# Constant or Variable? A Performance Analysis among Portfolio Insurance Strategies

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Portfolio Insurance Strategies: Definitions and Features

#### 2.1. Constant Proportion Portfolio Insurance Strategy

- The choice of a floor, ${F}_{{t}_{k}}$ which represents the minimum value of the portfolio which is acceptable for an investor at any instant of time during the management period $[0,T].$ Its initial value, ${F}_{{t}_{0}},$ capitalized at the non-risky rate, must be equal to a predetermined percentage of the initial capital deposit;
- The choice of a dynamic investment rule on the risky asset defined as follows: the total amount ${E}_{{t}_{k}}$ (the exposure) invested into the underlying asset ${S}_{{t}_{k}}$ is equal to a fixed proportion m (the multiplier) of the difference between the portfolio value ${V}_{{t}_{k}}^{CPPI}$ and the floor ${F}_{{t}_{k}}$. Such a difference is called the cushion and is denoted by ${C}_{{t}_{k}}$. Since the strategy results to be self-financing, the remaining amount, ${V}_{{t}_{k}}^{CPPI}-{F}_{{t}_{k}}$, is invested into the riskless asset ${B}_{{t}_{k}}$, such as a money market account or a government bond, with log-return ${r}_{{t}_{k}}$ for each period $[{t}_{k-1},{t}_{k}].$

#### 2.2. Time Invariant Portfolio Protection Strategy

#### 2.3. Exponential Proportion Portfolio Insurance Strategy

#### 2.4. Practical Issues for the Implementation of Portfolio Insurance Strategies

## 3. Simulation Setup

#### 3.1. Data and Design of Empirical Analysis

- We randomly draw a market index (S&P500, Hang Seng, Nikkei 225 or FTSE 100) with replacement;
- We draw with replacement a starting date;
- Starting from the initial date obtained in Step 2, we analyze the one-year performance of CPPI, TIPP, and EPPI strategies for the drawn market, i.e., the 252 days following the starting date are used to evaluate the different portfolio insurance strategies;
- The procedure (Step 1–Step 3) is repeated 20,000 times.

#### 3.2. Performance Measures and Statistical Tests

## 4. Performance Measurement Results

#### 4.1. Constant Proportion Portfolio Insurance vs. Its Generalizations

#### 4.2. Changing the Protection Level

#### 4.3. Changing the Rebalancing Frequency

## 5. Concluding Remarks

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Comparison between the payoff functions of CPPI and its generalizations, with daily rebalancing and $0.1$% transaction costs. The top chart compares CPPI and TIPP in terms of excess returns, and the bottom chart compares CPPI and EPPI in terms of excess returns. The dashed horizontal line indicates the floor values and the solid line is introduced to facilitate the interpretation.

Reference | Dynamic PI Strategies | Static PI | Methodology | ||
---|---|---|---|---|---|

CPPI | TIPP | EPPI | Strategies | (Model-Free) | |

Cesari and Cremonini (2003) | ✔ | ✔ | |||

Lee et al. (2008) | ✔ | ✔ | |||

Annaert et al. (2009) | ✔ | ✔ | ✔ | ||

Dichtl and Drobetz (2011) | ✔ | ✔ | ✔ | ||

Hamidi et al. (2014) | ✔ | ||||

Zieling et al. (2014) | ✔ | ||||

Ardia et al. (2016) | ✔ | ✔ | ✔ | ||

Dichtl et al. (2017) | ✔ | ✔ | ✔ | ||

Chen et al. (2022) | ✔ | ✔ | |||

Our work | ✔ | ✔ | ✔ | ✔ |

**Table 2.**Summary statistics for S&P500, Hang Seng, Nikkei 225, and FTSE 100 index returns. Each time series contains 7511 returns. The daily average returns are reported on an annual basis using ${\overline{r}}_{annual}={(1+{\overline{r}}_{daily})}^{252}-1.$ Daily standard deviations are transformed into annual standard deviations using ${\sigma}_{annual}={\sigma}_{daily}\xb7\sqrt{252},$ where we assume 252 trading days per year.

Series | Average Return (%) | Standard Deviation ($\mathit{\sigma}$) (%) | Skewness | Kurtosis | p-Value Autocorrelation (Ljung-Box Test) | p-Value Heteroscedasticity (Engle’s ARCH Test) |
---|---|---|---|---|---|---|

UKX | $6.044$ | $17.288$ | $0.120$ | $10.374$ | $0.000$ | $0.000$ |

HSI | $11.322$ | $25.268$ | $-0.120$ | $18.748$ | $0.000$ | $0.000$ |

NKY | $1.362$ | $23.415$ | $-0.006$ | $9.169$ | $0.001$ | $0.000$ |

SPX | $9.449$ | $17.518$ | $-0.142$ | $11.814$ | $0.000$ | $0.000$ |

**Table 3.**Comparison among CPPI, TIPP, and EPPI strategies. We denote by *, ** and *** the significant difference between the reference strategy (CPPI) and the TIPP (resp. EPPI) strategy, at a $10\%,\phantom{\rule{0.166667em}{0ex}}5\%$, and $1\%$ confidence level, respectively.

Portfolio Insurance Strategy | CPPI | TIPP | EPPI | EPPI | EPPI |
---|---|---|---|---|---|

(a = 5) | (a = 10) | (a = 20) | |||

Rebalancing discipline | Daily | Daily | Daily | Daily | Daily |

Protection level (%) | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ |

Multiplier | 14 | 14 | - | - | - |

$\eta $ | - | - | 14 | 14 | 14 |

Initial equity allocation (%) | $35.27$ | $35.27$ | $35.27$ | $35.27$ | $35.27$ |

Average excess return | $1.482$ | $0.213$ *** | $1.501$ | $1.519$ | $1.559$ |

Standard deviation | $15.616$ | $5.666$ *** | $15.828$ | $16.095$ *** | $16.660$ *** |

Sharpe ratio | $0.095$ | $0.038$ *** | $0.095$ | $0.094$ | $0.094$ |

% $<0$ | $75.115$ | $53.660$ *** | $76.010$ ** | $76.955$ *** | $77.910$ *** |

Average negative excess return | $-4.753$ | $-4.076$ | $-4.786$ | $-4.855$ *** | $-5.087$ *** |

VaR 5% | $-8.488$ | $-7.584$ *** | $-8.508$ * | $-8.644$ *** | $-9.456$ *** |

ES 5% | $-10.324$ | $-8.626$ *** | $-10.419$ *** | $-10.535$ *** | $-10.823$ *** |

Skewness | $3.525$ | $0.688$ | $3.493$ | $3.452$ | $3.368$ |

Omega measure | $1.413$ | $1.096$ | $1.410$ | $1.404$ | $1.391$ |

**Table 4.**Comparison among CPPI, TIPP, and EPPI strategies for different volatility market scenarios with daily rebalancing and assuming a fixed protection level. The volatility is $9.66$%, $17.90$%, and $27.28$% for the low-, medium-, and high-volatility scenarios, with 6667, 6666, and 6667 observations, respectively. We denote by *, ** and *** the significant difference between the reference strategy (CPPI) and the TIPP (resp. EPPI) strategy, at a $10\%,\phantom{\rule{0.166667em}{0ex}}5\%$, and $1\%$ confidence level, respectively.

Volatility Subgroup Rebalancing Discipline | Low Volatility Regime Daily | Medium Volatility Regime Daily | High Volatility Regime Daily | ||||||
---|---|---|---|---|---|---|---|---|---|

Portfolio insurance strategy | CPPI | TIPP | EPPI | CPPI | TIPP | EPPI | CPPI | TIPP | EPPI |

Protection level (%) | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ |

Multiplier | 14 | 14 | - | 14 | 14 | - | 14 | 14 | - |

$\eta $ | - | - | 14 | - | - | 14 | - | - | 14 |

a | - | - | 20 | - | - | 20 | - | - | 20 |

Initial equity allocation (%) | $64.25$ | $64.25$ | $64.25$ | $35.27$ | $35.27$ | $35.27$ | $95.28$ | $95.28$ | $95.28$ |

Average excess return | $3.683$ | $2.396$ *** | $3.657$ | $2.375$ | $0.687$ *** | $2.463$ | $-1.613$ | $-2.443$ *** | $-1.442$ |

Standard deviation | $11.286$ | $3.426$ *** | $12.319$ *** | $17.326$ | $5.494$ *** | $18.315$ *** | $16.996$ | $6.508$ *** | $18.202$ *** |

Sharpe ratio | $0.326$ | $0.699$ *** | $0.297$ *** | $0.137$ | $0.125$ | $0.134$ | $-0.095$ | $-0.375$ *** | $-0.079$ *** |

%$<0$ | $53.817$ | $24.269$ *** | $60.132$ *** | $79.283$ | $55.476$ *** | $81.263$ *** | $92.245$ | $81.238$ *** | $92.335$ |

Average negative excess return | $-3.238$ | $-2.036$ *** | $-3.835$ *** | $-4.792$ | $-3.427$ *** | $-5.182$ *** | $-5.604$ | $-5.128$ *** | $-5.819$ *** |

VaR 5% | $-6.736$ | $-2.783$ *** | $-7.335$ *** | $-8.877$ | $-7.216$ *** | $-9.779$ *** | $-8.776$ | $-8.350$ *** | $-9.989$ *** |

ES 5% | $-8.933$ | $-5.007$ *** | $-9.486$ *** | $-9.861$ | $-7.683$ *** | $-10.318$ *** | $-11.781$ | $-10.037$ *** | $-12.244$ *** |

Skewness | $1.924$ | $-0.078$ | $1.722$ | $2.511$ | $0.635$ | $2.390$ | $5.197$ | $1.812$ | $4.970$ |

Omega measure | $3.108$ | $5.836$ | $2.581$ | $1.622$ | $1.360$ | $1.583$ | $0.684$ | $0.412$ | $0.732$ |

**Table 5.**Comparison among CPPI, TIPP, and EPPI strategies for different protection levels with daily rebalancing. Within each strategy, the benchmark is the one with the highest protection level (PL = 97.5%). We denote by *, ** and *** the significant difference between the strategies with lower protection levels and the reference one at a $10\%,\phantom{\rule{0.166667em}{0ex}}5\%$, and $1\%$ confidence level, respectively.

Portfolio Insurance Strategy | CPPI | TIPP | EPPI | ||||||
---|---|---|---|---|---|---|---|---|---|

Rebalancing discipline | Daily | Daily | Daily | ||||||

Protection level (%) | 90 | 95 | $97.5$ | 90 | 95 | $97.5$ | 90 | 95 | $97.5$ |

Multiplier | 14 | 14 | - | 14 | 14 | - | 14 | 14 | - |

$\eta $ | - | - | 14 | - | - | 14 | - | - | 14 |

a | - | - | 20 | - | - | 20 | - | - | 20 |

Initial equity allocation (%) | $140.25$ | $70.27$ | $35.27$ | $140.25$ | $70.27$ | $35.27$ | $150.00$ | $75.28$ | $37.79$ |

Average excess return | $3.644$ *** | $2.369$ *** | $1.482$ | $2.015$ *** | $0.752$ *** | $0.213$ | $3.739$ *** | $2.480$ *** | $1.550$ |

Standard deviation | $23.647$ *** | $19.174$ *** | $15.616$ | $20.435$ *** | $10.474$ *** | $5.666$ | $23.941$ *** | $19.740$ *** | $16.352$ |

Sharpe ratio | $0.154$ *** | $0.124$ *** | $0.095$ | $0.099$ *** | $0.072$ *** | $0.038$ | $0.156$ *** | $0.126$ *** | $0.095$ |

% $<0$ | $60.365$ *** | $70.455$ *** | $75.115$ | $60.090$ *** | $56.995$ *** | $53.660$ | $59.635$ *** | $70.600$ *** | $77.290$ |

Average negative excess return | $-11.455$ *** | $-6.924$ *** | $-4.753$ | $-11.327$ *** | $-6.707$ *** | $-4.076$ | $-11.864$ *** | $-7.259$ *** | $-4.932$ |

VaR 5% | $-15.713$ *** | $-10.881$ *** | $-8.488$ | $-15.674$ *** | $-10.533$ *** | $-7.584$ | $-15.863$ *** | $-11.006$ *** | $-8.929$ |

ES 5% | $-17.232$ *** | $-12.506$ *** | $-10.324$ | $-17.229$ *** | $-12.046$ *** | $-8.626$ | $-17.473$ *** | $-12.798$ *** | $-10.656$ |

Skewness | $1.955$ | $2.793$ | $3.525$ | $1.828$ *** | $1.210$ *** | $0.688$ | $1.901$ | $2.712$ | $3.442$ |

Omega measure | $1.526$ | $1.484$ | $1.413$ | $1.295$ | $1.196$ | $1.096$ | $1.527$ | $1.482$ | $1.404$ |

**Table 6.**Comparison among CPPI, TIPP, and EPPI strategies for different rebalancing frequencies. Within each strategy, the benchmark is the one with the highest rebalancing frequency (daily). We denote by *, ** and *** the significant difference between the strategies with lower frequencies and the reference one at a $10\%,\phantom{\rule{0.166667em}{0ex}}5\%$, and $1\%$ confidence level, respectively.

Portfolio Insurance Strategy | CPPI | TIPP | EPPI | ||||||
---|---|---|---|---|---|---|---|---|---|

Rebalancing discipline | Daily | Weekly | Monthly | Daily | Weekly | Monthly | Daily | Weekly | Monthly |

Protection level (%) | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ | $97.5$ |

Multiplier | 14 | 14 | - | 14 | 14 | - | 14 | 14 | - |

$\eta $ | - | - | 14 | - | - | 14 | - | - | 14 |

a | - | - | 20 | - | - | 20 | - | - | 20 |

Initial equity allocation (%) | $35.27$ | $35.27$ | $35.27$ | $35.27$ | $35.27$ | $35.27$ | $37.79$ | $37.79$ | $37.79$ |

Average excess return | $1.482$ | $2.051$ *** | $2.383$ *** | $0.213$ | $0.495$ *** | $0.805$ *** | $1.559$ | $2.190$ *** | $2.480$ *** |

Standard deviation | $15.616$ | $16.866$ *** | $18.251$ *** | $5.666$ | $6.273$ *** | $7.408$ *** | $16.660$ | $17.462$ *** | $18.747$ *** |

Sharpe ratio | $0.095$ | $0.122$ *** | $0.131$ *** | $0.038$ | $0.079$ *** | $0.109$ *** | $0.094$ | $0.125$ *** | $0.132$ *** |

% $<0$ | $75.115$ | $71.665$ *** | $66.850$ *** | $53.660$ *** | $51.280$ *** | $47.900$ *** | $77.910$ *** | $72.735$ *** | $67.525$ *** |

Average negative excess return | $-4.753$ | $-5.327$ *** | $-6.619$ *** | $-4.076$ *** | $-4.443$ *** | $-5.256$ *** | $-5.087$ | $-5.527$ *** | $-6.845$ |

VaR 5% | $-8.488$ | $-9.975$ *** | $-12.866$ *** | $-7.584$ | $-8.009$ *** | $-9.319$ *** | $-9.458$ | $-10.157$ *** | $-13.383$ *** |

ES 5% | $-10.324$ | $-12.523$ *** | $-18.520$ *** | $-8.626$ | $-9.813$ *** | $-13.091$ *** | $-10.823$ | $-13.178$ *** | $-19.160$ *** |

Skewness | $3.525$ | $3.153$ | $2.626$ | $0.688$ | $0.600$ | $0.332$ | $3.368$ | $3.042$ | $2.529$ |

Omega measure | $1.413$ | $1.535$ | $1.536$ | $1.096$ | $1.216$ | $1.318$ | $1.391$ | $1.542$ | $1.535$ |

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**MDPI and ACS Style**

Mancinelli, D.; Oliva, I.
Constant or Variable? A Performance Analysis among Portfolio Insurance Strategies. *Risks* **2023**, *11*, 105.
https://doi.org/10.3390/risks11060105

**AMA Style**

Mancinelli D, Oliva I.
Constant or Variable? A Performance Analysis among Portfolio Insurance Strategies. *Risks*. 2023; 11(6):105.
https://doi.org/10.3390/risks11060105

**Chicago/Turabian Style**

Mancinelli, Daniele, and Immacolata Oliva.
2023. "Constant or Variable? A Performance Analysis among Portfolio Insurance Strategies" *Risks* 11, no. 6: 105.
https://doi.org/10.3390/risks11060105