forward rate, expected spot rate and par rate

see the statement:

For a three year maturity callable bond, the main driver of the call decision is the two-year forward rate one year from now. This rate is most significantly affected by changes in the one-year and three-year par rates.

I am confused as why the forward rate is affected by par rate, isnt forward rate decided by expected spot rate, par rate is just the rate that makes the bond discount to par level right?

The question is from example 7 of reading 45, question 7, it asks which par rate shift has the largest impact on a callable bond, how does par rate impact the price of a bond, i thought par rate is just a bench mark rate, shouldnt the interest rate and coupon rate the only rates impact bond price?

You may want to actually work this out in a spreadsheet. It helped make sense out of this for me. You will see how the affect eachother.

The Spot rates are bootstrapped (derived) from current par rates. Then forward rates are derived from those “bootstrapped” sport rates. This is why with an upward sloping par rate curve the spot rates will lie above the par rate curve and the forward rate curve will lie above both of these curves.

Breaking this down - to get forward rates you need to:

1. Gather the par rates

2. Bootstrap the spot rates from these par rates

3. Calculate the implied forward rates from these spot rates

Therefore if the par rates shift - so will spot rates (because they are derived from the par rate curve) and so will the forward rates (because they are derived from the spot rates which just shifted as well). This is how they are all intertwined.

For a three year maturity callable bond, the main driver of the call decision is the two-year forward rate one year from now. This rate is most significantly affected by changes in the one-year and three-year par rates.

Go through the logic above. The 1y2y is derived from the 3 year spot rate and the 1 year spot rate [(1+3y)^3/(1+1y)]^.5. Again these spot rates are derived from the current par rate curve. Thus if the 1 and 3 year par rates move then so will this forward rate.

Brilliant explanation. Kudos

This isn’t true.

The forward curve will lie above the spot curve only if the spot curve slopes upward. The fact that the par curve slopes upward isn’t sufficient to guarantee that the spot curve does as well.

Take a look at the last graph in this article I wrote on par curves, spot curves, and forward curves: http://financialexamhelp123.com/par-curve-spot-curve-and-forward-curve/

The par curve slopes upward everywhere. The spot curve does not. The forward curve is below the spot curve from about year 24 on.

Yea you are right - not 100% of the time will that happen but a good majority of the time it will especially on the short end where the slope doesnt tend to flatten off. It just depends on the curvature as well and where rates tend to flatten out - when rates start to flatten then the forward rate needed to “equalize” the spot rates will be lower. I guess you could come up with some numbers to show this isn’t always the case - I suppose thats where the saying “there are lies, damn lies, and statistics” comes from.

However, and correct me if I am wrong b/c you are the one with the charter, I think the text is wanting us to understand the relationships described above and the generalization that when par curves are upward sloping then the spot curve will be above that and the forward curve will be above that. When the par curve is downward sloping then the spot curve will be below it and the forward curve will be below that. At least thats the examples I have seen them go over.