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This paper compares the performance of the two main portfolio insurance strategies, namely the Option-Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI). For this purpose, we use the stochastic dominance approach. We provide several explicit sufficient conditions to get stochastic dominance results. When taking account of specific constraints, we use the consistent statistical test proposed by Barret and Donald [1]. It is similar to the Kolmogrov-Smirnov test with a complete set of restrictions related to the various forms of stochastic dominance. We find that the CPPI method can perform better than the OBPI one at the third order stochastic dominance.

The goal of portfolio insurance is to provide a guarantee against portfolio downside risk (usually 100% of the initial invested amount) while allowing to benefit from significant gains for bullish markets. The two standard portfolio insurance methods are the Option Based Portfolio Insurance (OBPI), introduced by Leland and Rubinstein [

To compare these two main portfolio strategies, we search for stochastic dominance (SD) properties since SD takes account of the entire return distribution. The major argument for stochastic dominance is that it does not require any specific knowledge about the preferences of investors. Indeed, the first stochastic dominance order is related to investors with an increasing utility function. Stochastic dominance of order two focuses on investors having an increasing and concave utility, meaning that they are risk-averse^{1}. However, at a given order (for example 1 or 2), the stochastic dominance criterion cannot always allow to rank all portfolios. There exist cases where no stochastic dominance is observable. But there exists a stochastic dominance criteria at each order and, the higher the order, the less stringent the criterion. Thus it is reasonable to expect that there exists an order for which a portfolio strategy dominates another one (or vice versa). De Giorgi [

For the portfolio insurance strategies, Bertrand and Prigent [^{2}. However, as proved theoretically by Zagst and Kraus [

The paper is organized as follows. In Section 2, we briefly introduce the basic properties of the CPPI and the OBPI strategies. In Section 3, we examine the stochastic dominance (SD) framework to compare portfolio insurance strategies. First, we provide several sufficient conditions to get stochastic dominance properties for the standard portfolio insurance methods. Second, to extend previous results, we introduce specific statistical tests and simulation methods for computing p-values when examining SD^{j} with j larger than one. We use the test considered by Barret and Donald [

We consider two basic assets that are traded in continuous time during the investment period

with initial value

with non negative initial value

a constant drift term parameterized by

To price options, we use the Black and Scholes formula while taking account of the spread between the empirical and the implied volatility^{3}.

The standard CPPI method consists in a simplified strategy to allocate assets dynamically over time so that its value

The excess of the portfolio value over the floor is called the cushion

Then, the amount allocated on the risky asset (called the exposure

The interesting case is when

Then the cushion value

By applying Itô’s lemma, we obtain:

By using the relation:

we deduce:

Substituting this expression for

with

We deduce that the value of the CPPI portfolio

Note that, for the CPPI method, the two key management parameters are the initial floor value

Remark 2.1. (Capped CPPI) The manager may want to increase his profits, from usual performances of asset S while potentially discarding very high values of S. In that case, the exposure is determined by:

where

In what follows, we describe the option-based portfolio insurance strategy. It provides a guarantee equal to

which implies that

This relation shows that the insured amount at maturity is the exercise price multiplied by the number of shares, i.e.

where

The portfolio value

which implies that:

Therefore, the strike K is an increasing function

Thus, for any investment value

In what follows, we price the option using the implicit volatility

We denote its price by

Remark 2.2. (Capped OBPI) If the manager wants to increase his profit while potentially discarding very high value of S, the options are capped at a level L, as follows. Consider a parameter L higher than the strike K.

The terminal value of the capped OBPI with strike K and parameter L is defined by:

In what follows, we provide several sufficient conditions to get stochastic dominance results as in Zagst and Kraus [

Mosler [

Theorem 3.1. (Mosler [

where

We deduce that, if

For example, we have:

And

The second order stochastic dominance depends on the values taken by the multiple m, the historical volatility

using theorem of Mosler [

Theorem 3.2. Let

Proof. See Appendix A1.

Remark 3.1. Condition

As mentioned by Zagst and Kraus [

Theorem 3.3. (Karlin-Novikov; Mosler [

Let

Denote

The validation of the third order stochastic dominance requires the analysis of the condition

Theorem 3.4. Assuming that

Proof. See Appendix A.2.

Using previous theorems, we deduce:

Theorem 3.5. Let

and

Then, we get:

Proof. Condition

m_{max} implies that

To illustrate these theoretical results, we consider the following numerical example:

Results of

Recall that, if the multiplier_{max} = 3.12,

we have

m = 1 | m = 2 | m = 3 | m = 4 | m = 5 | |
---|---|---|---|---|---|

2.99 | |||||

- | - | ^{*} | ^{*} | ^{*} | |

Condition | - | ^{*} | ^{*} | ^{*} | ^{*} |

3.12 | |||||

- | ^{*} | ^{*} | - | - | |

Third order SD if | - | - | ^{*} | - | - |

The value of the lower bound

The value of the upper bound

is almost always an increasing function of the implied volatility.

Previous stochastic dominance results have been established for the standard cases, i.e. the strategies are not capped. To deal with capped strategies as defined in Remarks (Capped CPPI) and (Capped OBPI), we have to conduct a numerical analysis. In a first step, we simulate the portfolios values using standard Monte Carlo methods; in a second step, we test the stochastic dominance properties.

In the empirical framework, the stochastic dominance has been pioneered for example by Kroll and Levy [

Due to the characterizations of stochastic dominance, it is convenient to represent the various orders of stochastic dominance using the integral operators,

and so on.

The general hypotheses for testing stochastic dominance of G with respect to F at order j can be written compactly as:

In this paper, we test for stochastic dominance using the empirical distribution functions estimated from simulation of the two insurance portfolio strategies. The test of Linton et al. [

The empirical distributions used to construct the tests are respectively given by:

where j denotes the order of dominance and

The statistical test

The linear operator

The second term of the linear operator is derived from Davidson and Duclos [

We have also to define a method in order to obtain the critical value of the test. The standard bootstrap does not work because we need to impose the null hypothesis in that case, which is difficult because it is defined by a complicated system of inequalities. According to Linton et al. [

subsamples

the subsample

with

The underlying rationale is that one can approximate the sampling distribution of

We consider that each of these sub samples is also a sample of the true sampling distribution of the original data.

Following Kläver [

Let

For

・ If

・ If

In this subsection, we apply the tests of stochastic dominance in particular to check if the interval

We note that, for all cases in which the implied volatility far exceeds the historical volatility, the CPPI strategy, with a multiplier equal to 2, dominates the OBPI one.

We can also study the effect of the drift on the third order stochastic dominance (values of drift

As shown in

For lower trend levels and implicit volatility

2 to 9 | 4.5% | 15% | NTSD | |||

2 | 4.5% | 27% | 15% | 0.0187 | TSD | |

[3,9] | 4.5% | 27% | 15% | NTSD | ||

2 | 4.5% | 30% | 15% | 0.2990 | TSD | |

[3,9] | 4.5% | 30% | 15% | NTSD |

0 | NTSD | |||||

NTSD | ||||||

NTSD | ||||||

NTSD |

4 | 1% | 17% | 15% | 0.0305 | TSD |

1% | 17% | 15% | 0 | NTSD | |

1% | 15% | 0 | NTSD | ||

3 | 2% | 18% | 15% | 0.3408 | TSD |

3 | 2% | 19% | 15% | 0.0190 | TSD |

2% | 15% | 0 | NTSD | ||

2% | 15% | 0 | NTSD | ||

2 | 3% | 18% | 15% | 0.0533 | TSD |

2 | 3% | 19% | 15% | 0.3913 | TSD |

15% | 0 | NTSD |

Remark 3.2. To summarize the numerical results:

-We have found that the CPPI method third order stochastically dominates the OBPI one for high implied volatility relatively to the empirical volatility;

-When the interval

-The implied volatility interval where the dominance relation is insured is larger for high values of implied volatility, for low values of the drift and for high values of the multiple.

-The TSD property of the CPPI strategy is rejected for the low values of

-Through this numerical study, we can detect cases of third order stochastic dominance beyond the theoretical cases.

-Finally, when strategies are capped, the TSD property is generally not satisfied^{4}.

In the present paper, we have compared the CPPI and OBPI strategies, mainly with respect to the third stochastic dominance (TSD). We find that the CPPI method third order stochastically dominates the OBPI one for high implied volatility relatively to the empirical volatility. We have checked the TSD of the CPPI method compared to the OBPI method for low values of the drift weighted by high values of the multiplier. We have shown that the relation of SDT is rejected for the low values of the implicit volatility with respect to the statistical one. Further extensions could be based on the use of almost stochastic dominance as defined by Leshno and Levy [

HelaMaalej,Jean-LucPrigent, (2016) On the Stochastic Dominance of Portfolio Insurance Strategies. Journal of Mathematical Finance,06,14-27. doi: 10.4236/jmf.2016.61002

The proof is similar to the proofs of Theorems 2, 3 and 4 of Zagst and Kraus [^{5}.

-The first step consists in proving the following equivalence:

which is also equivalent to:

The proof is straightforward, using usual computations of both

-The second step is to demonstrate that, for

For this purpose, we can note that both the cumulative functions

Therefore, we deduce in particular that the sign of H does change on

For

Then we get:

Therefore,

Now, we introduce the function

1) For

Therefore, if

Standard calculus leads to the following condition:

In that case, we have:

which implies that

2) For

This latter condition is equivalent to

The proof is similar to the proof of Theorem 6 of Zagst and Kraus [

We have to examine the condition

-For the CPPI strategy, we get:

-For the OBPI strategy, we get:

and

with

Then, we get:

from which we deduce:

We have also:

which obviously does not depend on the multiple m.

Introduce now the function g defined by:

The function

Therefore, assuming that

Finally, we deduce that:

Note that condition

corresponds to a whole investment on the risk free asset B.