Solving the dual-benchmark problem

Solving the dual-benchmark problem

Consider a pension fund. The fund seeks market returns but pays fixed-income liabilities. Given
the choice between 2 portfolios having the same active risk and return but different absolute
volatility, the fund would almost certainly (unless it is a hedge against other assets) be better off
with the less volatile portfolio. Or consider the business of portfolio management. A fund is
mandated to be around a benchmark, but it is performance against a peer group of competing
funds that determines whether the fund attracts and retains assets. In both these cases, the
standard mean-tracking variance objective does not fully reflect a portfolio manager’s
preferences. Adding a second benchmark, a risk-free asset in the first case, a composite of
competitors in the second, captures what a manager wants more accurately. The literature
1presents a number of compelling arguments for dual benchmark risk control ; at Northfield, client
requests have additionally convinced us of its practical importance.

The Framework

The starting point is the standard single-benchmark objective. A manager likes return and
dislikes active risk and transaction costs.

Utility(p) = α(p) – λTV(p,b) – f(p)

where
p = portfolio
b = benchmark
α = return
λ = tracking risk aversion
TV(p,b) = tracking variance between p & b
f(p) = convex function for additional terms – transaction costs, etc.


Suppose a manager’s preferences are known ...