Nonparametric estimation of 100(1 − p)% expected shortfall: p→ 0 as sample size is increased

ABSTRACT

Expected shortfall (ES) is a well-known measure of extreme loss associated with a risky asset or portfolio. For any 0 < p < 1, the 100(1 − p) percent ES is defined as the mean of the conditional loss distribution, given the event that the loss exceeds (1 − p)th quantile of the marginal loss distribution. Estimation of ES based on asset return data is an important problem in finance. Several nonparametric estimators of the expected shortfall are available in the literature. Using Monte Carlo simulations, we compare the accuracy of these estimators under the condition that p → 0 as n → ∞ for several asset return time series models, where n is the sample size. Not much seems to be known regarding the properties of the ES estimators under this condition. For p close to zero, the ES measures an extreme loss in the right tail of the loss distribution of the asset or portfolio. Our simulations and real-data analysis provide insight into the effect of varying p with n on the performance of nonparametric ES estimators.

KEYWORDS:

• Expected shortfall
• Nonparametric estimation

MATHEMATICS SUBJECT CLASSIFICATION:

• 62G05

Acknowledgment

We are thankful to the esteemed reviewer for the suggestions and important references which lead to significant improvement of the article.

Notes

1 Suppose a trader borrows money from a broker, takes a long position on a certain equity and also buys a put option (short position) of the market index future to hedge against any random fall in the stock market. The trader can adopt two strategies. In the event of any unforseen downward movement in the market, he may cover the gains in the put option and take delivery of the stocks by paying remaining dues to the broker in cash. Otherwise the trader can exit both the long and short positions at market price, and return the dues to the broker. In this example a sudden downward market movement is the event that causes default. The first strategy is not netted, as only positions with positive gains are used to meet the default obligation. The second strategy involves netting, where overall portfolio gain is used to meet the traders obligation to the broker. Our model (x) represents the loss in the second strategy at time t.