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Abstract:
We compare performances of the two standard portfolio insurance methods: the Option Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI). First we examine basic properties of these two strategies and compare them by means of various criteria: comparison of their payoffs, possible property of stochastic dominance, expectations, variances, skewness and kurtosis of their returns, and some of the quantiles of their returns. We prove that the OBPI method can be analyzed as a kind of CPPI where the multiple is allowed to vary. We then study the properties of this varying multiple.

In a second section, we analyze more deeply both method's dynamic properties. We turn our attention to the dynamics management involved by these two strategies. Although the pure OBPI do not require any management by the buyer (if the put or call option is available on the market), we can calculate the "greeks" of its call part. We derive the "greeks" of the CPPI and show the very different nature of the dynamic properties of the two strategies.

Portfolio Insurance, OBPI, CPPI

Abstract:
We compare the performances of the two standard portfolio insurance methods: the Option Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI), when the volatility of the stock index is stochastic. In this framework, we provide a quite general formula for the CPPI portfolio value. We use criteria such as comparison of payoffs functions at maturity and various quantiles. We emphasize in particular the role of the insured percentage of the initial investment.

Portfolio insurance, OBPI, CPPI, Stochastic volatility

Abstract:
This paper examines wether risk attribution process is consistent with portfolio optimizations under tracking-error constraints. Since Mina (2003), Bertrand (2005) and Menchero and Hu (2006), risk attribution has been widely used in the performance attribution process.

This article presents an extension of our previous work on risk attribution to others portfolio optimization contexts. It is shown that only optimization under the tracking-error constraint alone is consistent with the risk attribution process. Indeed, as soon as additional constraints (e.g. on total risk) are introduced, the risk attribution method conflicts with the performance attribution process, preventing us from legitimating all the optimal decisions taken by a portfolio manager.

risk attribution, performance attribution, tracking error, portfolio optimization, risk aversion

Abstract:
This paper examines whether the risk-adjusted performance attribution process is consistent with portfolio optimization under tracking-error constraints. Since Mina (2003), Bertrand (2005) and Menchero and Hu (2006), risk attribution has been widely used in the performance attribution process.

This article analyzes and discusses the information ratio decomposition proposed by Menchero (2007) in the light of the analysis of risk-adjusted performance attribution developed in Bertrand (2005). It is also shown that only optimization under the tracking-error constraint alone is consistent with the risk-adjusted performance attribution process. Indeed, as soon as additional constraints (e.g. on total risk) are introduced, the component information ratios of the decisions are no longer uniform nor equal to the information ratio of the whole portfolio. This means that no equilibrium between expected return and relative risk has been reached.

risk-adjusted performance, risk attribution, performance attribution, tracking error, portfolio optimization, risk aversion

Abstract:
Today, the use of a benchmark portfolio is common practice in the financial management industry. This setup allows the investor to evaluate the added value in line with the risks undertaken. But the relevant concept of risk is relative risk as defined by tracking-error volatility.

The problem of minimizing the volatility of tracking error was originally solved by Roll (1992). He noticed that the optimal portfolios obtained have several undesirable properties and then suggested introducing an additional constraint on the beta of the portfolio.

More recently, Jorion (2003) elegantly tackled this problem again, pointing out that constant-TEV portfolios are described by an ellipse. He showed that because of the flat shape of this ellipse, adding a constraint on total portfolio volatility can substantially improve the performance of the managed portfolio.

This paper looks at the problem from another angle. Instead of considering constant TEV frontiers as Jorion does, we allow tracking error to vary but we fix the risk aversion. It is shown that the resulting optimal portfolios have several desirable properties.

benchmark, tracking-error, portfolio optimization, risk aversion

Abstract:
This paper examines some properties of optimal portfolio positioning that are linked with the risk aversion and the prudence of the investor. It introduces the ratio of the degree of absolute prudence on the absolute risk aversion. This one allows the analysis of the degree of convexity/concavity of the optimal portfolio payoff. The role of the prudence is also analyzed.

Risk aversion, Prudence, Portfolio insurance

Abstract:
Portfolio insurance allows investors to recover, at maturity, a given percentage of their initial capital. This limits downside risk in falling markets and allows some participation in rising markets. Therefore, these properties prove the importance of such portfolio strategies. The two standard portfolio insurance methods are the Option Based Portfolio Insurance (OBPI) and the Constant Proportion Portfolio Insurance (CPPI).

The paper analyzes and compares their performances and risk characteristics by means of various criteria such as some of their quantiles. Their dynamic hedging properties are also examined in the Black and Scholes framework. In particular, the paper shows that the insured percentage of the initial capital plays a key role. It is
also proved that OBPI is a generalized CPPI.

optimal portfolio, portfolio insurance, OBPI, CPPI, dynamic hedging

Abstract:
One of the standard insurance portfolio method is the Constant Proportion Portfolio Insurance (CPPI). Using a quantile hedging approach, this paper provides an upper bound on the standard multiple m. This bound is statistically approximated by applying the extreme value theory to the study of extreme variations in rates of asset returns. Furthermore, we examine the distributions of interarrival times of these extreme variations. We analyze their impact on the portfolio guarantee. These results are illustrated on S&P 500 data.

Portfolio Insurance, CPPI, Extreme value theory