Recorded as the largest municipal loss in U.S. history, Orange County suffered a loss of $1.6 billion in December 1994 and went to bankruptcy shortly thereafter. The County Treasurer, who was also the wrecker of this financial disaster, Robert Citron, managed to build a $20.5 billion portfolio by leveraging 7.5 billion of investor equity. With years of success, the investment strategy exposed and multiplied the investor equity to the risk of interest rates, which finally rose six times in 1994 from 3.45% to 7.14% and led to the largest municipal failure.
Several tools of financial risk management, such as duration and VaR (value at risk) can be employed to analyze the investment failure. As a characteristic of a bond, duration measures sensitivity of price changes with changes in interest rates. As another prevailing measure, Value at Risk (VaR) is defined as "a loss that will not be exceeded at some specified confidence level and specified time horizon" (Hull, 2007). According to Jorion (2006), "VaR measures the worst expected loss over a given horizon under normal market conditions at a given level of confidence.
The objective of this paper is to try to investigate the dilemma which the Treasurer faced by using appropriate econometric techniques in the field of financial risk management. The remaining paper is organized as follows: in section 2, I present a review of the relevant literature. Section 3 describes the data and analyses the case questions. Finally, I discuss the findings and summarize the conclusions in section 5.
2. Literature review
Since Orange County declared the bankruptcy on Dec 6th 1994, it has been treated as classic case in the field of financial risk management. Following the first review "County in Crisis" by Richard Irving (1995), Philippe Jorion and Robert Roper published the book "Big Bets Gone Bad: Derivatives andBankruptcy in Orange County" (1995). Analyzing the bankruptcy in terms of VaR, Philippe Jorion maked the internet case study: "Orange County Case: Using Value at Risk to Control Financial Risk".
Markowitz (1959) firstly introduced "portfolio theory", which formed the basis of modern risk management and set the origin of measure of value-at-risk (VaR). J.P. Morgan Bank (1994) formally designed VaR as a measure of market risk and freely provided VaR to institutions at Risk Metrics. Although the concept of VaR has now been incorporated in the Basel II Capital Accord (2003), many articles, such as Bredow (2002) and Yamai and Yoshiba (2002), believe that VaR underestimates the risk of securities with fat-tailed properties and a high potential for large losses as well as disregards the tail dependence of asset returns.
3. Data and case analysis
3.1 Data
To show the portfolio structure, the balance sheet (Table 1) of Orange County as of 01 December 1994 is given by the assignment description. Of all the assets, the weight of structured notes accounts for 38% and the weight of fixed-income securities is 57.7%. the remaining part of assets contains cash(3.2%) and collateralized mortgage obligations(1.1%). In the column of liabilities, it can be seen that 7.5 billion of investor equity is leveraged into 20.5 billion of investments.
Table 1. The balance sheet of Orange County as of 01 December 1994
Assets ($)
Liabilities ($)
Structured notes (38%)
Inverse floating-rate notes (26.1%)
Others (dual index notes (0.7%), floating-rate notes (2.9%), index-amortizing notes (8.3%))
Fixed-income securities (57.7%)
Cash (3.2%)
Collateralized mortgage obligations (1.1%)
5.4 billion
2.4 billion
11.9 billion
0.6 billion
0.2 billion
ï‚® Liabilities
Reverse repurchase agreements (63.2%)
ï‚® Investor equity (36.8%)
13 billion
7.5 billion
20.5 billion
20.5 billion
The second part of data is used to measure the volatility of the change in yields and compute the monthly portfolio VaR, containing 5-year yields on current US Treasury issues between 1953 and 1994. This data is obtained from excel file attached behind assignment description.
3.2Duration and effective duration (for question 1)
As mentioned above, duration, which normally means the Macaulay Duration, is a measure that summarizes approximate response of bond prices to change in yields. Different with Macaulay duration, effective duration is often used to describe the response of price of bond with embedded options to yield change and dependent on option pricing model. Duration can be calculated as:
Where:
B is the present value of all payments of from the bond,
i indexes the period of the cash flows,
c is the amount of cash flow in every payment period,
is the discount factor,
D is the Macaulay Duration for the bond.
According to the formula, it can be intuitively found that greater yield (y) is associated with lower present value of bond and the direction of sensitivity between those should be negative. Furthermore, it can be proved that greater time to maturity is always followed by greater change in price to change in interest rates, namely greater duration. Besides the maturity, leveraging the portfolio can also increase the duration.
By reverse repurchase agreements, the investor equity (7.5billion) is leveraged into the whole portfolio (20.5 billion). The leverage rate is exactly shown as:
Leverage rate=20.5/7.52.7
it means the effect of any price change to investor equity will be enlarged 2.7 times. Considering the average duration of the securities in the portfolio is 2.74 years, the effective duration of the fund should be calculated as:
Effective duration =2.742.7=7.4 (years)
3.3 Convexity: difference between theoretic and actual loss (for question 2)
For a small change in yields y / d y, the corresponding change in bond price can be approximately expressed as:
=
Where:
is the dependent change in bond price,
B is the initial value of bond price,
is the modified duration of the bond,
is the change in yield.
On the other hand, for large changes in yield duration is less accurate in expressing the dependent changes in bond price. For example, two bonds with same duration can have different change in value of their portfolio for large changes in yields. It is basically because that in the first order approximation, duration cannot provide the information of bond convexity. Convexity for a bond, as a concept of second order, is a measure of the sensitivity of the duration of a bond to changes in interest rates. It is the weighted average of the 'times squared' when payments are made. Convexity for a bond is
Considering the convexity, the dependent change in bond price to yield change can be expressed in the following second order approximation:
=
Based on the above statement, the theoretical loss in this case using the duration approximation is as follows:
=2.74
Compared to the actual loss (1.64 billion), the loss in duration approximation is much larger. With the considerable change in interest rate (350bp), the difference between duration approximation and actual loss can be partly explained as the extra part in convexity approximation:
=0.22
C=17.52
3.4 volatility and VaR (for question 3)
As a traditional measure to qualify the risk of financial instrument, volatility refers to the standard deviation of returns of financial instruments in a specific period. A simple method to measure volatility is the simple-moving-average method, namely:
(Assume mean return=0)
Applying the simple-moving-average method, the volatility of the change in yields in December 1994 can be calculated as:
=0.402 %( Assume mean return=0)
As another measure to qualify the risk of financial instrument, value at risk (VaR) is nothing but the inverse loss distribution function, or quantile function, which can be denoted by
Pr (X ≤) = q}
There are several ways to compute the value of VaR, including variance-covariance method, Historical-Simulation method and Monte Carlo Simulation method. In variance-covariance method, the returns are assumed to be normally distributed. So the only two factors which we need to estimate are average (expected) return and standard deviation. Applying the fixed normal distribution curve, we can compute the value of VaR at a confidence level in a specific period.
Based on the attached data, the average change in 5-year monthly yield is 0.01%, the standard deviation of the change in yields is 0.40%. Under the assumption of normal distribution, the expected lowest and highest change in 5- year monthly yield at 95% confidence level is estimated as:
=0.01%; =0.40%
confidence
# of
calculation
Equals:
highest
95%
1.65*
0.01%+1.65*0.40%
0.67%
Lowest
95%
1.65*
0.01%-1.65*0.40%
-0.65%
It means that at 95% confidence level, the expected largest change in 5-year monthly yield will not exceed 0.67%.
In this case, it is assumed that the portfolio is only exposed to the change in interest rates. So the change in yield can fully explain the change in value of the portfolio by the concept of duration. Applying duration approximation, we can find the monthly portfolio VaR (by the variance-covariance method) in December 1994 at the 5% cut off point.
Va=2.74=0.358
Compared to variance-covariance method, the Historical-Simulation method does not make the assumption of normal distribution, but assumes that the history will repeat itself. By putting the historical returns in ascending order from lowest to highest, the Historical-Simulation builds its own distribution without estimating any variances or convariances. The histogram of the monthly change in yield is drawn in figure 1:
If we reorganize the data of monthly yield change in a new histogram (figure 2) which compare the frequency of the returns, it is much clearer to find the density of changes in yield at a specific interval. For example, at the highest point of the reorganized histogram (the highest bar), there are more than 35 times when the monthly yield change is between 0.05% and 0.08%. At the far right, we can see a bar at 2.03%; it shows the one single month (in Feb 1980) within the period of 40 years when the monthly yield change was as high as 2.01%.
From the cumulative density curve, we can find that with 95% confidence, the lowest monthly yield change will not exceed -0.56% and the highest monthly yield change will not exceed 0.66%.
Simply following the steps we take in the variance-covariance method, we can also calculate the monthly portfolio VaR (by the Historical-Simulation method) in December 1994 with 95% confidence level:
Va=2.74=0.352
It is interestingly noticed that the two results is quite similar to each other. But it does not mean that the two methods are actually identical or the change in yield is normally distributed. This point can be verified by the fact that the lowest yield change with 95% confidence level is -0.65% in the variance-covariance method while it is -0.56% in Historical-Simulation method. It remains us of the main difference between the two methods, namely the assumption of normal distribution.
3.5 converting the VaR (for question 4)
If we make the assumption that monthly returns at different period are normally distributed and uncorrelated to each other, the monthly VaR figures (from the above two methods) can be converted into an annual figure using the 'root-T' rule. Under the same confidence level, the montly VaR is converted as:
Va= Va=0.358
Va= Va=0.352
The estimated annual loss we get from the above calculation is not consistent with the actual loss (1.64 billion). One explanation should be that the assumption of normal distribution is not realistic. Specifically, the Durbin-Watson test can be applied to test the autocorrelation of the returns.
3.6 EWMA model and time-varying volatility (for question 5)
When we measure volatility by the simple-moving-average method, it is questionable to give all older data the same weight. It is more reasonable to give more recent data larger weight and older data smaller weight. The exponentially weighted moving average (EWMA) method fixes the problem by introducing the smooth parameter (ï¬). The formula of exponentially weighted moving average (EWMA) is shown as:
,
or rewritten as:
Where normally the smooth parameter (ï¬) is set as 0.94.
In this case, =0.30588%
=0.31533%
31567%
32165%
0.30977%
31887%.
The course of calculation is shown in attached calculations excel file. The actual change in interest rates and EWMA forecasts are compared in the below table.
period
Actual change (%)
(%)
Acceptance region (95%)
Within/outside
12.1994
-0.04
0.30588
(-0.4947 0.5147)
Within
11.1994
-0.31
0.31533
(-0.5103 0.5303)
Within
10.1994
-0.2
31567
(-0.5109 0.5309)
Within
9.1994
-0.47
32165
(-0.5207 0.5407)
Within
8.1994
-0.08
0.30977
(-0.5011 0.5211)
Within
7.1994
0.26
31887
(-0.5161 0.5361)
Within
It can be seen that all the actual changes in interest rates fall into the acceptance region, which is based on EWMA forecasting standard deviation with 90% confidence level.
If the EWMA forecasting volatility for December 1994 is used to calculate the VaR, the annual greatest loss at the 5% cut off point will be:
Va2.74
The new value of annual VaR is significantly smaller than what we get in Section 3.4. it is mainly because that the monthly volatility for Dec.1994 based on EWMA method is smaller than before.
3.7 Backtest EWMA model (for question 6)
The family of VaR models is enlightening only when they predict risk reasonably well. Backtesting is just one of the tools which check whether a model is adequate. Given a confidence level for back testing, too many exceptions mean the model underestimates risk. The simplest method of backtesting is to record the failure rate which is the exceeded proportion of exceptions compared to the expected number for given VaR level and given period.
In this case, we make the backtesting of the EWMA model on the last 100 months can see if at the 5% left tail cut off level (for the normal distribution) there are more than 5 'outliers'. The result is reported in the following figure (Figure 3).
Figure 3. the backtesting map for the last 100 months
The figure shows two facts: on one hand, there are two outliers which lie under the lower bound built by the EWMA forecasting VaR at 5% cut off level. On the other hand, there are five outliers which lie beyond the upper bound forecasted by EWMA method. Both of them support the validity of the EWMA model.
4. Conclusion
In the paper, the Orange County bankruptcy case is reviewed by applying the tools of financial risk management. After analysing the data of balance sheet and monthly yield change in the last 40 years, we discuss and try to answer all the case questions. As one of the root of the disaster, the effective duration of fund is enlarged by leveraging the investor equity nearly 3 times. Exposed to interest rate risk, the portfolio bears huge loss which can be calculated by duration approximation. The measure of volatility and VaR provide a more accurate way to weigh the expected worst loss at a confidence level. With different assumption, the variance-covariance method and historical simulation method report different results of VaR. considering the portfolio is only exposed to interest rate risk in the case, we can combine the VaR and duration approximation to estimate the monthly and annual loss of the portfolio at 5% cut off point. In the end, the backtest of 100 last monthly data shows the validity of the EWMA method.
