A recursive algorithm for multivariate risk measures and a set-valued Bellman's principle

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is calculated recursively backwards in time along the event tree. We motivate why the proposed procedure can be seen as a set-valued Bellman's principle, that might be of independent interest within the growing field of set optimization. We give conditions under which the backwards calculation of the sets reduces to solving a sequence of linear, respectively convex vector optimization problems. Numerical examples are given and include superhedging under illiquidity, the set-valued entropic risk measure, and the multi-portfolio time consistent version of the relaxed worst case risk measure and of the set-valued average value at risk.Comment: 25 pages, 5 figure

Taxation, Risk, and Portfolio Choice: The Treatment of Returns to Risk Under a Normative Income Tax

Many articles in the legal and economic literature claim that a pure Haig-Simons income tax cannot effectively tax investment income. This is because an investor can use leverage to gross up her investments in risky assets such that the increased gain (or loss) exactly offsets any income tax (or deduction) on the returns to risk-taking. This article argues, however, that while it is possible for an investor to make such portfolio shifts, she almost certainly will not because of the increased risk of doing so. Central to any discussion of the effects of taxation on investment risk-taking is the meaning of risk itself. The central claim of this article is that a better conception of investment risk is the risk of loss and not merely the variance of returns. Applying this notion of risk—one that is well supported in the finance literature but new to the taxation-and-risk literature—to an investor’s portfolio choice question shows that an investor will not increase her investment in risky assets by enough to offset the tax. As a result, there is an effective tax on investment risk-taking under a normative income tax

Discounting Rules for Risky Assets

This paper develops a rule for calculating a discount rate to value risky projects. The rule assumes that asset risk can be measured by a single index (e.g., beta), but makes no other assumptions about specific forms of the asset pricing model. It treats all projects as combinations of two assets: Treasury bills and the market portfolio. We know how to value each of these assets under any theory of debt and taxes and under any assumption about the slope and intercept of the market line for equity securities. Our discount rate is a weighted average of the after-tax return on riskless debt and the expected return on the portfolio, where the weight on the market portfolio is beta.

An Accurate Solution for Credit Value Adjustment (CVA) and Wrong Way Risk

This paper presents a new framework for credit value adjustment (CVA) that is a relatively new area of financial derivative modeling and trading. In contrast to previous studies, the model relies on the probability distribution of a default time/jump rather than the default time itself, as the default time is usually inaccessible. As such, the model can achieve a high order of accuracy with a relatively easy implementation. We find that the prices of risky contracts are normally determined via backward induction when their payoffs could be positive or negative. Moreover, the model can naturally capture wrong or right way risk.

A unified pricing of variable annuity guarantees under the optimal stochastic control framework

In this paper, we review pricing of variable annuity living and death guarantees offered to retail investors in many countries. Investors purchase these products to take advantage of market growth and protect savings. We present pricing of these products via an optimal stochastic control framework, and review the existing numerical methods. For numerical valuation of these contracts, we develop a direct integration method based on Gauss-Hermite quadrature with a one-dimensional cubic spline for calculation of the expected contract value, and a bi-cubic spline interpolation for applying the jump conditions across the contract cashflow event times. This method is very efficient when compared to the partial differential equation methods if the transition density (or its moments) of the risky asset underlying the contract is known in closed form between the event times. We also present accurate numerical results for pricing of a Guaranteed Minimum Accumulation Benefit (GMAB) guarantee available on the market that can serve as a benchmark for practitioners and researchers developing pricing of variable annuity guarantees.Comment: Keywords: variable annuity, guaranteed living and death benefits, guaranteed minimum accumulation benefit, optimal stochastic control, direct integration metho

Gauging risk with higher moments : handrails in measuring and optimising conditional value at risk

The aim of the paper is to study empirically the influence of higher moments of the return distribution on conditional value at risk (CVaR). To be more exact, we attempt to reveal the extent to which the risk given by CVaR can be estimated when relying on the mean, standard deviation, skewness and kurtosis. Furthermore, it is intended to study how this relationship can be utilised in portfolio optimisation. First, based on a database of 600 individual equity returns from 22 emerging world markets, factor models incorporating the first four moments of the return distribution have been constructed at different confidence levels for CVaR, and the contribution of the identified factors in explaining CVaR was determined. Following this the influence of higher moments was examined in portfolio context, i.e. asset allocation decisions were simulated by creating emerging market portfolios from the viewpoint of US investors. This can be regarded as a normal decisionmaking process of a hedge fund focusing on investments into emerging markets. In our analysis we compared and contrasted two approaches with which one can overcome the shortcomings of the variance as a risk measure. First of all, we solved in the presence of conflicting higher moment preferences a multi-objective portfolio optimisation problem for different sets of preferences. In addition, portfolio optimisation was performed in the mean-CVaR framework characterised by using CVaR as a measure of risk. As a part of the analysis, the pair-wise comparison of the different higher moment metrics of the meanvariance and the mean-CVaR efficient portfolios were also made. Throughout the work special attention was given to implied preferences to the different higher moments in optimising CVaR. We also examined the extent to which model risk, namely the risk of wrongly assuming normally-distributed returns can deteriorate our optimal portfolio choice. JEL Classification: G11, G15, C6

Extend the ideas of Kan and Zhou paper on Optimal Portfolio Construction under parameter uncertainty

In this dissertation, we extend the ideas of Raymond Kan and Guofu Zhou for optimal portfolio construction under parameter uncertainty. Kan and Zhou proved analytically that under parameter uncertainty, investing in the sample tangency portfolio and the riskless is not optimal. Based on this idea we will approach the portfolio construction under parameter uncertainty in a different way. We will optimise the expected out-of-sample performance of a portfolio using a numerical approach. Using Monte Carlo simulations we will develop an algorithm that calculates the expected out-of-sample performance of any portfolio rule. We will then extend this algorithm in order to be able to input new portfolio rules and test their performance.\ud \ud The new portfolio rules we introduce are based on shrinkages for the mean and covariance matrix of the assets returns. These shrinkages will have some parameters that will be chosen so that we optimise the expected out-of-sample performance of the input portfolio rule. A comparison is then done between the portfolio rules we introduce and Kan and Zhou portfolio rules

Assessing household credit risk: evidence from a household survey

This paper investigates the main individual driving forces of Hungarian household credit risk and measures the shockabsorbing capacity of the banking system in relation to adverse macroeconomic events. The analysis relies on survey evidence gathered by the Magyar Nemzeti Bank (MNB) in January 2007. Our study presents three alternative ways of modelling household credit risk, namely the financial margin, the logit and the neural network approaches, and uses these methods for stress testing. Our results suggest that the main individual factors affecting household credit risk are disposable income, the income share of monthly debt servicing costs, the number of dependants and the employment status of the head of the household. The findings also indicate that the current state of indebtedness is unfavourable from a financial stability point of view, as a relatively high proportion of debt is concentrated in the group of risky households. However, risks are somewhat mitigated by the fact that a substantial part of risky debt is comprised of mortgage loans, which are able to provide considerable security for banks in the case of default. Finally, our findings reveal that the shock-absorbing capacity of the banking sector, as well as individual banks, is sufficient under the given loss rate (LGD) assumptions (i.e. the capital adequacy ratio would not fall below the current regulatory minimum of 8 per cent) even if the most extreme stress scenarios were to occur.financing stability, financial margin, logit model, neural network, stress test.