18.2.3.1 The standard discrete-time case

Changes in asset prices are supposed to occur at discrete times along a whole time period (e.g., 10 years). Particular subperiods indexed by , with , can be introduced. They may correspond to several standard portfolio management periods (e.g., 1 week, 1 month, 1 year, etc.). Finally, for each subperiod , consider the sequence of deterministic prices variations times in , which for simplicity, for each , we denote by (e.g., daily variations).

In this framework, the CPPI portfolio value can be determined as follows. Denote by the opposite of the arithmetical return of the risky asset between times and . We have

Consider the maximum of these values. For any , define

Denote by the portfolio value at time . The guarantee constraint is to keep the portfolio value above the floor . The exposure invested in the risky asset is equal to , where the cushion value is equal to . The remaining amount is invested on the riskless asset with return for the time period . Therefore, the dynamic evolution of the portfolio value is given by (similar to the continuous-time case)

where the cushion value is defined by

Thus, we deduce both the cushion and the portfolio values.

For example, if the maximum drop is equal to −20% (the “Black Monday” on October 1987), then d = −0.2. Thus m must be less than 5.

18.2.3.2 The “truncated” discrete-time case

Because of the discrete-time estimation of stochastic processes or of the possible anticipation of the portfolio manager concerning the relative jumps during a given period (e.g., is considered as an actual jump if its value is higher than 3% or will never be higher than 20%, corresponding to the market crash in October 1987), we introduce also the truncated jumps defined by

where is equal to 1 if , and 0 otherwise.

In fact, when determining an upper bound on the multiple m, we have only to consider positive values of (see Section 18.3). Therefore, it is sufficient to take and equal to the maximum relative drop anticipated for the management period. Since the portfolio manager may take account of the occurrence probabilities of some given values of the relative jumps (corresponding, e.g., to maximum drops), we must introduce their arrival times which are random variables. Denote by the sequence of times at which takes values in the interval . The sequence is called a marked point process.³

18.3.1.1 The standard discrete-time case

In our framework, the portfolio manager must keep the cushion positive for each trading day. Thus, we measure stock market price fluctuations by the daily rates of returns. More precisely, we use the opposite of the arithmetical rate , which determines the conditions to impose on the multiple m, within the context of the CPPI method. If these variables are statistically independent and drawn from the same distribution, then the exact distribution of the maximum is equal to the power of the common distribution of .

Assume that the random variables are not truncated and the arrival times are the times themselves. Let us consider the maximum of the for all times in . Then, the previous quantile condition is equivalent to

Note that, since m is nonnegative, the condition is equivalent to .

If the N random variables are i.i.d. with cumulative distribution function (CDF) F, then we get the following result:

But, in most cases the CDF F is not exactly known, and some consecutive returns can be dependent. However, if block-maxima can be assumed to be independent by using a suitable length of the blocks, then the insurance condition on the multiple m can be analyzed by applying EVT.

Indeed, the fundamental result of EVT proves that there exists a normalization procedure of the maxima to get nondegenerate distributions at the limit (like the well-known central limit theorem for the sums of random variables).⁴ This is the well-known Fisher–Tippett theorem (1928). Indeed, we know that there exist scale and location parameters and such that converges in distribution to a generalized extreme distributions (GEV) defined by

where the parameter is called the tail index:

Consequently, from the condition , and by applying EVT to , we get . This leads to the result. Indeed, we get the following approximation of the upper limit on multiple m.

The statistical problem is to find the correct distribution of extremes of returns from the data and, in particular, to estimate the normalizing constants , and the tail index . For this purpose, when using a maximum likelihood method, we need the GEV distribution of a general, non-centered, non-reduced random variable, defined by

When examining the sequence , it is obvious that its distribution has a right end limit equal to 1. Therefore, the normalized maxima converge either to the Gumbel distribution or to a Weibull distribution . For the sequence of the opposite of log-returns , we get a Gumbel distribution or a Fréchet distribution.

For the determination of the limit, we can use the characterizations of Resnik (1987) for the maximum domains of attraction based on generalizations of the von Mises functions (see Embrechts et al., 1997).

18.3.1.2 The “truncated” discrete-time case

Consider now the truncated values ; denote the truncated jumps (), and the corresponding sequence of arrival times. As has been previously mentioned, the times are random variables and the range of their distributions can be the set of times or the whole management period if we consider jump-diffusion processes.

Define as the maximum of the in . The quantile hedging condition leads to

Denote by the random number of random variables during the period .

Then, by conditioning with respect to the random number , the quantile hedging condition becomes

which is equivalent to

Introduce the function

Since is increasing, has also the same property. Consider its inverse .

Then we get the following result:

As can be seen, this condition involves the joint conditional distributions of the marked point process . In the independent marking case, where and are independent, the CDF can be simplified as

Suppose, for example, that the number of times is sufficiently high so that we can consider that the set of seems like an interval (continuous time at the limit). Assume also that the sequence of inter-arrival times is an i.i.d. sequence and is exponentially distributed with a parameter . This case corresponds to the special case of an independent marked Poisson process. corresponds also to the parameter of the Poisson counting distribution. This implies that the expectation of the number of variations with values in during the period is equal to . We get

Furthermore, if are also i.i.d., then we get (see Prigent, 2001b)

This latter formula allows the explicit calculation of the upper bound.

This condition gives an upper limit on the multiple m, which is obviously greater than the standard limit (which is given in Proposition 18.4 with b = d if the distribution is not truncated). It takes account of both the distribution of the variations and of the inter-arrival times. Note that, if the intensity increases, then this upper limit decreases. So, as intuition suggests, if the frequency increases, then the multiple has to be reduced and, if goes to infinity, then the previous upper limit converges to the standard limit .

Again, if the distribution of the opposite of arithmetic returns is not known, we can apply EVT to get approximation of the upper bound, provided that the management period over which the risk is evaluated is sufficiently high and that the probabilities are small, for small values of k. Under the previous conditions, we get

From the practical point of view, condition “ small, for small values of k” implies that the set is not too small. Thus, we have to carefully calibrate this set in order to get rational upper bounds.

18.3.2.1 Estimations of the variations X_k

We examine the variations of the opposite of the arithmetic returns, , of the S&P 500 during both the period December 1983–December 2013 and the subperiod December 2003–December 2013. First we consider the whole support of . We refer to Longin (1996b) for details about estimation procedures when dealing with EVT. We begin with some usual descriptive statistics about daily variations (Table 18.1).

As expected, is far from normally distributed and has fat tails. We find evidence for heteroskedasticity and autocorrelation in the series. More precisely, the original series should be replaced by the innovation of an AR(5).⁵ Nevertheless, the estimates on the raw data do not differ qualitatively from the ones on innovations.⁶ As we need estimates on the raw data for the upper bound on the multiple, we conduct econometrics work on the original series.

We examine the behavior of maxima for various time periods. We first plot the maxima of over 20 and 60 days for the two time periods (Figures 18.2 and 18.3).

Notice, first, that on our sample the maximum values of are always positive. Knowing that we have to discriminate between the Weibull and the Gumbel distribution, the analysis confirms that the extremes of seems to follow a Weibull distribution.

Equivalently, we can introduce the opposite of the log-returns:

In that case, we get a Fréchet distribution.

We consider the maximum likelihood estimation of GEV distribution of the non-normalized series (see Martins and Stedinger, 2000). The density is given by

The likelihood function to maximize is defined as follows:

The estimation results for a Fréchet distribution and for various lengths of the subperiod are reported in Table 18.2.

18.3.2.2 Estimations of the upper bound

We are now able to give an estimation of the upper bound on the multiple. From Proposition 18.10, the upper bound is given by , where denotes the number of transaction dates during the management period. For the log-returns, we get an analogous expression of the upper bound:

where denotes the inverse of the extreme distribution of maxima of .

We illustrate numerically this upper bound in Table 18.3.

As expected, the upper bound on the multiple decreases when the probability level decreases, which corresponds to a more stringent gap risk control. It also decreases when the length of the selection period increases because at the same time the risk of significant drops increases. If bank “risk tolerance” to the probability level is very small, say , while the length of the rebalancing period is short, or if the length of the rebalancing period is high with , the upper bound on the multiple is in line with standard values of the multiple used by practitioners (between 4 and 8). As soon as bank risk tolerance is higher, , the corresponding multiple is above its usual values except for long periods. Note also that the upper bounds on the multiple are smaller when dealing with data on the past 10 years due to the more important impact of the financial crisis than for the longer time period corresponding to 30 years. However, for long management period (e.g., 1 year corresponding to trading days) and for (resp. ), the upper bound is small (resp. very small) compared to m = 5, which corresponds to the historical maximum daily drop of about 20%. In fact, when using the Fréchet distribution with parameter values in Table 18.3 for , we can check that the probability that the maximum daily drop is higher than 20% is about 25%, which is much higher. This is due to the weak convergence speed to the extreme distribution. On the contrary, using parameter values for the maximum on a week, this probability is much more in accordance with empirical frequency. Of course, it is possible to consider the dependence in the time series, such as the GARCH or stochastic volatility models. In that case, we can use results about the asymptotic of maxima. Note that, for example, using results of Davis and Mikosch (2009) for GARCH models, the upper bound involves the extremal index, which is the inverse of the mean cluster size of exceedances (see Ben Ameur et al., 2015). Another way to solve this problem is to introduce time-varying conditional multiples together with local quantile or expected shortfall risk measures to better take account of market fluctuations (see Ben Ameur and Prigent, 2014; Hamidi et al., 2014).