Clear Horizon

ClearHorizonTMTechnical Document

Jongwoo KimContributor: Jorge Mina

This draft: August 28, 2000

Forecasting Methodology for Horizons Beyond Two Years

ClearHorizonTM Technical Document, First Edition (June 2000)

© 2000 The RiskMetrics Group

Copyright © 2000 The RiskMetrics Group, LLC.All Rights Reserved. The RiskMetrics Group hereby grantsyou a non-exclusive, limited, revocable license to use theClearHorizonTM Technical Documentprovided thatyou comply with the following requirements. You must not edit, delete, supplement, alter or modify theClearHorizon Technical Documentincluding its text, charts, formulas, graphics, copyright notice, trademarknotice and/or the names and e-mail addresses of the authors. You acknowledge that the formulas that appearin the ClearHorizon Technical Documentare proprietary to RiskMetrics. You may not use, duplicate ordistribute any one or more of the formulas that appear in theClearHorizon Technical Documentfor anypurpose other than to evaluate the ClearHorizonTM measurement services, including how the ClearHorizonmethodology works. Any other use, copying and distribution of theClearHorizon Technical Documentisprohibited. The RiskMetrics Group reserves the right to terminate this limited license on notice to you.

ClearHorizonTM , FortuneManagerTM , PensionMetricsTM , RiskGradeTM , RiskGradesTM , RiskMetrics®, and theCircle Design Logo are trademarks and service marks owned by or licensed to The RiskMetrics Group, LLC.Without the express written approval of The RiskMetrics Group, you may not use the trademarks or servicemarks for any purpose other than to designate that these marks are owned by or licensed to The RiskMetricsGroup, LLC.

The RiskMetrics Group does not warrant any results obtained from the use of the ClearHorizon method-ology, RiskGrades measures, data, methodology, documentation or any information derived from the data(collectively the "Data") and does not guarantee its sequence, timeliness, accuracy, completeness or con-tinued availability. The Data is calculated on the basis of historical observations and should not be reliedupon to predict future market movements. The Data addresses market risk measurement and should not berelied upon to measure all of a company’s other risk types, such as credit, operational, business, legal, orreputational risks. The information contained in this document is believed to be reliable but The RiskMetricsGroup does not guarantee its completeness or accuracy. Opinions and estimates constitute our judgementand are subject to change without notice. Copyright 2000 The RiskMetrics Group, LLC.

Contents

1 Introduction 1

2 Forecasting Methodology 3

2.1 Two Basic Models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 A Hybrid Model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Forecasting Examples 15

3.1 Estimation of Parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.2 Degree of Mean Reversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Additional Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

4 Applications of ClearHorizon 23

4.1 FortuneManagerTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.2 Pension Risk Management. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

iii

iv CONTENTS

RiskMetrics

Chapter 1

Introduction

When RiskMetrics® [6] was first released in 1994 to introduce J.P. Morgan’s non-proprietary Value-at-Risk(VaR) methodology, it was accepted as a revolutionary risk measurement tool for the financial industry.RiskMetrics was originally invented for investment banks, whose day-to-day trading operations requirefrequent and quick adjustments of the banks’portfolios. The RiskMetrics VaR calculation therefore involvedshort horizons, from overnight to three months. Two of the key assumptions made in RiskMetrics are thatthe expected return on any asset is zero, and the asset’s volatility beyond a one-day horizon is scaled up fromthe RiskMetrics daily volatility according to the square root of time rule. These are innocuous assumptionsin the short-term forecasting context of RiskMetrics.

However, the demand for market risk measurement over long horizons continues to increase. For example,non-financial corporations are less sensitive to daily market moves and focus on long-term fluctuationswhen gauging their performance. Consequently, their budgeting and planning horizons extend to one yearand longer. To provide a dedicated and robust long-term risk measurement framework, we first developedLongRunTM [4], which focuses on market risk scenarios for time horizons up to two years. LongRun startswith the basic understanding that the random walk model, which forms the RiskMetrics framework, doesnot always provide the best explanation for the dynamics of financial returns over long horizons. LongRunthen provides two basic approaches to long-term forecasting of future asset returns: Forecasts based oncurrent market prices make intensive use of spot, futures, forwards and options price data as well as basicderivatives theory, while forecasts based on economic fundamentals rely on historical time series of financialand economic data and on econometric modeling of time series.

LongRun was launched in the spring of 1999 and created a deep impact on corporate treasuries. It alsostimulated the demand for a third type of market risk methodology, one that can generate forecasts for timehorizons beyond two years. The greatest demand for a longer-term forecasting method comes from themutual funds and pension plans industries. Their main source of risk is market price fluctuations, which theytrack for at least 30 years. In addition, pension plans and insurance companies have longer-term liabilities in

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2 CHAPTER 1. INTRODUCTION

the form of annuities or insurance claim payments. LongRun, however, is limited to a two-year forecastinghorizon; for the forecasts based on current market prices, it is difficult to find derivatives prices whoseexpiration dates exceed two years, while for the forecasts based on economic fundamentals, it is difficult tojustify how econometric modeling can remain valid beyond a two-year horizon.

In this technical document, we introduceClearHorizon, which we designed to provide robust forecastsfor generating scenarios to measure market risk beyond periods of two years. In contrast to LongRun,ClearHorizon is time-series specific, i.e., ClearHorizon uses only those properties that are specific to the timeseries of each asset included in the forecast. More precisely, ClearHorizon determines the optimal mixtureof the random walk and mean reversion properties of an asset’s price.

In Chapter 2, we explain the forecasting methodology of ClearHorizon. First, we estimate two extreme timeseries models: the random walk and mean reversion. We then determine the weighting factors for each modelin the optimal mixture in order to best calibrate the optimal mixture model to the historical behavior of anasset’s price. Finally, we calculate the expected returns and volatilities for the pre-specified set of future datesaccording to the optimal mixture model. The model’s expected-return forecasts capture the long-term trendthat is specific to a given asset’s price, and its volatility forecasts explicitly incorporate the mean reversionproperty of the asset’s price.

Chapter 3 presents 23 examples of asset price forecasting across several instruments: foreign exchange, shortand long-term government bonds, equity indices, and commodities. The degree of mean reversion variesacross asset prices, which the optimal mixture model is able to incorporate in the forecasting.

Chapter 4 presents two applications of ClearHorizon forecasting. One application is FortuneManagerTM ,which is a simulation-based risk management solution for long-horizon investment. While many individualinvestors use RiskGradesTM to manage their money for various investment horizons, FortuneManager providesmore specific risk measurements focusing on long-term investment, such as retirement plans. The otherapplication is pension risk management in which ClearHorizon is used to assess liability-related risks ofpension funds, as well as their asset-related risks.

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Chapter 2

Forecasting Methodology

2.1 Two Basic Models

We start our discussion by introducing two basic time series models: the random walk and mean reversion(see Metcalf and Hassett [5]).

Let Pt denote the asset price for which we will forecast the mean and volatility for a pre-specified set offuture dates. If the asset price follows the Geometric Brownian Motion (GBM) process, it can be describedby the following equation:

dPt = αPtdt + σPtdz, (2.1)

whereα denotes the trend coefficient,σ is the standard deviation, andz denotes a Wiener process with zeromean and unit variance.

Let pt denote the logarithm ofPt . To estimate parameters based on the discrete historical data ofPt , we canexpress Equation 2.1 as the following discrete version:

1pt = pt − pt−1 = α + σεt , (2.2)

whereεt denotes a random error term that follows the standard normal distribution, and parametersα andσ

in Equation 2.2 are equivalent toα andσ in Equation 2.1.

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4 CHAPTER 2. FORECASTING METHODOLOGY

Usually, we refer to1pt as a random walk process and topt as a difference stationary series.1 From thispoint, we simply call1pt a random walk (RW). Including the time parametert , the mean (µRW,k) andvariance (σ 2

RW,k) of pt+k are calculated as follows:

µRW,k = kα + pt (2.3)

σ 2RW,k = kσ 2

RW,1, (2.4)

whereα andσ 2RW,1 are the parameters that were estimated in Equation 2.2.

Next, if the asset price follows the Geometric Mean Reversion (GMR) process, it can be described by thefollowing equation:

dPt =[α + γ (P0e

αt − Pt)]Ptdt + σPtdz, (2.5)

where parameterγ has a positive value and represents the speed of mean reversion. When the trend coefficientα equals zero, Equation 2.5 degenerates to the Geometric Ornstein-Uhlembeck (GOU) process.

The GMR process of Equation 2.5 can be expressed as the following discrete form2

1pt = pt − pt−1 = α + γ [p0 + α(t − 1) − pt−1] + σ εt , (2.6)

then,

pt = γ (p0 + αt) + (1 − γ )(α + pt−1) + σ εt

= α + βt + γpt−1 + σεt , (2.7)

where parametersα, β, γ , andσ are equivalent toγp0 + (1 − γ )α, γ α, (1 − γ ), andσ in Equation 2.6,respectively.

Parameterγ (or γ ) has a value between 0 and 1, and represents the speed of mean reversion. Ifγ equals 1 (orγ equals 0), the process degenerates to the random walk; ifγ is significantly less than 1 (orγ is significantlylarger than 0), the process shows strong mean reversion properties.

1For a detailed discussion of stationary and nonstationary time series, see Chapter 15 of Hamilton [3].2It does not mean that the continuous form of Equation 2.5 and the discrete form of Equation 2.6 are equivalent.

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2.1. TWO BASIC MODELS 5

Usually, we refer topt as a mean reversion process, or a trend stationary series. From this point, we willsimply call it mean reversion (MR). Including the time parametert , the mean (µMR,k) and variance (σ 2

MR,k)of pt+k are calculated as follows:

µrm,k = α(1 − γ k)

1 − γ+ β

1 − γ

[(t + k) − γ k(t + 1) − γ (1 − γ k−1)

1 − γ

]+ γ kpt (2.8)

σ 2MR,k = σ 2

MR,1(1 − γ 2k)

1 − γ 2, (2.9)

whereα, β, γ andσ 2MR,1 are the parameters that were estimated in Equation 2.7.

As an example, we estimated Equation 2.2 (random walk model) and Equation 2.7 (mean reversion model)by using the USD month-end prices in the period January 1986 to April 2000. Parametersα, β, andγ wereestimated from the Ordinary Least Squares method (OLS) andσ was estimated from the standard deviationof residuals. For each model, the estimated results are presented in Table 2.1.

Table 2.1:Parameters Estimated by Random Walk and Mean Reversion

Random Walk Mean ReversionAsset α σ α β γ σ

USD S&P 500 0.0113 0.0450 0.2524 0.0005 0.9536 0.0444

The estimated mean reversion parameterγ is 0.9567, i.e., less than 1, which is important in determiningwhether a series is difference stationary or trend stationary, as discussed in the next section.3

In this section we demonstrate how the two models generate different forecasts of the S&P 500 in USD.

To determine how quickly variance changes with forecast horizon for each model, we calculate the varianceratio (VR) for each model from the following equation:

V Rk = σ 2k

kσ 21

. (2.10)

3The unit root test determines whether a time series is difference stationary or trend stationary, based on the estimatedγ . If theestimatedγ is significantly far away from unity, the test rejects the null hypothesis of the difference stationary time series (so-calledunit root process). For a detailed discussion of the unit root test, see Hamilton [3], Chapter 15.

ClearHorizonTM Technical Document


Figure 2.1 plots the variance ratio of USD S&P 500 forecasts up to ten years. The variance ratio of the randomwalk model is always 1 according to Equation 2.4, which means that the volatility or standard deviation ofthe random walk increases proportionally to the square root of its forecast horizon (the so-called square rootof time rule). For the mean reversion model, the Variance Ratio starts at 1 and steadily decreases as theforecast horizon increases. Equation 2.4 confirms thatσ 2

MR,k is less thankσ 2MR,1 as long asγ is less than 1.

The lower chart in Figure 2.1 plots the standard deviations calculated from the two models. The standarddeviation of the random walk is always greater than the standard deviation of the mean reversion model.Furthermore, it increases continuously while the standard deviation of the mean reversion model remains ata nearly constant value after a certain number of years within the time horizon. As a result, the gap betweenthe standard deviations of the two models widens as the forecast horizon increases.

Although the difference betweenγ and unity can be very small and can seem quantitatively trivial, it iscritical in forecasting the mean and confidence intervals. The reason is thatγ determines the choice of model— a γ equal to unity requires the use of the random walk, whileγ less than unity requires use of the meanreversion model. For a more detailed discussion, see Hamilton [3], Chapter 15.

Figure 2.2 compares the random walk and mean reversion models with respect to their mean and 90%confidence interval forecasts of the logarithmic value of USD S&P 500 from May 2000. The mean forecastsof the random walk model increase in parallel to the long-term path, and never die out. This means that anykind of price movement due to today’s information or shock persists and is continuously accumulated (i.e.,all shocks are permanent shocks). The future price, then, is nothing more than an accumulation of pricemovements across the forecasting horizon. Since information or shock is continuously generated and affectsthe price as time goes on, the standard deviation of the price should also increase continuously as time goeson. In addition, the size of the confidence interval is determined by the standard deviation forecasts, andtherefore the confidence interval also continuously increases.

By contrast, the mean forecasts of the mean reversion model coincide exactly with the long-term path, aftertheir initial deviations from the long-term path quickly die out. This means that any kind of short-term pricemovement that is caused by current information or shock disappears in the long run (the speed of disappearanceis determined by the mean reversion parameterγ ) and is not accumulated (i.e., all shocks are transitory).The future price is nothing more than the long-term price that is observed when the forecast horizon becomessufficiently long to capture price recovery; although new information or shock is continuously generated andaffects the price as time goes on, its effect disappears in the long run, and the price recovers to follow thelong-term path. Therefore, the standard deviation of the price does not increase continuously, which we havealready shown in Figure 2.1.

2.2 A Hybrid Model

One way to generate long-term forecasts of asset price is by selecting either the random walk or the meanreversion model according to the unit root test, and then applying the selected model to the mean and variance

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2.2. A HYBRID MODEL 7

Figure 2.1:Volatilities Computed from the Random Walk and Mean Reversion Models

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Figure 2.2:Forecasts of the Random Walk and Mean Reversion Models

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forecasts. However, a large number of time series studies have been reported to prove that any unit root testhas very low test power for finite samples. Furthermore, Cochrane [1] explains the low test power of theunit root test by the fact that any time series with a unit root can be decomposed into a stationary series anda random walk.

Therefore, we interpret the random walk and mean reversion models in the previous section as the extremesof a broad spectrum of possible models. At one extreme, we can consider the random walk to be a model thatprovides the maximum possible volatility, since it denies the existence of mean reversion properties for anytime series. (However, many time series studies have been reported to confirm the mean reversion propertyof asset prices in the long-term, especially interest rates (see Wu and Zhang [7].) At the other extreme, wecan consider mean reversion (specifically the GMR version) as a model that provides the minimum possiblevolatility. The small volatility is valid only when the parameters of the mean reversion model are correctlyspecified and constant through the forecasting horizon. In other words, today’s information and shock donot change the long-term trend of the price.

Based on this reasoning, we assume that the true model of an individual asset’s price is a mixture of randomwalk and mean reversion models (i.e., while some shocks are permanent, others are transitory). To confirmthis assumption, we need to design a way to determine the optimal weighting factors of the two models. Onepractical way is to determine the weighting factor for an individual series as the factor that best calibrates thedegree of mean reversion to the historical data of the same series.

To measure the degree of mean reversion, we generally use the variance ratio. The equation for the varianceratio for historical data is the same as Equation 2.10, except that the variance ofk-period forecasts is calculatedfrom the following equation (see Glen [2]):

σ 2k = n

(n − k)(n − k + 1)

n∑i=k+1

(pi − pi−k − kr1)2, (2.11)

wheren denotes the total number of observations, andr1 denotes the one-period historical average return.It is worth noting that since it is calculated from actual data, the variance ratio for historical data does nothave a geometric shape like the random walk (constant at 1) or the mean reversion (geometrically monotonicdecreasing) models.

Next, we determine the weighting factorsω and (1-ω) for the variance ratios of the random walk and meanreversion, respectively, such that the variance ratio of the optimal mixture best calibrates to the variance ratioof the historical samples. To estimate the weighting factor, we use the least squares method and minimizethe following equation with respect toω:

min

q∑k=1

[V RHS,k − (ωV RRW,k + (1 − ω)V RMR,k)

]2, (2.12)



whereV RHS,k denotes the variance ratio of historical data andq denotes the maximum forecast horizon.

Using historical data as well as the estimated random walk and mean reversion models for USD S&P 500in the previous section, we estimated the weighting factor for the optimal mixture to be 0.8013. This meansthat the degree of mean reversion in the historical data of USD S&P 500 is calibrated by 80% of the randomwalk and 20% of the mean reversion model.

The upper chart in Figure 2.3 plots the variance ratios of the optimal mixture, random walk, and meanreversion models. We can easily check that the variance ratio of the optimal mixture fits the historical dataquite well. In addition, it has a geometrically monotonic decreasing shape as the forecast horizon increases,which guarantees a smoothly increasing monotonic shape for the confidence interval.

The lower chart in Figure 2.3 shows the standard deviations estimated from the optimal mixture, randomwalk, and mean reversion models. In the case of USD S&P 500, the standard deviation of the optimal mixtureis close to that of the random walk because the estimated weighting factor for the optimal mixture is close tounity.

Figure 2.4 shows the mean and 90% confidence interval forecasts of the logarithmic value of USD S&P 500based on the optimal mixture, random walk, and mean reversion models. Because the optimal mixture modeluses the mean forecasts of the mean reversion model, it is classified as a type of modified mean reversionmodel. Actually, whether mean forecasts are selected from the random walk or the mean reversion model isunimportant in long-horizon forecasting, because the deviation of today’s price from the long-term path isdominated by the long-term path and its continuously increasing volatility.

One alternative choice in mean forecasting is to utilize current market data as an unbiased future expectation,as we do in LongRun. LongRun broadly uses forward prices for mean forecasting, and implied volatilitiesfrom option prices for volatility forecasting. However, the use of current market data in ClearHorizon isrestrictive because of the long horizon. To forecast interest rates and foreign exchange rates, we can useimplied forward interest rates and implied forward foreign exchange rates, which are calculated from currentyield curves and interest rate parity.

The size of the confidence interval of the optimal path is determined by the standard deviation forecasts andhas already been shown in Figure 2.4. Overall, the forecasted path and its confidence interval in the optimalmixture take the golden mean of two extremes.

Since we assumed that the logarithm of the asset pricept follows a normal distribution,

pt+k ∼ N(µt+k, σ2t+k), (2.13)

the asset pricePt follows the lognormal distribution.

Given the mean (µt+k) and standard deviation (σt+k) of the logarithm of the asset price, the mean [E(Pt+k)]and upper and lower confidence intervals [U(Pt+k, α) andL(Pt+k, α)] of the asset price are calculated as

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Figure 2.3:Volatility Computed from the Optimal Mixture Model

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Random Walk Actual Data Optimal MixtureMean Reversion



Figure 2.4:Forecasts of the Optimal Mixture Model, Logarithm of Price

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E(Pt+k) = exp

[µt+k + σ 2

t+k

2

], (2.14)

U(Pt+k, α) = exp[µt+k + Zασt+k

], (2.15)

and

L(Pt+k, α) = exp[µt+k − Zασt+k

], (2.16)

whereZα denotes the value of the standard normal distribution at (1− α) significance (e.g.,Z0.05 = 1.645).Note that the mean [E(Pt+k)] of the asset price is determined by the standard deviation (σt+k) as well as themean (µt+k) of the logarithm of the asset’s price.

Figure 2.5 shows the mean and 90% confidence interval forecasts of the price of USD S&P 500 based on theoptimal mixture, random walk, and mean reversion models. Across these models, we can see a much largerdifference in the confidence intervals of prices than in the logarithms of the prices (Figure 2.4).



Figure 2.5:Forecasts of the Optimal Mixture Model, Price

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Chapter 3

Forecasting Examples

3.1 Estimation of Parameters

In this chapter, we present 23 examples of asset price forecasting across several instruments: foreign ex-change; short and long-term government bonds; the equity indices of Germany, Japan, Mexico, and the U.S.;and the commodity prices of copper, WTI, heating oil, and gold. Generally, we estimated all three modelsby utilizing month-end prices from January 1986 to April 2000. Because of data restrictions, the parametersfor DEM DAX, DEM 3M and 10Y T-bonds, JPY 3M and 10Y T-bonds, MXN FX, MXN IPC, and MXN 3Mand 1Y swaps are estimated with less than ten years of historical data. The estimated parameters in the threemodels are shown in Table 3.1. Also, Figure 3.1 plots the variance ratios of the three models.

Parameter estimation results in very high fitness to data since theR2 of the mean reversion model exceeds0.9, except for the MXN 3M swap (0.87), MXN 1Y swap (0.89), WTI future (0.83), and Heating Oilfuture (0.75). This means that more than 90% of the price movement is explained by the mean reversionmodel. Furthermore, the mean reversion models of USD S&P 500, DEM DAX and 3M T-Bill, and MXNFX show exceptionally high fitness to the data because theR2 exceeds 0.99. Although theR2 of the randomwalk model is slightly lower than that of the mean reversion model, the random walk model still shows avery high degree of fitness to the data.1

It is worth noting that while all the equity indices and MXN FX have a strong time trend, almost all thegovernment bond yields lack a time trend (estimatedβ is near 0) and shrink to the Geometric Ornstein-Uhlembeck (GOU) process. When we estimate the mean reversion equation, Equation 2.7, for the interestrates, we need to remove the time-trend term for efficient estimation.2

1Since the number of unknown parameters in the mean reversion model is one more than in the random walk model, theR2 ofthe mean reversion model is usually higher than theR2 of the random walk model.

2In addition to efficient estimation, the GOU process is more reasonable for interest rate modeling from the theoretical point ofview because it is impossible for interest rates to maintain a strong time trend for a long period of time.

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16 CHAPTER 3. FORECASTING EXAMPLES

Table 3.1:Estimation of Parameters

Random Walk Mean Reversion Optimal MixtureAsset α σ α β γ σ ω

USD S&P 500 0.0113 0.0450 0.2524 0.0005 0.9536 0.0444 0.6857USD Gov’t 3M -0.0020 0.0479 0.0204 0.0000 0.9842 0.0475 1.0000USD Gov’t 1Y -0.0012 0.0556 0.0428 0.0000 0.9748 0.0552 1.0000USD Gov’t 10Y -0.0025 0.0416 0.1862 -0.0002 0.9142 0.0407 0.3195USD Gov’t 30Y -0.0032 0.0348 0.2169 -0.0003 0.9028 0.0339 0.2856EUR FX 0.0008 0.0327 0.0146 -0.0001 0.9518 0.0318 0.7042DEM DAX 0.0130 0.0582 0.2663 0.0005 0.9624 0.0575 0.9498DEM Gov’t 3M -0.0076 0.0421 -0.0599 0.0005 1.0167 0.0411 0.0000DEM Gov’t 1Y -0.0018 0.0644 0.0349 -0.0001 0.9807 0.0641 1.0000DEM Gov’t 10Y -0.0023 0.0346 0.0794 -0.0001 0.9587 0.0343 1.0000JPY FX -0.0019 0.0656 -0.8226 0.0003 0.9203 0.0631 1.0000JPY Nikkei 0.0019 0.0656 0.8226 -0.0003 0.9203 0.0631 1.0000JPY Gov’t 3M -0.0488 0.3979 0.5633 -0.0130 0.7273 0.3698 0.1466JPY Gov’t 1Y -0.0210 0.2884 0.2149 -0.0022 0.9108 0.2816 0.3126JPY Gov’t 10Y -0.0117 0.1295 0.3157 -0.0029 0.7980 0.1230 0.3268MXN FX 0.0115 0.0540 0.0569 0.0005 0.9607 0.0535 1.0000MXN IPC 0.0132 0.0959 1.4564 0.0031 0.8009 0.0907 0.2503MXN Swap 3M -0.0028 0.1620 0.2066 -0.0004 0.9384 0.1592 1.0000MXN Swap 1Y -0.0016 0.1398 0.1649 -0.0002 0.9503 0.1380 1.0000Copper Spot 0.0003 0.0733 0.2888 0.0000 0.9616 0.0726 0.6740WTI Future -0.0008 0.0979 0.2753 0.0000 0.9070 0.0955 0.2250Heating Oil 0.0007 0.1088 0.5192 0.0001 0.8696 0.1052 0.1654Gold -0.0015 0.0359 0.4723 -0.0002 0.9227 0.0350 0.3864

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3.2. DEGREE OF MEAN REVERSION 17

3.2 Degree of Mean Reversion

How much of the mean reversion property does an individual asset’s price have? If either the estimatedγ

of an asset price is close to 1or the estimatedω is close to 1 (Table 3.1), the variance ratio of the optimalmixture approaches or overlaps the variance ratio of the random walk, as shown in Figure 3.1.

Thus, the asset price has a weak mean reversion property or none at all. In our example, short-term interestrates and foreign exchange rates show weak mean reversion. Specifically, the USD 3M and 1Y T-bonds;the DEM 3M, 1Y, and 10Y T-bonds; JPY/USD Exchange, MXN/USD Exchange, and MXN 3M swap donot show any mean reversion properties. In this case, the variance ratio of the optimal mixture overlaps thevariance ratio of the random walk, as shown in Figure 3.1.

If the estimatedγ of an asset price is far less than 1and its estimatedω is far less than 1 (see Table 3.1),the variance ratio of the mean reversion model is far below that of the random walk model, and the varianceratio of the optimal mixture model comes close to that of the mean reversion model, as shown in Figure 3.1.Thus, the asset price has a strong mean reversion property. In our example, long-term interest rates andcommodity prices show strong mean reversion. Specifically, the USD 10Y and 30Y T-bonds, all maturitiesof JPY T-bonds, WTI future, Heating Oil future, and Gold spot all show strong mean reversion properties.

In the case of stock market indices, the degree of mean reversion varies by index. While JPY Nikkei andDEM DAX show no or very weak mean reversion, MXN IPC and USD S&P 500 show relatively strongmean reversion.

Our results are consistent with previous empirical studies of the mean reversion model. Many of thesestudies confirm strong mean reversion in long-term interest rates and commodity prices, but no or weakmean reversion in short-term interest rates and foreign exchange rates.

As we discussed in the previous chapter, different degrees of mean reversion lead to totally different long-termforecasts of the mean and confidence interval. We can use our observation of the degree of mean reversionacross asset prices to generate more robust forecasts. Since the process of generating forecasts, given ourestimated parameters, is tedious and lengthy, we omit it from this publication.

3.3 Additional Remarks

One of the most difficult aspects of long-term forecasting is the impossibility of implementing backtesting,due to the many data and regime changes within the sample period. Similarly, in this version of ClearHorizonwe could not implement any backtesting to prove that our optimal mixture model, consisting of random walkand mean reversion models, provides better forecasts than other models. Therefore, we are willing to use itas is. We emphasize, however, the fact that the optimal mixture model is a best fit to the historical behaviorof any given asset’s price.



ClearHorizon is not a stand-alone methodology for the calculation of VaR beyond a two-year horizon. Forthis special case, readers should consider ClearHorizon as an extension of LongRun. In most cases, theframework of LongRun, such as correlation structure and Level 1 and Level 2 Simulation, can also beimplemented at horizons exceeding two years, without significant modification.

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3.3. ADDITIONAL REMARKS 19

Figure 3.1:Variance Ratios

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Figure 3.1:Variance Ratios(continued)

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3.3. ADDITIONAL REMARKS 21

Figure 3.1:Variance Ratios(continued)

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Chapter 4

Applications of ClearHorizon

4.1 FortuneManagerTM

The RiskGradeTM statistic is a new measure of volatility recently devised by the RiskMetrics Group to helpinvestors better understand their market risk. RiskGrade measurements are based on the exact same data andanalysis as RiskMetrics® Value-at-Risk (VaR) estimates and, in fact, can be translated back intoVaR estimates.The RiskGrade measure, however, is scaled to be more intuitive and easier to use than VaR. RiskGradesTM aremeasured on a scale from 0 to 1000 or more, where 100 corresponds to the average RiskGrade value of majorequity market indices during normal market conditions from 1995 to 1999. You would expect cash to havea RiskGrade value of 0, while a technology IPO may have a RiskGrade value exceeding 1000.

The RiskGrade statistic is a good measure of risk. It is a representative and universal scaler of risk for alltypes of investment instruments. However, for specific situations, such as a relatively longer investmenthorizon, we need additional measurements in order to draw the whole picture of risk. For this purpose, theRiskMetrics Group devised FortuneManagerTM to provide a simulation-based risk management tool for thelong-horizon investment needs of individual investors and financial advisory companies.

This section briefly introduces the architecture of FortuneManager to illustrate the application of ClearHorizonforecasting methodology. FortuneManager consists of three engines: forecasting, simulation, and optimiza-tion engines, as shown in Figure 4.1.

1. Forecasting Engine

This engine forecasts the price movements of risk factors and thus provides a means of generatingreasonable scenario simulations. Since the forecasting horizon must extend up to 50 years in order tocover long-horizon goals (e.g., retirement funds), ClearHorizon is used as the forecasting methodologyby the engine.

23

24 CHAPTER 4. APPLICATIONS OF CLEARHORIZON

Figure 4.1:FortuneManager Flowchart

Input Engine Output

Asset Class Benchmark Index Expected Return and Standard DeviationEquity US Large Cap S&P 500

US Mid Cap Russell MidUS Small Cap Russell 2000International MSCI EAFE

Emerging IFCI ForecastingFixed Income US Long-term TB/Corp 10yr -

US Mid-term TB/Corp 4-9US Short-term TB/Corp 1-3

US Municipal LB 20yr Muni EngineUS High Yield SB High YieldInternational SB Non-USCash T-Bill 30 days

Real Estate REITs NAREIT All

Investor’s Information Mean and Probability of Shortfall for each goalPersonal InformationAge 30 yrIncome USD 60,000/yrCurrent Asset USD 10,000

Tax Bracket 20% SimulationScheduleSaving1 USD 500/mo up to 15 yrSaving2 USD 750/mo from 16 to 35 yr

Goal 1 College USD 200,000 in 15 yr EngineGoal 2 Retirement USD 1,000,000 in 35 yrAsset Allocation Commission FeeUS Large Cap 75% 2.0%US Long-term 25% 1.5% Year Mean Shortfall

Goal 1 Colleage 2020 309,154 4.5%Goal 2 Retirement 2035 1,479,664 8.7%

Mean and Probability of Shortfall for each allocation Goal 1: College Goal 2: Retirement

Alternative Asset Allocations mean shortfall mean shortfallAggressive 5 Small Cap 50 Mid Cap 50 Aggressive 5 497,896 3.3% 2,383,013 4.8%Aggressive 4 Mid Cap 50 Large Cap 50 Aggressive 4 452,633 2.8% 2,166,376 4.0%

Aggressive 3 Mid Cap 25 Large Cap 75 Optimization Aggressive 3 411,485 2.3% 1,969,432 4.5%Aggressive 2 Large Cap 95 Long-term 5 Aggressive 2 374,077 2.9% 1,790,393 5.6%Aggressive 1 Large Cap 85 Long-term 15 Aggressive 1 340,070 3.6% 1,627,630 7.0%Current Large Cap 75 Long-term 25 Current 309,154 4.5% 1,479,664 8.7%

Conservative 1 Large Cap 65 Long-term 35 Engine Conservative 1 278,239 5.4% 1,331,697 10.4%Conservative 2 Large Cap 45 Long-term 55 Conservative 2 250,415 6.5% 1,198,528 12.5%Conservative 3 Large Cap 20 Long-term 80 Conservative 3 225,374 7.8% 1,078,675 15.0%Conservative 4 Long-term 80 Short-term 20 Conservative 3 202,836 9.3% 970,807 18.0%Conservative 5 Long-term 50 Short-term 50 Conservative 4 182,553 11.2% 873,727 21.6%

USD TB 10yr Yield Forecasting

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4.1. FORTUNEMANAGERTM 25

The input to the forecasting engine consists of historical data for each benchmark index that the investorchooses to represent each asset type in any given asset class (see Figure 4.1, upper left). The specifiedasset classes must cover all of the investor’s asset allocation choices, and the benchmark index mustcontain more than ten years of historical data.

The engine provides up to 50 years of monthly forecasts consisting of the mean and standard deviationsof asset prices or returns. These forecasts are used as inputs to the simulation engine for constructingscenarios of the future. Investors can refer to the forecasts when they make asset allocation decisions.

2. Simulation Engine

The simulation engine generates scenarios of asset price movement and calculates the distribution ofportfolio values at each of the investor’s horizons. The distribution is summarized by two parameters:the forecasted mean value (the return measure) and the probability of shortfall from a predeterminedgoal (the risk measure). The probability of shortfall will be explained in detail in a later part of thissection. The simulation interval must be fixed to a monthly frequency to handle a monthly savingsschedule and near-future financial goals (e.g., down-payment for a new house in six months) as wellas long-horizon goals.

The investor needs to provide personal information (e.g., age, amount of current assets, and tax bracket),a savings schedule, a multiple goals schedule, and asset allocation decisions. The tax, commission feefor investment, and inflation are adjusted before and during simulation.1

The simulation methodology of LongRun is used for this multi-period simulation engine. For a moredetailed discussion, refer to Chapter 5 of theLongRun Technical Document[4].

3. Optimization Engine

Because we provide simulation-based risk measures, it is impractical for us to construct a globaloptimization engine to determine the investor’s best asset allocation strategy — the engine wouldrequire excessively long computational time. Instead, we developed a simple and practical optimizationalgorithm, the so-called pseudo-optimization engine, that is restricted to solving a local optimizationproblem and involves computing incremental improvements in the investor’s current asset allocations.

Based on the investor’s current asset allocations, age, and risk tolerance level, the algorithm constructsboth more aggressive and more conservative asset allocations. The construction of good alternativeasset allocations is a key element in obtaining a reasonable optimization solution that comes close tothe global optimization solution. Thus, FortuneManager’s optimization engine provides an entire risk-return profile of asset allocation alternatives. The investor can then choose the optimal asset allocationscheme based on her risk-return preference (the so-called utility function).

1While the commission fee is computed by applying a constant rate to the total amount of assets in each asset class, the futureinflation rate must be considered as another risk factor in the simulation.



Before concluding the discussion of FortuneManager, we define its risk measure, the probability of shortfall.The probability of shortfall shows the probability with which the value of an investment at a predeterminedhorizon can drop below the value that an investor predetermines as the goal.

We can calculate the probability of shortfall as

probability of shortfall= Prob[P&Lt,k < z] =∫ z

−∞f (xt,k)dx, (4.1)

where P&Lt,k = xt,k denotes thek-month P&L from timet , andf (·) is the P&L density function. Parameterz is the goal value of the investment.

The accuracy of the probability of shortfall depends on the accuracy of the P&L density function. As wedescribed in the sections on forecasting and simulation engines, we construct the P&L density function foreach horizon by Monte Carlo simulation with ClearHorizon forecasts of the expected returns and volatilitiesof the risk factors.

4.2 Pension Risk Management

ClearHorizon can be applied to any situation that requires the measurement of risk arising from long-termfluctuations in market prices. One such situation is that of defined benefit pension plans, where the mainsources of risk are the long-term fluctuations in prices driving the value of the plan’s assets, and the discountand inflation rates that determine the value of the liabilities.2

The most relevant risk for defined benefit plans is funding risk; that is, the risk that the plan’s assets will beinsufficient to fund the liabilities at a long horizon. One might be tempted to quantify the funding risk of apension plan by projecting the value of the assets calculated from a random walk model; but given the meanreverting nature of some assets (see Chapter 2), this approach is likely to overestimate the potential gains andlosses from the changes in asset prices. Under these circumstances, ClearHorizon’s optimal mixture modelcan provide better estimates of the funding risk faced by the plan.

As an example, consider a defined benefit plan with a current funded ratio (the ratio of assets to liabilities)of 100%. Suppose that the plan’s assets are allocated in the following way: 40% Lehmann Aggregate BondIndex, 30% S&P 500, 15% Russell 2000, 10% MSCI EAFE, and 5% cash. Let us also make the simplifyingassumption that the annual benefits paid to retired employees and the annual contributions are equal to 6%

2This section was written by Jorge Mina of RiskMetrics. He collaborated with J.P. Morgan Investment Management on thedevelopment of PensionMetricsTM .

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4.2. PENSION RISK MANAGEMENT 27

of the value of the liabilities. We set the initial discount rate to 7%, and, to simplify the exposition, we setthe normal cost of the plan at 8% of the value of the liabilities each year.3

Figure 4.2 compares 25 funded ratio paths generated by using a random walk and ClearHorizon’s optimalmixture model. The paths generated from the random walk model show unreasonably high funded ratios,which arise from the fast increase in variance, while the paths generated from the mixture model seem quitereasonable given that the data used to estimate the model was taken from a bull market period (monthlyreturns for the last ten years).

Figure 4.3 shows the distribution of the funded ratio at the end of ten years. The distribution is obviouslybounded at zero and also skewed mainly because of the large expected returns from the assets. The medianof the distribution corresponds to a funded ratio of 150%. We can also observe that with a probability of95%, the pension plan will not be underfunded after ten years. There is also a 1% probability that the fundedratio will end below 90% after ten years.

By repeating this exercise with different asset allocations and looking at the funding risk exposure of eachallocation, a pension plan can improve its investment decisions.

The example in this section shows an application of the optimal mixture model to estimating the fundingrisk of a defined benefit pension plan. The long-term risks of the plan are assessed by using a model thatincorporates the mean reversion property of asset prices and, hence, keeps their variances from blowing up.As the example shows, ClearHorizon’s optimal mixture model presents a more realistic picture of long-termrisk than the alternative random walk model.

3The normal cost is the present value of the increase in accrued benefits from year to year. Normal cost is one of the factors thatincreases liabilities from year to year; the other factor is the effect of the compound interest on the liabilities.



Figure 4.2:Funded Ratio over the Next Ten Years

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4.2. PENSION RISK MANAGEMENT 29

Figure 4.3:Distribution of the Funded Ratio Ten Years from Now

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RiskMetrics

Bibliography

[1] Cochrane, John H. (1991) A Critique of the Application of Unit Root Tests,Journal of EconomicDynamics and Control, Vol. 15, pp. 275–284.

[2] Glen, Jack D. (1992) Real Exchange Rates in the Short, Medium, and Long Run,Journal of InternationalEconomics, Vol. 33, pp. 147–166.

[3] Hamilton, James D. (1994)Time Series Analysis, Princeton, New Jersey: Princeton University Press.

[4] Kim, Jongwoo, Allan M. Malz, Jorge Mina (1999)LongRun Technical Document, RiskMetrics Group,New York.

[5] Metcalf, Gilbert E., Kevin A. Hassett (1995) Investment under Alternative Return Assumptions Com-paring Random Walks and Mean Reversion,Journal of Economic Dynamics and Control, Vol. 19,pp. 1471–1488.

[6] Morgan Guaranty Trust Company (1996)RiskMetrics Technical Document, 4th ed., New York.

[7] Wu, Yangru, Hua Zhang (1996) Mean Reversion in Interest Rates: New Evidence from a Panel ofOECD Countries,Journal of Money, Credit, and Banking, Vol. 28, pp. 604–621.

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