where

AIj = the fractional return of the irh underlying foreign investment component at a given time, r, and over a time horizon, At; /,- = /,( , t) = (It - It-At)/h-At, where /, are in units of the home currency,

at = the amount of the portfolio allocated to the ih currency in units of the home currency,

hf = the unitless static hedging ratio for the ih foreign currency, ARfx = the portfolio returns due to fluctuations in the static hedge positions, - the weights given to each trading model (in units of the home currency), and

ARTM = the trading model return values.

The risk of the portfolio is characterized by the variance of the portfolio returns :

= [( - [ ])2] (11.38)

11.8.3 Dynamic Hedging with Exposure Constraints

To compute the efficient frontier of the dynamic hedging strategy, we need to optimize the return given the risk or conversely optimize the risk given the return. Such an optimization is suitably done using the Lagrange multiplier technique. We can maximize the quantity16 [ ] - < or, more conveniently, minimize ° - ]. The parameter A. is the Lagrange multiplier and can be varied from 0 (considering risk only) to high positive values (considering mainly return while keeping risk under some control) to get the whole efficient frontier. Each value of X corresponds to a point on the efficient frontier. Let us call U the target function to be minimized,

U = - ] (11.39)

where [ ] and < have been defined and expressed in Equations 11.37 and 11.38.

Some components of the total portfolio have free coefficients, which can be determined with the goal of minimizing the target function U: the hedging ratios hi and the amounts of money (maximum exposures) coj allocated to the trading models. These coefficients are generally subject to constraints:

1. The trading model sizes coj must not be negative: toj > 0. In fact, such negative a>j would mean doing the opposite of the trading recommendations. In these cases, this will lead to new transaction costs, which make the actual returns much worse than the formal results suggest. Therefore, we exclude negative trading model coefficients.

16 This first quantity is very similar to Xeg, the risk-corrected return derived in Section 11.3.1. The optimization problem comes back to optimizing Xeg for different risk aversion constants.

2. The static hedging ratios / ,- are limited to an allowed range between hm\nj and hmaxj. Generally there are regulations or risk management rules that do not allow the investors to take too extreme positions, therefore hedging ratios that are above /imax,; or below /imin,/ are generally excluded.

The constraints make a direct solution of the allocation problem impossible. The minimization of U (a quadratic function of the coefficients) under linear constraints on the coefficients is a special case of quadratic programming. A technique developed by Markowitz (1959), based on the simplex method, can solve this problem; we follow Markowitz (1987). There is no local minimum of U in the space of the coefficients; this can be proven. The solution, once found, always represents the global minimum.

Currency-wise exposure limits may be less rigid than those on the hedging ratios. But exposure limits on foreign currencies are not simple, as they involve several assets together. The exposure constraints are linear in coj and / ,. As the simpler constraints, they can be fully accounted for in the framework of the solution method presented by Markowitz (1987).

The whole efficient frontier can be obtained by minimizing U for different choices of the Lagrange multiplier X. The left-hand side of the efficient frontier obviously starts at X = 0 where the return of the portfolio has no influence and only risk counts. The right-hand side limitation of the efficient frontier is less evident. For unconstrained dynamic strategies, the efficient frontier normally extends to infinity (when more and more money is allocated to trading models). A hedging strategy should obviously not deal with infinite risks. There should be a risk limit beyond which the strategies are considered unacceptable even if the formal exposure limits are not yet reached. But in the case of constraints, the efficient frontier generally has a genuine end on the right-hand side. Therefore, we use a "shooting" strategy for determining the X value that approximately leads to the desired maximum risk at the right end of the efficient frontier. The method by Markowitz (1987) helps us to find this value.

I 1.8.4 Concluding Remarks

This problem turns out to bc rather difficult and requires many different inputs and programs to solve it in a practical way. The algorithm needs the following main ingredients:

The user-defined investment goals such as primary investment and exposure limits.

Time series of returns of all assets in the portfolio: FX rates, interest rates (or their differentials), and typical primary investments such as stock indices and bond indices of many currencies.

Time series of returns of the trading models used for dynamic hedging (including transaction costs).

The computation of mean returns and the covariance matrix of all relevant assets.

A target function to be optimized, measuring risk and return, with a Lagrange multiplier that sets the balance between risk and return. This function depends on two types of parameters, which are static hedging ratios and trading model allocation sizes.

A method to solve the quadratic programming problem with linear constraints (exposure limits).

In conclusion, we are able to solve the dynamic overlay problem under exposure constraints with a quite complex but well-understood algorithm. The result is the efficient frontier of feasible solutions, each with a particular risk/return profile. It is the choice of the investor to decide which portfolio on the efficient frontier he/she wants to follow.