analytical var var mapping

LETTER FROM CRAIG HEATTERWelcome to the latest issue of the J.P. Morgan Investment Analytics and Consultingnewsletter, which aims to provide informative and thought-provoking articles ontopics relating to portfolio optimization. In this issue, we analyze the historicalreturns and future prospects of private equity investments, explore the potentialbenefits and pitfalls of an active currency management strategy, and provide anoverview of Value-at-Risk modeling, with a particular focus on Analytical VaR. We welcome your thoughts and suggestions, and hope that this issue provides youwith useful information.

CRAIG HEATTER

MANAGING DIRECTOR AND GLOBAL EXECUTIVE FOR INVESTMENT ANALYTICS AND CONSULTING

J.P. MORGAN WORLDWIDE SECURITIES SERVICES

[email protected]

September 2008 Edition

TABLE OF CONTENTS

Private Equity For InstitutionalInvestors 2

Can You Make Cash With Currency? 5

Value-at-Risk: An Overview ofAnalytical VaR 7

Global Capital Markets 10

Global Market Indices 12

ABOUT J.P. MORGAN INVESTMENTANALYTICS AND CONSULTING

J.P. Morgan Investment Analytics and Consulting (IAC) helps institutional clientsmake more informed investment decisions and optimize their portfolios throughcreating customized, innovative, and forward-looking solutions that address bothcurrent and future needs. IAC services over 230 clients globally with over 7,000institutional portfolios, representing approximately $2 trillion in assets. Its diverseclient list includes corporate and public DB/DC pensions, investment managers,endowments and foundations, corporate treasuries, insurance companies, centralbanks, and investment authorities.

Having the broadest and deepest product offering in the market, IAC offerssecurity-level, multi-currency performance measurement (monthly and daily) usingJ.P. Morgan or third party accounting; analytics and attribution at the asset class,sector, country, and individual security level; ex-ante risk management (includingRisk Budgeting and security-level VaR); investment manager analysis, universecomparison, and peer grouping; global consolidated reporting for multi-nationalplans; and consultative services in the areas of liability and plan allocation strategy,manager search, and liability-driven investments.

For further information, please visit www.jpmorgan.com/visit/iac or

Copyright ©2008 JPMorgan Chase & Co. All rights reserved.

Americas:Mark HuamaniExecutive [email protected]

Europe, Middle East, Africa:Romain BerryVice [email protected]

Asia Pacific:Stuart HoyVice [email protected]

Alex StimpsonVice [email protected]

PRIVATE EQUITYSPOTLIGHT

SEPTEMBER 2008 EDITION — 2

In our view, private equity involves a complicated risk andreturn proposition. Private equity investors may be attractedto the potential for impressive returns that are not highlycorrelated with traditional equity and fixed income invest-ments. Skeptics point to the multiple risks due to the illiquidand opaque nature of the funds.

The J.P. Morgan Investment Analytics and Consulting groupanalyzed more than 5,500 private equity funds for vintageyears from 1990 through 2005, including both U.S. andglobal funds and representing every major style. Ouranalysis indicates that there is a wide range of performancebetween top quartile and median funds among every majorstyle. Moreover, the relative performance of private equityfunds versus the public equity markets has been mixed.

In our view, institutional investors face several hurdles toinvestment success in private equity. Institutional investorshave to identify the funds, management teams, and dealstructures that are likely to result in positive results.Moreover, once attractive funds and strong managers areidentified, it is often difficult to gain access to the mostattractive funds. Finally, private equity partnerships typicallyinvolve annual fees of approximately 2% and carried interestof 20% of profits.

Nonetheless, investors seem willing to take their chances inthis challenging asset class. According to a recent survey

conducted by J.P. Morgan and Greenwich Associates, approx-imately 62% of current investors in private equity expect toincrease their allocations in the near term. In our view, plansponsors should only consider allocations to private equity ifthey believe they are able to identify, and gain access to,managers that are likely to be in the top quartile.

THE INVESTMENT CASE

In many ways, private equity holds the potential for hugegains. Clearly, high rates of return are possible, particularlyamong top-quartile funds. For example, top-quartile venturecapital funds that were raised between 1993-1997 generatedaverage internal rates of return of 52%. Likewise, buyoutfunds that were raised between 2001-2005 generatedaverage internal rates of return of 40% for the top quartile.In good times, private equity can generate outstandinginvestment results. In Exhibit 1, we summarize recent resultsfor private equity.

However, the risks are daunting. First, investment in privateequity is illiquid, and the money is often tied up for ten yearsor more. Second, the funds are often highly concentratedand involve significant company-specific risks. Third, privateequity funds are not transparent, and there is frequently alack of reliable, publicly-available information. Fourth, andperhaps most importantly, there is a wide dispersionbetween the top performing funds and the rest of the pack.

PRIVATE EQUITY FOR INSTITUTIONAL INVESTORSby Karl Mergenthaler, CFA, and Chad MotenJ.P. Morgan Investment Analytics and [email protected], [email protected]

Pension plans and other institutional investors are pouring money into private equity at an astounding rate. At this time, theprivate equity industry accounts for approximately $1.5 trillion in invested capital, and private equity firms raised $300 billionin fresh capital in 2007. Undoubtedly, there will be both winners and losers in this high-stakes, modern-day gold rush.

1 Year 3 Year 5 Year 10 YearU.S. Venture Capital Index 16.3% 14.1% 11.3% 35.2%

U.S. Private Equity Index 20.4% 25.1% 24.5% 14.1%

Global (ex-U.S.) Private Equity and Venture Capital 35.3% 36.0% 30.9% 19.7%

CSFB / Tremont Hedge Fund Index 6.7% 10.2% 10.8% 8.3%

Russell 2000 -1.6% 6.9% 16.3% 7.2%Source: J.P. Morgan Investment Analytics & Consulting estimates, Cambridge Associates, CSFB/Tremont, Bloomberg.

Exhibit 1 – Private Equity Performance (as of Dec. 31, 2007)

US Private Equity Fundraising Totals, 1993-2007

0

50,000

100,000

150,000

200,000

250,000

300,000

1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007

(US

$ M

illi

ons)

Venture Capital Other Mezzanine Capital Buyouts Fund of Funds

Source: J.P. Morgan Investment Analytics & Consulting estimates, Dow Jones Private Equity Analyst.

Exhibit 2 – Fundraising



-10.0%

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Vintage

IRR

Top Quartile Median

Source: J.P. Morgan Investment Analytics & Consulting estimates, Private EquityIntelligence.

Exhibit 3 – Venture Capital

The amount of capital raised in private equity funds hasskyrocketed over the past five years. As shown in Exhibit 2,the total amount of capital raised in private equity fundsincreased to approximately $300 billion in 2007.

With the huge amount of capital raised, private equity fundsare searching for new markets to deploy the cash. Forexample, emerging markets funds raised approximately $59billion in 2007. In our opinion, the large inflow of capitalinto private equity funds begs one question: Will futureresults be as attractive as they have been historically?

HISTORICAL RETURNS – THE EVIDENCE IS MIXED

Our analysis suggests that investment success in privateequity is not a “lay-up.” First, there is a wide dispersion ofreturns for similar funds, and top quartile funds tend toperform much better than the median. Also, the relative

performance of private equity and the public markets ismixed. Most importantly, our analysis suggests that theultimate returns of private equity funds that are raised inyears with high levels of fundraising are likely to be poor.

Clearly, there is a wide dispersion between top performingfunds and the median. Our analysis focuses on venturecapital and buyout funds, which together account for morethan 80% of assets invested in private equity. In Exhibits 3and 4, we illustrate the top quartile and median returns forventure capital and buyout funds with vintage years between1990 and 2005.

As indicated in Exhibit 3, top quartile venture capital fundsout-performed median funds by 1,750 basis points forvintage years between 1990 and 2005. Also, top quartilebuyout funds (see Exhibit 4) out-performed the median by1,230 basis points during this time period.

0.0%

10.0%

20.0%

30.0%

40.0%

50.0%

60.0%

70.0%

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Vintage

IRR

Top Quartile Median


Exhibit 4 – Buyout Funds

-5.0

0.0

5.0

10.0

15.0

20.0

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Ret

urn

s

-20,000

-10,000

0

10,000

20,000

30,000

40,000

50,000

60,000

70,000

80,000

Fun

dra

isin

g (

$M

)

Venture Capital - Median IRRRussell 2000 (Annualized)Fundraising


Exhibit 5 – Venture Capital

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005

Ret

urns

0

20,000

40,000

60,000

80,000

100,000

120,000

140,000Fu

ndra

isin

g ($

M)

Buyout - Median IRRRussell 2000 (Annualized)Fundraising


Exhibit 6 – Buyout Funds



The performance of private equity funds versus the publicequity markets has been mixed. We compared the internalrate of return for private equity funds versus the annualizedtime-weighted rate of return for the Russell 2000 overcomparable time periods. Although comparing resultsbetween public and private equity is problematic due to thedifferent performance measurement methodologies, webelieve the comparison is directionally correct and providessome useful information.

As indicated in Exhibit 5, venture capital funds that wereraised in the mid-1990’s posted returns in excess of theRussell 2000. In our opinion, it is likely that many venturesthat were funded in the mid-1990’s were harvested in thelate-1990’s technology bubble. It is interesting to note thatfunds raised in 1998-2002 (i.e., the strong fundraising years)produced internal rates of return that were materially lowerthan the public markets. In fact, based on our analysis, themedian venture capital fund has under-performed the Russell2000 for most of the past decade.

Buyout funds raised over the past decade have out-performed the Russell 2000 over the time period from 1995-2005 (see Exhibit 6). In recent years, fundraising for buyoutfunds sky-rocketed. Meanwhile, the differential betweenmedian buyout funds and the Russell 2000 has beendeclining in the most recent years.

DIVERSIFICATION

In order to assess the diversification benefits of privateequity, we analyzed the 20-year track record of theCambridge Venture Capital and Private Equity indicesrelative to several traditional and alternative asset classes. The results of our correlation analysis are summarized in Exhibit 7.

The correlation of private equity to other asset classes islow, which does indicate that there may be some diversifica-tion benefit to participating in this asset class. However, wewould note that many of the correlations for other alterna-tive assets, such as Timber and Funds of Hedge Funds, arelower than those experienced by private equity. Also, webelieve the correlations may be understated due to theinfrequency of reporting for private equity and use of esti-mates in calculating private equity returns. Therefore, diversification should not be the sole reason to investin private equity.

CONCLUSIONS

In our view, an allocation to private equity may make sensein the context of a large institutional portfolio. However,investors in private equity should be aware of the widedispersion in results among funds with similar strategies. Inour view, the high levels of fundraising in recent years mayhinder the ability of private equity firms to generate stellarresults going forward.

We believe investors in private equity should take a long-term view. It may make sense to spread out capital alloca-tions over many years to dampen the impact of fundraisingcycles. We believe plan sponsors must analyze the detailsof any individual private equity fund, and consider how thatfund is likely to perform in various market conditions.Furthermore, plan sponsors should only consider alloca-tions to private equity if they believe they are able to iden-tify, and gain access to, managers that are likely to be in thetop quartile. In all, we believe it is worth the effort toanalyze private equity funds and consider allocating apercentage of portfolio assets to this asset class.

For a copy of our complete white paper, “Private Equity forInstitutional Investors”, please contact Karl Mergenthaler [email protected].

U.S. SmallStocks

LehmanAGG EAFE

Wilshire5000

VentureCapital

PrivateEquity Timber Realty

Fund ofHedgeFunds

U.S. Small Stocks 1.00

Lehman AGG -0.20 1.00

EAFE 0.60 -0.14 1.00

Wilshire 5000 0.83 -0.12 0.76 1.00

Venture Capital 0.39 -0.17 0.33 0.44 1.00

Private Equity 0.58 -0.20 0.53 0.65 0.60 1.00

Timber -0.06 0.11 0.05 0.06 0.11 0.26 1.00

Realty -0.08 -0.18 0.10 -0.03 0.08 0.23 -0.21 1.00

Fund of Hedge Funds 0.37 -0.09 0.26 0.40 0.39 0.39 0.21 -0.13 1.00

Source: J.P. Morgan Investment Analytics & Consulting estimates, Cambridge Associates, Hedge Fund Research Inc.

Exhibit 7 – Diversification Benefits


Since the U.S. Dollar is currently near all-time lows, we areseeing an increasing number of U.S. institutional investorsrevising, or at a minimum, reviewing their currency strategy.In our opinion at this point in time, it is certainly possible thatthe U.S. Dollar will bounce back, and that a strengtheningdollar could have an adverse effect on the returns of interna-tional investments.

Foreign exchange rate risk and security valuation risk are thetwo main types of investment risk associated with foreigninvestments. Security valuation risk in foreign positions isdriven by a number of factors such as market, sector, industryand stock specific – similar to the risk associated withdomestic investments. Foreign exchange rate risk is the riskassociated with the ownership of foreign securities that aretraded in a foreign currency. The FX risk is due to the implicitcurrency exposure when holding these foreign positions.

Traditionally, when dealing with currency risk, institutionalinvestors have relied on three primary options: maintaincurrency exposure, passively manage foreign currency expo-sure, or actively manage foreign currency exposure. Recently,J.P. Morgan Investment Analytics and Consulting has seenmore plan sponsors treat currency as a separate asset class inan alpha-seeking strategy.

In this article, we will make a case for active currency manage-ment through a pure currency alpha strategy. The argumentfor an alpha-seeking currency mandate is threefold. First,currency’s low correlation to other traditional asset classesmakes it an ideal candidate for diversification. Secondly, anactive currency manager can capitalize on the trending natureof the currency market. Lastly, there are inefficiencies in thecurrency market that present opportunities to capture profit.

HEDGING STRATEGIES

Some institutional investors may choose to leave their portfo-lios un-hedged. In this instance, the portfolio will experiencethe full impact of the appreciation and depreciation of thelocal currency. Although the currency returns over a long timehorizon should approach zero, the portfolio may experience anincrease in volatility due to exchange rate fluctuations.

Passive currency management is intended to reduce foreigncurrency exposure and risk, not necessarily to increase return.The sole purpose of a passive currency program is to elimi-nate, to a degree, the impact from currency fluctuations on thevalue of foreign investments. The degree to which thecurrency risk is eliminated is dictated by the level of the hedgeratio that is employed1. A hedge ratio of 50%, for example,would indicate that half of the currency exposure has beeneliminated.

The goal of an active currency strategy is to capture gainswhile reducing risk on international investments. This can beachieved by altering the hedge ratio of the currencies to behedged. Depending on the guidelines and latitude providedby the plan sponsor, the currency manager may be able to takeadvantage of the movement and volatility in the foreignexchange market.

For example, if the currency managers have a view that theEuro will continue to appreciate relative to the U.S. Dollar,they could under-hedge the Euro (or leave it completely un-hedged) to capture the anticipated currency gain. Conversely,if the currency managers have a view that the Swiss Francwould depreciate relative to the U.S. Dollar, they could over-hedge (or completely hedge) the Swiss Franc to offset theimpact of the falling Franc. By partially hedging, institutionalinvestors can enhance their portfolio returns by activelymanaging their currency risk.

ALPHA-SEEKING STRATEGIES

Unlike traditional hedging programs, currency alpha strategiesare not constrained by the currency exposures of the portfolio.By managing currency as a separate asset class, plan spon-sors can make a pure alpha play by using forward contractsand eliminating the need for any initial funding. In this

CAN YOU MAKE CASH WITH CURRENCY?by Paul Ha and Carlos MarencoJ.P. Morgan Investment Analytics and [email protected], [email protected]

Pension plans, endowments, and foundations allocate a significant portion of their total portfolio to foreign assets in bothdeveloped and emerging markets. In a recent study of institutional investment strategies, the J.P. Morgan Investment Analyticsand Consulting group found that U.S. investors, on average, have targeted a 20% allocation to international investments.This sizeable allocation, if left un-hedged, has benefited from the weakness in the U.S. Dollar over the last several years. Forthe most recent year, the gain on currency has helped bolster these international equity returns by an estimated 9.64% (seeExhibit 1). Even longer term, the currency gains as a proportion of the total international return has been significant.

YTD 1 Yr 3 Yr 5 Yr 10 Yr

EAFE - USD -10.96 -10.61 12.84 16.67 5.83

EAFE - Local -15.70 -20.25 6.66 11.22 2.63

Return from Currency 4.74 9.64 6.18 5.45 3.20

Source: J.P. Morgan Investment Analytics & Consulting estimates.

Exhibit 1 - Foreign Currency Impact on Domestic Returns (as of June 30, 2008)

1 It is important to note that the effectiveness of the passive currency hedge is alsodependent upon the efficiency of the currency trade execution.

SPOTLIGHTCURRENCY MANAGEMENT


2 The correlation coefficients are based on 17.5 years of historical data ending June2008. The following indices were used as proxies for the asset classes shown inthe table. Domestic Equity - Russell 3000 Index; Foreign Equity - MSCI EAFE Index(Net Div); Fixed Income - L.B. Aggregate Bond Index; Real Estate - NCREIF Property;Commodities - Dow Jones AIG Commodities Index.

3 Source – Bank of International Settlements. 4 Source – Bloomberg.

context, the currency managers are expected to add alpha(i.e., increase the overall risk-adjusted returns of the port-folio).

DIVERSIFICATION

The asset allocation of a typical institutional investor mayinclude domestic and foreign equity, fixed income, real estate,and commodities. In our opinion, institutional investors maybe able to reduce the overall risk of the portfolio by incorpo-rating an alpha currency program into an established assetallocation. The addition of an asset class that is lowly or nega-tively correlated to the other major asset classes is a way toreduce the volatility of the overall plan and thus improve itsrisk/return profile.

In Exhibit 2, it is evident that a currency allocation can helpoffset some of the volatility that a “traditional” portfolio mayencounter. The currency markets, as measured by the U.S.Dollar Index (DXY), have a low to negative correlation to thestock, bond, real estate and commodity markets, making it anattractive portfolio diversifier.

OPPORTUNITIES

The argument against active currency management is that thelong-term investment in a currency of a developed country isessentially a zero-sum investment. The long-term expected returnsof the major currencies are zero. During times of economic pros-perity, the host nation’s currency will become stronger relative toother currencies. However, when the economic tides turn, it willeventually give back the previous gains.

Currencies typically will move according to the outlook of acountry’s economic data, and exchange rates tend to gaindirectional momentum and trend. In the short term, there isan inherent volatility as with any other asset class. However,over a longer time horizon, the economic health of the hostnation should keep the momentum moving in the same direc-tion. Exchange rate trends do persist for some time, resultingin an opportunity for skilled active managers to exploitcurrency swings.

The currency market is the world’s largest and most liquidmarket, with an average daily turnover of $3.2 trillion USD3.The daily currency turnover is more than ten times that of all ofthe world’s equity markets combined4. The volume can beattributed to a number of factors, including the different typeof market participants and the various objectives they have forcurrency exchange. Currency exchange is employed by centralbanks to implement monetary policy, by commercial banks tomanage cash flow, and by institutional investors to hedgeexposure and enhance return.

For example, many investment managers execute foreignexchange trades for the sole purpose of making funds avail-able in the local trading currency to execute foreign stock orbond trades. A central bank’s motivation for being players inthe currency market is to stabilize their domestic currency orstave off the forces of inflation. In fact, central banks mayhave to buy or sell currency at inopportune times to satisfytheir primary goals. Corporations can have significantcurrency exposure from foreign operations, trading, or debtservicing obligations. Their greater interest lies in neutralizingtheir currency exposure rather than maximizing profits.

With any active management strategy, there should be anexploitable market inefficiency in order for the strategy to besuccessful. Usually, this inefficiency comes at the cost ofliquidity. In the currency market, we find the rare case whereboth are available.

CONCLUSION

As institutional investors continue to increase their allocationto international investments, especially in the emergingmarkets, a clear currency mandate is becoming an increasinglyimportant part of portfolio management. Based on the goals,requirements and restrictions of the plan, the plan sponsorshould review the currency policy in place and determine theoptimal strategy.

Leaving the portfolio completely un-hedged may leave theportfolio exposed to unwanted volatility. While a passivestrategy will help neutralize the currency risk associated withforeign investments, on a long-term basis, a 100% hedge canonly be right 50% of the time. Given the portfolio diversifica-tion effects, the exploitable opportunities present in thecurrency market, and the potential to increase the risk/rewardprofile of the overall portfolio, institutional investors may wantto consider alpha-seeking strategies in order to cash oncurrency.

SPOTLIGHTCURRENCY MANAGEMENT

USD Index 1.00

Domestic Equity 0.03Foreign Equity -0.29

Fixed Income -0.21

Real Estate -0.07

Commodities -0.19Source: J.P. Morgan Investment Analytics & Consulting estimates.

Exhibit 2 - Correlation of U.S. Dollar Index (DXY) withTraditional Asset Classes2

RISK MANAGEMENTSPOTLIGHT


This article aims at giving an overview of one of the most wide-spread models in use in most of risk management departmentsacross the financial industry: Value-at-Risk (or VaR)2. VaR calcu-lates the worst expected loss over a given horizon at a givenconfidence level under normal market conditions. VaR esti-mates can be calculated for various types of risk: market, credit,operational, etc. We will only focus on market risk in this article.Market risk arises from mismatched positions in a portfolio thatis marked-to-market periodically (generally daily) based onuncertain movements in prices, rates, volatilities and other rele-vant market parameters. In such a context, VaR provides asingle number summarizing the organization’s exposure tomarket risk and the likelihood of an unfavorable move. Thereare mainly three designated methodologies to compute VaR:Analytical (also called Parametric), Historical Simulations, andMonte Carlo Simulations. For now, we will focus only on theAnalytical form of VaR. The two other methodologies will betreated separately in the upcoming issues of this newsletter.Part 1 of this article defines what VaR is and what it is not, anddescribes the main parameters. Then, in Part 2, we mathemati-cally express VaR, work through a few examples and play withvarying the parameters. Part 3 and 4 briefly touch upon two crit-ical but complex steps to computing VaR: mapping positions torisk factors and selecting the volatility model of a portfolio.Finally, in Part 5, we discuss the pros and cons of AnalyticalVaR.

PART 1: DEFINITION OF ANALYTICAL VARVaR is a predictive (ex-ante) tool used to prevent portfoliomanagers from exceeding risk tolerances that have been devel-oped in the portfolio policies. It can be measured at the port-folio, sector, asset class, and security level. Multiple VaRmethodologies are available and each has its own benefits anddrawbacks. To illustrate, suppose a $100 million portfolio has amonthly VaR of $8.3 million with a 99% confidence level. VaRsimply means that there is a 1% chance for losses greater than$8.3 million in any given month of a defined holding periodunder normal market conditions.

It is worth noting that VaR is an estimate, not a uniquely definedvalue. Moreover, the trading positions under review are fixedfor the period in question. Finally, VaR does not address thedistribution of potential losses on those rare occasions whenthe VaR estimate is exceeded. We should also bear in mindthese constraints when using VaR. The ease of using VaR is alsoits pitfall. VaR summarizes within one number the risk exposureof a portfolio. But it is valid only under a set of assumptions

that should always be kept in mind when handling VaR.

VaR involves two arbitrarily chosen parameters: the holdingperiod and the confidence level. The holding period corre-sponds to the horizon of the risk analysis. In other words, whencomputing a daily VaR, we are interested in estimating theworst expected loss that may occur by the end of the nexttrading day at a certain confidence level under normal marketconditions. The usual holding periods are one day or onemonth. The holding period can depend on the fund’s invest-ment and/or reporting horizons, and/or on the local regulatoryrequirements. The confidence level is intuitively a reliabilitymeasure that expresses the accuracy of the result. The higherthe confidence level, the more likely we expect VaR to approachits true value or to be within a pre-specified interval. It is there-fore no surprise that most regulators require a 95% or 99%confidence interval to compute VaR.

PART 2: FORMALIZATION AND APPLICATIONS

Analytical VaR is also called Parametric VaR because one of itsfundamental assumptions is that the return distributionbelongs to a family of parametric distributions such as thenormal or the lognormal distributions. Analytical VaR cansimply be expressed as:

(1)

where

• VaRαα is the estimated VaR at the confidence level100 × (1 – αα)%.

• xα is the left-tail αα percentile of a normal distribution is described in the expression

where R is the expected return. In order for VaR to be mean-ingful, we generally choose a confidence level of 95% or99%. xα is generally negative.

• P is the marked-to-market value of the portfolio.

The Central Limit Theorem states that the sum of a large numberof independent and identically distributed random variableswill be approximately normally distributed (i.e., following aGaussian distribution, or bell-shaped curve) if the random vari-ables have a finite variance. But even if we have a large enoughsample of historical returns, is it realistic to assume that thereturns of any given fund follow a normal distribution? Thus, weneed to associate the return distribution to a standard normaldistribution which has a zero mean and a standard deviation of

VALUE-AT-RISK: AN OVERVIEW OF ANALYTICAL VARby Romain BerryJ.P. Morgan Investment Analytics and [email protected]

In the last issue, we discussed the principles of a sound risk management function to efficiently manage and monitor thefinancial risks within an organization. To many risk managers, the heart of a robust risk management department lies inrisk measurement through various complex mathematical models. But even one who is a strong believer in quantitative riskmanagement would have to admit that a risk management function that heavily relies on these sophisticated models cannotadd value beyond the limits of understanding and expertise that the managers themselves have towards these very models.Risk managers relying exclusively on models are exposing their organization to events similar to that of the sub-prime crisis,whereby some extremely complex models failed to accurately estimate the probability of default of the most senior tranches ofCDOs1. Irrespective of how you put it, there is some sort of human or operational risk in every team within any given organ-ization. Models are valuable tools but merely represent a means to manage the financial risks of an organization.

1 CDO stands for Collaterized Debt Obligation. These instruments repackage a portfolio ofaverage- or poor-quality debt into high-quality debt (generally rated AAA) by splitting aportfolio of corporate bonds or bank loans into four classes of securities, called tranches.

2 Pronounced V’ah’R.

RISK MANAGEMENTSPOTLIGHT


one. Using a standard normal distribution enables us to replacexα by zα through the following permutation:

(2)

which yields:

(3)

zα is the left-tail α percentile of a standard normal distribution.Consequently, we can re-write (1) as:

(4)

EXAMPLE 1 – ANALYTICAL VAR OF A SINGLE ASSET

Suppose we want to calculate the Analytical VaR at a 95% confi-dence level and over a holding period of 1 day for an asset inwhich we have invested $1 million. We have estimated3 μμ(mean) and σσ (standard deviation) to be 0.3% and 3% respec-tively. The Analytical VaR of that asset would be:

This means that there is a 5% chance that this asset may lose atleast $46,347 at the end of the next trading day under normalmarket conditions.

EXAMPLE 2 – CONVERSION OF THE CONFIDENCE

LEVEL4

Assume now that we are interested in a 99% Analytical VaR ofthe same asset over the same one-day holding period. Thecorresponding VaR would simply be:

There is a 1% chance that this asset may experience a loss of atleast $66,789 at the end of the next trading day. As you cansee, the higher the confidence level, the higher the VaR as wetravel downwards along the tail of the distribution (further lefton the x-axis).

EXAMPLE 3 – CONVERSION OF THE HOLDING

PERIOD

If we want to calculate a one-month (21 trading days onaverage) VaR of that asset using the same inputs, we can simplyapply the square root of the time5:

(5)

Applying this rule to our examples above yields the followingVaR for the two confidence levels:

EXAMPLE 4 – ANALYTICAL VAR OF A PORTFOLIO

OF TWO ASSETS

Let us assume now that we have a portfolio worth $100 millionthat is equally invested in two distinct assets. One of the mainreasons to invest in two different assets would be to diversifythe risk of the portfolio. Therefore, the main underlying ques-tion here is how one asset would behave if the other asset wereto move against us. In other words, how will the correlationbetween these two assets affect the VaR of the portfolio? As weaggregate one level up the calculation of Analytical VaR, we

replace in (4) the mean of the asset by the weighted mean ofthe portfolio, μμp and the standard deviation (or volatility) of theasset by the volatility of the portfolio, σσ p. The volatility of aportfolio composed of two assets is given by:

(6)

where• w1 is the weighting of the first asset• w2 is the weighting of the second asset• σσ1 is the standard deviation or volatility of the first asset• σσ2 is the standard deviation or volatility of the second asset• ρ1,2 is the correlation coefficient between the two assetsAnd (4) can be re-written as:

(7)

Let us assume that we want to calculate Analytical VaR at a 95%confidence level over a one-day horizon on a portfoliocomposed of two assets with the following assumptions:• P = $100 million• w1 = w2 = 50%6

• μμ1 = 0.3%• σσ1 = 3%• μμ2 = 0.5%• σσ2 = 5%• ρ1,2 = 30%

(8)

EXAMPLE 5 – ANALYTICAL VAR OF A PORTFOLIO

COMPOSED OF N ASSETS

From the previous example, we can generalize these calcula-tions to a portfolio composed of n assets. In order to keep themathematical formulation handy, we use matrix notation andcan re-write the volatility of the portfolio as:

(9)

where• w is the vector of the weights of the n assets• w’ is the transpose vector of w• Σ is the covariance matrix of the n assets

Practically, we could design a spreadsheet in Excel (Exhibit 1) tocalculate Analytical VaR on the portfolio in Example 4.

3 Note that these parameters have to be estimated. They are not the historical parame-ters derived from the series.

4 Note that zα is to be read in the statistical table of a standard normal distribution.5 This rule stems from the fact that the sum of n consecutive one-day log returns is the n-

day log return and the standard deviation of n-day returns is √n × standard deviation ofone-day returns.

6 These weights correspond to the weights of the two assets at the end of the holdingperiod. Because of market movements, there is little likelihood that they will be thesame as the weights at the beginning of the holding period.

SPOTLIGHT


RISK MANAGEMENT

It is easy from there to expand the calculation to a portfolio of nassets. But be aware that you will soon reach the limits of Excelas we will have to calculate n(n-1)/2 terms for your covariancematrix.

PART 3: RISK MAPPING

In order to cope with an increasing covariance matrix each timeyou diversify your portfolio further, we can map each security ofthe portfolio to common fundamental risk factors and base ourcalculations of Analytical VaR on these risk factors. This processis called reverse engineering and aims at reducing the size ofthe covariance matrix and speeding up the computational timeof transposing and multiplying matrices. We generally considerfour main risk factors: Spot FX, Equity, Zero-Coupon Bonds andFutures/Forward. The complexity of this process goes beyondthe scope of this overview of Analytical VaR and will need to betreated separately in a future article.

PART 4: VOLATILITY MODELS

We can guess from the various expressions of Analytical VaR wehave used that its main driver is the expected volatility (of theasset or the portfolio) since we multiply it by a constant factorgreater than 1 (1.6449 for a 95% VaR, for instance) – asopposed to the expected mean, which is simply added to theexpected volatility. Hence, if we have used historical data toderive the expected volatility, we could consider how today’svolatility is positively correlated with yesterday’s volatility. Inthat case, we may try to estimate the conditional volatility of theasset or the portfolio. The two most common volatility modelsused to compute VaR are the Exponential Weighted MovingAverage (EWMA) and the Generalized AutoregressiveConditional Heteroscedasticity (GARCH). Again, in order to beexhaustive on this very important part in computing VaR, wewill discuss these models in a future article.

PART 5: ADVANTAGES AND DISADVANTAGES OF

ANALYTICAL VARAnalytical VaR is the simplest methodology to compute VaR andis rather easy to implement for a fund. The input data is ratherlimited, and since there are no simulations involved, thecomputation time is minimal. Its simplicity is also its maindrawback. First, Analytical VaR assumes not only that the histor-ical returns follow a normal distribution, but also that thechanges in price of the assets included in the portfolio follow anormal distribution. And this very rarely survives the test ofreality. Second, Analytical VaR does not cope very well withsecurities that have a non-linear payoff distribution like optionsor mortgage-backed securities. Finally, if our historical seriesexhibits heavy tails, then computing Analytical VaR using anormal distribution will underestimate VaR at high confidencelevels and overestimate VaR at low confidence levels.

CONCLUSION

As we have demonstrated, Analytical VaR is easy to implementas long as we follow these steps. First, we need to collecthistorical data on each security in the portfolio (we advise usingat least one year of historical data – except if one security hasexperienced high volatility, which would suggest a shorterperiod of time). Second, if the portfolio has a large number ofunderlying positions, then we would need to map them againsta more manageable set of risk factors. Third, we need to calcu-late the historical parameters (mean, standard deviation, etc.)and need to estimate the expected prices, volatilities and corre-lations. Finally we apply (7) to find the Analytical VaR estimateof the portfolio.

As always when building a model, it is important to make surethat it has been reviewed, fully tested and approved, that a UserGuide (including any potential code) has been documented andwill be updated if necessary, that a training has been designedand delivered to the members of the risk management team andto the recipients of the outputs of the risk management function,and finally that a capable person has been allocated the over-sight of the model, its current use, and regular refinement.

Exhibit 1 – Excel Spreadsheet to calculate Analytical VaR fora portfolio of two assets

Analytical VaR

Expected parameters

p 100,000,000 Asset 1 Asset 2

w1 50%Standard Deviation 0.03 0.05

w2 50%

μμ1 0.3%Correlation Matrix 1 0.3

σ1 3% 0.3 1μμ2 0.5%σ2 5%

p1,2 30%Covariance Matrix

Σ 0.00090 0.00045

μμp 0.40% 0.00045 0.00250

σ p 3.28%

Exposures

Confidence level w1 0.5 0.5

95% -1.6449Σw 0.00068

0.00148

σ 2=w’Σw 0.00108

σ 0.03279

VaR 4,993,012 .77

Grey = input cellsSource: J.P. Morgan Investment Analytics & Consulting.

Opinions and estimates offered in this Investment Analytics and Consulting newsletter constitute our judgment and are subject to change without notice, as are statementsof financial market trends, which are based on current market conditions. We believe the information provided here is reliable, but do not warrant its accuracy or complete-ness. References to specific asset classes, financial markets, and investment strategies are for information purposes only and are not intended to be, and should not beinterpreted as, recommendations or a substitute for obtaining your own investment advice.

This document contains information that is the property of JPMorgan Chase & Co. It may not be copied, published, or used in whole or in part for any purposes other thanexpressly authorized by JPMorgan Chase & Co.

www.jpmorgan.com/visit/iac

• The U.S. Dollar recovered some of the losses of the past few years against both the Euro and Yen, as the Euro-zone and Japaneseeconomies showed signs of weakening and the U.S. Federal Reserve hinted at monetary tightening in the future.

U.S. CURRENCYby Manpreet Hochadel, CFAJ.P. Morgan Investment Analytics and [email protected]

U.S. FIXED INCOMEby Manpreet Hochadel, CFAJ.P. Morgan Investment Analytics and [email protected]

GLOBAL CAPITAL MARKETSCORNER


0.80

0.90

1.00

1.10

1.20

1.30

1.40

1.50

1.60

Dec-98 Dec-99 Dec-00 Dec-01 Dec-02 Dec-03 Dec-04 Dec-05 Dec-06 Dec-07 Dec-080.0070

0.0080

0.0090

0.0100

0.0110US Dollar vs. EURO and Japanese Yen

US

Dollar vs. JPY

US

Dol

lar v

s. E

URO

EURO Left Scale

JPY Right Scale

1.00%

2.00%

3.00%

4.00%

5.00%

30 Yr.201052

US Treasury Yield Curve

August 31, 2008

December 31, 2007

June 30,2008

• The U.S. Treasury yield curve steepened as rates continued to fall across the entire yield curve.

Source: J.P. Morgan Investment Analytics and Consulting

Source: J.P. Morgan Investment Analytics and Consulting, Bloomberg

AS OF AUGUST 2008

AS OF AUGUST 2008

EUROPEAN AND ASIAN CURRENCIESby Simreet GillJ.P. Morgan Investment Analytics and [email protected]

GLOBAL CAPITAL MARKETSCORNER


1

1.2

1.4

1.6

1.8

2

2.2

2001 2002 2003 2004 2005 2006 2007

GBP/USD GBP/EUR

20080.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

2001 2002 2003 2004 2005 2006 2007

CHF/GBP CHF/EUR CHF/USD

2008

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

2001 2002 2003 2004 2005 2006 2007

NOK/EUR NOK/USD NOK/GBP

2008

Norwegian Krone vs EUR, USD & GBP

• The uptrend in EUR/GBP over the past six monthsthrough July reflected the risk of an independentimplosion in the UK economy. The economy iscertainly buckling under the post-credit crisis strainbut so too is the Euro area, a story which is much morerecent. (Morgan Markets)

0.000

0.003

0.006

0.009

0.012

0.015

0.018

2001 2002 2003 2004 2005 2006 2007

JPY/GBP JPY/EUR JPY/USD JPY/AUD

20080.2

0.4

0.6

0.8

1

2001 2002 2003 2004 2005 2006 2007

AUD/GBP AUD/EUR AUD/USD

2008

0.06

0.08

0.1

0.12

0.14

0.16

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9

10

11

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CNY/EUR CNY/USD CNY/JPY

CNY/

JPY

CNY/ U

SD

& EU

R

20082007200620052004200320022001

Japanese Yen vs GBP, EUR, USD & AUD

Chinese Yuan vs EUR, USD & JPY

Australian Dollar vs GBP, EUR & USD

• There has been a growing divergence between the fallingequity markets and the lagged response of Asian FXlately. The key concern is whether Asian currencies willcatch up with falling equities. At a superficial level, thisequity:FX correlation is being driven by international port-folio equity flows. However, the ultimate drivers for theseflows are expectations for the global growth cycle andhow these may impact export and corporate earnings inthe future. (Morgan Markets)



AS OF JULY 2008

Pound Sterling vs USD and EUR Swiss Franc vs GBP, EUR & USD

ASSET CLASS RETURN COMPARISON (INCLUDING U.S.)by William PomettoJ.P. Morgan Investment Analytics and [email protected]

GLOBAL MARKET INDICESCORNER


AS OF AUGUST 2008

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

Current MonthReturn

3 Months Year to Date 1 Year 2 Year 3 Year 5 Year 10 Year

MSCI EMERGING MARKETS FREEMSCI-Eafe (Net)

M.L. HIGH YIELD INDEXL.B. AGGREGATE BOND INDEX

RUSS-Russell 1000 Growth (Gross)S & P 500 - CAP. WEIGHTED

RUSS-Russell 3000 (Gross)RUSS-Russell 2000 Value (Gross)

IndexMonthlyReturn

Trailing 3Months

Year ToDate

1 Year 2 Year 3 Year 5 Year 10 Year

L.B. AGGREGATE BOND INDEX 0.95 0.79 2.00 5.86 5.56 4.26 4.61 5.58

M.L. HIGH YIELD INDEX 0.32 (3.91) (2.62) (1.42) 2.48 3.42 6.87 4.23

MSCI EMERGING MARKETS FREE (7.95) (20.18) (21.67) (9.83) 13.96 19.38 23.89 17.72

MSCI-Eafe (Net) (4.05) (14.73) (17.31) (14.41) 0.80 8.08 13.86 6.34

RUSSELL 1000 GROWTH (Gross) 1.08 (7.99) (9.83) (6.77) 4.76 4.39 6.10 2.59

RUSSELL 2000 VALUE (Gross) 4.75 (0.44) (0.71) (7.52) (0.69) 3.59 10.25 11.28

RUSSELL 3000 INDEX (Gross) 1.55 (7.57) (10.39) (10.22) 1.58 3.92 7.57 5.52

S & P 500 - CAP. WEIGHTED 1.45 (7.89) (11.39) (11.14) 1.15 3.67 6.93 3.11

U.S. EQUITY

• U.S. Stock Markets avoided a third straight down month,posting modest gains in August.

• Much of the upward momentum was driven by declining oilprices.

• Value stocks experienced the biggest lift with the Russell2000 Value Index posting a 4.75 percent gain for the month.

• The S&P 500 and NASDAQ Composite added 1.45 percentand 1.92 percent respectively.

INTERNATIONAL EQUITY

• International Markets performed poorly in August. • Investor concern continues over the slumping United States

economy and how it will affect international markets. • The strengthening U.S. dollar has also negatively impacted

international market returns. • The Emerging Markets were hit the hardest, as evidenced by

the 7.95 percent loss for the MSCI Emerging Markets FreeIndex in August.

FIXED INCOME

• Fixed Income Markets were a safe bet for August. • Government bonds continued to post respectable gains. • The L.B. Government Long Term Index posted a 2.32 percent

return for the month, putting it up 3.75 percent year to date.

REAL ESTATE

• Real Estate Markets haven't improved as mortgage concernsstill exist.

• Home sales continued to slow.

Europe

GLOBAL MARKET INDICESCORNER


GLOBAL EQUITIES (EXCLUDING U.S.)by Simreet GillJ.P. Morgan Investment Analytics and [email protected]

• Resource-consuming emerging markets such as India andChina, which had been sharply sold off owing to growinginflationary concerns on rising commodity prices, picked up.The Japanese market had the second largest fall amongmajor developed nations after England. However, while theJapanese market has tended to be as highly volatile asemerging markets recently, we think movement in theJapanese market in July was only slight compared with thesharp movement in emerging markets. (Morgan Markets)

0

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0

50

100

150

200

250

2000 2001 2002 2003 2004 2005 2006 2007 2008


• The financial markets have remained highly unstable in July,owing in part to the troubles at GSEs such as Freddie Mac.Equities have been slowly grinding through the cyclical slow-down triggered by a housing bust, credit crunch, andcommodity price pressures.

• In July, equities saw a month of two halves with sharp falls inthe first half and a rebound of equal magnitude and intensityin the second half. The downshift in global growth indicators

earlier in the month triggered further declines in globalequity markets. In particular, the abrupt weakening inEuropean and Japanese growth into midyear, combined withsigns that global industrial activity is now contracting,marked a sharper downshift in momentum than theconsensus anticipated. Later in the month, sharply lower oilprices and short covering in financials helped equity marketsto more than retrace their losses seen in the first two weeksof July. (Morgan Markets)

Australia, Hong Kong, Singapore

0

0.3

0.6

0.9

1.2

1.5

1.8

2000 2001 2002 2003 2004 2005 2006 2007 2008


Korean (KOSPI)

Japan (Nikkei 225)

AS OF JULY 2008



analytical var var mapping

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