The instance of bankruptcy of Orange County in 1994 stress the importance of utilizing continuance and Value at hazard ( VAR ) to measure portfolio hazard and avoid hereafter bankruptcy. Duration and VAR analysis provide deeper understanding about the underlying hazard of the Orange County Investment Pool which was to a great extent leveraged and interest-pledged through contrary redemption understandings and other derived functions in the pool. Some VAR appraisal, including historical simulation method, delta-normal method and Monte Carlo simulation will be used to cipher worst possible loss. The EWMA will be used to supply more accurate appraisal of the volatility to better the truth of VAR appraisal.

## Background:

On Dec 6, 1994, Orange County declared bankruptcy after enduring losingss of around $ 1.6billion from a ‘wrong manner stake on involvement rates ‘ 7.5 billion investing pool. This pool was intended to derive some returns from the puting the money which is raised from revenue enhancements and other authorities incomes. It was implemented a stake that the involvement would worsen or remain low by Citron ( the portfolio director ) . Because of the steadily worsening involvement rates from 1989 to 1992, the portfolio performed highly good before 1994 and earned 1000000s of above mean net income. However, in 1994, the authorities all of a sudden declared policies which included raise the involvement rates from 3.45 % to 7.14 % to forestall high rising prices and overheating economic system. This addition in involvement rate caused the portfolio suffer 1.6 billion loss and farther lead the bankruptcy of Orange County.

## Section 1: The heavy leveraged and interest-pledged portfolio

In order to prolong above norm returns, several investing tools are used by Citron to leverage the $ 7.5 billion financess into $ 20.5billion investing. In item, change by reversal redemption understandings allow Citron to utilize the securities which had already purchased as collateral on farther adoption and so reinvested the hard currency into new securities ( Jameson, 2001 ) . Besides the to a great extent leveraged hazard, the portfolio besides encounters important hazard from the unexpected involvement motion. First, these repurchase understandings ‘ values significantly depend on the alteration in involvement rate. In item, its value lessening as the involvement rate addition and increase as the involvement rate lessening ( P= ) . Second, $ 2.8 billion of derived functions, including reverse floating-rate notes, double index notes, floating-rate notes, index-amortizing notes and collateralized mortgage duties, are used to increase the portfolio ‘ stake on the term construction of the involvement rate ( Jorion, 2009 ) . Third, average term adulthoods which had higher outputs ( 5.2 % ) than the short term investings ( 3 % ) were used to increase the return of the portfolio ( Jorion, 2009 ) . However, by utilizing longer term adulthoods, the portfolio ‘s sensitiveness to involvement alteration will significantly increase. Clearly, by making these, the portfolio ‘s value will be significantly impacted by the motion of the involvement.

## Section 2: Duration of the portfolio and its application

Duration of the portfolio

Hull ( 2009 ) defines the continuance as ‘a step of how long, on norm, the holder of the instrument has to wait before having hard currency payments ‘ . It measures sensitiveness of monetary value alterations with alterations in involvement rates. Duration can be calculated by burdening norm ( the weight is the proportion of portfolio ‘s entire present value of hard currency flow received at clip T ) of the times. In this instance, the portfolio was to a great extent bet on the involvement, hence, continuance might be a good step for the portfolio.

In the $ 7.5 billion portfolio, average term adulthoods ( 5 old ages ) , instead than short term adulthoods ( 1-3 old ages ) , were used to increase the return. By making this, the continuance of the portfolio important increased. In other words, the portfolio exposed higher hazard of involvement rate motions. In December 1994, the mean continuance of the securities in the portfolio was 2.74 old ages. It means 1 % alteration in involvement would do 2.74 % alteration in portfolio ‘s monetary values.

Furthermore, Citron leveraged $ 7.5 billion equity into a $ 20.5 billion portfolio. This means that a 2.73 purchase ratio ( 20.5/7.5 ) . In other words, for every dollar of the pool invested, the pool borrowed excess $ 1.73. For a leveraged portfolio, the effectual portfolio continuance = ordinary continuance * purchase ratio. Therefore, the effectual portfolio continuance of the portfolio is 7.4 ( 2.74*2.7 ) .

Appraisal by utilizing continuance

The response of portfolio monetary values to alter in involvement rate:

In 1994, the involvement rates went up by about 3.5 ( and the 5 old ages bond output was 5 % , hence, the loss of the portfolio equals 1.85 ( 7.5*7.4*3.5 % /1.05 ) which is somewhat larger than the existent loss of 1.64 billion. This somewhat difference between the loss estimated by continuance and the existent loss might be caused by that the continuance applies to merely little alterations in involvement rate. As a first order estimate, continuance can non capture the information that two bonds with same continuance can hold different alteration in monetary value for big alteration in involvement rate ( different convexness ) . So, convexness ( 2nd order estimate ) which can capture this information should be added into the gauging theoretical account.

Through adding this ( convexness factor ) , the estimated loss will somewhat less than earlier, and will more near to the existent loss ( 1.64 billion ) . Therefore, continuance seems to hold the ability to accurate step the portfolio ‘s sensitiveness to involvement rate alteration.

## Section 3: Value at hazard ( VAR )

Value at hazard ( VAR )

In order to gauge the underlying hazard of the portfolio, VAR which measures the worst expected loss over a given skyline under normal market conditions at a given assurance degree could be used ( Jorion, 2001 ) . Because the portfolio was to a great extent bet on the involvement rate, its return and hazard are significantly depending on the alteration of involvement rate. In other words, the alteration of involvement output multiplies the modified continuance and portfolio value could be used as an estimate of the alteration of portfolio ‘s value. Therefore, the alteration of involvement output could be used in the 3 simulation methods as the lone factor that contribute the alteration of portfolio ‘ value.

Non-parametric attack ( no demand to place variance-covariance matrix )

Historical simulation attack

The historical simulation histories for non-linearity, income payments, and even clip decay effects through utilizing marking-to-market the whole portfolio over a big figure of realisations of implicit in random variables. VAR is calculated from the percentiles of the full distribution of final payments ( Jorion, 2001 ) . By utilizing existent monetary value, the method captures Grecian hazard ( gamma, vega hazard etc. ) and corrections of securities ( already exist in the existent historical informations ) in the portfolio, and it does non trust on some specific premise, such as the implicit in stochastic construction of the market ( the pre-requests of gauging volatility and mean ) . Furthermore, it can account for fat dress suits distribution besides normal distributions ( Jorion, 2001 ) .

( Figure 1 )

The root-T attack will be used to reassign the monthly VAR to annually VAR in all the 3 attacks. Its success significantly relies on the some specific premises, including the monthly output alterations of the portfolio are identically and independently distributed ( iid distribution ) and the return has a changeless discrepancy ( Cuthbertson and Nitzsche, 2001 ) . However, in the existent universe, stock returns ever has clip changing discrepancy and there are some autocorrelation factors exist ( therefore, non independent ) . Therefore, as the T addition, the mistake of the transmutation will significantly increase.

The VAR will be calculated through screening the monthly output alteration and picking the worst day-to-day output alteration at 5 % percentile ( see inside informations in Cadmium ) . However, in this instance, the addition in output will do lessening in portfolio return, hence, the worst day-to-day output alteration should be picked at the right manus side of the histogram ( see figure 1 ) . The VAR equals 1.24 billion yearly ( 0.36 billion monthly ) which is less than the existent loss ( 1.64 billion ) . This inaccuracy might be caused by the jobs exist in historical simulation method. First, the success of the method significantly relies on the premise that the past monetary value can stand for the future monetary value information. However, the premise is non realistic to some extent because of the being of market efficient. Second, simple historical simulation method may lose the information of temporarily elevated volatility, such as structural interruptions and utmost value ( Butler and Schachter, 1996 ) . In this instance, the historical simulation method can non capture the utmost value ( 1.64 billion loss ) which is caused by 6 suddenly lessenings of involvement rate.

Parametric attack ( need to necessitate to place variance-covariance matrix )

Delta normal attack

The delta normal method is peculiarly simple attack to implement. It takes account simple variance-covariance matrix and so calculate the entire discrepancy of the portfolio ( volatility ) . Then, The VAR can be calculated through the expression: VAR = MD*Portfolio Value*=7.4*7.5*0.4 % *1.65/ ( 1.005 ) =0.35 billion ( monthly ) = 1.21 billion ( yearly ) .

Delta normal method is somewhat less accurate than the historical in the instance. This might caused by that the alteration in output does is a fat tail distribution ( Kurtosis =6.9, Skewness = -0.44 ) instead than a normal distribution ( Kurtosis =6.9, Skewness = -0.44 ) . Therefore, the theoretical account based on the normal distribution will undervalue the proportion of outliers and therefore the value at hazard ( Jorion, 2001 ) . In add-on, the portfolio contains a batch of derived functions instrument. This will do the method inadequately measures the hazard of nonlinearity.

Monte Carlo simulation ( MCS ) ( the theoretical most powerful method )

Unlike historical simulation, through specifying and exciting a stochastic procedure for fiscal variables, Monte Carlo simulation covers a broad scope of fiscal variables ( volatility and stochastic variables ) and to the full gaining controls correlativities of securities ( unlike HS, need to specify the matrix ) in the portfolio ( Jorion, 2001 ) . It does non merely history for a broad scope of hazards, such as nonlinear monetary value, volatility and theoretical account hazards ( the same as historical simulation ) , but besides integrated clip fluctuation of volatility ( structural interruptions and utmost values ) , and fat dress suits. Furthermore, it can capture the construction alterations in the portfolio as the clip base on balls ( Jorion, 2001 ) . In theoretical manner, MCS should be the best method in gauging VAR.

The MCS VAR is about 0.295 monthly, through utilizing the root-T regulation, the yearly VAR is about 1 billion ( see item computation in Cadmium ) . There are besides some restrictions of Monte Carlo simulation cause the estimated mistake between the estimated loss and existent loss. Its success significantly relies on the specific pricing theoretical account for underlying assets and stochastic procedures for the implicit in hazard factors. In this instance, the pricing expression is Brownian attack without impetus may non accurately capture the existent value alteration of the portfolio. This might be one possible ground that the estimated loss is non equal to the existent loss. Furthermore, the jobs may be in the sample used to derivate the implicit in hazard factors. For illustration, MCS will bring forth less accurate estimations so delta normal method when the hazard factors are jointly normal and all final payments are liner ( Cuthbertson and Nitzsche, 2001 ) .

Why MCS ( theoretical best method ) shows the worst appraisal in this instance

MCS seems to hold the least accurate appraisal ( more nearer to the existent loss ) in this instance. This might be caused by the portfolio used in MCS are treated as one plus which is merely impacted by the involvement output. Three factors, including the correlativity between all the securities in the portfolio, the implicit in hazard factors of these securities and the different monetary value expression should be used for each security, are ignored in the powerful attack ( Tardivo, 2002 ) . On the other manus, compared with the MCS, historical simulation does non necessitate to specify the correlativity matrix, because the information has already captured the information. In add-on, underlying hazard factors besides contains in the existent information. Thus, in the instance with limited information, historical simulation provides more accurate appraisal.

## Section 4: EWMA

In realistic universe, the discrepancy of the clip series is changing overtime. Therefore, the simple unconditioned discrepancy ( simple variance/standard divergence ) may non supply indifferent appraisal of the volatility. This will further ensue in inaccuracy appraisal of the VAR. in the instance, In the instance, the simple discrepancy ( volatility ) are ciphering through delegating the same weight on all observations during Jan 1953 and Dec 1994. This may take to colored prognosiss of VAR because the Fed dramatically increased/decreased the involvement rate during this clip period. In order to better the truth of gauging VAR, Exponentially weighted traveling norm ( EWMA ) will be used to supply more accurate appraisal to the volatility at a specific clip ( conditional criterion divergence ) ( Cuthbertson and Nitzsche, 2001 ) .

EWMA method allows more recent observations to hold stronger impact on the prognosis of volatility than the old observations. In practical manner, the recent informations are given more weights than the old information. By using this theoretical account, volatility in pattern will be more wedged by recent events and the impacts on volatility will worsen as clip base on balls ( smaller weights apple to the event ) ( Brooks, 2002 ) .

Through using the EWMA theoretical account, the monthly criterion divergence for the six months before December 1994 is 0.348 % . The following 6 months ‘ volatility could be forecasted through utilizing the expression: . In add-on, the existent monthly volatility could utilize the alteration in output as estimate. Harmonizing to RiskMetrics, the optimalshould be 0.97 ( Brock, 2002 ) .

## aˆˆ

Forecast volatility ( % )

Actual volatility ( % )

Scope of the possible volatility at 5 % assurance degree

## aˆˆ

## Volatility at June 1994

0.35

## aˆˆ

Left side ( -1.65 )

Right side ( 1.65 )

## Forecasted volatility

## aˆˆ

## aˆˆ

## aˆˆ

## aˆˆ

## Jul-94

0.35

-0.26

-0.57

0.57

## Aug-94

0.34

0.08

-0.56

0.56

## Sep-94

0.35

0.47

-0.57

0.57

## Oct-94

0.34

0.20

-0.56

0.56

## Nov-94

0.34

0.31

-0.56

0.56

## Dec-94

0.34

0.04

-0.55

0.55

By and large, the EWMA attack does non to the full capture unnatural volatility alteration in 1994. In item, the existent volatility alteration more volatile than the prognosis one ( table 1 ) . The inaccuracy involve in gauging the volatility may ensue in that the deliberate VAR is significantly different from the existent possible loss of the portfolio ( table 2 ) . If the prognosis volatility is used to cipher VAR, director should aware that the deliberate VAR is merely an estimate and it can non capture all the volatility alteration information. For illustration, in this instance, the existent volatility in Sep-94 is significantly larger than the prognosis 1. This may do director to undervalue the hazard in the clip period and so keeping the portfolio unchanged as earlier. It is besides support by Mahoney ( 1996 ) who through empirical observation support that the EWMA volatility has inaccuracy jobs.

Table 1:

## aˆˆ

Forecast volatility ( % )

Actual volatility ( % )

Left side ( -1.65 )

Right side ( 1.65 )

## Volatility at June 1994

0.35

## aˆˆ

## aˆˆ

## Forecasted volatility

## aˆˆ

## aˆˆ

## aˆˆ

## aˆˆ

## Jul-94

0.35

-0.26

-0.57

0.57

## Aug-94

0.34

0.08

-0.56

0.56

## Sep-94

0.35

0.47

-0.57

0.57

## Oct-94

0.34

0.20

-0.56

0.56

## Nov-94

0.34

0.31

-0.56

0.56

## Dec-94

0.34

0.04

-0.55

0.55

On the other manus, VAR calculated based on EWMA volatility can still be used as a benchmark to measure the portfolio ‘s hazard. All of the existent volatility is in the boundary of the prognosis volatility ‘s 5 % tail cut off ( on both sides *1.65 ) . That is to state, although there are important differences between the prognosis and the existent volatility in this instance, portfolio director may still non undervalue the implicit in hazard at 5 % assurance degree ( normal distribution ) . In add-on, if better theoretical accounts are used, including GARCH, EGARCH, and GJR, the VAR can supply more precise appraisal of the worst possible loss.

Table 2:

## aˆˆ

## Forecasted VAR ( *-1.65 ) monthly

## Actual VAR monthly

## Forecasted VAR yearly

## Actual VAR yearly

## Jul-94

-0.302

0.227

-1.045

0.786

## Aug-94

-0.297

-0.070

-1.030

-0.242

## Sep-94

-0.301

-0.410

-1.044

-1.420

## Oct-94

-0.298

-0.174

-1.034

-0.604

## Nov-94

-0.298

-0.270

-1.031

-0.937

## Dec-94

-0.293

-0.035

-1.015

-0.121

## Section 5: Backtest EWMA theoretical account

In order to prove whether VAR can be used as s a benchmark to measure the portfolio ‘s hazard, the backtest should be used to prove whether EWMA can capture the existent alteration in involvement output at the 5 % left tail cut off degree ( normal distribution ) . Practically, if all of the existent alterations in involvement output are within the prognosis volatilities boundary ( the prognosis volatility multiply 1.65 at right manus side and -1.65 at the left manus side ) , the EWMA theoretical account can be considered as supplying accurate appraisal at 5 % assurance degree. Harmonizing to figure 2, there are 4 outliers ( Aug-89, Jan-92, Feb-94 and Mar-94 ) are outside the prognosis. This will do director to over/under gauge the underlying hazard of the portfolio.

Figure 2: prognosis volatilities boundary and existent alteration in involvement output

## Section 6: Whether the portfolio should be liquidated in December 1994

Miller and Ross ( 1997 ) recommend that the portfolio should non be liquidated until the adulthood of the structural notes. This is because after the Orange County ‘ bankruptcy, the involvement rate fell from 7.8 % to 5.25 % during Dec 1994 to Dec 1995. If it did non denote the bankruptcy, this lessening in involvement rate could assist the County to retrieve 7.4*7.5*2.55 % /1.05= 1.32 billion losingss.

However, the job is that ‘in Dec 1994, how the directors would cognize that there would be a lessening in involvement in 1995 ‘ . Jorion ( 1997 ) suggest that because it is impossible to foretell all of a sudden involvement rate lessening, keeping the assets in order to retrieve value in the following old ages is bad and hazardous. Given this alteration in output is a normal distribution, the chance of 2.55 % lessening in involvement can be calculated through P ( =P ( -6.223 ) . Harmonizing to the normal statics table, the chance of such big lessening in involvement is less than 1 % . Therefore, the rational directors would non anticipate all of a sudden big lessening in involvement rate. In order to minimise to farther loss, it is sensible to neutralize the portfolio on Dec 1994.

In add-on, as the portfolio is involvement pledged, some involvement hereafters, such as the T-bond hereafters, could be shorted to fudge the portfolio in Dec 1993. Long cap could besides a good pick to bring forth net income when involvement rate exceeds the work stoppage rate. This could partly counterbalance the monolithic loss.

## Decision

The orange county ‘s heavy leveraged and interest-pledged portfolio suffer monolithic loss in 1994 because of the suddenly addition of involvement rate. Through analyzing this instance survey, the Duration and VAR are of import measuring of hazard to avoid future bankruptcy. Compare the continuance estimated loss with the existent loss, Duration ( plus convexness ) of the portfolio seems to hold the ability to accurately mensurate the portfolio ‘s sensitiveness to the alteration in involvement rate. In add-on, all of the VARs calculated through three attacks, including historical simulation, delta normal, and MCS, are less than the existent loss. The theoretical best attacks ( MCS ) does non supply the most accurate appraisal because of ignorance of some of import factors, such as the correlativity between all the securities in the portfolio, the implicit in hazard factors of these securities and the different monetary value expression should be used for each security. The backtest of EWMA ( 4 outliers ) suggest that there are some hazard in utilizing VAR to mensurate the worst possible loss in the existent universe.