The case of bankruptcy of Orange County in 1994 emphasize the importance of using duration and Value at risk (VAR) to assess portfolio risk and avoid future bankruptcy. Duration and VAR analysis provide deeper understanding about the underlying risk of the Orange County Investment Pool which was heavily leveraged and interest-pledged through reverse repurchase agreements and other derivatives in the pool. Some VAR estimation, including historical simulation method, delta-normal method and Monte Carlo simulation will be used to calculate worst possible loss. The EWMA will be used to provide more accurate estimation of the volatility to improve the accuracy of VAR estimation.

Background:

On Dec 6, 1994, Orange County declared bankruptcy after suffering losses of around $1.6billion from a ‘wrong way bet on interest rates’ 7.5 billion investment pool. This pool was intended to gain some returns from the investing the money which is raised from taxes and other government incomes. It was implemented a bet that the interest would decline or stay low by Citron (the portfolio manager). Because of the steadily declining interest rates from 1989 to 1992, the portfolio performed extremely well before 1994 and earned millions of above average profit. However, in 1994, the government suddenly declared policies which included raise the interest rates from 3.45% to 7.14% to prevent high inflation and overheating economy. This increase in interest rate caused the portfolio suffer 1.6 billion loss and further lead the bankruptcy of Orange County.

Section 1: The heavy leveraged and interest-pledged portfolio

In order to sustain above average returns, several investment tools are used by Citron to leverage the $7.5 billion funds into $20.5billion investment. In detail, reverse repurchase agreements allow Citron to use the securities which had already purchased as collateral on further borrowing and then reinvested the cash into new securities (Jameson, 2001). Besides the heavily leveraged risk, the portfolio also encounters significant risk from the unexpected interest movement. Firstly, these repurchase agreements’ values significantly depend on the change in interest rate. In detail, its value decrease as the interest rate increase and increase as the interest rate decrease (P=). Secondly, $2.8 billion of derivatives, including inverse floating-rate notes, dual index notes, floating-rate notes, index-amortizing notes and collateralized mortgage obligations, are used to increase the portfolio’ bet on the term structure of the interest rate (Jorion, 2009). Thirdly, median term maturities which had higher yields (5.2%) than the short term investments (3%) were used to increase the return of the portfolio (Jorion, 2009). However, by using longer term maturities, the portfolio’s sensitivity to interest change will significantly increase. Clearly, by doing these, the portfolio’s value will be significantly impacted by the movement of the interest.

Section 2: Duration of the portfolio and its application

Duration of the portfolio Hull (2009) defines the duration as ‘a measure of how long, on average, the holder of the instrument has to wait before receiving cash payments’. It measures sensitivity of price changes with changes in interest rates. Duration can be calculated by weighting average (the weight is the proportion of portfolio’s total present value of cash flow received at time t) of the times. In this case, the portfolio was heavily bet on the interest, therefore, duration might be a good measure for the portfolio. In the $7.5 billion portfolio, median term maturities (5 years), rather than short term maturities (1-3 years), were used to increase the return. By doing this, the duration of the portfolio significant increased. In other words, the portfolio exposed higher risk of interest rate movements. In December 1994, the average duration of the securities in the portfolio was 2.74 years. It means 1% change in interest would cause 2.74% change in portfolio’s prices. Moreover, Citron leveraged $7.5 billion equity into a $20.5 billion portfolio. This means that a 2.73 leverage ratio (20.5/7.5). In other words, for every dollar of the pool invested, the pool borrowed extra $1.73. For a leveraged portfolio, the effective portfolio duration = ordinary duration * leverage ratio. Thus, the effective portfolio duration of the portfolio is 7.4 (2.74*2.7). Estimation by using duration The response of portfolio prices to change in interest rate: In 1994, the interest rates went up by about 3.5 ( and the 5 years bond yield was 5%, therefore, the loss of the portfolio equals 1.85 (7.5*7.4*3.5%/1.05) which is slightly larger than the actual loss of 1.64 billion. This slightly difference between the loss estimated by duration and the actual loss might be caused by that the duration applies to only small changes in interest rate. As a first order approximation, duration cannot capture the information that two bonds with same duration can have different change in price for large change in interest rate (different convexity). So, convexity (second order approximation) which can capture this information should be added into the estimating model. Through adding this (convexity factor), the estimated loss will slightly less than before, and will more close to the actual loss (1.64 billion). Thus, duration seems to have the ability to accurate measure the portfolio’s sensitivity to interest rate change.

Section 3: Value at risk (VAR)

Value at risk (VAR) In order to estimate the underlying risk of the portfolio, VAR which measures the worst expected loss over a given horizon under normal market conditions at a given confidence level could be used (Jorion, 2001). Because the portfolio was heavily bet on the interest rate, its return and risk are significantly depending on the change of interest rate. In other words, the change of interest yield multiplies the modified duration and portfolio value could be used as an approximation of the change of portfolio’s value. Thus, the change of interest yield could be used in the 3 simulation methods as the only factor that contribute the change of portfolio’ value. Non-parametric approach (no need to identify variance-covariance matrix) Historical simulation approach The historical simulation accounts for non-linearity, income payments, and even time decay effects through using marking-to-market the whole portfolio over a large number of realizations of underlying random variables. VAR is calculated from the percentiles of the full distribution of payoffs (Jorion, 2001). By using actual price, the method captures Greek risk (gamma, vega risk etc.) and corrections of securities (already exist in the real historical data) in the portfolio, and it does not rely on some specific assumption, such as the underlying stochastic structure of the market (the pre-requests of estimating volatility and mean). Moreover, it can account for fat tails distribution besides normal distributions (Jorion, 2001). (Figure 1) The root-T approach will be used to transfer the monthly VAR to yearly VAR in all the 3 approaches. Its success significantly relies on the some specific assumptions, including the monthly yield changes of the portfolio are identically and independently distributed (iid distribution) and the return has a constant variance (Cuthbertson and Nitzsche, 2001). However, in the real world, stock returns always has time varying variance and there are some autocorrelation factors exist (thus, not independent). Therefore, as the T increase, the error of the transformation will significantly increase. The VAR will be calculated through sorting the monthly yield change and picking the worst daily yield change at 5% percentile (see details in CD). However, in this case, the increase in yield will cause decrease in portfolio return, therefore, the worst daily yield change should be picked at the right hand side of the histogram (see figure 1). The VAR equals 1.24 billion annually (0.36 billion monthly) which is less than the actual loss (1.64 billion). This inaccuracy might be caused by the problems exist in historical simulation method. Firstly, the success of the method significantly relies on the assumption that the past price can represent the future price information. However, the assumption is not realistic to some extent because of the existence of market efficient. Secondly, simple historical simulation method may miss the information of temporarily elevated volatility, such as structural breaks and extreme value (Butler and Schachter, 1996). In this case, the historical simulation method cannot capture the extreme value (1.64 billion loss) which is caused by 6 suddenly decreases of interest rate. Parametric approach (need to need to identify variance-covariance matrix) Delta normal approach The delta normal method is particularly simple approach to implement. It takes account simple variance-covariance matrix and then forecast the total variance of the portfolio (volatility). Then, The VAR can be calculated through the formula: VAR = MD*Portfolio Value*=7.4*7.5*0.4%*1.65/(1.005)=0.35 billion (monthly) = 1.21 billion (annually). Delta normal method is slightly less accurate than the historical in the case. This might caused by that the change in yield does is a fat tail distribution (Kurtosis =6.9, Skewness = -0.44) rather than a normal distribution (Kurtosis =6.9, Skewness = -0.44). Thus, the model based on the normal distribution will underestimate the proportion of outliers and hence the value at risk (Jorion, 2001). In addition, the portfolio contains a lot of derivatives instrument. This will cause the method inadequately measures the risk of nonlinearity. Monte Carlo simulation (MCS) (the theoretical most powerful method) Unlike historical simulation, through specifying and stimulating a stochastic process for financial variables, Monte Carlo simulation covers a wide range of financial variables (volatility and stochastic variables) and fully captures correlations of securities (unlike HS, need to define the matrix) in the portfolio (Jorion, 2001). It does not only account for a wide range of risks, such as nonlinear price, volatility and model risks (the same as historical simulation), but also incorporate time variation of volatility (structural breaks and extreme values), and fat tails. Moreover, it can capture the structure changes in the portfolio as the time pass (Jorion, 2001). In theoretical way, MCS should be the best method in estimating VAR. The MCS VAR is about 0.295 monthly, through using the root-T rule, the annually VAR is about 1 billion (see detail calculation in CD). There are also some limitations of Monte Carlo simulation cause the estimated error between the estimated loss and actual loss. Its success significantly relies on the specific pricing model for underlying assets and stochastic processes for the underlying risk factors. In this case, the pricing formula is Brownian approach without drift may not accurately capture the actual value change of the portfolio. This might be one possible reason that the estimated loss is not equal to the actual loss. Moreover, the problems may exist in the sample used to derivate the underlying risk factors. For example, MCS will generate less accurate estimates then delta normal method when the risk factors are jointly normal and all payoffs are liner (Cuthbertson and Nitzsche, 2001). Why MCS (theoretical best method) shows the worst estimation in this case MCS seems to have the least accurate estimation (more closer to the actual loss) in this case. This might be caused by the portfolio used in MCS are treated as one asset which is only impacted by the interest yield. Three factors, including the correlation between all the securities in the portfolio, the underlying risk factors of these securities and the different price formula should be used for each security, are ignored in the powerful approach (Tardivo, 2002). On the other hand, compared with the MCS, historical simulation does not need to define the correlation matrix, because the data has already captured the information. In addition, underlying risk factors also contains in the actual data. Thus, in the case with limited information, historical simulation provides more accurate estimation.

Section 4: EWMA

In realistic world, the variance of the time series is varying overtime. Thus, the simple unconditional variance (simple variance/standard deviation) may not provide unbiased estimation of the volatility. This will further result in inaccuracy estimation of the VAR. in the case, In the case, the simple variance (volatility) are calculating through assigning the same weight on all observations during Jan 1953 and Dec 1994. This may lead to biased forecasts of VAR because the Fed dramatically increased/decreased the interest rate during this time period. In order to improve the accuracy of estimating VAR, Exponentially weighted moving average (EWMA) will be used to provide more accurate estimation to the volatility at a specific time (conditional standard deviation) (Cuthbertson and Nitzsche, 2001). EWMA method allows more recent observations to have stronger impact on the forecast of volatility than the old observations. In practical way, the recent data are given more weights than the old data. By applying this model, volatility in practice will be more impacted by recent events and the impacts on volatility will decline as time pass (smaller weights apple to the event) (Brooks, 2002). Through applying the EWMA model, the monthly standard deviation for the six months before December 1994 is 0.348%. The next 6 months’ volatility could be forecasted through using the formula:. In addition, the actual monthly volatility could use the change in yield as approximation. According to RiskMetrics, the optimalshould be 0.97 (Brock, 2002).

A£â‚¬â‚¬

Forecast volatility (%) Actual volatility (%) Range of the possible volatility at 5% confidence level

A£â‚¬â‚¬

Volatility at june 1994

0.35

A£â‚¬â‚¬

Left side (-1.65) Right side (1.65)

Forecasted volatility

A£â‚¬â‚¬

A£â‚¬â‚¬

A£â‚¬â‚¬

A£â‚¬â‚¬

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 Generally, the EWMA approach does not fully capture abnormal volatility change in 1994. In detail, the actual volatility change more volatile than the forecast one (table 1). The inaccuracy involve in estimating the volatility may result in that the calculated VAR is significantly different from the actual possible loss of the portfolio (table 2). If the forecast volatility is used to calculate VAR, manager should aware that the calculated VAR is only an approximation and it cannot capture all the volatility change information. For example, in this case, the actual volatility in Sep-94 is significantly larger than the forecast one. This may cause manager to underestimate the risk in the time period and then holding the portfolio unchanged as before. It is also support by Mahoney (1996) who empirically support that the EWMA volatility has inaccuracy problems. Table 1:

A£â‚¬â‚¬

Forecast volatility (%) Actual volatility (%) Left side (-1.65) Right side (1.65)

Volatility at June 1994

0.35

A£â‚¬â‚¬

A£â‚¬â‚¬

Forecasted volatility

A£â‚¬â‚¬

A£â‚¬â‚¬

A£â‚¬â‚¬

A£â‚¬â‚¬

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 hand, VAR calculated based on EWMA volatility can still be used as a benchmark to assess the portfolio’s risk. All of the actual volatility is in the boundary of the forecast volatility’s 5% tail cut off (on both sides *1.65). That is to say, although there are significant differences between the forecast and the actual volatility in this case, portfolio manager may still not underestimate the underlying risk at 5% confidence level (normal distribution). In addition, if better models are used, including GARCH, EGARCH, and GJR , the VAR can provide more precise estimation of the worst possible loss. Table 2:

A£â‚¬â‚¬

Forecasted VAR(*-1.65) monthly

Actual VAR monthly

Forecasted VAR annually

Actual VAR annually

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 model

In order to test whether VAR can be used as s a benchmark to assess the portfolio’s risk, the backtest should be used to test whether EWMA can capture the actual change in interest yield at the 5% left tail cut off level (normal distribution). Practically, if all of the actual changes in interest yield are within the forecast volatilities boundary (the forecast volatility multiply 1.65 at right hand side and -1.65 at the left hand side), the EWMA model can be considered as providing accurate estimation at 5% confidence level. According to figure 2, there are 4 outliers (Aug-89, Jan-92, Feb-94 and Mar-94) are outside the forecast. This will cause manager to over/under estimate the underlying risk of the portfolio. Figure 2: forecast volatilities boundary and actual change in interest yield

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

Miller and Ross (1997) recommend that the portfolio should not be liquidated until the maturity of the structural notes. This is because after the Orange County’ bankruptcy, the interest rate fell from 7.8% to 5.25% during Dec 1994 to Dec 1995. If it did not announce the bankruptcy, this decrease in interest rate could help the County to recover 7.4*7.5*2.55%/1.05= 1.32 billion losses. However, the problem is that ‘in Dec 1994, how the managers would know that there would be a decrease in interest in 1995’. Jorion (1997) suggest that because it is impossible to predict suddenly interest rate decrease, holding the assets in order to recover value in the next years is speculative and risky. Given this change in yield is a normal distribution, the probability of 2.55% decrease in interest can be calculated through P(=P(-6.223). According to the normal statics table, the probability of such large decrease in interest is less than 1%. Thus, the rational managers would not expect suddenly large decrease in interest rate. In order to minimize to further loss, it is reasonable to liquidate the portfolio on Dec 1994. In addition, as the portfolio is interest pledged, some interest futures, such as the T-bond futures, could be shorted to hedge the portfolio in Dec 1993. Long cap could also a good choice to generate profit when interest rate exceeds the strike rate. This could partially compensate the massive loss.

Conclusion

The orange county’s heavy leveraged and interest-pledged portfolio suffer massive loss in 1994 because of the suddenly increase of interest rate. Through examining this case study, the Duration and VAR are important measurement of risk to avoid future bankruptcy. Compare the duration estimated loss with the actual loss, Duration (plus convexity) of the portfolio seems to have the ability to accurately measure the portfolio’s sensitivity to the change in interest rate. In addition, all of the VARs calculated through three approaches, including historical simulation, delta normal, and MCS, are less than the actual loss. The theoretical best approaches (MCS) does not provide the most accurate estimation because of ignorance of some important factors, such as the correlation between all the securities in the portfolio, the underlying risk factors of these securities and the different price formula should be used for each security. The backtest of EWMA (4 outliers) suggest that there are some risk in using VAR to measure the worst possible loss in the real world.