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Value at Risk An Advanced Discussion of a Tool to Assist Advisors in Measuring and Communicating Portfolio Risk

By John Nersesian, CIMA®, CIS, CPWA®, CFP®

osses in investment portfolios 1. Help manage client expectations later mandated disclosure of risk to over the past two years have had regarding downside potential. investors and VaR became the de facto L a sobering eff ect on investors 2. Help consultants construct appropri- measure.1 Within the past few years, as they have come to understand the ate portfolios that accurately refl ect advisors to high-net-worth individuals two-sided nature of investment deci- the risk preference of their clients. who wanted to take a more institutional sions: return and risk. With these losses 3. Position the advisor as an institu- approach to portfolio construction and top-of-mind, clients have a renewed tional-caliber consultant with an management began using VaR. interest in fully understanding their advanced skill set. portfolio risks and a renewed desire for Th e extraordinary recent market Value at Risk Defined their advisors to demonstrate how they decline has brought into focus the short- VaR quantifi es the maximum downside manage the potential for loss. comings of traditional risk management loss exposure under normal market Most investors have a good under- tools, including VaR. Proponents of risk conditions within a specifi ed period of standing of the concept of return. measures, including VaR, have cau- time, in dollar terms, with a confi dence Advisor-client conversations about risk, tioned about the importance of realizing level of occurrence (generally ranging however, can be challenging because that these measures are intended to be from 90–99 percent). often each has a diff erent perspec- quantitative tools used in concert with While many VaR models are propri- tive or defi nition of risk. Investment due diligence and quantitative judgment. etary, they all incorporate the statistical professionals often seek to quantify risk Critics highlight the importance of concepts of probability distributions, using standard deviation. Th ey see the viewing risk management as a continu- confi dence intervals,2 asset class corre- downside potential of risk as well as the ous process, not a static event. Working lation, and volatility. Th ese are the three upside, i.e., the return opportunities with their clients, advisors need to methods of calculating VaR: available for taking on a certain level of establish a target risk for the portfolio Historical models use real market risk. Th is upside is an abstract concept and measure risk on an ongoing basis. data from prior market phases, allowing to most investors, who view risk simply VaR can help investors understand the for nonnormal distribution patterns. as the potential to lose money (and, risk associated with their investment Many consultants favor this approach by extension, the potential for falling decisions as part of a process that also because it accurately refl ects the mar- short of objectives or having to reduce uses other appropriate risk measures ket’s kurtosis (the fatness of a distribu- spending, etc.) One way to quantify the including beta, capture ratio, downside tion tail).3 However, this approach is potential downside risk is Value at Risk deviation, and tracking error. dependent on historical time series and (VaR), which calculates the maximum assumes that future conditions will be expected loss over a certain time period Value at Risk’s Evolution the same as historical conditions so it and confi dence level. Clients are looking VaR has roots in the institutional world. may not always be statistically relevant. for the answer to the question, “How Its development is widely credited to Monte Carlo models use a statisti- much money can I potentially lose in a JP Morgan in the late 1980s and early cally signifi cant number of simulations given period?” In their quest to answer 1990s. At that time the risk of indi- of randomly sequenced returns to that question, some advisors have found vidual trading desks could be known, project a range of future results and the the concept of VaR to be helpful. but JP Morgan’s then-chairman, Dennis probability of achieving them. Because Th is paper discusses how VaR can be Weatherstone, was seeking a measure this model is unlimited by historical a useful tool to assess risk exposure and to understand the fi rm’s overall risk results, it provides a forward-looking to communicate with clients. Th ere are every day. Over time VaR was embraced VaR with potentially greater predic- three benefi ts to understanding, embrac- by others in the institutional world; the tive ability than linear methods. Monte ing, and employing the VaR calculation: Securities and Exchange Commission Carlo simulations do, however, have

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their limitations and the outcomes are not guaranteed or FIGURE 1: STANDARD DEVIATIONS IN A NORMAL assured in any respect. DISTRIBUTION Both the Historical and Monte Carlo methods use large data sets. Parametric VaR (or Delta Normal) uses the known properties of standard deviation to help determine maximum 1.65σ downside risk. It assumes a normal distribution of returns 95% and a linear relationship to the various risk factors, which 2.33σ seldom occur, making the tool simple but somewhat limited. 99% Th is is the method used in this paper. Each method relies on historical results. Past performance σ ⊗ σ⊗ σ σ σ σ -3 -2 -1 mean +1 +2 +3 Return is no guarantee of future results. VaR does not measure all 84% of returns 97.5% of returns forms of risk. 99.5% of returns

Hypothetical example does not refl ect the performance of an actual investment; for Levels of Value at Risk illustrative purposes only. Th e fl exibility of the VaR measure is another reason it has be- come an important tool for many practitioners. In addition to being able to specify risk for various horizons and confi dence Figure 1 illustrates standard deviations in a normal intervals, it also can be expressed in both absolute and rela- distribution. In a normal distribution, 50 percent of returns tive terms. Th ere are three related VaR measures outside the will be greater than the mean and 50 percent will be less traditional absolute measure. than the mean. For the purposes of VaR, which measures the Relative VaR measures the risk of underperformance potential for loss, the 95-percent confi dence level is at 1.65 relative to a pre-determined index. Most institutional inves- standard deviations to the left of the mean and the 99-percent tors use this measure because their performance is frequently confi dence level is at 2.33 standard deviations to the left of evaluated against a target benchmark. the mean.4 Marginal VaR measures how much risk a position or manager adds to the portfolio. Moreover, this metric mea- A Hypothetical Example—Portfolio Scenario sures how much overall portfolio VaR would change by elimi- Th is hypothetical example does not refl ect the performance nating the position completely from the portfolio. Marginal of an actual investment; it is for illustrative purposes only. VaR is useful for measuring which position or manager is the largest contributor to portfolio risk. Investment A Investment B Incremental VaR measures how a small percentage change Allocation $5 million $5 million in an individual position weight aff ects the overall portfolio. Average Return (X) 0.15 0.09 It is closely related to marginal VaR because both metrics measure the change in overall VaR caused by changing the Weight (Wt) 0.50 0.50 construct of the portfolio. Incremental VaR, however, is useful Variance (2) 0.04 0.02 for identifying positions for gradual risk reduction over time. Covariance (COV) 0.05 0.05 of AB Calculating Value at Risk Th e formula for a parametric VaR calculation (using 95-percent and 99-percent probability) can be expressed as: Calculating Value at Risk Assuming One-Year Time Period VaR @ 95% = PV$ X – 1.65σ To determine the client’s maximum exposure to loss in any N one-year holding period (with a 95-percent confi dence level), we use the following calculation (results of Step 2 are rounded VaR @ 99% = PV$ X – 2.33σ to two decimal places): N (1) Compute the mean weighted return of the portfolio: PV$ = Portfolio value in dollars ERp = WA X + WB X σ = Standard deviation of the portfolio — = (0.50 0.15) + (0.50 0.09) = 0.12 X = Mean return of the portfolio for holding period N = Number of years in projected holding period

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(2) Compute the variance of the portfolio: is especially true if the portfolio or manager measured has 2 σ2 2 σ2 high turnover. Also note the assumed level of risk and return Vp = W A A + W B B + 2WAWBCOVAB = (0.502 0.04) + (0.502 0.02) + (2 0.50 0.50 0.05) remain constant over varying time horizons. Th is assumption may lead to an unreliable VaR outcome over very short or very = 0.0100 + 0.005 + 0.0250 = 0.04 long time horizons. To measure the potential for loss over a (3) Compute the standard deviation of the portfolio: six-month horizon or a ﬁ ve-year horizon the calculations are: σ = Vp 0.12 – (1.65 0.20) VaR = $10,000,000 = –$2,969,848 = 6mo 0.04 = 0.20 0.50

0.12 – (1.65 0.20) (4) Compute the VaR of the portfolio: VaR5yr = $10,000,000 = –$939,149 5.0 VaR @ 95% = PV$ X – 1.65σ N Calculation Summary = $10,000,000 (0.120 – 1.65 (0.20)) Th e VaR calculation depends largely on two variables: the probability level and the time period. Given a particular dollar 1 value of potential loss, however, there is a positive relation- = $10,000,000 (–0.21) = –$2,100,000 ship between these inputs. Th at is, if the holding period is in- creased (decreased) and the dollar loss value is constant, the Th is VaR number helps the advisor understand the portfo- probability of that loss increases (decreases). Tables 1 and 2, lio’s potential risk exposure at the 95-percent confi dence level. which summarize the calculations from the examples above, It also can be used to manage expectations during the portfolio illustrate that the larger the probability and/or the shorter the construction process by helping a client estimate what the holding period, the greater the maximum potential loss. potential loss may be. In this example, there is a 95-percent probability the client’s largest potential loss in the one-year Risk Budgeting and Risk Decomposition holding period would have been no more than $2.1 million. VaR also is useful in the risk budgeting and decomposi- Increasing the confi dence level of the calculation to 99 tion process. Th e process of risk budgeting helps determine percent expands the scope of the analysis to include a wider whether an investor is being rewarded for the risk being range of possible outcomes. Th is is accomplished by replacing taken. It establishes a method for calculating the expected the 1.65 in the numerator with 2.33. Th e result is a 99-percent contribution of each asset on a return basis as well as in terms probability the client’s largest potential loss in the one-year of risk assumed. By providing an enhanced view of the initial holding period would have been no more than $3.46 million. risk and return assumptions, risk budgeting helps refi ne the process of optimizing a client’s portfolio. 0.12 – (2.33 .020) VaR @ 99% = $100,000,000 = –$3,460,000 Risk decomposition is used to determine the contribu- 1 tion of the individual components to total portfolio risk that is used in risk budgeting. Because risk is not additive, total Th is example can be extended further to consider diff er- portfolio risk cannot be determined by simply adding the risk ent holding periods. Note that the VaR calculation becomes levels of the individual components. In this analysis, covari- less relevant the longer the period of time modeled. Th is ance between the securities plays a central role in determin-

TABLE 1: MAXIMUM POTENTIAL LOSS ASSUMING ONE-YEAR TIME PERIOD OVER VARIOUS PROBABILITY LEVELS Probability Expected Return Standard Deviation Holding Period VaR 95% 0.12 0.20 1 year –$2,100,000 99% 0.12 0.20 1 year –$3,460,000 Hypothetical example does not reﬂ ect the performance of an actual investment; for illustrative purposes only.

TABLE 2: MAXIMUM POTENTIAL LOSS ASSUMING 95-PERCENT PROBABILITY OVER VARIOUS TIME PERIODS Probability Expected Return Standard Deviation Holding Period VaR 95% 0.12 0.20 6 months –$2,969,848 95% 0.12 0.20 1 year –$2,100,000 95% 0.12 0.20 5 years –$939,149

Hypothetical example does not reﬂ ect the performance of an actual investment; for illustrative purposes only.

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ing the total portfolio risk. When adding assets to a portfolio, VaR tells the maximum po- the amount of accompanying risk is a function of the variance of that component plus its covariance with the other securi- “tential loss at a particular confi- ties in the portfolio. dence level as well as the prob- A Hypothetical Example—Risk Decomposition ability of losses beyond that To determine the VaR contributions of Investment A and Investment B, we use the following calculation: maximum loss (tail probability),

(1) Calculate the contribution of Investment A: but it doesn’t tell the severity of Cont = X W – 1.65 2 W2 + COV W W A A A A A AB A B loss in the tail. √ P ” = (0.15)(0.50) – 1.65 (0.04) (0.50 2) + (0.05) (0.50) (0.50) Th e VaR calculation also assumes a normal distribution, √0.2 and in reality the world is not necessarily normal. Returns are = –0.110625 subject to skewness,6 or returns may be otherwise not normally distributed. A distribution may have outlying returns (2) Compute the portfolio eff ect: in “fat tails.” Th is is the basis for one of the major critics of = PortfolioEff ectA ContA PV$ VaR the use of VaR, Nassim Nicholas Taleb, author of the best- 7 = –0.110625 $10,000,000 $2,100,000 selling book, Th e Black Swan. Taleb’s criticism highlights an = 52.7% important consideration about the confi dence interval used. With a 99-percent confi dence level there is still the 1-percent (3) Calculate the contribution of Investment B and portfolio probability at the extreme edge of the curve about which to eff ect: be concerned. Th ese are the “black swans”8 or unexpected 2 ContB = (0.09) (0.50) – 1.65 (0.02) (0.50 )+(0.05) (0.50) (0.50) events that Taleb says many never consider but happen more 9 √0.2 frequently than expected. = –0.099375 $10,000,000 = $993,750 Complements to Value at Risk = $993,750 ÷ $2,100,000 = 47.3% Stress testing can be a complement to VaR because it “at- tempts to gauge the vulnerability of portfolios to hypothetical (4) Further proving the portfolio eff ect calculation of Invest- events.”10 Scenario analysis and factor push analysis are two ments A and B: approaches to stress testing. + Total Risk = PortfolioEff ectA PortfolioEff ectB = 100% Scenario analysis, as its name implies, tries to estimate 52.7% + 47.3% = 100% the impact of particular scenarios, e.g., a loss in an extreme occurrence such as a market crash, or sudden changes in cor- Understanding the respective contribution to portfo- relation. VaR tells the maximum potential loss at a particular lio risk and return is critical in risk budgeting and in the confi dence level as well as the probability of losses beyond portfolio construction process. In the example, Investment that maximum loss (tail probability), but it doesn’t tell the A provided 62.5 percent of the portfolio return [0.15(0.50) ÷ severity of loss in the tail. Scenario testing helps overcome 0.12] while contributing 52.7 percent of the risk. Investment this limitation because it focuses on those extreme events that B provided 37.5 percent of the portfolio return [0.09(0.50) ÷ generally are found in the tail. It usually consists of imposing 0.12] while contributing 47.3 percent of the risk.5 extreme scenarios on the portfolio and observing the reaction to performance within the entire portfolio. Th e simplest form Important Considerations about Value at Risk involves “bumping” up or down each risk factor one at a time Risk is an ambiguous concept for investors and VaR can be a (e.g., moving interest rates up 100 basis points and seeing useful tool to help them understand it. However it is not the how the bond portfolio reacts or depreciating a position in single perfect solution. an equity portfolio by 20 percent and seeing how the overall Th e VaR calculation depends heavily on past experience, portfolio reacts). Th is measure also has limitations. Its result which may or may not be the future experience of the market is dependent on the scenarios chosen, and future events can or investor. If the data used for the calculation is from a be very diffi cult to predict. Also, the number of scenarios to period that is generally positive and does not consider periods analyze can be overwhelming, and they can be complex and of stress, then the risk measure will not refl ect stress. require signifi cant computation.

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Factor push analysis diff ers from scenario analysis because 8 A black swan is an event or occurrence that deviates beyond what is it results in a range of possibilities rather than specifi c sce- normally expected of a situation and that would be extremely diffi cult narios. Th is type of analysis involves “pushing” each security to predict. Black swans are typically random and unexpected. Th e price or risk factor to the extreme “worst case” and determin- term was popularized by Nassim Nicholas Taleb, a fi nance professor ing the combined eff ect on the value of the portfolio. Th e and former Wall Street trader. Source: investopedia.com. limitations of this type of stress testing include the need to 9 Joe Nocera, Risk Mismanagement, New York Times Magazine take into account the diff ering sensitivity of each position to (January 4, 2009), available at http://www.nytimes.com/2009/01/04/ the diff ering underlying risk factors, which can be compli- magazine/04risk-t.html. cated and also require signifi cant computation.11 10 Kevin Dowd, Stress Testing, chapter 6 in Beyond Value at Risk: Th e New Science of Risk Management (Chicester, UK: John Wiley & Sons Summary Ltd, 1998): 16–22. VaR, a tool used for more than 15 years by institutions, now 11 Ibid is being embraced by advisors to high-net-worth investors. Th ese advisors are seeking a fuller understanding of clients’ Disclosure: Th is educational and informational report is risk exposure and additional ways to meaningfully commu- provided for investment professionals only, and should not be nicate risk to clients. As with most metrics, however, VaR construed as specifi c fi nancial planning or investment advice

has its limitations, but the ability to frame risk in terms of for clients. All opinions and views are subject to change without the maximum potential dollar loss at any given probabil- notice. Hypothetical examples are shown for illustrative and ity allows an advisor to address risk in terms most clients educational purposes only, and are not intended to predict or understand. depict performance of any actual investments. Th e analyses con- tained herein are based on numerous assumptions. Diff erent John Nersesian, CIMA®, CIS, CPWA®, CFP®, is manag- assumptions could result in materially diff erent results. Certain ing director of wealth management services at Nuveen information was obtained from third party sources, which Investments in Chicago, IL. He is a member of IMCA’s we believe to be reliable but not guaranteed for accuracy or board of directors and serves on the Wealth Management completeness. Calculations in this report are based on the Committee. He earned a BS in business and economics Parametric VaR (or Delta Normal) method, a tool which uses from Lehigh University. Contact him at john.nersesian@ known properties of standard deviation to create assumptions nuveen.com. of maximum downside risk. It assumes a normal distribution of returns and a linear relationship to the various risk factors, Endnotes which seldom occur in reality. Value at Risk does not provide 1 Joe Nocera, Risk Mismanagement, New York Times Magazine the worst case scenario and does not measure losses under (January 4, 2009), available at http://www.nytimes.com/2009/01/04/ particular market conditions or address cumulative losses. All magazine/04risk-t.html. methods of calculating Value at Risk use past performance, 2 Confi dence interval is the probability that a population parameter lies which is no guarantee of future results. Th e outcomes of Value within an estimated range of value; the most common probabilities at Risk are not guaranteed or assured in any respect, and should used for a confi dence interval are 95 percent or 99 percent. Source: not be relied upon solely for any investment decision. Value at investopedia.com Risk is a statistical tool for illustrative purposes and is designed 3 Kurtosis is a parameter that describes the shape of a probability to be used in concert with other risk measures. Other tools may distribution around the mean. A high kurtosis describes a probability produce substantially diff erent success and failure outcomes. distribution that has fat tails. A low kurtosis describes a distribution Investing entails risk, including possible loss of principal. Neither with thinner tails that is more concentrated around the mean. Source: Nuveen nor any of its affi liates, directors, employees or agents investopedia.com accepts any liability for any loss or damage arising out of the use 4 Professor Rangarajan Sundaram, New York University. Certifi ed of all or any part of this report.” Investment Strategist (CIS) Program sponsored by IMCA, “Risk Budgeting and Value at Risk,” New York University Leonard N. Stern See page 49 for a link to the online School of Business, April 23, 2002. CE quiz. 5 Cont = XAWA ÷ ERp. 6 Skewness risk is the risk that a model assumes a normal distribution of data when in fact data is skewed to the left or right of the mean. Source: investopedia.com 7 Nassim Nicholas Taleb, Th e Black Swan: eTh Impact of the Highly Improbable (New York, NY: Random House, 2007).

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