# Duration and parallel shift in the yield curve

I don’t understand why the parallel shift assumption exists in the first place. I even checked my level 1 notes and I am still clueless. Pls help!

It’s the easiest way of translating the impact of duration to bond price. And its inline with how we interpet duration (%change on bond price given a change in yields)

Example, if the modified duration of a bond is 5 and there was 100 bps parallel shift upwards in YTM, then bond price is expected to decline by 5%.

We use a parallel shift so that if we have a portfolio of bonds of various maturities, the yield change is the same for all such bonds.

For a single bond, the shift doesn’t have to be parallel; all that matters is the change in the par rate at that bond’s maturity; i.e., the change in that bond’s YTM.

First and foremost thanks a lot! Your answer gave me a whole new perspective. So can I regard duration of each bond in a portfolio of bonds somewhat similar to that of a key rate of a bond?

I am confused about the concept of “parallel shift” and “shaping risk”.
So does that mean you don’t have to have a parallel shift of the yield curve of a bond to calculate ModDur, EffDur, etc.?

Please correct me if I am wrong. So parallel shift in the yield curve of a bond portfolio means yields of all of the bonds shift in the same magnitude and direction. On the other hand, if we only have a single bond we don’t have to shift all of the yields of every single maturity on the yield curve of the bond but rather only the yield corresponding to that bond’s maturity.

But then isn’t that similar to Key Rate Duration? So how does Key Rate Duration apply to a single bond and a portfolio of bonds?

Thanks for the reply! I do fully understand the duration and its concept regarding price sensitivity. What I am confused about is why do you have to shift the entire yield curve of a bond to calculate the change in price when a bond would inevitably have a single maturity? e.g. a bond with 5 year maturity would only be affected by a change in yield which has the same maturity.

But that kind of overlaps with the definition of Key Rate Duration.

I see.

Nothing wrong with this statement that a 5-yr bond is affected by a change in the 5-year YTM (for that we measure the price sensitivity using modified duration).

But to calculate the bond’s no-arbitrage value, we can also discount its cash flows using spot rates from the spot yield curve.

E.g. for the 5yr coupon bond, the:

• Year 1 coupon is discounted by 1-year spot rate
• Year 2 coupon is discounted by 2-year spot rate
• Year 3 coupon is discounted by 3-year spot rate
• Year 4 coupon is discounted by 4-year spot rate
• Year 5 coupon is discounted by 5-year spot rate
• Year 5 face value is discounted by 5-year spot rate

So when we are using the effective duration, we are assuming a parallel shift in the 1yr, 2yr, 3yr, 4y and 5y spot rates.

If you want to assume that, say, only the 3yr spot rate changes while the rest stays constant, then use key rate duration.

FTFY

The point is that it will, for example, be affected by changes in spot yields of different (specifically, shorter) maturities.

I conclude that 5-year bond is affected by a change in the 5-year YTM or 5-year Par rate and we don’t care any other maturities par rates WHEN CALCULATING EFFECTIVE DURATION AND MODIFIED DURATION.
Because 5-Year Par rate is composed of any SHORTER-TERM SPOT RATES (e.g., 1-year, 2-year, 3-year,… SPOT RATES) → parallel shift of spot curve, for example increase in 1%, will result in increase in 1% 5-YEAR PAR RATE.
But in the curriculum, it states that we need an assumption to calculate EFFECTIVE DURATION :" parallel shift of PAR CURVE"