

where d denotes a partial derivative.

Applying the formula to the case at hand gives the following.

Marginal Risk Cash Manager 0 Bond Manager 150 Stock Manager 450

The stock manager has three times the marginal risk of the bond manager. Recall that the expected excess return of the stock manager is also three times that of the bond manager. As we will see, this is not a coincidence.

A Markowitz-efficient portfolio offers the greatest expected return for a given level of risk. To find a set of such portfolios it is usually more computationally efficient to solve the following problem:

Maximize: EU = EER

_{p}- V_{p}/ rt

Subject to: S

_{i}X_{i}= 1

where EER_{p} and V_{p} are respectively the expected value and
variance of portfolio excess return.

For a given level of rt (risk tolerance), the solution will provide an efficient portfolio. By solving with different levels of rt, all efficient portfolios can be found, and the one with the most desirable risk and expected return selected based on the preferences of the decision-makers. In our example, the selected portfolio maximizes EU (expected utility) for a risk tolerance (rt) of 75.

In practice, optimization problems are formulated with inequality constraints. The amounts invested in at least some asset classes are constrained to lie above a lower bound (e.g. 0) and/or below an upper bound (e.g. .25, or 25% of the portfolio). We discuss this later. For now, we assume that such constraints are absent or, if present, none is binding in the solution.

Consider the *marginal expected utility* (MEU) of a position in a portfolio,
defined as the rate of change of expected utility (EU) per unit change in the amount
invested in that position. This will equal:

MEU_{i} = dEU/dX_{i} = dEER_{p}/dX_{i}
- ( dV_{p}/dX_{i})
/ rt

But under the assumption that the expected return of an asset is the same regardless of
the amount invested in it, the derivative of EER_{p} with respect to X_{i}
will equal EER_{i} Moreover, the derivative of V_{p} with respect to X_{i}
is the value that we have defined as the manager's marginal risk, MR_{i}. So we
can write:

MEU

_{i}= EER_{i}- MR_{i}/ rt

Imagine a portfolio in which the marginal expected utilities of two managers differ. Clearly the portfolio is not optimal. Why? Because one could take some money away from the manager with the lower MEU and give it to the manager with the higher MEU, thereby increasing the expected utility of the portfolio. It thus follows that a condition for portfolio optimality in the absence of binding constraints is that:

MEU

_{i }= k for all i

where k is a constant

This is the first-order condition for portfolio optimality. *It provides the
economic basis for risk management systems of the type we study here*. To see
why, consider cash, the risk-free asset. Its expected excess return (EER_{i})
is zero, as is its risk (V_{i}) and marginal risk (MR_{i}). Thus its MEU_{i}
value will equal zero. But this requires that k=0. Thus for each manager (i):

MEU

_{i}= EER_{i}- MR_{i}/ rt = 0

so that:

EER

_{i}= MR_{i}/ rt

The previous equation is central to the motivation for a risk management system based
on mean/variance analysis. One way to solve an optimization problem in the absence
of inequality constraints is to find the set of proportions (X_{i}'s) summing to
one for which each manager's marginal risk (MR_{i}) is equal to its expected
return times the fund's risk tolerance. In other words, solve the set of simultaneous
equations that will make

MR

_{i}= rt * EER_{i}

for every asset i. This process is termed *portfolio optimization*.

For our purposes, it is more instructive to reverse this process. Assume that a portfolio is optimal and that the covariances of its components are known, as is the risk tolerance of the fund. Then one can find the expected excess returns for the components using the first-order conditions

EER

_{i}= MR_{i}/ rt

for every asset i.

This is generally termed *reverse optimization*. It is also sometimes described
as finding the *implied views* (of expected excess returns) for a portfolio.

Assume that the covariance matrix for a fund's managers is known. To compute implied expected excess returns one only needs an estimate of the fund's risk tolerance (rt). But this can be found if one manager's expected excess return is known. Recall our example. Using only the covariance matrix, we computed the marginal risks for the managers. To find the implied expected excess returns we need only know the expected excess return of one component or combination of components. For example, the marginal risk of the stock manager was 450. If the expected excess return on stocks is 6%, then rt = 450/6, or 75 and the implied expected excess returns are those shown below.

Marginal Risk Implied EER Cash Manager 0 0 Bond Manager 150 2 Stock Manager 450 6

Not surprisingly, the implied values of the asset EER's are identical to those that were used in the optimization process.

Given expected excess returns, we can compute dollar expected excess returns as before
as well as the proportions of total dollar expected excess return. However, the
latter will be the same regardless of risk tolerance, and can be computed solely from the
covariance matrix and portfolio composition. Defining P$EER_{i} as the
proportion of dollar expected excess return provided by manager i:

P$EER_{i }= X_{i}
EER_{i }/ S_{i} ( X_{i}
EER_{i} )

= ( X_{i} MR_{i }/ rt ) / S_{i}
( X_{i} MR_{i} / rt )

= X_{i} MR_{i }/ S_{i}
( X_{i} MR_{i })

This relationship both explains and justifies the computations that lie behind
mean/variance risk budgeting and management. Marginal risks act as surrogates for expected
excess returns. An overall fund should be managed so that the marginal risk of each
of its components is commensurate with the expected excess return of that component. Given
expectations about returns, a risk budget can be established, with each component allowed
to have a marginal risk (MR_{i}) and fund allocation (X_{i}) that give it
an appropriate contribution (P$EER_{i}).

Recall from our earlier discussion that the marginal risk of a portfolio component will equal twice its covariance with the portfolio. From the properties of covariance we know that:

S_{i}
X_{i} MR_{i} = S_{i} X_{i}
2 C_{ip }= 2 S_{i} XC_{ip }= 2
V_{p}

Thus the sum of the weighted marginal risks of the portfolio components will equal
twice the variance of the overall portfolio. This leads some to define the risk
contribution of a component as half its marginal risk (that is, its covariance with the
portfolio) so that a weighted average of these values will equal the variance of the
overall portfolio. Clearly, this will have no effect on the P$EER_{i}
values. However, it sometimes leads to an incorrect view that it is possible to
decompose portfolio risk into a set of additive components and to incorrect statements of
the form "this manager contributed 15% to the total risk of the portfolio".

There is a case in which computations based on marginal risks do provide an additive
decomposition of total portfolio risk. If all component returns are independent, the
marginal risk of manager i will equal 2X_{i}V_{i }and the product X_{i}MR_{i}
will equal 2X_{i}^{2}V_{i}. Summing these values over
managers will give an amount equal to twice the portfolio variance. In this special case,
defining a manager's risk contribution as half its marginal risk thus makes the product X_{i}MR_{i}
equal precisely its contribution to total portfolio risk.

We will see that the assumption of independence may be appropriate for the portion of a manager's return that is not related to the factors included in the underlying factor model and that this interpretation of risk decomposition can be applied to that portion of overall portfolio variance. Much of the literature on risk budgeting and monitoring focuses on such non-factor risk and thus is justified in claiming that the procedures provide an allocation of portfolio risk. However, this is not applicable for the correlated components of a pension fund, which generate most of its risk.

Humans best process information when relationships are linear. Expected returns are linear, and the expected return of a portfolio can be decomposed into portions provided by each of the portfolio components. This is not generally the case for risk. For this reason, the computations we have described, which are utilized in many pension fund mean/variance risk budgeting and monitoring systems are best viewed in terms of implied expected excess return budgets and deviations therefrom.

In practice pension fund risk budgeting and monitoring involves three related but somewhat separate phases.

In the first phase, the fund selects a* policy portfolio *in which dollar
amounts are allocated to managers. As indicated earlier, this often involves two
stages: (1) an asset allocation or asset/liability study using optimization analysis that
allocates funds among asset classes, and (2) the subsequent allocation of funds among
managers using procedures which may be quantitative, qualitative or a combination of both.
Rarely is the policy portfolio determined entirely by mean/variance optimization, and even
the optimization analysis utilized as part of the process often involves binding
constraints so that the first order conditions we have described will not hold
strictly for every component. In any event, we term this entire process the *policy
phase*. Part of the process will use expected returns and covariances, which we term
the *policy expected returns* and *policy risks and correlations*. It will
also use a (possibly implicit) factor model which we will term the *policy factor
model. *The output of this phase is the* policy portfolio*, which allocates
specific amounts of capital to each of the components of the fund.

The second phase is the establishment of the fund's risk budget. This makes the
(possibly heroic) assumption that the policy portfolio is optimal in a mean/variance sense
at the time of its designation. Ideally this would be interpreted using the policy
factor model and policy expected returns and covariances. However, the very nature
of the policy phase may make this impossible, since there are typically insufficient
estimates of risks and returns, and in most cases the policy portfolio is not created
directly from a formal unconstrained optimization. In practice, therefore, a more
complete factor model and set of risks and correlations, typically provided by an outside
vendor, is used to perform a reverse optimization based on the policy portfolio as of the
date of its formation. This yields a series of implied expected excess returns and
proportions of the portfolio's dollar expected excess return attributable to each
component. Such values represents the components' *risk budgets* (RB).
From our previous formulas this can be stated most directly as:

RB

_{i}= X_{i}C_{ip}/ V_{p}

for each manager i.

Even if the policy portfolio is implemented precisely when the policy phase is
completed, market movements, manager actions and changing risks and correlations will lead
to changes in many or all aspects of the situation. This leads to the third phase.
Current estimates of risks and correlations, typically provided by the outside vendor,
along with manager positions and/or return histories, are used to compute a new set of
values for investment proportions, covariances and portfolio variance. This provides the
current set of* risk proportions* (RP). Letting primes denote current
values, we have:

RP

_{i}= X_{i}'C_{ip}' / V_{p}'

The monitoring phase involves comparisons of the current risk proportions with the risk budgets. Significant disparities lead to evaluation, analysis and in some cases action.

Not surprisingly, this process can easily be misunderstood and misused. The risk budget
figures are actually surrogates for proportions of expected value added (over cash) at the
time of the policy analysis. The risk proportion figures are surrogates for the
proportions of implied expected value added, based on the current situation. The
presumption is that large disparities need to be justified by changes in estimates of the
abilities of the manager to add value. Lacking this, some sort of action should be
taken. Note, however, that a change in a manager's RP value may be due to events
beyond his or her control. Moreover, there are many ways to change a manager's RP value if
such a change is needed. The amount invested (X_{i}) may be adjusted, as may
the covariance of the manager's retrun with that of the portfolio. The covariance
may, in turn, be changed by altering the manager's investment strategy or by changing the
allocation of funds among other managers and/or the strategies of other managers.

In any event, the comparison of RP values with RB values provides a discipline for monitoring an overall portfolio to insure that it remains reasonably consistent with original or modified estimates of the abilities of the managers. While the actions to be taken in the event of significant disparities are not immediately obvious, it is important to know when actions of some sort are desirable

The diagram below shows the three phases.

The policy phase is performed periodically, followed by the risk budgeting phase. The monitoring phase is then performed frequently, until the process begins anew with another policy phase.

Thus far we have ignored the presence of liabilities, assuming that all three phrases focus on the risk and return of the fund's assets. However, this may not be appropriate for a fund designed to discharge liabilities. Fortunately, the procedures we have described can be adapted to take liabilities into account.

We continue to utilize a one-period analysis. Current asset and liability values are A_{0}
and L_{0}, respectively. At the end of the period the values will be A_{1}
and L_{1}, neither of which is known with certainty at present. We define the
fund's *surplus* as assets minus liabilities. Thus S_{0}=A_{0}-L_{0}
and S_{1}=A_{1}-L_{1}. We assume that the fund is concerned with
the risk and return of its future surplus, expressed as a portion of current assets, that
is S_{1}/A_{0}. Equivalently,

S

_{1}/ A_{0}= (A_{1}/ A_{0 }) - ( L_{0}/ A_{0}) * ( L_{1}/ L_{0})

The first parenthesized expression equals 1 plus the return on assets, while the last
parenthesized expression can be interpreted as 1 plus the return on liabilities.
Defining the current ratio of liabilities to assets (L_{0}/A_{0}) as the
fund's *debt ratio* (d), S_{1}/A_{0} may be written as:

(1-d) + R

_{A}- d*R_{L}

The parenthesized expression is a constant and hence cannot be affected by the fund's
investment policy. We thus consider only the difference between the asset return ( R_{A
}) and the liability return multiplied by the debt ratio (d*R_{L} ).

We are now ready to write the expected utility as a function of the expected value and
risk of ( R_{A} - d*R_{L} )Let rt be the fund's risk tolerance for
surplus risk. Then:

EU = E( R

_{A}) - E( d* R_{L}) - V( R_{A}- d*R_{L }) / rt

Expanding the variance term gives:

EU = E( R

_{A}) - E( d* R_{L}) - V( R_{A}) / rt + 2*d*C_{AL }/ rt - d^{2}*V_{L}/ rt

Since the decision variables are the asset investments, we can ignore terms that are not affected by them. Neither the expected liability return nor the variance of the liability return is affected by investment decisions. Hence for optimization and reverse optimization purposes we can define expected utility as:

EU = E( R

_{A}) - V( R_{A}) / rt + 2*d*C_{AL }/ rt

Note that this differs from expected utility in the case of an asset-only optimization
only by the addition of the final term, which includes the covariance of the asset return
with the liabilities. Moreover, covariances are additive, so that the covariance of
the asset portfolio with the liabilities will equal a value-weighted average of the
covariances of the components with the liabilities. This implies that the marginal
expected utility of component i will equal:

MEU

_{i}= EER_{i}- MR_{i}/ rt + 2*d*C_{iL}/rt

where C_{iL }is the covariance of the return of asset i with that
of the liabilities.

We can now write the first order condition for optimality in an asset/liability analysis as:

MEU

_{i}= EER_{i}- MR_{i}/ rt + 2*d*C_{iL}/ rt = k for each asset i

But the risk-free asset (here, cash) has zero values for all three components. Hence, as before, k=0 so that:

EER

_{i}- MR_{i}/ rt + 2*d*C_{iL}/ rt = 0

and:

EER

_{i}= MR_{i}/ rt - 2*d*C_{iL}/ rt

To compute implied expected excess returns we need only subtract the covariance of an
asset with the liability from its marginal risk, then divide by risk tolerance.
Alternatively, recalling that MR_{i}= 2*C_{ip} , we can write:

EER

_{i}= (2 / rt ) * ( C_{ip}- d*C_{iL})

All the procedures described in the asset/only case can be adapted in straightforward ways to incorporate liabilities. For example, the risk budgets and risk positions can be determined using the following formulas:

RB

_{i}= [ X_{i}* ( C_{ip}- d*C_{iL}) ] / [ V_{p}- d*C_{pL}]

RP

_{i}= [ X_{i}' * ( C_{ip}' - d'*C_{iL}' ) ] / [ V_{p}' - d'*C_{pL}' ]

where, as before, the variables without primes reflect values at the time of the policy
analysis and the variables with primes reflect values at the current time. Due to the
properties of variances and covariances, the values of RB_{i} will sum to one, as
will the values of RP_{i}.

As indicated earlier, most risk estimation procedures employ a factor model to provide more robust predictions. Generically, such a model has the form:

R

_{i}= b_{i1}F_{1}+ b_{i2}F_{2}+ .... + b_{in}F_{n}+ e_{i}

where b_{i1}, b_{i2}, ...., b_{in} are the sensitivities
of R_{i} to factors F_{1},_{ }F_{2}, ...,F_{n} ,
respectively and e_{i} is component i's residual return. Each e_{i} is
assumed to be independent of each of the factors and of each of the other residual
returns.

A risk model of this type requires estimates of the risks (standard deviations) of each of the factors and of each of the residual returns. It also requires estimates of the correlations of the factors with one another.

Note that in this model each return is a linear function of the factors. We may thus aggregate, using the proportions held in the components to obtain the portfolio's return:

R

_{p}= b_{p1}F_{1}+ b_{p2}F_{2}+ .... + b_{pn}F_{n}+ e_{p}

where each value of b_{p }is the value-weighted average of the corresponding b_{i}
values and e_{p }is the value-weighted average of the e_{i} values.

It is convenient to break each return into a factor-related component and a residual
component. Defining R_{Fi }as the sum of the first n terms on the right-hand
side of the equation for R_{i}, we can write:

R

_{i}= R_{Fi}+ e_{i}

Similarly, for the portfolio:

R

_{p}= R_{Fp}+ e_{p}

Now consider the covariance of component i with the portfolio. By the properties of covariance, it will equal

C

_{ip}= cov( R_{Fi}+ e_{i , }R_{Fp}+ e_{p}) = cov( R_{Fi}, R_{Fp}) + cov ( R_{Fi}, e_{p}) + cov( e_{i}, R_{Fp}) + cov( e_{i}, e_{p})

By the assumptions of the factor model, the second and third covariances are zero. Hence:

C

_{ip}= cov( R_{Fi}, R_{Fp}) + cov( e_{i}, e_{p})

Recall that:

e_{p }= S_{i} X_{i} e_{i}

Since the residual returns are assumed to be uncorrelated with one another, the
covariance of e_{i} with e_{p} is due to only one term. Let v_{i}
be the variance of e_{i }(that is, component i's *residual variance*).
Then:

cov( e

_{i}, e_{p}) = X_{i}v_{i}

and

C

_{ip}= cov( R_{Fi}, R_{Fp}) + X_{i}v_{i}

Substituting this expression in the formula for the implied excess return in the presence of liabilities we have:

EER

_{i}= (2 / rt ) * ( cov( R_{Fi}, R_{Fp}) + X_{i}v_{i}- d*C_{iL})

This can be regrouped into two parts -- one that would be applicable were there no residual risk, and one that results from such risk:

EER

_{i}= (2 / rt ) * [ cov( R_{Fi}, R_{Fp}) - d*C_{iL}] + (2 / rt )*X_{i}v_{i}

The final term is often termed the manager's *alpha value* -- that is the
difference between overall expected return and that due to the manager's exposures to the
factors (and here, covariance with the fund's liability). Thus we have:

EER

_{i}= Factor-related EER_{i}+ a_{i}

where:

Factor-related EER

_{i }= (2 / rt ) * ( cov( R_{Fi}, R_{Fp}) - d*C_{iL})

and

a

_{i }= (2 / rt ) * X_{i}v_{i}

Just as implied expected returns can be decomposed into factor-related and residual components, so too can risk budgets and risk proportions. For example, a manager could be given a budget for factor-related contributions to risk and a separate budget for residual risk. Many systems concentrate on the latter, which has substantial advantages since, as indicated earlier, the contributions to portfolio residual risk do in fact add to give the total portfolio residual variance. However, they cover only a small part of the total risk of a typical pension fund.

Expected excess returns are additive. Thus the expected excess return for a group of managers will equal a weighted average of their expected excess returns, using the relative values invested as weights. Covariances are also additive. Thus the marginal risk of a group of managers can be computed by weighting their marginal risks by relative values invested. This makes it possible to aggregate risk budgets and risk proportions in any desired manner. For example, a fund may be organized in levels, with each level's risk budget allocated to managers or securities in the level below it, and so on. Thus there could be a risk budget for equities, with sub-budgets for domestic equities and international equities. Within each of these budgets there may be sub-budgets for individual managers, and so on.

The following tables provide an example for a large pension fund. These results were obtained using an asset-class factor model with historic risk and correlation estimates. The relationships of the managers to the factors were found using returns-based style analysis. Residual variances are based on out-of-sample deviations from benchmarks based on previous returns-based style analyses. After the managers were analyzed, they were combined into groups based on the fund's standard classifications. Implied expected excess returns were calibrated so that a passive domestic equity portfolio would have the expected excess return used in thef und's most recent asset allocation study. Although the fund does take liabilities into account when choosing its asset allocation, the figures shown here are based solely on asset risks and correlations.

**Implied Expected Excess Returns (EER)**

Factor-related Alpha Total Cash Equivalents 0.02 0.00 0.02 Fixed Income 1.53 0.00 1.53 Real Estate 4.01 0.36 4.36 Domestic Equity 5.66 0.03 5.69 International Equity 4.82 0.03 4.84 Int'l Fixed Income 1.42 0.00 1.42 Global Asset Allocation 3.79 0.00 3.79 Special Assets 3.13 0.26 3.40

Note that the implied alpha values are all small, including four that are less than 1/2 of 1 basis point and thus shown as 0.00. This is not unusual for funds with many managers. A high degree of diversification is consistent with relatively low expectations concerning managers' abilities to add value.

As we have shown, the implied expected excess returns can be combined with the amounts allocated to the managers to determine the implied expected values added over a cash investment, which we have termed the dollar expected excess returns ($EER). These can be divided by the total $EER for the portfolio to show the relative contribution to excess expected return for each manager or aggregate thereof. The final three columns of the following table shows the results for the fund in question, broken into the factor-related component and residual-related component (alpha).

**
Percents of Implied Portfolio Dollar Expected Excess Return**

% of $ % of $EER: Factor-related % of $EER:

Alpha% of $EER:

TotalCash Equivalents 1.72 0.01 0.00 0.01 Fixed Income 21.30 7.69 0.02 7.71 Real Estate 5.15 4.88 0.43 5.31 Domestic Equity 44.74 59.84 0.31 60.14 International Equity 19.11 21.77 0.11 21.88 Int'l Fixed Income 3.15 1.06 0.00 1.06 Global Asset Allocation 0.00 0.00 0.00 0.00 Special Assets 4.84 3.59 0.30 3.89 TOTAL 100.00 98.83 1.17 100.00

In this case, by far the largest part of the implied added value (98.83 %) is
attributable to the managers' factor exposures. This has a natural interpretation as the
proportion of portfolio variance explained by factor risks and correlations plus the
portfolio's exposures to those factors. This follows from the fact that the alpha values
are derived from contributions to residual variance, each of which equals X_{i}^{2}v_{i},
making the sum equal to the portfolio's residual variance.

Reports such as this can be valuable when allocating pension fund staff resources. Note, for example, that the fund has allocated slightly less than 45% of its money to domestic equity managers, but this analysis indicates that such managers should be expected to provide over 60% of the added value over investing the entire fund in cash. This might lead to the conclusion that 60% of staff resources might be assigned to this part of the portfolio, instead of 45%.

We have chosen to present the results of this analysis in terms of implied dollar expected excess returns. However in most risk budgeting systems the terms "risk budget" and "risk contribution" would typically be used instead. For example, assume that the prior report was produced using the fund's policy portfolio. Then the percentages in the final column would constitute the "risk budgets" for the aggregate groups. At subsequent reporting periods the same type of analysis could be performed, giving a new set of results, which could be compared with those obtained at the time the policy phase was completed. The resulting report would have the following appearance, with the final two columns filled in based on the current situation.

**
Risk Management Report**

Risk Budget Risk Proportions Difference Cash Equivalents 0.01 Fixed Income 7.71 Real Estate 5.31 Domestic Equity 60.14 International Equity 21.88 Int'l Fixed Income 1.06 Global Asset Allocation 0.00 Special Assets 3.89 TOTAL 100.00

In many systems each part of the portfolio is given both a risk budget and an accompanying set of ranges. Often the latter are broken into a "green zone" (acceptable), a "red zone" (unacceptable), with a "yellow zone" (watch) between.

While risk budgeting and monitoring systems can prove very useful in a pension fund context, some issues associated with their implementation need to be addressed.

As we have shown, the central principle behind the use of risk budgets based on mean/variance analysis is the assumption that a particular portfolio is optimal in the sense of Markowitz, with no binding inequality constraints. This may be inconsistent with the procedures used to allocate funds among managers at the time of a policy study (or at any time thereafter). It is true that asset class allocations are typically made with the assistance of optimization analysis. However, the formal optimization procedure often includes bounds on asset allocations, some of which are binding in the solution. Moreover, the results of the optimization study provide guidance only on allocation across broad asset classes and the study typically assumes that all funds are invested in pure, zero-cost index funds, each of which tracks a single asset class precisely. Actual implementations involve managers that engage in active management and often provide exposures to multiple asset classes. Since the eventual allocation of funds across managers is made using a variety of procedures, some quantitative, others qualitative, the resulting allocation may not be completely optimal in mean/variance terms.

Potential problems may also arise when the asset allocation model uses one set of factors (the asset class returns), while the risk budgeting and monitoring system uses another. Even if the policy portfolio is optimal using the policy factor model, expected returns and risk and correlation assumptions, it may not be optimal using the risk budgeting system's factor model, manager factor exposures and risk and correlation estimates. Yet this may be assumed when the risk budgets are set.

Finally, there is the problem of choosing an appropriate action when a risk proportion (RP) diverges unacceptably from a previously set risk budget (RB). Consider a case in which the risk proportion exceeds the risk budget. Should money be taken away from the manager or should the manager be asked to reduce his or her contribution to portfolio risk? If the latter, what actions should the manager take? One alternative is to reduce residual risk, but this may not be sufficient, and may lower the manager's chance of superior performance. The manager could be asked to change exposures to the underlying factors, but such changes could force a manager to move from his or her preferred "style" or investment habitat, with similar ill effects on overall performance.

Some of these problems are mitigated if the risk budgeting and monitoring system deals only with residual (non-factor) risks. But, for a typical pension fund such risks constitute a small part of overall portfolio risk, which is consistent with low implied expectations for added return (alpha). To provide a comprehensive view of a portfolio it is important to analyze both the small (uncorrelated) part of its risk and the large (correlated) part.

We have shown that a great many results can be obtained by combining a risk model with attributes of a fund's investments. A portfolio based on a policy study and its implementation can be used to set targets, or risk budgets. These can be used to allocate effort for manager oversight, selection, and monitoring. Subsequently, actual portfolios can be analyzed to determine the extent to which risk computations based on current holdings differ from those obtained using policy holdings. Significant differences can then be used to initiate changes, as needed.

At this date, the use of risk budgeting and monitoring by defined benefit pension funds is limited. As more funds implement such procedures we will find the strengths and weaknesses in this context and deal with issues associated with their implementation . There is no doubt that risk budgeting and monitoring systems can produce large amounts of data. In time we will learn how to insure that they produce the most useful information.

* There is an extensive literature on Risk Budgeting and Monitoring Systems. An
excellent source is* Risk Budgeting, A New Approach to Investing*, edited by Leslie
Rahl, (Risk Books, 2000). In describing and interpreting some of the procedures used in
risk budgeting systems I have drawn on a great deal of work done by others as well as some
of my earlier results. The idea of computing implied views of expected excess returns
based on portfolio composition and covariances can be found in William F. Sharpe,
"Imputing Expected Returns From Portfolio Composition," *Journal of Financial
and Quantitative Analysis*, June 1974. The relationship between an asset's
expected return and its covariance with a set of liabilities is described in William F.
Sharpe and Lawrence G. Tint, "Liabilities -- A New Approach,", *Journal of
Portfolio Management*, Winter 1990, pp. 5-10.