Didier Maillard

A User’s Guide to the Cornish Fisher Expansion

Using the Cornish Fisher expansion is a relatively easy and parsimonious way of dealing with non-normality in asset price or return distributions, in such fields as insurance asset liability management or portfolio optimization with assets such as derivatives. It also allows to implement portfolio optimization with a risk measure more sophisticated than variance, such as Value-at-Risk or Conditional Value-at-Risk The use of Cornish Fisher expansion should avoid two pitfalls: (i) exiting the domain of validity of the formula; (ii) confusing the skewness and kurtosis parameters of the formula with the actual skewness and kurtosis of the distribution.This paper provides guidelines for a proper use of the Cornish Fisher expansion.

More on Cornish Fisher: Distribution Density and Boundary Conditions

The Cornish Fisher expansion is a mean to transforming a Gaussian distribution into a non-Gaussian distribution, the skewness and the kurtosis of which can be controlled if the transformation is properly implemented.This paper displays the characteristics of the density of the transformed distribution, which may be obtained analytically by reversing the Cornish-Fisher transformation.It also studies what happens when nearing the borders of the domain of validity for the transformation.

Where’s the Value of Wealth Management: Tax Planning or Asset Management?

For savers, the quality of an investment is measured by the efficiency with which it defers purchasing power to the future. This efficiency can in turn be measured by the real rate of return after tax (discounting for inflation) obtained on the investment.Two main factors affect the real rate of return after tax: the allocation and management of assets, and tax planning.Using a simple but realistic model, we show that, given the generally high levels of taxation and the nominal nature of the tax system, tax planning represents a source of value creation of roughly the same importance as the allocation and management of assets.That does not mean that tax considerations should play an exclusive or even dominant role in wealth management, as many examples of misleading tax incentives show. But the high levels of taxation on wealth and its yield now imposed in many countries mean that tax aspects cannot be overlooked.

Cornish-Fisher versus Gramm-Charlier: An Appraisal

Both Cornish-Fisher and Gramm-Charlier expansions are means to transforming a Gaussian distribution into a non-Gaussian distribution, the skewness and the kurtosis of which can be controlled if the transformations are properly implemented. This may be useful for modelling distributions for a wide range of issues, especially in risk assessment and asset pricing.The expansions differ in their nature: Cornish-Fisher is a transformation of a random variable, or of quantiles, meanwhile Gramm-Charlier is a transformation of a probability density. Both transformations must be implemented with care, as their domain of validity does not cover the whole range of possible skewnesses and kurtosis. It appears that the domain of validity of Cornish-Fisher is much wider that the domain of validity of Gramm-Charlier.This, and the fact that Cornish-Fisher provides easily the quantiles of the distribution, gives it an advantage over Gramm-Charlier in several configurations.

Manipulation-Proof Performance Measure and the Cost of Tail Risk

This paper builds on the seminal Goetzmann, Ingersoll, Spiegel and Welch research on Manipulation-Proof Performance Measures (MPPM), with a different purpose. Manipulation of usual performance measures generally goes through taking risk which is not reflected in the second moment measure of return distribution, variance or volatility. The MPPM corrects the impact of tail risk – negative skewness and kurtosis – taken by a fund manager (not necessarily with the explicit aim to manipulate the performance measures). In our paper, we try to quantify, using a Cornish Fisher technology allowing us to control for tail risk, the impact of such risk on the MPPM. In that framework, we find that the MPPM effectively does impose a penalty on tail risk. This penalty increases nearly linearly with return kurtosis and return negative skewness. The size of the penalty is rather benign when return volatility is low or the risk parameter is low. It increases substantially for high volatilities and/or high risk parameters.

Sovereign Risk: A Reappraisal

The Greek drama of the late 2000s has returned sovereign risk awareness to centre stage. The default affected a country with a relatively developed economy. It resulted in huge losses for the value of domestic assets: public debt, but also private debt, equity, real estate and furthermore pension rights and human capital. The burden has, not entirely but importantly, fallen on residents. The questions that arise from the possibility of sovereign default impacting the sovereign’s subjects are how to properly assess the risk, what the fallout from its occurrence would be and what precautionary measures should be taken. International diversification is part of the answer.

Portfolio Management under Constraints on Ex-Ante and Ex-Post Tracking-Errors

Investment fund managers are judged by ex post or realized tracking error rather than the ex ante tracking error, which is unobservable. The main objective of this paper is to examine the relation between the theoretical tracking error that the manager defines ex ante and the realised tracking error that is actually measured ex post. We determine the ex ante tracking error that must be set in order for the ex post tracking error to remain within a given tolerance margin.

The Smallest General-Equilibrium Model in the World - Major Lessons for Tax Policy

General-Equilibrium (GE) models were revived in the mid 1980s, in the computable form allowed by the progresses of information technology. They have been applied to the assessment of many economic policies, especially taxation, international trade and intergenerational transfers. A new recent development consists of dynamic stochastic general-equilibrium models. This paper builds on work done at the time of the emergence of computable GE models, and aims at presenting as simply and pedagogically as possible the concept of general equilibrium, its limitations and some of its lessons, in the field of taxation and tax policy. Those lessons do not always get the attention they deserve.

Stock-Options and Portfolio Management

For many managers who have been granted them, stock-options represent a significant share of their wealth, in certain cases several times the value of other assets. This tends to create a serious unbalance in the structure of their total portfolio, with an excessive exposure to their firm’s stock value, and also an excessive exposure to its volatility. The result of that lack of diversification is that stock-options, unless hedged, may be worth less to the holder than their market, or model, value. The value loss may be quantified. With reasonable parameters for risk aversion, stock-options are worth to the holder 50 percent of their market value if that market value is equivalent to half the value of other assets, 30 percent if it is equivalent to the value of other assets, and 20 percent if it is equivalent to twice that value. The loss is greater the higher the exercise price is: stock-options worth the equivalent of the other assets in market terms are worth to the holder 30 percent for options just at the money, 10 percent for a strike twice the share value, and 85 per cent for very low strikes which make options equivalent to the underlying stock. The gap between market value and value to the holder also increases with stock volatility, whether it comes from a high beta or from a high specific volatility (the latter has a little more influence). It increases with the holder’s risk aversion. Those findings certainly point to efficiency problems in the use of stock-options as a management tool. In addition, it is shown that the loss comes as well from the exposure to the stock’s volatility as from the exposure to the stock’s value. This means that the mitigation of risk that could be expected, in the managed part of wealth, by reducing the exposure to (or even shorting) the risky assets, or better reducing the exposure to the firm’s stock, will be limited, except if the options are largely out of the money. However, a full coverage of options would conflict with their incentive objectives. There is quite an important literature on stock-options and stock grants and their efficiency in terms of compensation tool, and on their tax treatment. This paper builds on that literature but focuses on a portfolio management issue.