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!

Campisi Fixed Income Attribution - Price Contribution Explained

The CIPM curriculum does not give much of an explanation of bond pricing in relation to interest rates, so here is a brief primer.

Price change on bonds can be explained by how interest rates change during the period.

Bonds are a lending agreement:  the purchaser is the lender, and the issuer of the bond is the borrower.  The coupon rate on the bond is the interest rate on the loan. 

When interest rates rise over an evaluation period, that means that the cost of borrowing for bond issuers has increased.  But it also means that newly issued bonds are more attractive to investors than existing bonds of the same time to maturity.  Because existing bonds are less attractive due to rising interest rates, their price drops, making their market value drop over the period - which resullts in a negative rate of return.

Conversely, if interest rates drop over an evaluation period, existing bonds look more attractive, making their price rise, thus their market value increases and they have a positive rate of return over the period. 

That is the inverse relationship between bonds and interest rates.  When rates rise, bond prices drop, and vice versa.

The Treasury (or risk-free) yield curve is the basis for bond prices.  On the X-axis is time to maturity, and on the Y-axis are the interest rates.  Thus, the combination of interest rates at various times-to-maturity create a curve.  Thus, how the yield curve changes over the period of evaluation gives us information we can use to approximate price change due to changes in interest rates.

Price change can be approximated by the following equation:

Thus, the portion of return that is due to price change over the evaluation period is equal to the change in interest rates over the period  multiplied by the negative of duration.  Or, said differently, we need to know the price change for a given bond or set of bonds.  Let's say we are talking Treasury bonds (i.e., risk free bonds).  That price change corresponds to how much interest rates changed during the period for a particular point on the risk-free yield curve that is equal to the duration of the bond in question.  For Treasury bonds of a given duration, how much did interest rates change during the period (i.e, from the start of the period T, to the end of the period T+1).

The yield curve could change over the period in different ways.  When there is a parallel shift in the yield curve, interest rates change by the same amount over the evaluation period at all points on the yield curve (i.e., at all time-to-maturity/duration points)

Or, there may be some sort of structural change in the yield curve over the evaluation period where the change in interest rates is different at one time-to-maturity or duration compared to at other times-to-maturity or duration:

Thus, if we know how much interest rates changed over the evaluation period for a particular time-to-maturity or duration, we can quantify how much return occurred for bonds of that duration that corresponds to that change in interest rates.

If we are talking about a class of bonds or a portfolio of bonds, we can use the market value weighted average of the durations of the individual bonds in the class or portfolio, and use the change in interest rates that corresponds to that.

In a subsequent blog post, we'll apply this concept in the Campisi model for attribution.

Happy studying!

Campisi Fixed Income Attribution - Income Contribution Explained

Recall from my previous posting, the steps to executing the Campisi fixed income attribution model are:

1. Decompose the benchmark return into:
   - income contribution
   - Treasury contribution (i.e., price change due to changes in Treasury rates)
   - spread contribution (i.e., price change due to changes in the average spreads of a risky bond class
    
2. Decompose the index portfolio return into:
   - income contribution
   - Treasury contribution (i.e., price change due to changes in Treasury rates)
   - spread contribution (i.e., price change due to changes in the average spreads of a risky bond class
3. Calculate the index portfolio spread change.  This is the change in interest rates that will be used to calculate the spread contribution of the portfolio (more on this in a subsequent blog post).
4. Decompose the portfolio return into:
   - income contribution
   - Treasury contribution (i.e., price change due to changes in Treasury rates)
   - spread contribution (i.e., price change due to changes in the average spreads of a risky bond class
   - security specific contribution
5. Calculate the attribution effects as the value added contributions:
   - income effect = portfolio income contribution minus the benchmark income contribution
   - Treasury effect = portfolio Treasury contribution minus the benchmark Treasury contribution
   - spread effect = portfolio spread contribution minus the benchmark spread contribution
   - selection effect = portfolio selection contribution (note - the benchmark has no selection contribution)

The first step under items 1, 2 and 4 above are to calculate an income contribution (for the benchmark, index portfolio and portfolio, respectively).

The income contribution is how much of the return comes from the income paid by the bonds in the portfolio or benchmark.  We express this income contribution as a rate of return, which has a numerator and a denominator.  The formula for this contribution is simply:

For example, in your reading, the portfolio's weighted average coupon is 7.118% and the  portfolio's weighted average price is 98.1.  Thus, the contribution of income to the total return of the portfoli may be calculated as:

To say it differently, out of the portfolio return of 0.31%, the portion that comes from income is the weighted average coupon divided by the weighted average price which is 7.26% in total.  Thus, obviously the sum of the other contributions must be negative.

The income contribution can also be calculated from money amounts:

Essentially, the income contribution is a form of current yield, comparing interest earned by the manager's portfolio to the market value invested to earn that income. 

Given, then, that the portfolio return may be decomposed into income and price change, we have decomposed the income portion of the return.  All remaining elements are part of the price change component.  We'll tackle that subject next!

Happy studying!

Decomposing the Campisi Fixed Income Attribution Model

For many CIPM Expert Level candidates, fixed income attribution is the most difficult topic.  This is evident when I teach The Spaulding Group's CIPM prep classes, as we devote an entire afternoon to the subject.  Among the three models that candidates are required to learn is the Campisi model, which is based around the idea of decomposing bond performance according to the picture above.

Over the next few days, I will present various points on the Campisi model, in order to simplify it for candidates.  This model is actually not very complicated - it is fairly intuitive - but candidates might benefit by having a roadmap to guide them through the process of calculating the attribution.

The diagram above is a decomposition of bond performance.  Thus, it can represent the decomposition of a single bond, a group of bonds, a portfolio of bonds and/or a benchmark of bonds.

The first level of decomposition applies to any asset one may own in a portfolio.  Return on an asset comes from two sources:

• income and expense (interest, dividends, and other expenses)
• price change (i.e., gains  and losses, both realized and unrealized)

In our situation, of course, we are dealing with bonds, so the income is from interest.

The next level of decomposition, which breaks up the sources of price change, is specific to bond investments.  Price change on bonds comes from three sources:

• price change that is due to changes in interest rates on the Treasury yield curve
• price change that is due to changes in spreads that non-Treasuries of a specific class (bond class and/or ratings class) pay above Treasuries of the same duration (average spreads)
• price change that is due to security specific traits of bonds that cause them to perform differently than the average for their class (nominal spreads)

A Treasury bill, note or bond could have return from the first and third sources.

A non-Treasury fixed income security could have return from all three sources.

Thus, the terminal nodes in the tree represent the lowest level of the decomposition:

• income
• price change due to changes in Treasury rates
• price change due to changes in the average spreads of a risky bond class
• price change due to security specific traits (i.e., due to nominal traits)

In order to calculate the Campisi attribution effects, the return of both the portfolio and the benchmark must be decomposed into contributions from these sources.  Thus, in order to calculate the Campisi attribution effects, the following steps must be taken (i.e., your roadmap):

1. Decompose the benchmark return into:
   - income contribution
   - Treasury contribution (i.e., price change due to changes in Treasury rates)
   - spread contribution (i.e., price change due to changes in the average spreads of a risky bond class
    
2. Decompose the index portfolio return into:
   - income contribution
   - Treasury contribution (i.e., price change due to changes in Treasury rates)
   - spread contribution (i.e., price change due to changes in the average spreads of a risky bond class
3. Calculate the index portfolio spread change.  This is the change in interest rates that will be used to calculate the spread contribution of the portfolio (more on this in a subsequent blog post).
4. Decompose the portfolio return into:
   - income contribution
   - Treasury contribution (i.e., price change due to changes in Treasury rates)
   - spread contribution (i.e., price change due to changes in the average spreads of a risky bond class
   - security specific contribution
5. Calculate the attribution effects as the value added contributions:
   - income effect = portfolio income contribution minus the benchmark income contribution
   - Treasury effect = portfolio Treasury contribution minus the benchmark Treasury contribution
   - spread effect = portfolio spread contribution minus the benchmark spread contribution
   - selection effect = portfolio selection contribution (note - the benchmark has no selection contribution)

The picture below shows my scribblings from doing this in a recent class in the form of an attribution effects scoreboard - I recommend candidates mimic this during the exam to keep track of where they are:

 More details to come!

Expert Level - Sample Exam Question #15

Question #15 from the Expert Level Sample Exam is with respect to the data in the table shown above, and reads:

15. Tom Styles, the head of performance measurement at Signal Investment Management, uses a sector allocation/security selection attribution model for both equity and fixed-income portfolios. The bond portfolio managers ask Styles to make his department’s fixed-income attribution analysis more meaningful. Styles uses the Campisi methodology to analyze the performance of a portfolio that contains US Treasury notes and bonds, corporate bonds, and high yield bonds. He prepares Exhibit 1 for the fixed-income portfolio managers’ review.

Which statement is most accurate? Over the evaluation period, the yield curve:

A. fell, and the portfolio’s duration was longer than the benchmark’s duration.
B. fell, and the portfolio’s duration was shorter than the benchmark’s duration.
C. rose, and the portfolio’s duration was longer than the benchmark’s duration.

There are two things you need to determine here:

1. Did interest rates rise or fall, which would cause a price change that was negative or positive, respectively?
   
   
2. Did the portfolio have a better yield curve positioning than the benchmark, given the change in interest rates?

You can tell that interest rates fell because the Treasury return is positive (for both the portfolio and benchmark). The Treasury return is the portion of the price return that is due to changes in Treasury interest rates. The fact that this is a positive number means that Treasury interest rates declined during the period, making existing Treasuries look more attractive, resulting in their prices going up, and affecting a positive return. This concept is covered in the second reading in Study Session VI, which covers the Campisi attribution model.

The fact that the benchmark’s Treasury effect is higher than the portfolio’s Treasury effect means that the benchmark was better positioned (from a duration standpoint) to respond to the changes in Treasury interest rates over the period than the portfolio. From the first reading in Study Session VI, which covers the Fong-Pearson-Vasicek model, candidate should understand that holding long duration portfolios during periods of decreasing interest rates will add value, as will holding short duration portfolios during periods of increasing interest rates. Thus, the benchmark’s higher Treasury effect while interest rates decreased is a sign that the portfolio’s duration was less than (shorter than) the benchmark’s.

Thus, the correct answer is B.