# How risk managers should fix tracking error volatility and value-at-risk constraints in asset management

## Luca Riccetti

#### Need to know

• A consistent set of limits that risk management offices should impose is determined.
• A limit on maximum VaR is usually better than a limit on maximum variance.
• The set of limits includes lower and upper limits on TEV, and upper limit on VaR.

#### Abstract

ABSTRACT

Investors usually assign part of their funds to asset managers, who are given the task of beating a benchmark. Asset managers usually face a constraint on maximum tracking error volatility (TEV), which is imposed by the risk management office. In the mean-variance space, Jorion, in his 2003 paper "Portfolio optimization with tracking-error constraints", shows that this constraint determines an ellipse containing all admissible portfolios. However, many admissible portfolios have problems in mean-variance terms, for example, because of an overly high variance. To overcome this problem, Jorion also fixes a constraint on variance, while, in their 2008 paper "Active portfolio management with benchmarking: adding a value-at-risk constraint", Alexander and Baptista fix a constraint on value-at-risk (VaR). In this paper, I determine an optimal value for a set of limits composed of the lower limit on TEV, the upper limit on TEV and the upper limit on VaR. To fix the upper limit on VaR, I use the TEV constrained efficient frontier developed in Palomba and Riccetti's 2013 paper "Asset management with TEV and VAR constraints: the constrained efficient frontiers", which is the set of portfolios that is on Jorion's ellipse and not dominated from the mean-variance perspective. In particular, I develop a strategy to impose on asset managers a set of portfolios that contains as many TEV constrained efficient portfolios and as few inefficient portfolios as possible. Moreover, I show that a limit on maximum VaR is usually better than a limit on maximum variance.