Introduction

• By now, value-at-risk (𝑉𝑎𝑅) has spread well beyond the Wall Street trading departments where it originated. The investment management industry has also discovered the benefits of 𝑉𝑎𝑅 systems.

• 𝑉𝑎𝑅 has many benefits
  • 𝑉𝑎𝑅 is a forward-looking measure of the risk profile of a fund based on current positions.
  • The more traditional returns-based approach, in contrast, is purely historical; it does not offer timely measurement of risk.
  • 𝑉𝑎𝑅 can be used to measure, control, and manage risk.
  • VaR is comprehensive because it accounts for leverage, volatility, and diversification.
  • 𝑉𝑎𝑅 is a simple measure of risk that can be explained easily to portfolio managers and investors.
  • 𝑉𝑎𝑅 systems also can be used to set consistent guidelines that improve over traditional guidelines using limits on notionals or sensitivity measures.
  • As a bonus, comprehensive risk management systems provide some protection against rogue traders, thereby helping to avoid embarrassing financial losses.
• VaR has led to the development of risk budgeting. Risk budgeting is the process of allocating and managing risk using a top-down approach to different aspects of the investment process. Risk budgeting builds on 𝑉𝑎𝑅 measures that are applied to asset classes, asset managers, and even securities.
• Risk budgeting is fast spreading as a best practice method to manage risk.

VaR Applications To Investment Management

• The investment management industry usually is called the “buyside” of Wall Street, in contrast with banks, the “sell side” that developed 𝑉𝑎𝑅. Whereas 𝑉𝑎𝑅 has been widely, and rather quickly, accepted by the banking industry, it has spread more slowly to the investment management industry. Perhaps this is so because investment management differs in many fundamental respects from the fast-paced trading environment of dealing banks.

• Consider first bank trading portfolios
  • Horizon is short, turnover rapid, and leverage high. 𝑉𝑎𝑅 is particularly appropriate for such an environment. In this case, historical measures of risk basically are useless because yesterday’s portfolio profile may have nothing to do with todays.
  • Bank trading portfolios are also highly leveraged, which makes it particularly important to control their risk. A sequence of adverse events easily could bankrupt the institution, as shown by the Barings crisis. In contrast, pension funds, whose positions are guided by a “prudent investor” philosophy, do not allow much leverage. Thus there is a less crucial need to control the downside risk.
• Hence, the daily application of 𝑉𝑎𝑅 measures has become a requirement of bank trading portfolios owing to short horizons, rapid turnover, and high leverage. Risk is controlled through position limits, VaR limits, and stop-loss rules. Although the investment management industry operates with different risk parameters, the proper measurement of risk is also a critical function.
• In an investment environment, in contrast, the horizon, as measured by the
  portfolio evaluation period, is much longer, monthly or quarterly. Positions change more slowly.

Investment Process

• Generally, this process consists of two steps.
• In the first step, a consultant provides a strategic, long-term asset-allocation study usually based on mean-variance portfolio optimization, that balances off expected return against risk. This study determines the amounts to be invested in various asset classes, for example, domestic stocks, domestic bonds, foreign stocks, foreign bonds, and perhaps additional classes such as emerging markets, real estate, venture capital, and total-return funds, also known as hedge funds. The asset allocation relies on benchmarks, or passive indices, that represent a feasible investment strategy.

• In the second step, the fund may delegate the actual management of funds to a stable of active managers. These managers are reviewed periodically for performance relative to their benchmark, measured in terms of their tracking error. Risk typically is controlled through a list of investment guidelines defining the universe of assets they can invest in, with some additional restrictions such as duration, maximum deviations from equity-sector weights, or maximum amounts of foreign currency to hedge or cross-hedge. Generally, risk is measured ex post, that is, from historical data.
• Institutions exposed to a diversity of risks, to complex financial instruments, and to changing positions should benefit from 𝑉𝑎𝑅 risk management systems. These criteria apply to the investment management industry because of the following reasons.
  • First, investments are becoming more global in nature, creating a need for risk measures that take diversification into account. Before 1974, for example, few pension funds invested in foreign markets. By now, funds invest all over the world. They also invest in new asset classes, such as hedge funds.
  • Second, financial instruments are becoming more complex over time. This creates a need for stronger, centralized risk management systems.
  • Third, most investment portfolios are dynamic, with changing positions. Because the assets of the fund typically are dispersed over several managers, it is difficult to create a current picture of the overall risk of the fund. In addition, money managers sometimes change their investment strategy, either deliberately or inadvertently. If so, the fund should be able to detect and correct such changes quickly.

Hedge Funds

• Hedge funds pose special risk measurement problems. This group is very heterogeneous.

• Most hedge funds have leverage. Some groups have greater turnover than traditional investment managers. Long Term Capital Management is an extreme example of a hedge fund that went nearly bankrupt owing to its huge leverage.

• Another category of funds, however, invests in illiquid assets, such as convertible bonds, which are traded infrequently, even within a month. When this is the case, risk measures based on monthly returns give a misleading picture of risk because the closing net asset value (NAV) does not reflect recent transaction prices. This creates two types of biases.

• First, correlations with other asset classes will be artificially lowered, giving the appearance of low systematic risk. This can be corrected using enlarged regressions with additional lags of the market factors and summing the coefficients across lags.
• Second, volatility will be artificially lowered, giving the appearance of low total risk. Such illiquidity, however, will show up in positive serial autocorrelation in returns. Biases in volatility measures can be corrected by taking this autocorrelation into account when extrapolating risk to longer horizons.
• Finally, hedge funds can pose special problems owing to their lack of transparency. Many hedge funds refuse to reveal information about their positions for fear of others taking advantage of this information. For clients, however, this makes it difficult to measure the risk of their investment both at the hedge-fund level and in the context of their broader portfolio.

Absolute And Relative Risks

• Risk can be clearly defined for a bank trader. It is the risk of loss on the marked-to-market position. Investment asset managers, however, can have different perceptions of risk.

• Risk can be defined as the possibility of losses measured in the base currency, dollar or other. This is the most common definition of risk. For managers who have a mandate to beat a benchmark, however, risk must be measured in relative terms. We can distinguish between two definitions:
  • Absolute risk, which is the risk of a dollar loss over the horizon. This is the usual definition of risk in a trading environment. Sometimes this is called asset risk. The relevant rate of return is 𝑅_asset
  • Relative risk, which is the risk of a dollar loss in a fund relative to its benchmark. This shortfall is measured as the dollar difference between the fund return and that of a like amount invested in the benchmark. The relevant return is the excess return of the asset over the benchmark

𝐸 = 𝑅_asset — 𝑅_b

If this is normally distributed, VaR can be measured as 𝑉𝑎𝑅 = 𝛼𝑊₀𝜎_E

Policy Mix And Active Management Risk

• Consider next a fund that allocates its investment to a pool of active managers in various asset classes. The absolute performance of the fund can be broken down into two components:

1. Policy-mixrisk, which is the risk of a dollar loss owing to the policy mix selected by the fund. Since the policy mix generally can be implemented by investing in passive funds, this risk represents that of a passive strategy.
2. Active-management risk, which is the risk of a dollar loss owing to the total deviations from the policy mix. This represents the summation of profits or losses across all managers relative to their benchmark. Thus there may be diversification effects across managers, depending on whether they have similar styles or not. In addition, the current asset-allocation mix may deviate temporarily from the policy mix.

• Hence total asset risk can be attributed to two sources, the risk of the total policy mix and the risk of active manager deviations from the policy mix:
• The absolute risk can be measured from fund returns and can be defined as
• 𝑅asset=Σ 𝑤i𝑅i

where

𝑤_i is the weight on fund 𝑖 with return 𝑅_i

This return can be decomposed into

\( R_{\text{asset}} = R_{\text{policy mix}} + R_{\text{active mgt}} = \sum_{i} w_i^b R_i^b + \sum_{i} \left( w_i R_i – w_i^b R_i^b \right) \)

where

\( R_i^b \) represents the return on the benchmark for fund 𝑖, and

\(w_i^b\) is its policy weight.

If the pension plan deviates from its policy mix \( (w_i \neq w_i^b) \) the active-management portion can be decomposed further into a term that represents policy decisions and manager performance.

Policy Mix And Active Management Risk

• The funds’ total 𝑉𝑎𝑅 can be obtained from the policy-mix VaR, the active-management VaR. and a cross-product term. As an example, the Ontario Teachers’ Pension Plan Board (𝑂𝑇𝑃𝑃𝐵) estimates that its annual 𝑉𝑎𝑅 at the 99 percent level of confidence can be decomposed as follows (in percent of the initial fund value):

Source of Risk	VaR
Policy-mix VaR	19.6%
Active-mgt. VaR	1.6%
Asset VaR	19.3%

Source: table 7-2 2019 Financial Risk Manager Exam Part II: Risk Management

Seventh Edition by Global Association of Risk Professionals

• Most of the risk is due to the policy mix since most of the variation in portfolio performance can be attributed to the choice of asset classes. In other words, the choice of mix of stocks and bonds will have more effect on the portfolio performance than the choice of a particular equity or bond manager.
• Active management 𝑉𝑎𝑅 is rather small. Apparently, the fund diversifies away much of the risk of managers deviating from their benchmarks through a careful choice of various styles or many managers. Another explanation is that most of the assets are invested in indexed or closely indexed funds.
• Policy-mix 𝑉𝑎𝑅 and active management 𝑉𝑎𝑅 do not add up to the total-asset VaR . In fact, there is a slightly negative correlation between the two, leading to a lower overall asset VaR . If this occurs, active managers could take greater deviations from their benchmark without affecting the plan’s total 𝑉𝑎𝑅 .

Funding Risk

• Funding risk is the risk that the value of assets will not be enough to cover the liabilities of the fund.

• The relevant variable is the surplus 𝑆, defined as the difference between the value of assets 𝐴 and liabilities 𝐿. The change then is ∆𝑆 = ∆A — ∆𝐿.
• While the value of assets can be measured by marking to market, liabilities are more difficult to evaluate.
• For pension funds, the liabilities represent accumulated-benefit obligations, which measure the present value of pension benefits owed to employees discounted at an appropriate interest rate.
• When liabilities consist mainly of nominal payments, their value in general will behave like a short position in a long-term bond. Thus decreases in interest rates, while beneficial for equities on the asset side, can increase even more the value of liabilities, thereby negatively affecting the surplus. If liabilities are indexed to inflation, they behave like inflation-protected bonds.
• The minimum-risk position then corresponds to an immunized portfolio, where the duration of the assets matches that of the liabilities. In practice, it may not be possible to immunize the liabilities completely if the existing pool of long-term bonds is insufficient. More generally, immunization carries an opportunity cost if other asset classes generate greater returns over time.
• This funding risk represents the true long-term risk to the owner of the fund. If the surplus turns negative, it will have to provide additional contributions to the fund. Sometimes this is called surplus at risk (𝑆𝑎𝑅).

Sponsor Risk

• This notion of surplus risk can be extended to the risk to the owner of the fund, the plan sponsor, who ultimately bears responsibility for the pension fund. One can distinguish between the following risk measures:
• Cash-flow risk, which is the risk of year-to-year fluctuations in contributions to the pension fund. Plan sponsors that can absorb greater variations in funding costs, for instance, can adopt a more volatile risk profile.
• Economic risk, which is the risk of variation in total economic earnings of the plan sponsor. The surplus risk may be less of a concern, for instance, if falls in the surplus occur in an environment where the firm enjoys greater operating profits.
• From the viewpoint of the plan sponsor, risk is measured not only by movements in the assets, or even the surplus, but also by the ultimate effect on the economic value of the firm. Thus pension-plan management should be integrated with the overall financial goals of the plan sponsor. This is in line with the trend toward enterprise wide risk management.

Using VaR To Monitor And Control Risks

• 𝑉𝑎𝑅 systems can be used to measure and control market risks. This also applies to the investment management industry. 𝑉𝑎𝑅 systems allow investors to check that their managers comply with guidelines and to monitor their market risks. Credit risk usually is controlled through limits on exposures on a name-by-name basis. 𝑉𝑎𝑅 systems provide some protection against operational risk, which is also controlled by policies and procedures.

Using VaR To Check Compliance

• The impetus for centralized risk management in the investment management industry came from the realization that the industry is not immune to the “rogue trader” syndrome that has plagued the banking industry. The lessons from such losses are applicable to any “manager of managers,” that is, a manager who delegates the actual investment decisions to a stable of managers.
• Even though rogue traders are rare, but minor violations of investment guidelines occur quite often. Some securities may be prohibited because of their risks or for other reasons (e.g., political or religious). Bank custodians, however, indicate that fund managers sometimes trade in and out of unauthorized investments before the client realizes what happened. With monthly reporting, it is hard to catch such movements. Centralized risk management systems, in contrast, can monitor investments in real time.
• Such occurrences have moved the pension-fund industry toward centralized risk management. 𝑉𝑎𝑅 systems provide a central repository for all positions. Independent reconciliation against manager positions makes fraud a lot more difficult. 𝑉𝑎𝑅 systems also allow users to catch deviations from stated policies quickly.

Using VaR To Monitor Risk

• With a 𝑉𝑎𝑅 system in place, investors can monitor their market risk better. This applies to both passive and active allocations.
  • Passive allocation, or benchmarking, does not keep risk constant because the composition of the indices can change substantially.
  • Active portfolio management can change the risk profile of the fund.
• A manager taking more risk – VaR allows dynamic risk monitoring of managers, who are given a 𝑉𝑎𝑅 limit or risk budget. Any exceedance of the 𝑉𝑎𝑅 limit will be flagged and should be examined closely. Sometimes, there may be good reasons to increase the risk profile. Perhaps the risk increase is temporary or justified by current conditions. In any event, it is important to understand the reason behind the change.
• Different managers taking similar bets – This can happen, for instance, when managers increase their allocation to a particular sector, which is perhaps becoming more attractive or has performed well in the recent past. Because active managers operate in isolation, such a problem can be caught only at the portfolio level. To decrease the portfolio risk, managers can be given appropriate instructions.
• More volatile markets– 𝑉𝑎𝑅 can increase if the current environment becomes more volatile, if time variation in risk is explicitly modeled, such as with GARCH models. The plan sponsor then will have to decide whether it is worth accepting greater volatility. If the risks are deemed to be too large, positions can be cut. Increased volatility, however, often is associated with falls in asset prices leading to correspondingly higher expected returns. Thus the rebalancing decision involves a delicate trade-off between risk and return.
• More generally, 𝑉𝑎𝑅 can be reverse engineered to understand where risk is coming from using 𝑉𝑎𝑅 tools. Measures of marginal and component 𝑉𝑎𝑅 can be used to identify where position changes will have the greatest effect on the total portfolio risk. This approach assumes that all the relevant risks are captured by the risk management system.
  • But risk cannot be measured easily for some important asset classes such as real estate, venture capital, and some categories of hedge funds owing to illiquidity.
  • Other series may have very short histories, such as emerging markets, or no history at all, such as initial public offerings.
• In some cases, the missing series can be replaced by a proxy, using a mapping approach. The risk manager should be aware of the limitations of the system.

Using VaR To Manage Risks

• 𝑉𝑎𝑅 systems can be used to manage risk, which is an active application. 𝑉𝑎𝑅 can be used to improve investment guidelines for active managers and to help with the investment process. In theory, 𝑉𝑎𝑅 also could be used to compute the risk-adjusted performance of investment managers, as is done for bank traders.

Using VaR To Design Guidelines

• 𝑉𝑎𝑅 systems can be used to design better investment guidelines. Managers’ guidelines generally are set up in an ad hoc fashion to restrict the universe of assets in which the managers can invest and, to some extent, to control risk. Typically, guidelines include limits on notionals, for example, maximum sector weight deviations for equities and maximum currency positions, or limits on sensitivities, such as duration gaps between fixed-income portfolios and their benchmarks.
• Banking institutions, however, have learned the hard way that limits on notionals and sensitivities are insufficient. Limits on notionals work best with simple portfolios with no derivatives and leverage. They do not account for variations in risk nor correlations. Limits on sensitivities are an improvement but still have blind spots, such as for hedged portfolios. In contrast, 𝑉𝑎𝑅 limits are comparable across assets and account for risk, diversification, leverage, and derivatives.
• Another problem is that the spirit of these limits can be skirted with new financial instruments. For example, a manager may not be allowed to trade in futures that may be viewed as too “risky,” such as futures contracts. Instead, investments may be allowed in high-grade medium-term notes, often viewed as safe because they have no credit risk. The problem is that these notes can be designed as structured notes with as much market risk as futures contracts.
• Hence detailed guidelines, like government regulations, are one step behind continuously changing financial markets. Traditional guidelines cannot cope well with new instruments or leverage. They also totally ignore correlations.
• This is precisely what 𝑉𝑎𝑅 attempts to measure. Instead of detailed guidelines, plan sponsors could specify that the anticipated volatility of tracking error cannot be more than 3 percent, for instance. Position limits can be set consistently across markets.

Using VaR For The Investment Process

• A good risk management system can be used to improve the investment process, starting with the top-level asset allocation process all the way down to trading decisions for individual stocks.
• As explained earlier, the strategic asset-allocation decision is the first and most important step in the investment process for pension funds. It is usually based on a mean variance optimization that attempts to identify the portfolio with the best risk-return trade-off using a set of long-term forecasts for various asset classes.
• In practice, the optimization usually is constrained to obtain solutions that look “reasonable.” This adjustment, however, partly defeats the purpose of portfolio optimization and fails to recognize the effects of marginal adjustments from the selected portfolio.
• Since 𝑉𝑎𝑅 is, after all, perfectly consistent with a mean-variance framework, 𝑉𝑎𝑅 tools can be used to allocate funds across asset classes.
• Risk management systems are also useful at the trading level. While expected returns can be estimated on an individual basis, assessing the contribution of a particular stock to the total portfolio risk is much less intuitive. Even if analysts could measure the individual risk of the particular stock they are considering, they cannot possibly be aware of the relationships between all existing positions of the fund. This is where 𝑉𝑎𝑅 systems help.
• For each asset to be added to the portfolio, analysts should be given a measure of its marginal 𝑉𝑎𝑅. If two assets have similar projected returns, the analyst should pick the one with the lowest marginal 𝑉𝑎𝑅, which will lead to the lowest portfolio risk. Assume, for instance, that the analyst estimates that two stocks, a utility and an Internet stock, will generate an expected return of 20 percent over the next year. If the current portfolio is already heavily invested in high-tech stocks, the two stocks will have a very different marginal contribution to the portfolio risk. Say that the utility stock has a portfolio beta of 0.5 against 2.0 for the other stock, leading to a lower marginal 𝑉𝑎𝑅 for the first stock. With equal return forecasts, the utility stock is clearly the preferred choice. Such analysis is only feasible within the context of a portfolio wide 𝑉𝑎𝑅 system.

Risk Budgeting

Advances in 𝑉𝑎𝑅 have led to risk budgeting, which is spreading rapidly in investment management. This concept is equivalent to a top-down allocation of economic risk capital starting from the asset classes down to the choice of the active manager and even to the level of individual securities.

Budgeting Across Asset Classes

• This approach can be refined further if assumptions are made about the expected performance of active managers. Active managers usually are evaluated in terms of their tracking error (𝑇𝐸), defined as the active return minus that of the benchmark. Define 𝜇 as the expected 𝑇𝐸 and 𝜔 as its volatility (𝑇𝐸𝑉). The information ratio then is defined as

\( IR = \frac{\mu}{\omega} \)

• Managers are commonly evaluated on the basis of their 𝐼𝑅. Grinold and Kahn (1995), for example, assert that an 𝐼𝑅 of 0.50 is “good,” meaning in the top quartile of the active managers. Logically, a greater risk budget should be allocated to managers with better performance, as measured by the 𝐼𝑅 criterion.

Budgeting Across Active Managers

• The optimization problem for active manager allocation attempts to maximize the 𝐼𝑅 for the total portfolio subject to a 𝑇𝐸𝑉 constraint. Define 𝑥_i as the fraction invested in manager 𝑖, who has a tracking error of 𝜔_i; and excess return of 𝜇_i The value added for the total portfolio 𝑃

\( \mu_P = \sum_{i} x_i \mu_i = \sum_{i} x_i (IR_i \times \omega_i) \)

• It is assumed that the deviations for each manager are independent of each other. The portfolio 𝑇𝐸𝑉 is fixed at

\( \omega_P = \sqrt{ \sum_{i} x_i^2 \omega_i^2 } \)

• Maximizing the portfolio information ratio subject to a fixed 𝑇𝐸𝑉 gives the following solution:

\( x_i \omega_i = IR_i \left( \frac{1}{IR_P} \omega_P \right) \)

• Thus the relative risk budgets should be proportional to the information ratios.