Dynamic portfolio insurance strategies: risk management under Johnson distributions

Abstract

The purpose of this paper is to analyze the gap risk of dynamic portfolio insurance strategies which generalize the “Constant Proportion Portfolio Insurance” (CPPI) method by allowing the multiple to vary. We illustrate our theoretical results for conditional CPPI strategies indexed on hedge funds. For this purpose, we provide accurate estimations of hedge funds returns by means of Johnson distributions. We introduce also an EGARCH type model with Johnson innovations to describe dynamics of risky logreturns. We use both VaR and Expected Shortfall as downside risk measures to control gap risk. We provide accurate upper bounds on the multiple in order to limit this gap risk. We illustrate our theoretical results on Credit Suisse Hedge Fund Index. The time period of the analysis lies between December 1994 and December 2013.

Appendix

1.1 Proofs of Propositions 8 and 9

In what follows, the cushion at time $t_{k-1}$ is strictly positive ($C_{t_{k-1}}>0$).

1.1.1 Case 1: $C_{t_{k-1}}>L_{t_{k-1}}$ (Proposition 8)

The local quantile condition is the following:

$$\begin{aligned} {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ 1+m_{t_{k-1}}\times \frac{\Delta S_{t_{k}}}{S_{t_{k-1}}}>\frac{L_{t_{k-1}}}{C_{t_{k-1}}}\right] \ge \left( 1-a\right) , \end{aligned}$$

which is equivalent to:

$$\begin{aligned} {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}}-1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] \ge \left( 1-a\right) . \end{aligned}$$

At time $t_{k}$, we have two cases:

$$\begin{aligned} \left\{ \begin{array} [c]{c} R_{t_{k}}<1:S_{t_{k}}<S_{t_{k-1}}\text { (}S\text { decreases),}\\ R_{t_{k}}>1:S_{t_{k}}>S_{t_{k-1}}\text { (}S\text { increases).} \end{array} \right. \end{aligned}$$

We assume that the multiple $m_{t_{k-1}}\$is higher than 1 , (thus, it is non negative).

Then, we deduce:

$$\begin{aligned}&{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}}-1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] \\&\quad ={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\cap \left( R_{t_{k}}-1\right) >0\right] \\&\quad \quad +\, {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}}-1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\cap \left( R_{t_{k}}-1\right) <0\right] . \end{aligned}$$

1.1. If $C_{t_{k-1}}>L_{t_{k-1}},$ then:

$$\begin{aligned}&{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}}-1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] \\&\quad = {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ R_{t_{k}}-1>0\right] { } + {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}}-1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\cap \left( R_{t_{k}}-1\right) { <0}\right] . \end{aligned}$$

Note that we have:

$$\begin{aligned} m_{t_{k-1}}\left( R_{t_{k}}-1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\Leftrightarrow R_{t_{k}}>\frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1. \end{aligned}$$

Then, if the condition $\frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}} }+1>0$ is fulfilled, we have: $R_{t_{k}}>\frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}} }-1}{m_{t_{k-1}}}+1.$ ⁶

1.1.1. If we have $\frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1<0,$ then the condition $R_{t_{k}}>\frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}} -1}{m_{t_{k-1}}}+1$ is satisfied and:

$$\begin{aligned} {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ 1+m_{t_{k-1}}\times \frac{\Delta S_{t_{k}} }{S_{t_{k-1}}}>\frac{L_{t_{k-1}}}{C_{t_{k-1}}}\right] =1>(1-a). \end{aligned}$$

1.1.2. Examine the case $C_{t_{k-1}}>0$, $C_{t_{k-1}}>L_{t_{k-1}}$ and $\frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1>0.$

We have:

$$\begin{aligned}&{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] \\&\quad ={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ R_{t_{k}}-1>0\right] +{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ 1>R_{t_{k}}>\frac{\frac{L_{t_{k-1} }}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right] \\&\quad ={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ R_{t_{k}}>\frac{\frac{L_{t_{k-1} }}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right] , \end{aligned}$$

which is equivalent to:

$$\begin{aligned}&{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ \exp \left[ h_{t_{k}}^{-1}\left( \varepsilon _{t_{k}}\right) \sigma _{t_{k}}+\mu _{t}\right] >\frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right] \ge \left( 1-a\right) \\&\quad ={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ \varepsilon _{t_{k}}>h_{t_{k} }\left( \frac{1}{\sigma _{t_{k}}}\ln \left( \frac{\frac{L_{t_{k-1}} }{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right) -\frac{\mu _{t_{k}}}{\sigma _{t_{k}} }\right) \right] \ge \left( 1-a\right) \\&\quad =1-{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ \varepsilon _{t_{k}}<h_{t_{k} }\left( \frac{1}{\sigma _{t_{k}}}\ln \left( \frac{\frac{L_{t_{k-1}} }{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right) -\frac{\mu _{t_{k}}}{\sigma _{t_{k}} }\right) \right] \ge \left( 1-a\right) \\&\quad ={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ \varepsilon _{t_{k}}<h_{t_{k} }\left( \frac{1}{\sigma _{t_{k}}}\ln \left( \frac{\frac{L_{t_{k-1}} }{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right) -\frac{\mu _{t_{k}}}{\sigma _{t_{k}} }\right) \right] \le 1-\left( 1-a\right) . \end{aligned}$$

Denote by N the cdf of the standard Gaussian distribution (the cdf of $\varepsilon _{t_{k}}$). We have:

$$\begin{aligned} N\left[ h_{t_{k}}\left( \frac{1}{\sigma _{t_{k}}}\ln \left( \frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right) -\frac{\mu _{t_{k}} }{\sigma _{t_{k}}}\right) \right] \le 1-(1-a), \end{aligned}$$

which is equivalent to:

$$\begin{aligned} h_{t_{k}}\left( \frac{1}{\sigma _{t_{k}}}\ln \left( \frac{\frac{L_{t_{k-1}} }{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right) -\frac{\mu _{t_{k}}}{\sigma _{t_{k}} }\right) \le (N)^{-1}\left( 1-(1-a)\right) . \end{aligned}$$

and also equivalent to:

$$\begin{aligned} \frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}\le \exp \left[ \sigma _{t_{k}}h_{t_{k}}^{-1}\left( N^{-1}\left( 1-(1-a)\right) \right) +\mu _{t_{k}}\right] -1. \end{aligned}$$

Recall that $q_{(\epsilon ,T)}=N^{-1}\left[ \left( 1-(1-a)\right) ^{\frac{1}{T}}\right] .$

Finally, we have:

(i) If $\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] <1,$

If $\frac{L_{t_{k-1}}}{C_{t_{k-1}}}<\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] ,$ then $m_{t_{k-1}}$must satisfy the following constraint:

$$\begin{aligned} m_{t_{k-1}}\le \frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{\exp \left[ h_{t_{k} }^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k} }\right] -1}. \end{aligned}$$

If $\frac{L}{C_{t_{k-1}}}>\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] ,$ then the previous constraint must be satisfied but also involves more that $m_{t_{k-1}}<1.$

(ii) If $\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] >1,$ then there is no constraint for $m_{t_{k-1}}.$

1.1.2 Case 1: $0<C_{t_{k-1}}<L_{t_{k-1}}$ (Proposition 9)

As previously, the quantile condition is given by:

$$\begin{aligned} {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ 1+m_{t_{k-1}}\times \frac{\Delta S_{t_{k}}}{S_{t_{k-1}}}>\frac{L_{t_{k-1}}}{C_{t_{k-1}}}\right] \ge (1-a), \end{aligned}$$

which is equivalent to:

$$\begin{aligned} {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}}-1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] \ge (1-a) \end{aligned}$$

At time $t_{k}$, we have two cases:

$$\begin{aligned} \left\{ \begin{array} [c]{c} R_{t_{k}}<1:S_{t_{k}}<S_{t_{k-1}}\text { (}S\text { decreases),}\\ R_{t_{k}}>1:S_{t_{k}}>S_{t_{k-1}}\text { (}S\text { increases).} \end{array} \right. \end{aligned}$$

We assume that the multiple $m_{t_{k-1}}\$is non negative.

$$\begin{aligned}&{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] \\&\quad ={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\cap \left( R_{t_{k}}-1\right) >0\right] \\&\quad \quad + {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\cap \left( R_{t_{k}}-1\right) <0\right] . \end{aligned}$$

Therefore, we have:

$$\begin{aligned}&{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] \\&={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\cap \left( R_{t_{k}}-1\right) >0\right] \\&={\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ m_{t_{k-1}}\left( R_{t_{k}} -1\right) >\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1\right] . \end{aligned}$$

Consequently, we get the equivalence between the two following conditions:

$$\begin{aligned}&{\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ 1+m_{t_{k-1}}\times \frac{\Delta S_{t_{k}}}{S_{t_{k-1}}}>\frac{L_{t_{k-1}}}{C_{t_{k-1}}}\right] \ge (1-a),\\&\quad {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ R_{t_{k}}\ge \frac{\frac{L_{t_{k-1} }}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right] \ge (1-a). \end{aligned}$$

This is equivalent to:

$$\begin{aligned} {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ R_{t_{k}}\ge \frac{\frac{L_{t_{k-1} }}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right] \ge (1-a). \end{aligned}$$

Thus, the quantile condition is equivalent to:

$$\begin{aligned} {\mathbb {P}}^{{\mathcal {F}}_{t_{k-1}}}\left[ \varepsilon _{t_{k}}>h_{t_{k}}\left( \frac{1}{\sigma _{t_{k}}}\ln \left( \frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}} -1}{m_{t_{k-1}}}+1\right) -\frac{\mu _{_{t_{k}}}}{\sigma _{t_{k}}}\right) \right] \ge \left( 1-a\right) \end{aligned}$$

And also to:

$$\begin{aligned} h_{t_{k}}\left( \frac{1}{\sigma _{t_{k}}}\ln \left( \frac{\frac{L_{t_{k-1}} }{C_{t_{k-1}}}-1}{m_{t_{k-1}}}+1\right) -\frac{\mu _{t_{k}}}{\sigma _{t_{k}} }\right) \le N^{-1}\left( 1-(1-a)\right) . \end{aligned}$$

We obtain:

$$\begin{aligned} \frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{m_{t_{k-1}}}\le \exp \left[ h_{t_{k} }^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k} }\right] -1. \end{aligned}$$

Finally, we have:

2-1) If $\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] <1,$ there is no positive solution for $m_{t_{k-1}}$

2-2) If $\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] >1,$ then:

If $\frac{L_{t_{k-1}}}{C_{t_{k-1}}}<\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] ,$ then there is no constraint for $m_{t_{k-1}}$

If $\frac{L_{t_{k-1}}}{C_{t_{k-1}}}>\exp \left[ h_{t_{k}}^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k}}\right] ,$ then $m_{t_{k-1}}$must satisfy the following constraint:

$$\begin{aligned} m_{t_{k-1}}\ge \frac{\frac{L_{t_{k-1}}}{C_{t_{k-1}}}-1}{\exp \left[ h_{t_{k} }^{-1}\left( q_{_{(a,T})}^{\varepsilon }\right) \sigma _{t_{k}}+\mu _{t_{k} }\right] -1}. \end{aligned}$$