Philippe Artzner

Coherent multiperiod risk adjusted values and Bellman’s principle

Starting with a time-0 coherent risk measure defined for “value processes”, we also define risk measurement processes. Two other constructions of measurement processes are given in terms of sets of test probabilities. These latter constructions are identical and are related to the former construction when the sets fulfill a stability condition also met in multiperiod treatment of ambiguity as in decision-making. We finally deduce risk measurements for the final value of locked-in positions and repeat a warning concerning Tail-Value-at-Risk.

Term structure of interest rates: The martingale approach

Martingale methods are used to study interest rate risk in a market with two fundamental assets: savings accounts and zero coupon bonds. Discounted prices of bonds have to be a martingale for a risk-neutral probability. Specifications are given when the instantaneous rate of interest is adapted to a Brownian motion or follows a diffusion.

Credit Risk and Prepayment Option

The paper examines a type of insurance contract for which secondary markets do exist: default risk insurance is implicit in corporate bonds and other risky debts. It applies risk neutral martingale measure pricing to evaluate the option for a borrower with default risk, to prepay a fixed rate loan. A simple “matchbox” example is presented with a spreadsheet treatment.

Risk Measures and Efficient use of Capital

This paper is concerned with clarifying the link between risk measurement and capital efficiency. For this purpose we introduce risk measurement as the minimum cost of making a position acceptable by adding an optimal combination of multiple eligible assets. Under certain assumptions, it is shown that these risk measures have properties similar to those of coherent risk measures. The motivation for this paper was the study of a multi-currency setting where it is natural to use simultaneously a domestic and a foreign asset as investment vehicles to inject the capital necessary to make an unacceptable position acceptable. We also study what happens when one changes the unit of account from domestic to foreign currency and are led to the notion of compatibility of risk measures. In addition, we aim to clarify terminology in the field.

Multiperiod Insurance Supervision: Top-Down Models

We describe a top-down procedure for the supervisory accounting of insurance companies with special emphasis on market impacts. The technical tools are a multiperiod risk assessment, a market consistent best estimate and an eligible asset. First, to avoid supervisory arbitrage by financial market instruments, the risk assessment is bounded by a market consistent best estimate. Applied to the risk bearing capital, i.e. asset value minus best estimate of obligations, the risk assessment immediately gives the free capital which has to be positive for acceptability. Next, optimal hedging of the obligation process by suitable asset portfolios yields the supervisory provision as the minimal initial value of a portfolio acceptable with respect to the given obligations. The problem to attain this minimal value leads to the definition of an optimal replicating portfolio. A further task of supervision is the determination of the “Fremd”-capital in the supervisory balance sheet. This is formalized by the cost-of-capital method, i.e. a fictitious standardized transfer of the obligations to new investors on the market. The regulated price of such a transfer leads to the technical provision and the risk margin as “Fremd”-capital items. Finally, the additional financial risks within the insurance’s real asset portfolio are taken care of by the solvency capital requirement defined as the minimal acceptable “Eigen”-capital for a given business plan. It measures the adequacy or inadequacy of the trading risks incorporated in the portfolio with respect to the obligation risks. An optimal replicating portfolio is characterized by a minimal solvency capital requirement. Solvency II and the Swiss Solvency Test (SST) are defined as bottom-up models. In the forthcoming paper Eisele and Artzner (2011), we shall show how bottom-up and top-down models can be made congruent.

‘Finem Lauda’ or the risks in swaps☆

Abstract The assertion about swaps being mutually beneficial is examined. A simple example of interest rate swaps is first built, to emphasize the point overlooked in usual studies: the parties to a swap may well exchange (interest payments) default risks of different values. Fair prices of swaps and swaptions are then given, in a default free context, as well as in the case of possible default. In the latter case, credit insurance with level premiums requires the study of mathematical reserves.