The Cosmos Global Optimizer

by Peter Conkey

Previous BARRA newsletter articles¹ have described the Risk and Scenario functionality of The BARRA Cosmos System™ for international fixed income portfolio managers (Cosmos-Global) and its underlying risk model. This article illustrates, by means of a case study, how the recently released Optimizer module of Cosmos-Global can be used to construct global fixed income portfolios.

The basic approach the Cosmos Global Optimizer takes is mean-variance optimization. The Optimizer maximizes an objective function:

Portfolio return - Risk aversion
x Portfolio risk penalty
- Transaction costs - Penalties

Investors may not have a precise estimate for their respective risk aversions. Fortunately, as we'll see, the Optimizer can also generate an efficient frontier of portfolios with different risk aversions, each of which has the highest level of return for its level of risk.

In addition to risk and return parameters, the case discussed below also includes transaction costs and constraints. The penalties term in the objective function refers to user-defined penalties for deviation of the portfolio from targets for particular characteristics, such as duration or currency exposures. These targets can be defined in absolute terms (e.g., duration of five years) or relative to a benchmark (e.g., same duration as the benchmark). Let's make all of this more concrete by looking at a case study.

Return forecasts

Table 1 summarizes the exchange and interest rate forecasts that will be used in this case study for a subset of the markets in Cosmos-Global. In this example, exchange rate movements are expressed relative to the U.S. dollar, so the expectation is for a weakening of the dollar.

Table 1. Exchange and interest rate forecasts used in the Cosmos-Global case study.

Market	Term structure	Currency Movement (%)
Australia	No change	1
Belgium	Flattening	3
Canada	No change	No change
Denmark	Flattening	3
France	Flattening	3
Germany	Flattening	3
Ireland	No change	-2
Italy	Up 25 basis points	3
Japan	No change	5
Netherlands	Flattening	3
New Zealand	No change	1
Spain	Up 25 basis points	2.5
Sweden	Flattening	3
United Kingdom	Short rates up	5
United States	No change	No change

A common flattening of the yield curve is expected in European markets, with long rates falling. Figure 1 shows the exact specification of the movement within the Scenario module of Cosmos-Global.

The asset-level returns to be used in the optimization will be generated by the Scenario module by repricing each security with the forecast market conditions as of a June 1 horizon, about three months from the March 3 analysis date.

Initial portfolio

Figure 1. Cosmos-Global: German term structure forecast in the Scenario module.

Figure 2. Cosmos-Global: Currency and duration exposures of the initial portfolio.

This portfolio already reflects to some extent the market views described above. For example, the portfolio is long duration in the user-defined Core Europe block and is underweight U.S. dollars. However, it has some legacy positions, such as the holding in Ireland, which are clearly unattractive given these views.

Figure 3 shows that the portfolio will outperform the benchmark by just 8 basis points to the three-month horizon, if the manager's expectations prove accurate. We could analyze this further in Cosmos-Global to see where the return comes from at the market or asset level, but instead let's see how this portfolio can be improved by using the Optimizer.

Figure 3. Cosmos-Global: Initial portfolio and benchmark returns.

Optimizer case

Figure 4. Main Settings window of the Cosmos Global Optimizer

We already know something about the currently managed portfolio that will be the starting point in this example. The manager has also created a universe portfolio containing benchmark bonds at the various maturities in each market, bond futures in those markets, and synthetic currency forwards. This is intended to be representative of the available investment opportunities.

Other options in the Main Settings window determine what transactions to allow. For example, if there is a cash flow into the portfolio, the manager can specify "Sell None" as the transaction type, which would force the Optimizer to retain existing holdings and invest only the new cash flow.

The remaining options in the window simply specify the form the Optimizer output will take. This can be a single portfolio, chosen based on either a risk aversion parameter, a required return, or an acceptable level of risk. Alternatively, you can ask for an efficient frontier of the maximum return portfolio at each level of risk. This is the option we'll use in this case study.

Figure 5 shows a Scenario dialog where the "June97" Scenario we saw earlier (in Figure 1) has already been selected. The probability-weighted expected returns are being used here as the returns in the objective function; these are the same as the main returns, since there is only one sub-scenario. The manager could have specified alternative market views in other sub-scenarios and assigned them probabilities to come up with the expected returns. For example, he might have looked at the outcome for no rate rise in the U.K. Another option is to specify a minimum return target under several alternative sub-scenarios: for example, don't underperform the benchmark by more than 50 basis points in any sub-scenario. For the purposes of this article, we'll stick to the simple case of using one set of returns.

Figure 5. Scenario selection in the Cosmos Global Optimizer.

The next step is to specify some transaction costs. As shown in Figure 6, the Optimizer lets us easily specify transaction costs reflecting bid-offer spreads country by country. If this is not sufficiently detailed, we can also specify transaction costs on a bond-by-bond basis. In our case study, a default transaction cost of 10 basis points in price has been applied to all markets.

Figure 6. Entering transaction costs in the cosmos Global Optimizer.

The two other tabs in the Settings window (Figure 6) control risk and asset-level settings. In the Risk tab, you can specify the benchmark and, if generating a single portfolio based on a risk/return trade-off, your risk aversion. In the Asset Data tab, you can specify information at the asset level, such as returns (instead of using Scenario), transaction costs, and constraints (limits) on individual asset holdings.

A wide variety of higher-level constraints on portfolio characteristics can be specified in the Constraints window, shown in Figures 7 and 8. These constraints are organized for convenience into various types (see Figure 7). Aggregate Characteristics refers to standard quantitative measures of portfolio risk or value, such as duration, yield, and option-adjusted spread. BARRA Risk Measures refers to quantities relating to the Cosmos-Global risk model such as shift, twist, and butterfly exposures. Other constraint types are categorical, such as constraints on portfolio weight in sectors or rating buckets.

Figure 7. Entering constraints in the Cosmos Global Optimizer.

Figure 8. Cosmos-Global: Currency constraint used in the case study.

These constraints can be applied to the portfolio as a whole or to the component in a particular currency, such as a constraint on the duration of the Deutschmark component of the portfolio. They can even be applied to the portfolio component in a user-defined currency block, such as the Core Europe block mentioned above. They can be defined in absolute terms (for instance, duration between three and five years in every market) or relative to the benchmark (e.g., within one year of the benchmark's duration in each market, and convexity within plus or minus 10% of the benchmark's convexity).

Figure 8 shows the one constraint we are going to apply to this optimization: That the total currency exposure in each currency must be positive. This implies that while a currency can be hedged in the optimization, this can only be to the extent of long positions in bonds in that currency.

The "Permit Short Sales For" box near the middle of the Constraints window contains another setting important for currency hedging: that short sales are being permitted for cash assets, in this case synthetic generic currency instruments that will be used for hedging. The Optimizer's default lower bound on any asset holding is zero, meaning short sales are not permitted. It is possible to override this default by creating a file of asset-by-asset lower bounds. For futures and currency derivatives, however, you can permit short sales simply by clicking the check boxes in the Constraints window.

Optimizer output

Figure 9 shows the efficient frontier output by the Optimizer. The upper part of the display shows a graph of the efficient frontier; the bottom part shows a scrollable spreadsheet summary of the portfolios. You can double-click on either a point on the graph or a row in the spreadsheet to view the portfolio constituents, which can then be exported to the Cosmos-Global Risk module for further analysis.

Figure 9. Efficient frontier created by the Cosmos Global Optimizer.

We'll focus on portfolio number 26 in the spreadsheet, which has a tracking error of 38 basis points against the J.P. Morgan benchmark and a return after transaction costs of 4.64%. It has a slightly lower tracking error than the initial portfolio and a return—net of transaction costs—that is 66 basis points higher (compare Figures 2 and 3).

Figure 10 shows the market allocation and duration for this portfolio relative to the benchmark. As expected, in those European markets where a flattening of the term structure was forecast, the new portfolio is longer in duration and has greater interest rate exposure. Inspection of the cash flows (not shown) shows that the portfolio's U.K. component is heavily concentrated in a five-year issue, avoiding the increase in yield forecast at the short end, but at the same time providing some counter to the long duration elsewhere in Europe.

Figure 10. Cosmos-Global: Market allocation and duration of the optimized portfolio and benchmark.

Figure 10, like Figure 2, shows weights and durations in each bond market. In the new portfolio we also have some currency forwards. Figure 11 shows the explicit currency exposure resulting from these positions. In Japan, for example, although the value weight in JGBs is slightly less than the index, this is more than made up for by a purchase of yen.

Figure 11. Cosmos-Global: Explicit currency exposures of the optimized portfolio.

Summary

In this article we have seen how the Cosmos-Global Optimizer can quickly rebalance an actively managed fixed income portfolio, accounting for the manager's market expectations. And beyond this case study, of course, it can easily handle the more complex and realistic forecasts and constraints that managers work with every day.

Footnotes