Orange County is a prosperous district in California. In December 1994, due to a large loss ($1.64 billion), Orange County declared bankruptcy. This report will first introduce the trading strategy employed by Orange County. Then the relationship between the portfolio and the leverage ratio will be introduced. In part three, the loss will be calculated by duration approximation to compare with the actual loss. Furthermore, two methods will be used to calculate the monthly value at risk (VaR). By using root-T rule, the monthly VaR will be transformed to annually VaR. Then the exponentially weighted moving average (EWMA) model will be introduced. In the next section, backtesting will be used to estimate whether the VaR forecast based on the EWMA is accurate. Finally, two recommendations will be given.
Trading Strategy
Due to the successful performance of trading strategy employed by Robert Citron in 1992 and 1993, his trade took a significantly important role in Orange County's budget. In 1994, Robert Citron then was re-elected by Orange County as its Treasurer. Robert Citron raised the investment fund of $13 billion from Reverse Repurchase Agreements (repo market), which is on a short term. With initial investment pool of $7.5 billion, Robert Citron had a total fund of $20.5 billion, which mainly had been invested in fixed income securities ($11.9 billion) and structured notes ($7.8 billion). And both of them are considered as the long maturity instruments. Therefore, the trading strategy of Robert Citron is to use the proceeds from short the short short-term fixed-income instrument to long the long-term instrument (Hull, 2009). From the balance sheet, he invested heavily in inverse floaters, which pay a rate of interest equal to a fixed rate (Fixed-income securities) minus a floating rate (Structured notes). Therefore, it can be seen that he employed the trading strategy of "a yield curve play". The aim of this strategy is to capture the large differences of yield curve between short term and long-term instruments. This trading strategy can only work when the term structure of interest rate is flat or upward sloping, so that Orange County can borrow at a low rate of interest and invest at a substantially higher rate. From 1989 to 1992, the steady decline of U.S. interest rates is the main reason that the Orange County's portfolio can have a large gain. However, the value of longer maturity debt instruments can change dramatically if interest rates changes, thus a rise in the long-term rates could create a substantial capital loss for Orange County on the invested funds (Kolb and Overdahl, 2003).
Leveraged Bond Portfolio and its Duration
This investment strategy was also combined with substantial leverage, which was leveraged by a factor of 2.7 (). This leverage ratio indicates that the investor need borrow an additional dollar of , when every extra dollar is raised in the investment pool. This is relative higher degree of financial leverage. A higher leverage ratio has a positive impact on the earning per share, which can increase the shareholder's value. However, the company with higher leverage ratio will also struggle with high risk. Therefore, with the position (leverage) of borrowing money in the repo market, Robert Citron's investment portfolio can be considered as a leveraged bond portfolio, which will be much more sensitive to the change of the interest rates. The reason will be discussed in the followings.
The duration of a bond is a measure of how long on average the holder of the bond has to wait before receiving cash payment. It is the way to measure the sensitivity of price changes (volatility) with the changes in interest rates. Moreover, the duration is also a measure that summarizes approximate response of bond prices to change in yields. In December 1994, the average duration of the securities in Orange County's portfolio was 2.74. Thus, the effective duration of the portfolio was ordinary duration multiplied by leverage ratio (Babble, Merrill and Panning, 1997)
.
This large duration denotes that there will be a high sensitivity of the price changing in bond when the interest rate changes.
Duration Approximation
According to Hull (2009), there is a equation to the relationship between bond price B and bond yield y: . This equation will be approximately true if a small change . Therefore, the key duration relationship can be derived: , and D denotes the Bond duration. This equation is based on the assumption that y is expressed with continuous compounding. If y is expressed with annual compounding, will be the duration approximate relationship between bond and its yield, and is the modified duration. Both equations show that there is a negative relation between B and y. When bond yields increase, bond prices decrease (vice versa).
In 1994, the Fed then tightened its credit and raised the interests by 3.5%. We can use the modified duration to calculate how much money that Orange County will loss in December 1994 with the yield 7.83%. The result is showing below: . The Orange County will loss $1.80 billion. Compared with its actual loss of $1.64 billion, the difference between them is $0.16 billion, which is very close. The difference can be explained by the term of convexity. The duration rule is a good approximation for only a small change in bond yield, but less accurate for larger changes (Bodie, Kane and Marcus, 2008). Figure 1 shows that the duration approximation always underestimates the value of the bond. The reason is the curvature of the true price-yield relationship. Therefore, duration approximation will understate the increase in bond price when the yield falls, and overstate the decrease in price when the yield rises. In conclusion duration approximation is an accurate way to calculate the price change of the bond.
Actual Price
Duration Approximation
Change in YTM
Percentage Change in Bond Price
Figure 1: Convexity
Value at Risk
Volatility
In this report, we first use the formula to transform the discrete compound yields () to continues compound yields (). Then we can generate the monthly changes in yield: . Then we can use the simple measure of the volatility () to calculate the volatility of the monthly change in yields in December 1994, which is 0.003689. The volatility refers the spread of all likely outcomes of an uncertain variable. Figure 2 has shown the monthly yield change from January 1953 to December 1994.
Concept of the VaR
The Value at Risk (VaR) is the risk that provides a single number to summarize the total risk in a portfolio of financial assets. It can be simply described as "it is X percent certain that there will not be a loss of more than V dollars in the next days." In short, the VaR helps to predict the worst loss during a certain period by providing the confidence interval. The VaR assumes a situation that the change of the value in the portfolio is approximately normally distributed (Hull, 2009). There are three methods to calculate the VaR, which are variance-covariance method, historical simulation method and Monte Carlo simulation method. In this section, we will use variance-covariance method and historical simulation method to solve the case of Orange County.
Variance-covariance Method
The basic formula for calculate VaR at 95% confidence level is , 1.65 is the critical value in 5% significant level. When we know the initial position () in the portfolio and the portfolio volatility, and VaR can be calculated. Therefore, the aim of Variance-covariance Method is to estimate the volatility or the correlation of the assets in the portfolio. Cuthbertson and Nitzsche (2001) also provide the relationship of volatility between price change and yield change for bond:
,
where n is maturity of cash flow (duration). Therefore,, which means that it is confidence that only 5% of the time will the loss be more than $0.3378 billion.
Historical Simulation Method
Historical Simulation Method uses past data to build a scenario for tomorrow's price changes of the portfolio. In this process, a portfolio will be created based on the actual historical data, and the returns will also be calculated. Then the VaR for this portfolio is estimated by creating a hypothetical time series of its returns. Due to the VaR is determined by the actual price movement in historical simulation, this method does not need the assumption of normality (Kodapani, 2005). However, historical simulation assumes that an equal weight has been carried by each day in the time series, and historical data will repeat itself (ibid). However, in reality, some specific portfolios do not follow the assumptions.
Here, we have 505 monthly data from 1953-1994, we plot these data into the bin, which is the range from -0.025 to 0.018 with the small moving average of 0.001. Then, we generate each frequency in each small moving average, and the Histogram has shown in Figure 3. We have already known the negative relationship between bond price and bond yield, hence we estimate the 95th percentile of monthly increase in the 5-year yield, which is 0.00606. By using this figure with the equation (), we can get . This means that we are 95 percent certain that there will not be a loss of more than $0.3118 billion in our portfolio in the following month.
Root T-rule and the VaR
We can use root T-rule to forecast the volatility during a longer period based on a shorter period. For example, we can use monthly volatility to generate yearly volatility. The equation is , where ï³ is the forecast of the daily standard deviation and T is the number of trading days in the forecast horizon.
The assumption for root T-rule is that daily (log) price changes are identically and independently distributed (i.i.d), which mean that the prices follow random walk and are not serially correlated (Cuthbertson and Nitzsche, 2001). However, in the reality, the assumption sometimes is not true. Firstly, stock price tends to be mean reverting, which means that the high stock price and low stock price are temporary, and stock price tend to have an average price over time. The other one is that the volatility of the price change tends to be serial correlated. For instance, the term of volatility clustering indicates that large changes tend to be followed by large changes, and small changes are followed by small changes.
When we adopt the root T-rule in the VaR, we need an extra assumption of normality to assume our portfolio returns follow a normal distribution. Therefore, from root T-rule, we can have the transformation of the VaR between different time horizons:
.
Therefore, for Variance-covariance Method, the monthly VaR will be
.
For Historical Simulation Method, it will be
.
However, these forecast losses are much smaller than the actual loss ($1.64 billion), but it is still meaningful that this a figure denotes a relative large loss and give the investor a warning for their futures investment.
The Exponentially Weighted Moving Average Model
The exponentially weighted moving average (EWMA) model is widely used to estimate the conditional volatility of asset returns, because of its simple and rapid computations. The EWMA model requires the normality assumption. The formula is
.
is the estimator of the volatility of the variable for day n, is the percentage change in that variable, and is the decay factor which is between 0 and 1. The formula can also be transformed into
If m is large enough, the last term can be ignored, and then the equation will be
(Hull, 2009)
Therefore, the weights will decline exponentially. For instance, If , then the weights will follow the sequence of 0.94, 0.88, 0.83, etc, which means that past volatility will be given less weight than current forecast variance (Cuthbertson and Nitzsche, 2001). This makes the EWMA forecast react fast to recent changes in volatilities by giving the higher weight to recent events. Moreover, the volatility forecast from the EWMA model will decline gradually after a sudden shock. plays a vital role in the EWMA model. If the variable experiences a large change on day , it can cause a large . Therefore, an upward-move of the current volatility will be estimated by the EWMA model. governs how responsive the estimate of the daily volatility is to the most recent daily percentage change (Hull, 2009).
However, empirical studies show that asset returns are not normally distributed. The conditional distribution of returns for short horizon asset is tend to be leptokurtic, which shows that the tails are significantly fatter than normal distribution (Liu, Wu, and Lee, 2004). Therefore, the EWMA estimator will lose its power and underestimate the true value of volatility for asset returns.
In this section, we will first use the EWMA model to compute the monthly standard deviation of yield change for the six months before December 1994. J.P. Morgan's RiskMetrics model uses factor value () as of 0.94 for daily and 0.97 for monthly volatility estimations. The result is showing in Table 1. Therefore, the actual change in interest rates is outside the EWMA forecasts. From Figure 4 and 5, the VaR will change due to the changes of the volatilities. If we can get the accurate volatility, the VaR will also give the accurate forecast.
Table 1: Volatility Comparison
Monthly Yield Change
Volatility of Monthly Yield Change
Volatility Forecasted by EWMA
VaR
VaR based on EWMA
1994-6
0.001590
0.003701
0.003283
0.338941
0.300632
1994-7
-0.002433
0.003699
0.003245
0.338759
0.297161
1994-8
0.000749
0.003696
0.003224
0.33843
0.295203
1994-9
0.004391
0.003697
0.003178
0.33855
0.290984
1994-10
0.001863
0.003694
0.003221
0.338289
0.294926
1994-11
0.002880
0.003693
0.003188
0.338143
0.291967
1994-12
0.000371
0.003689
0.003179
0.337809
0.29116
Figure 4: Comparisons of the Volatility
Figure 5: VaR (95%) by Different Volatility
Moreover when we take the EWMA model as the methodology of the VaR, plays a vital role. The higher the , the lesser the process of the information will decay. If is high, the VaR movement will be relative low. Then market shock will give the impact on the VaR for a long period. The high can also reduce the failures of back-testing for the VaR.
Back-testing
Back-testing is the process to evaluate the strategy to see how well and accurate its performance when it is applied to the historical data. However, back-testing does not allow predicting that strategy for their future conditions. In this section, the VaR is our strategy based on the EWMA model to forecast the monthly loss, and compares it with the actual loss (Berry, 2009).
Due to the change of the volatility of the monthly yield change, the monthly VaR forecast will also change. Therefore, the idea of back-testing is to compare the forecast from the VaR with the actual historical profit or loss, and to see whether the VaR estimates would have actually performed in the past. We first calculate the volatility of monthly yield change by the EWMA model for last 100 months, so that we can calculate the monthly VaR at 95% confidence level. Moreover, by using the formula () with existing historical monthly yield change, we can derive the monthly actual profit or loss. Finally we plot to above data into graph to compare these figures. The result is showing in Figure 6 and 7, where we can see that only 4 spots (4% of spots) are outside the line (forecast loss by the VaR). This means that it is only 4% chance that the losses in a month exceed loss forecasted by the 1-month 95% VaR. Therefore, we can conclude that the VaR forecast based on the EWMA model is reasonable. The study from Bredin and Hyde (2001) on Irish currency risk also finds that the EWMA model is the more appropriate compared to other methodologies for the VaR.
Figure 6: Back-testing
Figure 7: Comparisons of the Volatility
Recommendations
Liquidation
It is not a good choice for Orange County to liquidate the portfolio in December 1994. One reason is that, at that time, the 5-year yields on US Treasury issues had already almost been at the peak, and started to decline from that time. Figure 8 has shown this trend. Therefore, the losses from this bond portfolio would also decrease. Moreover, in 1995, the level of yield reached the level during1992 and 1993 when the portfolio began to invest. If Orange County held on its position for more several months or half a year, and did not declare bankruptcy and liquidate the portfolio in December 1994, then the initial "paper" loss would be not realized. Therefore, it may be possible for it to 'ride out of storm', and the portfolio may also recovered to its formal level.
Source: US Department of Treasure.
Hedging Strategies
We have already known that the cause of the loss for Orange County is the rising interest rate. Therefore, to avoid the loss and bankruptcy, Orange County can employ three Hedging Strategies in December 1993, which are interest rate futures, interest rate swaps, and interest rate caps.
Orange County can take a short position in interest rate futures. One of them is Treasury bond futures contract, which allow the delivery any government bond that has more than 15 years to maturity and is not callable within 15 years from that day (Hull, 2009). If the interest rate is increasing, the value of the contract will fall. Therefore, Orange County can receive the amount of money for its short position, to offset the loss in the bond portfolio.
Orange County can also enter into the interest rate swaps provided by over-the-counter market. The idea is that it can pay the floating rate with the aim to receive the fixed rate. Therefore, the value of the portfolio will be locked and would not change by the change of interest rate.
Orange County can employ another interest rate derivative which is called interest rate cap. It can take a long position in interest rate caps. Therefore they would receive the amount of insurance if the rate of interest on the floating-rate note is rising above a certain level (the cap rate) (Hull, 2009).
Conclusion
In conclusion, the trading strategy employed by Orange County is "a yield curve play", which shorts the short short-term fixed-income instrument and long the long-term instrument. It will only be profit if the interest rate declines. Duration approximation gives the accurate loss figure compared to the actual loss. The monthly VaR based on Variance-covariance Method and Historical Simulation Method has been calculated. By using root-T rule, we transform the monthly VaR to annually VaR, and the differences between the VaR and actual loss is large. The EWMA model helps to measure the volatility of monthly changes in the yield. By using backtesting, we can conclude that the VaR forecast based on the EWMA model is an accurate method to forecast future loss for the portfolio. We also suggest that Orange County did not declare bankruptcy because of the declining yields in the following months. Moreover, they can employ some hedging strategies as suggested in December 1993.
