You are an active investment manager of a global balanced portfolio looking back at the investment decisions you made during the past year, such as asset allocation, country allocation and stock

selection. You are sure that during this period you made more good decisions than bad ones, yet your portfolio still did not outperform its benchmark. How is this possible? There can only be one reason;

something went terribly wrong in the translation of your investment views (forecasts). To illustrate how this can happen, consider the following

example:

We have a benchmark consisting of 50% global equities with open

currency exposures and 50% global bonds with hedged currency

exposures. We have investment views regarding the equity country

allocation, the bond country allocation and the currency allocation. With respect to all other investment

decisions, such as asset allocation and stock selection, we hold benchmark neutral positions. One way to

implement our views is to use bets of equal percentage size for each investment decision. The figures we will use for this implementation are summarised in Exhibit 1.

Note that the combination of the first two bets in exhibit 1 can be interpreted as overweighting Japanese equities and underweighting US equities within the equity sub portfolio with open currency exposures. However, it makes sense to treat this as two separate investment decisions: an equity country allocation decision and a currency allocation decision. Within the bond sub portfolio we assume that the country allocation bet is implemented on a currency hedged basis, just as the benchmark of the bond sub portfolio is completely

currency hedged.

Now suppose Japanese equities underperform US equities by 20%,

the yen rises 8% against the dollar and US bonds outperform Euro bonds by 4%. These returns are comparable on standard deviation moves, roughly based on actual market data.

Although our positions are correct for two out of the three investment

decisions, the overall active return is negative as shown in this equation:

(50% x 10% x -20%) + (50% x 10% x 8%) + (50% x 10% x 4%)= -0.4%

In this example, we forgot two important aspects when placing our bets. First, we should have considered the tracking errors (volatility of the relative performance of the portfolio against the benchmark) resulting from each decision. Because equity markets are much more volatile than bond markets and currencies, the tracking error of the 10% equity country allocation bet is much higher than that of the 10% bond country and currency allocation bets. In other words, we used equal percentage bets for each decision, although the risk associated with the bets was definitely not equal. The effect of the equity country allocation bet therefore outweighed the two other (successful) bets.

Secondly, we should have taken the degree of confidence in each of the forecasts into account. Clearly, the more confidence we have in our ability to forecast a particular investment decision, the higher the tracking error allocated to that decision should be. We quantify forecasting capabilities by formulating expected information ratios for each investment decision. An expected information ratio is the ratio between expected outperformance and the corresponding tracking error.

We will continue our exposition by describing how to determine expected information ratios. Next we will show how these expected

information ratios should be used for

allocating a tracking error to each investment decision. We then take another look at the example and show the consequences of optimal risk budgeting.

One way to establish expected information ratios is to use the historical track record of individual investment decisions. If quantitative models play an important role in the investment process, the long-term backtest results of these models might be examined. However, care should be taken when extrapolating historical realisations into the future.

What ultimately matters is the expectation of strengths and weaknesses. In particular, the ability to forecast a decision relative to other decisions should be quantified. Actually, the magnitude of the expected information ratios is not relevant for the translation of the investment views into the portfolio. Only the ratio of each expected information ratio with respect to the other decisions influences the risk budgeting. Therefore, it is essential that the expected information ratios express the relative forecasting ability for each decision.

After the expected information ratio of each investment decision has been determined, an optimal tracking-error allocation can be established. The question is how much tracking error to allocate to each decision. Should, for example, all tracking error be allocated to the investment decision which has the highest expected information ratio? Or is it more sensible to diversify over the different investment

decisions?

It can be shown that the tracking error allocation among investment decisions should be set in proportion to the expected forecasting power

for each decision. For example, if

the expected information ratio for equity country allocation is twice

that of bond country allocation, the corresponding tracking error level should be twice as high as well.

To illustrate how these insights influence the translation of investment views, we reconsider the earlier example.

Suppose we have equal confidence in our forecasts for each of the three investment decisions (equity country allocation, bond country allocation and currency allocation). Each decision should then have an equal tracking-error contribution. An example of such an allocation is shown in Exhibit 2.

Note that by decreasing the size of the equity country allocation bet and by simultaneously increasing the size of the bond country allocation bet we establish a tracking error contribution of 0.8% for each investment

decision.

For example, because the equity country allocation is five times more risky than the bond country allocation bet, it follows that the percentage bet size is now five times smaller. It is also worth noting that the equity

country allocation and currency

allocation bets are no longer of

equal percentage size, which

implies that currency derivatives

are required to achieve this

implementation.

With the new allocation, the resulting returns, for the same one-standard deviation return realisations as used earlier, are equal for each investment decision. This leads to a positive active return as illustrated in the

following equation:

(50% ¥ 4% ¥ -20%) + (50% ¥ 10% ¥ 8%) + (50% ¥ 20% ¥ 4%) = +0.4%

The equity bet no longer dominates the overall performance, because instead of using equal percentage overweights and underweights, we have allocated equal tracking errors to each decision.

This article is based on the research paper: ‘Risk Budgeting’ by the authors. In this paper they extend the intuitive insights that were described in this article into a formal risk-budgeting framework.

Jouke Hottinga and David Blitz are both team managers at Robeco’s

quantitative research department in Rotterdam

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