Value at Risk Forecasts

Value-at-Risk (VaR) has emerged as one of the most prominent tools in the finance and insurance industries. Many regulatory bodies, financial institutions and insurance companies adopt VaR as a measure for risk. Technically speaking, VaR is a statistical estimation of a portfolio’s loss, where the owner of the portfolio expects to incur that loss or more with a given probability level over a specified time horizon for risk measurement (see for example, [17, 81, 127]). For example, if the daily VaR at a 95 % confidence level is US$ 1 million, then there is a 0.95 chance that the actual loss will not exceed the amount of US$ 1 million in the next day. For the practical implementation of VaR, refer to J.P. Morgan’s RiskMetrics—Technical Document and the monograph by McNeil et al. [157]. Basically, there are two common approaches to the VaR implementation, namely, (a) the historical simulation and (b) the model-based approach. The historical simulation is to calculate VaR based on the empirical distribution of historical data by bootstrapping. This method is non-parametric in the sense that stringent assumptions for the profit/loss distribution are not required. The model-based method assumes a parametric form of the profit/loss distribution and estimates the unknown parameters using historical data.

Although VaR is a popular tool in the practice of risk measurement and management in the finance and insurance industries, [5, 6] have pointed out the theoretical shortcomings of VaR. In particular, VaR does not satisfy the sub-additive property saying that the merge of two risky positions does not reduce risk. This is counter-intuitive from the perspective of diversification, which is a key theme in modern finance. Furthermore, VaR remains silent about the severity of losses when the losses exceed a certain threshold level. Despite these theoretical shortcomings, VaR still remains a popular measure of risk in practice due to its simplistic interpretation. It provides both an easy-to-understand risk measure in financial reporting and computational tractability under some specific parametric assumptions, such as the multivariate normality assumption. It is also worth noting that VaR satisfies the sub-additive property when the profit/loss distribution is in the elliptical class (i.e., symmetrical), which includes normal distributions as a special case. In what follows, we discuss the evaluation of VaR forecasts using the HMRS model presented in the last subsection.

SupposeqtC1jt.˛/is the˛-quantile of the predictive distribution ofYtC1 given F_t^Y underP. By definition,

FY_tC1.qtC1jt.˛/jF_t^Y/D˛; (6.48)

whereFY_tC1.yjF_t^Y/is the predictive distribution ofYtC1givenF_t^Y underP in the second-order HMRS model derived in the last subsection.

LetP Vt be the market value of the portfolio at time t. The VaR, denoted by VaRtC1jt.˛/, for the long position of the portfolio with probability level˛is defined as the˛-percentile, (in practice,˛can be1% or5%), of the loss distribution. It is easy to see that

VaRtC1jt.˛/DP VtŒ1exp.qtC1jt.˛/r/: (6.49) In [189], the performance of VaR forecasts implied by the second-order HMRS model were evaluated using backtesting, which is a standard procedure to evaluate the performance of VaR forecasts used in the finance and insurance industries.

The numerical results of the backtesting reveal that the degree of long-term memory described by the order of the HMRS model has a significant impact on the accuracy of VaR forecasts. For details, please refer to Sect. 3 of [189].

6.7 Summary

In this chapter, a higher-order Markov chain model is proposed with efficient estimation methods for the model parameters. A further extension of the model is also discussed. The higher-order Markov chain model and its extension are then applied to diverse fields including sales demand predictions, web page predictions, the Newsboy problem and financial risk management.

176 6 Higher-Order Markov Chains

6.8 Exercise

1. Consider the following categorical sequence of three states:

1; 1; 2; 3; 2; 1; 1; 1; 2; 2; 3; 3; 2; 1; 2; 3; 2; 1; 2; 2; 2; 1; 2; 3; 1:

Build a third-order Markov chain model, as discussed in Sect.6.2, by using the Euclidean normjj:jj₂in solving the parameters.

2. Prove Proposition6.7.

3. Suppose the demand sequence of the product discussed in Sect.6.5is given as follows:

1; 2; 5; 2; 3; 4; 4; 3; 4; 5; 1; 2; 2; 2; 4; 4; 4; 5; 4; 5; 3; 2; 2; 2; 3; 4; 4; 5; 5; 5; 4; 4; 3; 2:

(a) Construct a second-order Markov chain model by using the Euclidean norm jj:jj₂in solving the parameters.

(b) Suppose the following non-symmetric cost matrix (compare with (6.38)) is employed:

C D 0 BB BB B@

0 200 400 800 2000 100 0 200 400 800 300 100 0 200 400 700 300 100 0 200 1500 700 300 100 0

1 CC CC CA:

Find the optimal production policy when the second-order Markov chain model is adopted.

4. Derive the semi-martingale dynamics for the chainXQ in (6.46).

5. Derive the recursive filterXQ_t^Y in (6.47).

6. Show that the filter estimateXQ_t^Y in (6.47) is optimal over the space of all linear estimates in the mean-square-loss sense.

7. Write a Matlab program to compute the filterXQ_t^Y in (6.47).

Multivariate Markov Chains

7.1 Introduction

By making use of the transition probability matrix in Chap. 6, a categorical data sequence ofmstates can be modeled by anm-state Markov chain model. In this chapter, we extend this idea to model multiple categorical data sequences. One would expect categorical data sequences generated by similar sources or the same source to be correlated to each other. Therefore, by exploring these relationships, one can develop better models for the categorical data sequences and hence better prediction rules.

The outline of this chapter is as follows. In Sect.7.1, we present the multivariate Markov chain model with estimation methods for the model parameters. In Sect.7.3, we apply the model to the multi-product demand estimation problem. In Sect.7.4, an application to credit rating is discussed. In Sect.7.5, we extend the model to a higher-order multivariate Markov chain model. Section7.6discusses an improved model with application to dependency rating transition. Finally, a summary is given in Sect.7.7to conclude the chapter.