A Comparative Study of VaR Estimation for Structured Products

1. Introduction

Structured products include reverse convertible bonds, discount certificates, and principal-protected notes. They can prevent investors from market risk and credit risk. Telpner [1] illustrates that depending on volatility of asset returns and the spread of long rate and short rate, investors can choose a suitable investment tool. Principal-protected notes are widespread securities in structured products. These notes can protect investors’ final payoffs in recession and become an important instrument in portfolio selections.

Principal-protected note prices can be linked to stock index, interest rate, exchange rate, and commodities. Salomon Brothers Inc. first issued stock index-linked and principal-protected notes in 1986, namely SPIN, whose value was linked to S&P500. A year later, Chase Manhattan Bank launched a market index linked to certificates of deposit, called MICD, which was the first one of its kind by a commercial bank. In 1991, Goldman Sachs & Co. developed SIGN also linked to S&P500 in Australia.

In the 1990s, the fashion of principal-protected notes is increasing in portfolio selections. Hong Kong Investment Funds Association reported that their increase percentage had accounted from 2.94% to 84% in the 2001~2002 period. In Taiwan, the notes started in 2003 due to legal restrictions but began to pick up momentum by Winbond Electronics.

As for academic research of the principal-protected notes, the seminal work by Brennan and Schwartz [2] ushered in the concept of convex products and protected put products. The research of convex products can be traced to Gerber and Pafumi [3] and Imai and Boyle [4]. They investigate the dynamic hedging strategy in continuous time.

The literature in the framework of a protected put strategy follows the line of Chance and Broughton [5], Chen and Kensinger [6], and Chen et al. [7]. Chance and Broughton [5] and Chen and Kensinger [6] set the floor for stock growth rate and calculated the fair price of the MICD in the absence of the maximal payoff. Chen et al. [7] examine the value of investment funds linked to the Asian and Latin American stock indices.

Alternatively, the majority of the empirical studies focus on the relationship between the theoretical and actual prices of financial products. For example, Chen and Sears [8] analyze the difference of the two prices on SPIN. Burth et al. [9] computed the difference of the two Swiss structured products based on the root mean squared error. Additionally, Wilkens et al. [10] estimate the impact of actual price differentials and maturity date on the average price differential by means of regression analysis on reverse convertible bonds and discount certificates.

Above all, we find that no literature of structured products focuses on the management of market risk. In general, the VaR measure is commonly used to quantify market risk. Some available methods are the following: Riskmetrics, the GARCH approach, quantile estimation, and extreme value theory (for instance, McNeil and Frey [11] and Zivot and Wang [12]). In spite of any approach, forecast volatility is important to VaR calculation. Previous studies, such as the works of Akgiray [13], Pagan and Schwert [14], Baillie and Bollerslev [15], West and Cho [16], Brooks [17], and Taylor [18], generally evaluated asset returns’ volatility by different time series models and measured the accuracy by mean absolute squared errors (MAEs) or mean squared errors (MSEs). In all these studies, the errors are represented as the absolute gap between actual volatility (measured by a squared daily return) and model volatility. A general conclusion from these studies is that GARCH models tend to perform better in measuring volatility than the other time series models. In contrast to MAE and MSE tests, the Diebold and Mariano [19] test allows for forecast error that can be non-Gaussian, nonzero mean, serially correlated, and contemporaneously correlated for nested models. It can satisfy a real life.

In addition, Jorion [20] shows that VaR measure only can be efficiently used in a normal market. But, in the recent years, the world economy is not stable. One of the reasons is owing to oil price fluctuation, which causes stock price fluctuation. Many previous studies show that the impact of oil price changes on the world economy is large such as the studies of Adelman [21] and Driesprong et al. [22]. Driesprong et al. [22] find that stock returns tend to be lower after oil price increases and higher if the oil price falls in the previous month. Since West Texas Intermediate (WTI) crude oil price broke the record on August 30, 2005, oil price behaved like the prices of other commodities. Therefore, in order to accurately calculate VaRs, we use August 30, 2005, to breakdown the whole samples into the low oil price period from May 13, 2004, to August 29, 2005, and the high oil price period from August 30, 2005, to April 30, 2008.

The purpose of this paper is to reexamine the forecast accuracy of GARCH by means of the multivariate extension of the Diebold and Mariano [19] test, which is proposed by Mariano and Preve [23] for nonnested models, and backtesting method using index-linked and principal-protected notes as samples, which are traded on the American Stock Exchange from May 13, 2004, to April 30, 2008. The time break of August 30, 2005, is chosen to make a comparative study in low oil price and high oil price periods. The last section contains a conclusion.

2. Properties of Index-Linked and Principal-Protected Notes

Index-linked and principal-protected notes are one of structured products. They have some properties. We can take an example to describe them. Assume that the initial (notional) par value of the notes is N₀ per unit. Issuers promise at expiration (T) to deliver capital gains (N_T − N₀) and (1 + λ) portions of the initial par value if the par value at maturity N_T is no fewer than N₀, where λ is a constant guaranteed rate. Otherwise, issuers only pay (1 + λ) portions of the initial par value if the par value at maturity is less than the initial par value.

3. Model Checking

Consider the RUD, DJQ, and SPO notes, which are index-linked and principal-protected notes issued by Structures Products Corporation in USA. They are linked to Russel 2000 index, Dow Jones Industrial Average index, and S&P500, respectively. The samples in this study span from May 13, 2004, through April 30, 2008, or 997 daily log returns. Furthermore, WTI oil price is used to breakdown all of the samples into the low oil price years from May 13, 2004, to August 29, 2005, the high oil price years from August 30, 2005, to April 30, 2008, and the whole period from May 13, 2004, to April 30, 2008, because the time break occurred on August 30, 2005.

In the above models, r_t denotes the daily log returns at time t, ε_t stands for the shock at time t, and μ₀, μ₁, α_i, β_i, and γ represent constant coefficients.

Then, the order of these models is determined by means of AIC and SBIC. AIC and SBIC denote Akaike information criterion and Schwartz Bayesian information criterion, respectively. As stated in Table 2, AR(1)-ARCH(5), AR(1)-GARCH(1,1), AR(1)-GARCH(1,4), and AR(1)-EGARCH(1,1) are adequate for RUD because their AIC or SBIC is the smallest. Similarly, AR(1)-ARCH(5), AR(1)-GARCH(1,1), AR(1)-GARCH(2,5), and AR(1)-EGARCH(1,1) are suitable for SPO; AR(1)-ARCH(5), AR(1)-GARCH(2,1), AR(1)-GARCH(2,5), AR(1)-EGARCH(2,1), and AR(1)-EGARCH(2,2) are suitable for DJQ.

The estimation of the selected models is shown in Table 3 during the three periods—the whole period, the low oil price period, and the high oil price period. In next section, the results of the estimation of these selected models are used to measure VaRs of index-linked and principal-protected notes.

4. Accuracy Evaluation

This section will conduct accuracy evaluation of selected time series models for VaR forecast by backtesting method, the multivariate extension of Diebold and Mariano test, RMSE, and MAE. Before backtesting method, it is necessary to calculate VaRs for these three kinds of securities.

4.1. VaR Measurement and Implication

Daily returns of RUD, SPO, and DJQ are used in these samples. They are computed using the following formula:

$$ (7)

where P_t is the market value of the notes at time t.

VaR can be defined as the maximal loss over a target horizon (l) with a given probability (α). Assuming normal returns of the asset, it can be written as

$$ (8)

where E[r_t(l)] denotes the expected return over an l-period horizon, Z_α stands for a critical value with a given probability α, and σ_t(l) is the forecast volatility of returns. In this paper, the VaRs are for a one-day-ahead horizon and a 99% confidence level for losses such that l = 1, α = 1%, and Z_α = −2.33. Applying Table 3, one can compute VAR(1) summarized in Table 4.

Table 4. Daily VaRs of selected time series models for alternative notes during various periods.

Time series model	The whole period			The low oil period			The high oil period
	E(rt(1))	$$	VaR	E(rt(1))	$$	VaR	E(rt(1))	$$	VaR
Panel A: RUD notes
AR(1)-ARCH(5)	0.000084	0.000052	−0.014279	0.000088	0.000121	−0.025542	0.000283	0.000033	−0.013179
AR(1)-GARCH(1,1)	0.000101	0.000038	−0.014262	0.000265	0.000124	−0.025681	0.000247	0.000017	−0.009498
AR(1)-GARCH(1,4)	0.000041	0.000041	−0.014878	0.000212	0.000106	−0.023770	0.000252	0.000018	−0.009531
AR(1)-EGARCH(1,1)	0.000512	0.000036	−0.013468	0.000312	0.000115	−0.024674	0.000502	0.000016	−0.008866
Panel B: SPO notes
AR(1)-ARCH(5)	0.000512	0.000541	−0.053682	0.000112	0.000112	−0.024546	0.000412	0.000211	−0.033433
AR(1)-GARCH(1,1)	0.000426	0.000521	−0.052757	0.000312	0.000512	−0.052410	0.000408	0.000312	−0.040748
AR(1)-GARCH(2,5)	0.000451	0.000212	−0.033474	0.000122	0.000552	−0.053312	0.000410	0.000224	−0.034462
AR(1)-EGARCH(1,1)	0.000554	0.000312	−0.040602	0.000613	0.000126	−0.025541	0.000511	0.000266	−0.037490
Panel C: DJQ notes
AR(1)-ARCH(5)	0.000405	0.000062	−0.017941	0.000056	0.000460	−0.015747	0.000671	0.000056	−0.016745
AR(1)-GARCH(2,1)	0.000311	0.000094	−0.022279	0.000013	0.000049	−0.016297	0.000524	0.000085	−0.020957
AR(1)-GARCH(2,5)	0.000377	0.000093	−0.022093	0.000086	0.000033	−0.013299	0.000495	0.000074	−0.019496
AR(1)-EGARCH(2,1)	0.000293	0.000110	−0.024144	0.000366	0.000038	−0.013997	0.000505	0.000092	−0.021838
AR(1)-EGARCH(2,2)	0.000364	0.000095	−0.022346	0.000258	0.000040	−0.014478	0.000536	0.000096	−0.022247

• Note that the VaRs are for a one-day-ahead horizon and a 99% confidence level for losses. E[r_t(1)] and $$ denote the expected value and the forecast variance of asset returns over a one-period horizon, respectively.

Table 4 indicates that the VaRs based on GARCH-type models are the smallest for RUD notes in the whole period and low oil price period; the VaRs in AR(1)-ARCH(5) model are the smallest in high oil price period. This implies that in the GARCH-type models, it is necessary for financial investors to keep the greatest need of a minimum solvency safety margin because the worst potential losses of the models are the largest than the other models in the whole period and low oil price period. Meanwhile, the result is low profitability of the return on equity (ROE) and downward pressure on the bank’s stock price during the whole and low oil price periods. Similarly, the phenomena occur in the AR(1)-ARCH(5) model in the high oil price period.

As for SPO notes, the minimum VaRs happen in the GARCH-type models during the low and high oil price periods; the VaR of the AR(1)-ARCH(5) model is the lowest among those of the other models in the whole period. Hence, it is necessary for a bank to maintain the maximum capital amount in order to prevent from default risk in terms of the GARCH-type models during the low oil price and high oil price periods. However, the need of a minimum solvency safety margin under the AR(1)-ARCH(5) model is the greatest during the whole period.

Again, for DJQ notes, the VaRs measured by the EGARCH-type models are the smallest in the whole and high oil price periods; the capital requirement of the AR(1)-GARCH(2,1) model is the largest in the low oil price period.

To sum up, there exists a consistent phenomenon that the VaRs of GARCH-type models are the smallest among those of the selected models during the low oil price period. When financial risk managers employ GARCH-type models to measure VaRs during the low oil price period, the firm’s default risk can fall because of sufficient capital requirement. In general, the worst potential losses of ARCH-type models are fewer than those of GARCH-type models during the three periods. On account of the increase of costs during the high oil price period, it would be better for banks to use ARCH-type models to manage market risk in order not to worsen the profitability.

On the other hand, there is also an interesting phenomenon that the amounts of VaRs are independent of economic situations as illustrated in Figure 4. That is different from common sense that the worse the economic situation is, the larger VaRs are to prevent from bankruptcy.

5. Conclusion

Because VaR is a function of volatility, volatility forecast is the main issue of VaR. In this paper, using backtesting method, the multivariate extension of Diebold and Mariano test, RMSE, and MAE, we find that there seems to be no forecasting model that consistently has superior performance based on these criteria. But, the performance of the GARCH-type models is probably better than that of ARCH and EGARCH in the whole period, while it is not in the low price period and the high oil price periods. Despite in the low oil price and the high oil price period, ARCH model seems to have superior performance for stock index-linked and principal-protected notes.

Additionally, it is worthy of note for banks to manage market risk that the VaRs of GARCH-type and EGARCH-type models are generally smaller during these three periods. It also means that GARCH and EGARCH produce higher capital requirements with respect to ARCH model in order to avoid default risks. As a result, the profitability in the GARCH-type and EGARCH-type models is lower. Thus, GARCH-type and EGARCH-type models further dampen a bank’s financial statement during the high oil price period. We suggest that in a boom economy, GARCH-type and EGARCH-type models can be provided to manage market risk for financial institutions whereas ARCH model would be adequate to a recession economy.

Alternatively, we find that the amounts of VaRs are independent of economy scenarios as illustrated in Figure 4. That is different from common sense that the worse the economic situation is, the larger VaRs are to prevent from bankruptcy.