The bankruptcy of the Orange County in California has been the largest municipal failure in U.S. history. The $1.6 billion loss of its investment pool was the largest loss ever recorded by a local government investment pool. The portfolio, managed by its Treasurer, Robert Citron, consisted of fixed income securities and structured notes. Additionally, it was very risky since it was highly leveraged by issuing Reverse Repurchase Agreements. The investment strategy worked excellently until 1994, but when interest rates rose, it plummeted. As a result, the county declared bankruptcy and decided to liquidate the portfolio.
This report aims to explain how a municipality can lose $1.64 billion in financial markets. Moreover, the concept of "Value at Risk" (VAR) is also introduced, which is a simple method to measure the risk of a portfolio, and also show how VAR is calculated using different methods. Finally, this report attempts to illustrate the use of interest rate derivatives might be a way to hedge the risk of the portfolio.
Duration -- measurement of exposure to risk
The duration of a bond is used to measure the average length of time the bond holder has to wait before their receipt of cash payments(Hull,2008). In detail, it is the time weighted average of the cash flows of a bond, with the weight being equal to the present value of the cash flow as a proportion of the bond value(Stulz,2003). Moreover, if each bond in the portfolio has the same yield to maturity, the duration of the portfolio equals the weighted average of the portfolio's bond's durations, with the weights being proportional to the bond prices(Hull,2008). It reflects the sensitivity of the value of a portfolio to changes in yields(Stulz,2003). In the orange country's case, the average duration of the securities in the portfolio was 2.74 years in December 1994.
In additional, Robert Citron, the county treasurer, the investment strategy he used is to take short position in short-term fixed-income instruments, and use the proceeds to take long position in long-term fixed-income instruments. As a result, he leveraged $7.5 billion of investor equity into a $20.5 billion portfolio. Specifically,
Change in the total value of assets
Change in Liabilities (reverse repurchase agreements 63.2%)
First round
7.5 billion
4.74 billion
Second round
+4.74 billion
+3.00 billion
Third round
+3.00 billion
+1.893 billion
Fourth round
+1.893billion
+1.197 billion
:
:
：
：
：
：
total
20.5 billion
13 billion
Aapted from McConnell and Brue (2008)
The table above shows that the initial change in liabilities set off a chain, although it diminishes successively, it will cumulate to a multiple change in the total value of assets. That is to say, Orange County borrowed the money (13 billion) by issuing Reverse Repurchase Agreements to invest in securities with longer duration to take advantage of the slope of the term structure. Consequently, it implies that there is a leverage ratio, which equals to 2.7 (=20.5/7.5). That is to say, for every dollar of pool investor's funds, the poor borrowed an additional $1.7.
However, this strategy is not able to work when the interest rate increase. From the equation below:
It can be known that the value of the portfolio will decrease with an increase in the yields(Hull,2008). And also, since longer maturity bonds have higher duration, the duration of the long position side is higher than that of the short side. Consequently, the long position side will be exposed to higher interest rate risk. Therefore, the effective duration, which measures the actual changes in the value of the portfolio relative to the changes in yields, is equal to:
D (effective) =
= 2.74 * 2.7
= 7.4
Moreover, the interest rates began to rise in February 1994 resulting in the loss of the Orange County's portfolio. Here, the duration of the pool was estimated at 7.4, the yield of the five-year US treasury went from 5.22% in December 1993 to 7.83% in December 1994 and the portfolio value is $7.5 billion. Through the formula below, the loss using the duration approximation is computed:
(Here, is the modified duration)
Compared with the actual loss, this figure is close to $1.64 billion. It illustrates that duration estimates exactly the impact on the value of the portfolio of an infinitesimal change in the interest rate but for larger changes, it is only used to approximate the effect(Stulz,2003).
The Value at Risk of the Orange County pool
Value-at-Risk is probably the most widely used risk measure in financial institutions and it is used to set minimum capital adequacy requirements to cover market risk(McNeil et al.2005). "It is defined as the decline in the portfolio market value that can be expected within a given time interval with a probability not exceeding a given number"(Fabozzi,1997). For example, if the daily VaR reported is $100 million at a 5% critical value, it demonstrates that the financial institution expect to lose more than $100 million only 1 day in every 20 days (Cuthbertson and Nitzsche,2001). Moreover, there are several ways to measure VaR. In this report, the monthly VaR of the Orange County pool is computed using both the Variance-covariance method and the Historical Simulation method first.
To begin with, the 'variance-covariance' approach is used. Suppose that the return on a portfolio of assets is linear in the individual returns, and also, the asset returns are identically, independently and normally distributed so that the risk can be measured by variance or standard deviation(Cuthbertson and Nitzsche,2001) . Furthermore, under the normality assumption, the quantiles of the normal distribution is used to determine the VaR(Cuthbertson and Nitzsche,2001). More specifically, for 90% certainty, we expect the actual return to be between the range {μ-1.65σ,μ+1.65σ}(the 5% quantile cut-off point is 1.65). Additionally, if we make the additional assumption that the mean return of the portfolio is zero, '1.65σ' is used as a measure of 'downside risk'(Cuthbertson and Nitzsche,2001). Therefore, the formula for VaR is
VaRp=$V0p(1.65σp )
(Here, Vop is the initial market value of the portfolio, σp is the standard deviation of the returns on the portfolio.)
For more than two assets in the portfolio, the formula above is represented as:
VaRp=$V0p [zCz'] 1/2
Here, z = [w1(1.65σ1) w2(1.65σ2) ….. wn(1.65σn)] , C is the correlation matrix.
From the formulas, it is shown that the standard deviation of the portfolio returns is required to be calculated. Let's assume that changes in yields are normally distributed. Using monthly data from January 1953 to December 1994, the yield volatility is calculated first. Initially, the volatility for the change in the yield of the five-year US Treasury is 0.4037% per month. After the yield volatility in December 1994 is known, the standard deviation of the returns on the portfolio is easily obtained from the duration formula:
σ(dP/P) = -nσy
where σ(dP/P) is the standard deviation of the portfolio returns
σy is the standard deviation of the (continuously compounded) spot yield
n is its duration
As a result, using the formula for VaR, the monthly portfolio VaR in December 1994 at the 5% cut off point is:
=$0.351(billion)
However, the above approach in calculating portfolio VaR assumes that returns are normally distributed. "If returns are actually non-normal then this will produce biased estimates or forecasts"(Cuthbertson and Nitzsche,2001). Hence, a non-parametric approach is instead used, such as the Historical Simulation method.
To start with, given the historical 5-year yield monthly data from 1953 to 1994, the monthly change in yield is calculated. Then, I first plot the monthly change in yield data:
In the graph above, it can be seen that there are positive and negative outliers, namely, some monthly changes in yield are very large. Furthermore, I take the monthly change in yield data (using the whole data set), sort and plot them into bins, and then create a histogram of the monthly change in yield.
After that, the 95th percentile of worst monthly increase in the five-year yield is estimated using this distribution, which is equal to 0.647%. This implies that the VaR of the Orange County portfolio over a one-month horizon at the 5% probability level is:
=$0.341 billion
Overall, the VaR obtained using the two methods are close to each other. But the one calculated using the Historical Simulation method might be superior. For the reason that we do not need to estimate variances and covariances, there is no model risk or parameter estimation risk(Cuthbertson and Nitzsche,2001). Moreover, the empirical histogram is constructed by ourselves, so the normality does not have to be assumed to find the lower tail cut-off point, and VaR estimates encapsulate any fat tails or autocorrelation implied in the historic data(Cuthbertson and Nitzsche,2001). Additionally, "the method can also be used to estimate the VaR of non-linear positions"(Cuthbertson and Nitzsche,2001). Therefore, although the Historical Simulation approach has some drawbacks, for example, it requires a long time series of accurate data to estimate the accurate percentile while large standard errors on VaR might occur with the longer historic data set, it is found that the non-parametric approach could be more accurate than parametric ones(Cuthbertson and Nitzsche,2001). But in practice, they are complementary to each other.
Furthermore, after the monthly VaR figures are calculated using two different methods, we convert them into an annual figure using the 'root-T' rule. First, the monthly VaR obtained from the Variance-Covariance method is converted:
=$1.22 billion
And the one from the Historical Simulation method:
=$1.18 billion
It is worth noting that the 'root-T' rule only works under very rigorous assumptions that are the return volatility is independently and identically distributed(Cuthbertson and Nitzsche,2001). However, in practical applications, these assumptions seem to be not realistic. For example, "the generalization of exchangeable random variables is often sufficient and more easily met"(Independent and identically distributed random variables,2010). Moreover, if the return volatility exhibits mean reversion, technically, we violate the identical distribution conditions for the use of the square root rule. In fact, long-run volatility for most financial assets series, like stocks, does exhibit mean reversions(Allen et al.2004). Additionally, another condition--the independence， it applies that the covariances between periodic returns are equal to zero. If returns are serial correlated, that is to say, the return volatility is sticky, we violate another condition for the application of the square root rule. In one word, the 'root-T' rule is so convenient that requires a couple of assumptions that empirically we know these might be violated.
Moreover, if we compare the two figures for the annual VaR with the $1.64 billion loss, it can be seen that they are not consistent with the actual loss. This highlights that one of the limitations of VaR, which fails to accommodate extreme cases of highly volatile bond market(Stulz,2003).
In sum, in spite of its weakness, VaR is the most attractive and popular measure of risk and is widely used among regulators and risk managers(Hull,2008).
EWMA model and Back-testing
As is stated in the previous part, we need forecast the standard deviation of the returns on the portfolio, namely, its volatility, to calculate portfolio VaR. However, if the variance of the portfolio returns is not constant, an alternative method is used to forecast the time-varying volatility, which is the exponentially weighted moving average model (EWMA) (Cuthbertson and Nitzsche,2001).
Looking at the equations below:
σ2t+1|t=(1-λ)λiR2t-i)
(Assuming the mean return is zero)
Equivalently,
σ2t+1|t=λσ2t|t-1+ (1-λ)R2t
This forecasting scheme assumes that monthly volatility is a weighted moving average of past squared returns, where the weights (1-λ)λi decrease exponentially as we move back through time. This means that "squared returns which occur further in the past are given less weight in the current forecast of the variance", so it is known as exponentially weighted moving average(EWMA) (Cuthbertson and Nitzsche,2001). In addition, the decay factor, λ, is lying between 0 and 1. More specifically, the RiskMetrics optimal value for the decay factor λ is 0.97 (for monthly volatility) (Cuthbertson and Nitzsche,2001).
In addition, one attractive feature of the EWMA approach is that the computer does not have to store a lot of data since given an initial value for σ02 to start off the recursion in the second equation above, the estimate of the variance can be updated(Cuthbertson and Nitzsche,2001). Moreover, "the EWMA approach is designed to track changes in the volatility"(Hull,2008). For example, if λ is close to 1, that means that the estimates of the volatility respond slowly to the changes of R2t-i(Hull,2008).
For Orange County case, using the first formula above, we calculate the one-month ahead forecast of the standard deviation of changes in yields for the six months before December 1994. But we truncated the infinite sum in the equation to a finite sum (of 504 months). The results are as follows:
σ2t+1|t
Volatility
EWMA
Volatility EWMA(%)
Jul-94
0.0000121
0.003482825
0.34828251
Aug-94
0.0000120
0.00345962
0.34596198
Sep-94
0.0000116
0.003410147
0.34101466
Oct-94
0.0000119
0.003455854
0.3455854
Nov-94
0.0000117
0.003421204
0.34212043
Dec-94
0.0000116
0.003412008
0.34120082
In the table, σ2t+1|t can also be obtained using the second formula, for example, 0.0000120:
σ2t+1|t=λσ2t|t-1+ (1-λ)R2t
0.0000120=0.97*0.0000121+(1-0.97)*0.000007
It is worth noting that the expected value of R2t is σ2t|t-1, here, the realized value of R2t (0.000007) is less than the expected value (0.0000121), and consequently, the volatility estimate decreases (from 0.0000121 to 0.0000120). Namely, the estimates of the volatility would not increase unless the interest rates swings are greater than their expected values.
Furthermore, we are required to assess whether our forecasts of the volatility are accurate. To do so, assuming the changes in yields are conditionally normally distributed with zero mean, we compare the monthly forecasts of the standard deviation obtained by using the EWMA method with the actual change in interest rates and to see whether the actual changes exceed -1.65σ or +1.65σ(for the 5th percentile). This is done in the figure as follows:
The forecast error bands are the two solid lines and the actual change in interest rates are the small triangles. We expect 5% of actual changes to be in each 'tail'. From the figure above, we know that our measures of the volatility of changes in yields are adequate. It is because there is no 'triangle' outside either of the two lines.
Moreover, the volatility of the change in yields in December 1994 measured using the EWMA is equal to 0.3412%, so the VaR can be calucated:
VaR(one month)=
=$0.297(billion)
If we convert this monthly VaR figure into an annual one using the 'root-T' rule, we obtain the annually VaR which equals to $1.029 billion. Compared with the one ($1.18 billion) obtained from Historical Simulation method, $1.029 billion is less accurate referring to the actual loss $1.64 billion. In fact, Cuthbertson and Nitzsche(2001,p.669),citing Mahoney's view(1996), found that the non-parametric approach is more accurate than the parametric EWMA model. "Although there might be an inconsistency in using the 'root-T'rule which assumes returns have a constant variance, whereas the EWMA forecast assumes a time varying variance"(Cuthbertson and Nitzsche,2001).
From the previous calculation, we obtained three annual VaR using different methods, they are close to each other, but neither of them is consistent with the actual loss. There are several reasons why a VaR estimate might not be very good(Stulz,2003). For instance, VaR could be biased for the reason that one of the assumptions might be inappropriate(Stulz,2003).
Additionally, using the first EWMA formula above, we also calculate the one-month ahead forecast of the standard deviation of changes in yields for the last 100 months. Then, we take the actual change in interest rates for the last 100 months and the EWMA forecasts using the same back-testing procedure as in the previous analysis. The results are as follows:
We expect 5% of actual changes to be in the left 'tail', namely, 5 'outliers' is less than -1.65σ. Nevertheless, from the figure, it shows that no observation exceeds -1.65σ except two 'triangles' are on the lower line. As a result, our forecasts are adequate.
Overall, back-testing is often used to assess the accuracy of the accuracy of forecasts of volatility. "To the extent that the error fraction is within or outside acceptable ranges determines the validity of the risk model"(Das,1997). Through backtesting the EWMA model, the forecasts overstate the risk at the 5th percentile.
Hedging
For the Orange County case, although the portfolio plummeted as the interest rates began to increase in February 1994, the portfolio should not have been liquidated in December 1994. It was possible to 'ride out the storm' by the use of derivatives to hedge its interest rate exposure. For instance, develop a strategy for hedging the portfolio, using (i) interest rate futures, (ii) interest rate swaps, and (iii) interest rate caps or floors. Alternatively, there was a possibility that the interest rates would go back down in the following year and the portfolio would have recovered to its former level. In fact, the interest rates fell down in 1995.
Moreover, in December 1993, if the portfolio manager had hedged the bond portfolio using interest rate futures, the story would have been different. More specifically, the hedging instrument can be Eurodollar futures, Treasury bond or Treasury note futures contracts(Hull,2008). Generally speaking, "Eurodollar futures are used for exposures to short-term interest rates while T-bill and T-bond futures contracts are used for exposures to longer-term rates"(Hull,2008). Moreover, "we tried to choose the futures contract so that the duration of the underlying asset is as close as possible to the duration of the asset being hedged"(Hull,2008). Additionally, due to the interest rates move in opposite directions to the movements of the futures prices, we should hedge by taking a short futures position since the Orange County's portfolio would make a loss when interest rates increase. Therefore, we assume that parallel shifts in the yield curve and the number of contracts needed to hedge against an uncertain change in yields is:
(where P is the value of the portfolio today
D(P) is the portfolio's duration at the maturity of the hedge
F(C) is the price for the interest rate futures contract
D(F) is the duration of the underlying asset at the maturity of the futures contract)
By doing so, we can make the duration of the total position zero.
Conclusion
In conclusion, although the VaR has limitations, it can be used to understand risk and to avoid big mistakes(Hull,2008). If the officials of Orange County had known the VaR, they would have not used that investment strategy(Stulz,2003). A lesson can be learned from this case is that it is of great importance to measure portfolio risks and to hedge them. Additionally, the interest rate derivatives could be helpful.
