# Optimization

## 1. Introduction

OPTI+ is an optimization tool for hedging the risk of a multi-currency position with the assumption that the user’s view of the exchange rates is different from the forward rates implied by the market. (Hence, the user expects some outcome - positive or negative - for open currency positions which will be realized if the user’s view is correct.).

Setting a combination of the expected return and volatility as its target function, OPTI+ defines the optimal hedging portfolio which will maximize the return and minimize the volatility of the position left open after the hedging.

The optimization is constrained by a constant average hedging level, i.e. a constant percentage of the whole currency position (measured in terms of the home currency) that is being hedged. Also, the user can set arbitrary boundaries for the hedging of each individual currency position. The user can regulate his risk aversion by a coefficient which determines how many units of return are required for each excessive unit of risk (i.e. volatility). OPTI+ also calculates efficient frontiers as a function of this risk aversion coefficient.

OPTI+, version 1.0, is a stand alone solution which can be operated in the standard Mathematica interface, i.e., notebooks. (However, it is possible to modify the version so that it can be integrated to a VAR system. This document is written about OPTI+, version 1.0.)

## 2. OPTI+ Methodology

OPTI+ is an optimizing tool which minimizes the volatility and maximizes the return on a given portfolio. When running the optimization procedure, OPTI+ searches for the best possible combination of open positions, where we dislike volatility and like return. In order to combine two contradicting objectives, we assign specific weights for the two different target functions (the return function and the risk function) and add them up to form the optimization function.

The trade-off between volatility and return is captured by lambda (l), which is equal to the concept of market price of risk developed in the context of Capital Asset Pricing Model in the modern portfolio theory. For a given value of lambda, the optimal portfolio is such that the marginal change in volatility will result in a marginal change in return with multiplier lambda. In other words, if lambda = 1/3, the optimum is characterized by saying that we receive (marginally) one third of percent additional return, whereas we accept one percent additional volatility.

The set of efficient choices with different values of lambda is depicted in the efficient frontier. OPTI+ can draw the efficient frontier by solving the optimization problem for different values of lambda. In normal use, it is likely that, if we allow lambda to vary from 0 to 1, we obtain all solutions that can be considered given a certain size of the currencies. Different frontiers can be set for different currency hedging levels. Scenarios can be seen within a certain frontier, if the hedging level is fixed; if the hedging level of each currency is flexible, then scenarios can be studied also for different frontier platforms.

The special feature of OPTI+ is that it can take into account two different types of limits for hedging:

• the limit for overall hedge ratio, and
• the individual limits for currencies.

The overall limit is defined as a percentage of the total position: if we have an original exposure of 1 000 units, we can set a limit that the hedge ratio is 60 %, implying that the sum of open positions is 400.

The second class of restrictions is constituted by the individual lower and upper boundaries for currencies. The individual boundary shows what hedging ratio is selected for each currency of the portfolio. Individual boundaries are given as percentages of the original position. In the default configuration, OPTI+ determines automatically and for all currencies that the lower limit is 0 % and the upper limit is 100 % of the original position.

The upper and lower limits and their default values can be changed.

The user can ‘freeze’ variables i.e., set them as constants, if the upper and lower limits are equal. This helps to drop out currencies that are too small to be covered, but allows for their effect on the risk-return characteristics to be taken into account.

Another feature of OPTI+ is its ability to automatically freeze currencies where the exposure is below the given limit.

## 3. Market and portfolio data

### 3.1 Market data

OPTI+ reads all market and portfolio data from ASCII text files. If needed, the user may change any data during the analysis session by using Mathematica commands. The essential market input data consists of:

• Correlation matrix of foreign exchange rates;
• Volatility vector of exchange rates;
• Spot rate vector;
• Forward rate vector;
• Forecast exchange rates.

All input data should reflect an equal time frame, i.e., the volatilities should be calculated for the same time period as the forward rates and forecast rates.

It is essential to note, that if the user considers forward rates as best estimates for future rates, then the optimization problem is only to minimize risk. However, if other estimates than forward rates are available, the objective of the optimization has two dimensions: risk and return. The analysis will be carried out as a standard portfolio theory, where efficient frontier shows the set of optimal choices.

### 3.2 Portfolio data

We use the following terms:

• ‘gross’ to refer to the original exposure;
• ‘open’ to refer to the current exposure with the existing hedging transactions;
• ‘hedge’ to refer to any hedging vector OPTI+ suggests (not taking current hedges into account);
• ‘change’ to refer to the difference between the hedge strategy suggested by OPTI+ and current hedging transactions.

The system needs to know original exposures or, in other words, the gross position. In excess, it needs to know the vector of current hedges to show the proposed action.

## 4. User interface

OPTI+ is written with Mathematica development tool and is run under the Mathematica User Interface (MUIF). In order to use OPTI+ one needs to have a Mathematica system installed in the computer. OPTI+ toolbox will run on any system that runs Mathematica, but, since the file access modules are operating system specific, they may need to be tailored for the customer. The default operating system for OPTI+ installation is Windows NT.

The MUIF is a command interpreter and, after loading the OPTI+ package, the system is able to understand the commands of OPTI+ toolbox (in excess to the standard Mathematica commands). This means that the user of version 1.0 of OPTI+ needs to have the basic skills in Mathematica. Nevertheless, the use of Mathematica provides great flexibility and endless possibilities for further development by the user.

## 5. Note

For all our products, full demonstrations along with case studies are available on an individually tailored basis. For more information please contact:

CD Financial Technology
Aikatalo
Mikonkatu 8
FIN-00100 HELSINKI
FINLAND

E-mail: sales@cdgroup.fi
Tel: +358 9 612 3322