## Tuesday, April 9, 2013

### Campisi's Index Portfolio Explained

A common question I get is what is the purpose of the "index portfolio" used in the Campisi fixed income attribution model.

Before answering, let me first give an outline of the steps in calculating attribution in the Campisi framework.

Step 1:  Decompose the benchmark return
1.1 Calculate the contribution of income to the benchmark return
1.2 Calculate the contribution of Treasury to the benchmark return
1.3 Calculate the contribution of spread to the benchmark return

Step 2:  Decompose the index portfolio return

2.1 Calculate the contribution of income to the index portfolio return
2.2 Calculate the contribution of Treasury to the index portfolio return
2.3 Calculate the contribution of spread to the index portfolio return

Step 3:  Decompose the portfolio return
2.1 Calculate the contribution of income to the portfolio return
2.2 Calculate the contribution of Treasury to the portfolio return
2.3 Calculate the contribution of spread to the portfolio return
2.4 Calculate the contribution of selection to the portfolio return

Convenient fact #1:  Knowing the benchmark return and having calculated the contribution of income and the contribution of Treasury to the benchmark return, we can back into the spread contribution - it's everything that's left.

Convenient fact #2:  Knowing the index portfolio return and having calculated the contribution of income and the contribution of Treasury to the index portfolio return, we can back into the spread contribution - it's everything that's left.

Inconvenient situation:  We can't back into the spread contribution of the portfolio, knowing the income and Treasury contributions, because spread contribution is not all that is left... there is also the selection contribution.  Thus, we need a specific formula for the spread contribution to the portfolio return.

Spread contribution is an element of price return.  And price return is always dictated by the change in interest rates during the period.  Specifically, if we know the duration of the given portfolio (or benchmark, etc) at the start of the period, and if we know how much interest rates have changed during the period for bonds of that duration, then we can calculate the price return caused by changes in interest rates as:

return contribution = (-1) * (duration) * (change in interest rates)

Note:  the (-1) is because of the inverse relationship between interest rate change and bond prices.

We use this formula in various situations in the Campisi model.  When calculating Treasury contribution, for example, the relevant change in interest rates to use in the formula is the change in Treasury interest rates for the given duration.  But when calculating spread contribution, we are trying to calculate the return contribution from the non-Treasury securities (in excess of the Treasuries).  Thus, the relevant interest rate to consider is the interest spread paid by non-Treasuries in excess of Treasuries of the same duration.  And the change in interest rates, then, is the amount that the spread changed over the evaluation period.  A positive number indicates that the spread paid by non-Treasuries over Treasuries (of the given duration) increased during the period (i.e., the spread widened).  A negative number indicates that the spread paid by non-Treasuries over Treasuries decreased (i.e., the spread narrowed).

So then, back to the original question, what is Campisi's index portfolio?

The relevant change in interest rates to calculate spread contribution for the portfolio is based on the index portfolio.  The index portfolio is a hypothetical portfolio based on the manager's (portfolio) sub-sector weights but benchmark sub-sector returns.  I often describe it to students in our classes as being analogous to Brinson's semi-notional portfolio used in stock attribution, reflecting the manager's weighting decision but retaining the security selection of the benchmark.  The index portfolio, by using the sub-sector weights of the portfolio and the benchmark sub-sector returns:
• reflects the income contribution of the portfolio
• reflects the Treasury contribution of the portfolio (i.e., the manager's duration decision)
• reflects the manager's allocation to sub-sectors
• retains the security selection of the benchmark
Thus, by using the formula:

spread contribution = (-1) * (duration) * (change in index portfolio spread)

we can calculate a spread contribution for the portfolio that is free from the manager's security selection.

In reality, we could also use the last formula above to calculate spread contribution for the benchmark and for the index portfolio (steps 1.3 and 2.3), if we knew the appropriate change in spread rates to use.  But, we make use of the "convenient facts" stated above to do so more efficiently.

Happy studying!