The bankruptcy of the Orange County in California in December 1994, due to the huge loss of $ 1.6 billion of its investment, was a typical case that people searching high return without investment supervision. It was the largest loss recorded in the history of local government investments in the USA. The key operator in this issue was the County Treasurer Robert Citron. He managed an investment pool of $7.5 billion that mainly invest in fixed-income securities and structure notes. With the confidence that interest rates will still fall or at least keep low, he heavily invested in derivatives securities and leverage $7.5 billion to 20.5 billion by issuing Reverse Repurchase Agreements for gain higher returns on the capital pool. Definitely, the performance of this portfolio was outperformed until 1994, when the interest rates increased dramatically under the Fed policy for reducing inflation and against economic overheating in American market. The unpredictable policy contributes to huge loss, but, the major factor here is the lacking of complete risk management system and rigorous investment regulations. (Jorion, 1995)
1. Portfolio
1.1 Portfolio leverage
The Orange County has an investment pool with $7.5 billion that mainly invested in fixed-income securities and structured notes. From the balance sheet of Orange County at December 1th 1994, we can found that the total assets have a value of 20.5 billion, which is greater than the capital value mentioned above. The amount of funds exceed $7.5 billion are collected from by issuing Reverse Repurchase Agreements. This action leveraged the portfolio value by 2.7 times. The leverage factor can be calculated by total assets 20 billion divided by investor equity 7.5 billion.
RRA is a purchase of securities with an agreement to resell back to issuers at a higher price at a specific future date. (investorwords, n.d) Because risk is part of RRA, only parties with good creditworthiness can enter Reverse Repurchase transactions. It is a common measure used by government for borrowing money from public. Reverse repurchase transactions may result in higher fluctuation in net value of the portfolio. It directly contributes the increasing risk of huge loss.
Figure 1. Term Structure on December 1993
C:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\{V}K1[6U%N753CQSAYA%)K8.jpg
Source: Jorion (1996)
Investments with longer maturities are always coupled with higher yields, its true at least at the end of 1993. (Jorion, 1996) However, uncertainty over longer horizon is always higher than that in short time, and raising investment risks. In other words, an extra yield is always coming at the cost of a higher risk. For increasing current income, Citron increased duration to purchased 5-year yields rather than short-term investment yields for the Orange County portfolio. Hull (2008) pointed out that Portfolio value will more sensitive to interest rate change if the portfolio has higher duration at a given coupon. So the strategy that leverage duration exacerbate the risk of excess loss if the interest rate increasing.
1.2 Duration and Convexity
Jorion (1996) indicates that duration has ability to measure interest rate sensitivity of portfolio value in fixed-income market. In December 1994, the average duration for the Orange County portfolio was 2.74 years and the leverage factor of the portfolio was 2.7. As the maturity increase, the sensitivity of portfolio value to interest rate change increase. But, this is Macaulay duration and cannot be used for calculating returns for the portfolio. The modified duration can be obtained by using formula: D*=D/(1+y/m). Consistent with Jorion's orange county case in 1996, this report use 5% yield (y) and 1 frequency of payment (m) to calculate duration approximation. The yield change is 3.5% increase in 1994. Following the formula: B=-BD*, we can see that the computed loss $1.85 billion is greater than actual loss $1.64 billion. It seems that duration relationship cannot service well in Orange County case with huge changes in interest rates during short period.
Jorion (1996) argued that the relationship between prices and yields is simplified as a linear one, under the concept of duration. But, actually, the relationship is curved in real world. Also, Hull (2008) provides support idea that the duration relationship is only appropriate for situations in which changes in yield are small. The idea of convexity, which is consistent with duration relationship in the cases of small changes in yields, was introduced to explain the different behaviours of different portfolios with same duration for large yield changes. It advanced the equation B/B=-Dy to B/B=-Dy+1/2C(y)2, where D is the modified duration and C is the convexity of the portfolio. In other words, portfolios have same duration will result in different VaR based on different convexities. Definitely, the difference between computed VaR and actual loss are reasonable by using duration approximation; even computed loss is not consistent with actual loss of $1.64 billion. For getting precise result, the impacts of convexity should be considered.
Figure 2. Convexity
C:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\DCOS}GYYQV@$WRHV1EBTHJJ.jpg
Source: Hull, 2008
2. Value at Risk
2.1 Concept
The concept of Value at Risk (VaR) is adopted in this report to measure portfolio's market risk. The statement of VaR is making up by confidence level (X %) and time horizon (N days): no more than V dollars will be loosed in the next N days at X% confidence level, where V is the VaR to the portfolio. (Hull, 2008) From the Orange County case, we can found that its portfolio value was directly and highly influenced by the changes in interest rates. So, in this report it will use the change in 5-year monthly yields from 1953 to 1994 instead of actual volatility of portfolio returns for calculating VaR values.
2.2 Delta-normal approach
The variance-covariance approach (also you can call Delta-normal method) is first introduced to measure monthly VaR of portfolio in December 1994 at 95% confidence level. This approach is based assumptions that portfolio returns are linear correlated with returns of individual assets in the portfolio and the portfolio return are normally distributed. (Cuthbertson and Nitzsche, 2001) Such assumptions simplified VaR calculation. Definitely, in the cases of doing VaR for stocks, bonds and derivatives, the Delta-normal approach did very good works indeed. In the Orange County case, the VaR of the portfolio can be obtained by the formula: $V(-1.65σ) with the assumption that mean of monthly return is equal to zero. In the equation of VaR, $V is the portfolio value, -1.65 is the critical value at 5% cut off point in normal distribution and the Volatility σ is collected from the change in interest rates by using descriptive statistics (a data analysis tool in Excel). Here, the calculated VaR is positive value 0.36964 billion for monthly, even it is a loss.
Drawback of Delta-Normal approach
The great drawback of delta-normal approach is that returns are not unconditionally belongs to normal distribution. Even if returns of certain assets are normally distributed, the derivative securities are not belongs to normal distribution if derivative price does not correlate with underling asset in a linear relationship. (Cuthbertson and Nitzsche, 2001)
2.3 Historical Simulation approach
Pritsker (2001) argued that Historical simulation method is an improvement of VaR approaches with assumption of normal distribution. Simply, it uses historical data directly to forecast volatility in the future based on assumption of the 'tomorrow' is the repeat of the 'yesterday'. The starting point in this method is identifying the market variables influence portfolio value (Hull, 2008); here is the change in interest rates in the Orange County case. The database of monthly change in yield on 5-year monthly yields from 1953 to 1994 is created in Excel. The list of change in portfolio value which equal to the arithmetic product of monthly change in yield, value of portfolio and effective duration. In this report, it is 95% sure that monthly loss will not huger than 0.32967 billion based on the distribution of change in portfolio value. The VaR obtained by using historical-simulation method is not consistent with VaR computed based on delta-normal approach. VaR of portfolio under delta-normal approach is higher in absolute value.
Limitation of Historical Simulation method
The principle disadvantage of this approach is it weighted past data equally. It means that this approach assumes that historical changes in yields are independent and identically distributed over time, subconsciously. Bollerslev (1986) questioned the truth of assumption and, indicates that there are serial relationships among changes in yields (or returns) in a given period and data are not identically distributed in different horizons. Based on HS approach, the suddenly change in variables will not reflected in VaR immediately, but in a longer horizon. If the changes of variables disappeared quickly, such changes may even have no chance to reflect in VaR. In the Orange County case, sudden and huge interest rate changes destroyed the usefulness of historical information.
Table 1. Portfolio VaR
Delta-normal approach
Historical-Simulation method
Monthly
-0.36964
-0.32967
Annual
-1.28048
-1.14201
2.4VaR conversion based on Root-T Rule
The -RULE is a tool used to forecast the volatility over a long horizon by converting short term volatility based on formula: σT= Tσ. (Cuthbertson and Nitzsche, 2001) In this section, the -RULE is adopted to convert monthly volatilities calculated by using two approaches mentioned above to annual volatilities. This process directly converts monthly VaRs to annual ones. According to Root-T rule, the annual VaR is 1.28048 billion for delta-normal approach and 1.14201 for historical-simulation method. Overall, neither calculated VaR is consistent with actual loss in 1994 of the Orange County.
2.5 Assumptions and actuality
The foundation of -RULE is that the variance of returns is a constant figure over time. (Cuthbertson and Nitzsche, 2001) It is unrealistic issue in real world. Independence of returns is another important assumption for Root-T rule. While, as mentioned above, historical data may help to predict future in many cases. It must be some autocorrelations between returns, even if these relationships are small over short horizon. A longer horizon is accompanied by a huger bias in volatility forecasting. Obviously, rule is only appropriate for data conversations in situations that the forecast horizons are reasonable short. One year is a very long horizon rather than a reasonable short one. In addition, a 'regime change' like interest rate change will cause invalidation of historical data. Comprehensibly, the computed VaRs are derivate from the actual loss of the portfolio, and annual VaR value that converted from delta-normal approach is outperformed than the one using historical data.
3. Exponentially Weighted Moving Average (EWMA)
3.1 EWMA volatility forecast
EWMA model is a good approach to improve performance of Historical Simulation method by allocating greater weights to recent information for volatility forecasting in vary horizons. The weights of data decreased exponentially as the distance between the any given time and older dates becomes to lager. (Cuthbertson and Nitzsche, 2001; Hull, 2008) The key issue in using EWMA formula is to find optimal λ (which is selected to minimize the RMSEs) and the appropriate number days 'n'. The optimal λ for monthly volatility forecasting is 0.97 in RiskMetrics. (Cuthbertson and Nitzsche, 2001) In this report, it will compare actual volatility of monthly changes in 5-year yields for 6 months before December 1994 and volatilities obtained by using EWMA model. In EWMA forecast, the monthly change in yield is used as actual volatility, EWMA volatility is obtained by using the formula of σ2t+1|t=λσ2t|t-1+(1-λ)R2t. However, EWMA forecast provides no useful information with 6 months volatilities because the sample is too small that any conclusion may result in bias. The only thing we can confirm is the differences between volatilities determined the gap between VaRs.
Figure 3. Volatilities based on EWMA and actual volatilities
Source: Excel
3.2 Back-testing
Test the accuracy of EWMA is called 'back-testing'. In this section, this report will compare monthly forecast of volatility with actual volatilities from Sep 1986 to Dec 1994. It will check whether or not forecasted volatility is consistent with actual volatility at 95% confidence level with 100 monthly observations. Consistent with above, the monthly change in yield is used as actual volatility and forecasted volatilities are got from EWMA model. May be not very clear from the graph below, there are 4 actual volatilities outside the zero of EWMA forecast. It means that the EWMA volatility forecast reflect recent information immediately and is applicable in the Orange County case at 95% confidence level.
Figure 4. EWMA Back-testing
Source: Excel
3.3VaR based on EWMA forecast
Based on the equation: VaR=V*1.65*σ, the volatility is the key determinant for VaR estimation if the value of portfolio is known. If the volatility is not outside the EWMA forecast, the VaR value should also consistent with EWMA forecast at 95% confidence level. It may provide some evidence that the huge loss is predictable and reasonable under EWMA forecast. If the Orange County can recognize the huge risk of excess loss quickly, they can avoid such deficit or at least reduce the amount of loss. This viewpoint is similar with Jorion's idea in 1995 that the case of huge loss in the Orange County is a typical example of non-supervision.
4. Decision on liquidation or not
According to the interest rate changing trend in the USA, we can found that interest rate started to decrease in a few months later after December 1994. The portfolio value was getting up from the ground, even it cannot back to the formal level in short term. If the Orange County can get the information that interest rate will fall-back after the rise, the portfolio should not be liquidated at the point with bottom value. With an assumption of all things still same, the portfolio should be hold until December 1995, for short-term strategy. For long term, the portfolio can recovered to its formal level or even higher level (we can see than the interest rates in new millennium are lower on average than 1990s). However, no one can know what will happen in the future. Based on the historical data in 1994 of downward trend on portfolio value (or increasing loss), it is a possibility that situation in 1995 will worse. As we known, a huge amount of funds are collected from by issuing RRA and the remaining $7.5 billion is belongs to schools, other public insinuations and the county itself. (Jorion, 1996) If the loss is keeping increase, those participants may face the problem of fund shortage for their daily operation. People may prefer liquidate portfolio with known loss rather than unpredictable futures. In short, it is really difficult to make a choice between liquidation and hold.
Figure 5. Loss of Orange County if it did not liquidated
C:\DOCUME~1\ADMINI~1\LOCALS~1\Temp\RTCWUF~0C2B116XMD}`7(_4.jpg
Source: Jorion (1996)
Figure 6. Interest rates in the USA from 1954 to 2010
FEDFUNDS_Max_630_378.png
Source: Federal Reserve Bank of St. Louis: http://research.stlouisfed.org/
5. Hedging interest rate risk
5.1 Interest rate risk
When the durations between assets and liabilities are different, it will cause higher volatility in equity value. (Belongia and Santoni, 1984) At a given change in yield, assets with longer duration suffered huger than liabilities. And value of assets is sensitive to interest rate changes, it directly changes equity values. In the case of the Orange County case, interest rate determined the portfolio value. The portfolio will do well with declining yield or at least keep low as ones in December 1993. The little bitter increase in interest rates will make the leveraged portfolio value decrease dramatically. People can reduce the risk of investment by using interest rate futures and derivatives.
5.2 Hedging with futures
Futures are rights but not obligations that sell or buy securities at a certain price at a specific future date. (Hull, 2008) Here, it will long underlying securities in portfolio and short future contracts to against risk from interest rate fluctuation. When the interest rate increase, the value of portfolio decrease and the Orange County suffered from loss; while, Orange County can benefit from shorting futures that the securities can still sale at agreed prices in contract. The loss of portfolio value from interest rate increase can offset by the gain from the close out on futures. When the yields decrease, we do not execute short futures and the portfolio can benefit from value appreciation. In conclusion, futures against the portfolio from loss, but still allow it benefit from gains.
5.3 Hedging with Swap
Interest rate Swap is an exchange of interest payments on specific investments, generally the payments are fixed in amount. It can protect portfolio value against yield increase, but also limited the benefit from declining rates. (Hull, 2008) In the case of Orange County, people can against portfolio value from the increasing rates by long an interest rate swap contract.
5.4 Hedging with Interest Rate Caps
The Cap is a portfolio of European interest rate call options to protect the securities values from rising interest rates above Cap rate and still allow securities benefit from future decrease in yields.(Hull, 2008) In 1994, the interest rate went up by 3.5% and the portfolio loosens about $1.64 billion. On average, $0.47 billion of value will decrease with 1% increase in interest rates. The Orange County should employ Cap to offset the loss and keeping benefit from interest rate declining.
6. Conclusion
The case of huge loss in the Orange County is a typical example that seeking excess return without continuing monitoring on the performance. The portfolio value was leveraged by 2.7 times and directly changed the duration of the portfolio. The increased duration result in higher sensitivity of portfolio value to the interest rates changes. The concept of VaR was introduced in this report. Delta-normal approach, historical simulation method, Root-T rule and EWMA forecast were adopted to measure the VaR. It contains the advantages and disadvantages of each approach. The EWMA model is the best approach in all and well applied in the Orange County case. The information from EWMA forecast indicates that the huge loss is avoidable if the County pays great attention on the portfolio investment. if the Orange County can got information that interest rate would decreased in the near future after December 1994, the portfolio should not be liquidated at the point with bottom value. But, the future is unpredictable and a continuing loss may result in a shortage of daily operation funds for the portfolio participants. With the awareness of unexpected on interest rate changes, the Orange County can hedge the interest rate risks by using interest rate futures or interest rate derivatives in December 1993.
