Replicating the return of a hedged benchmark
E.1 INTRODUCTION
This appendix outlines a way to construct a set of FX hedging positions for a hedged benchmark so that its overall return is identical to its vendor-supplied return.
Benchmark providers supply security-level returns in both local and base currency returns, from which it is straightforward to calculate the aggregate, unhedged return of the benchmark.
However, things become less clear when hedging is taken into account. Hedging involves adding a number of forward or FX holdings so that the currency risk is hedged away. Unfortunately, the details of the calculation (and the forward and FX rates) are seldom made available by the benchmark vendor, who will typically only supply the top-level hedged benchmark return. To model hedged benchmarks for attribution requires the user to supply a set of hedge trades so that the overall return is correct.
E.2 THE RE-HEDGING ALGORITHM
The algorithm works by constructing an artificial currency hedge at each date, and by adjusting the returns of the currencies by a small amount.
- Calculate the weights of the various securities in the benchmark by currency sector. The sum of these sector weights will be 100%.
- Set the weights of the currency hedges to be the negative of the currency sector weight, for all except the base currency sector, which is assigned a weight of 100% − (sector weight). The sum of these hedge weights will be 0%.
- Calculate the FX returns for each sector. This is the ratio of the base currency and the local currency returns.
- Calculate the overall return of the hedge as the sum-product of the hedge weights and the FX weights. This return, plus the unhedged return of the benchmark, should be close to the hedged benchmark return.
- Adjust the returns of all sector weights up or down by a small multiplicative factor until the overall return of the hedged benchmark is correct.
E.3 WORKED EXAMPLE
As an example, Table E.1 shows the calculation on the Citigroup World Global Bond Index, hedged into GBP.
On the 17 November 2011, the details of the calculation were as follows:
Table E.1 Benchmark forward hedging
The return of the unhedged benchmark was 0.00305% and the return of the artificial hedge was 0.07496%, implying a total return of 0.07801%. This was close, but not identical, to the published return of 0.07870%.
To correct this, define a quantity f, which is close to one, and multiply all currency returns used in the hedge calculation by this amount until the sum of the unhedged return and the hedge portfolio match the published return.
On any given day, denote the return of the unhedged benchmark as u, the return of the artificial hedge as h, and the published hedge return as H. Then
which implies
In the above example u = 0.00305%, h = 0.07478%, H = 0.07870%, giving a value of
To ensure that the return of the hedged benchmark equals the published return on any particular date, simply multiply the hedge holdings on that day by f.
E.4 FRACTIONAL HEDGING
A manager sometimes has a mandate to use a partially hedged benchmark. In this case, part but not all of the currency risk is hedged away. The market convention is to talk ‘50% hedging’, ‘100% hedging’ and so on.
The above example demonstrates 100% hedging. To apply 50% hedging to a portfolio, simply multiple the factor f by the desired fraction. For instance, the above benchmark can be 50% hedged by multiplying each of the synthetic forward lines by 0.5 × f.
E.5 DISCUSSION
This approach makes some fairly sweeping assumptions – in particular, that we can recreate the hedge using spot FX positions instead of forwards, that the hedge is rebalanced against the benchmark every day, and that there are no interest effects in the hedge.
Nevertheless, given that index vendors often supply minimal information about the methodology or instruments used for hedging, these assumptions are generally permissible, especially as the simulated hedge returns are close to the true returns.