## Saturday, September 8, 2012

### 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!