Managing yield curve risk

CFA Level II Fixed Income
#1

Consider a portfolio of 2 years and 5 years zero coupon bond with equality weighted. The effective duration of 2 years zero coupon bond is 2 and the effective duration of 5 years zero coupon bond is 5. The effective duration of the portfolio is 2*0.5+5*0.5=3.5. Suppose only 2 years YTM increase by 1%, how the value of portfolio will change? According to the book page 51, the value of portfolio will fall by 1%. Because as 2 year YTM increase by 1%, price of 2 years zero coupon bond will decrease by 2%. 2 years zero coupon bond has 50% weight. So the value of portfolio will decrease by 1%.

But I think the effective duration of 2 years zero coupon bond is 2. It does mean the 2 years key rate duration is 2. If only 2 years YTM increase by 1%, we should consider key rate duration, not effective duration. Therefore, if only 2 years YTM increase by 1%, price of 2 years zero coupon bond may not decrease by 2%.

On the other hand, changing in 2 years YTM will not only affect the price of 2 years zero coupon bond, it also affect the price of 5 years zero coupon bond. Because 2 years YTM is also a key rate of 5 years zero coupon bond. Therefore, changing in 2 years YTM will change both the price of 2 years and 5 years zero coupon bond.

I think, for a bond portfolio, each bonds’ weighted effective durations are the key rate durations of the portfolio. However, each individual bonds also have their own key rate duration. The sum total of an individual bond’s key rate duration equals to its effective duration.

Is what I think correct?

#2

When you say that the 2 years YTM increases by 1%, do you mean that the 2-year par rate increases by 1%, or that the 2-year spot rate increases by 1%?

That’s the issue, in a nutshell.

#3

I think par rate is always YTM.

And also, when we talk about key rate duration, it refers to YTM, the change in yield curve. It has nothing to do with spot rate.

Right?

#4

For zeros, the spot rate is the YTM.

The par rate is the YTM for coupon-paying bonds.

#5

What if the bonds are coupon bonds. And the YTM is benchmark par yield. What I said is correct?

#6

It’s _ not correct _ to say that it has nothing to do with spot rates; when you change one par rate (and leave all other par rates unchanged), the spot rate at that maturity will change (in _ the same _ direction, and by a greater amount), and the spot rates at longer maturities will change (in _ the opposite _ direction, and by a much smaller amount); the spot rates at shorter maturities _ will not change _.