Why do par-yield shifts grow faster across the curve than spot-rate shifts when looking at key-rates?

Question

Consider the following 10y key-rate shifts of bond par yields and its implied shift of bond spot rates:

Assume we have the key-rates for 2y, 5y, 10y and 30y.

The y-axis is in basis points, and the x-axis is the term on the curve. The red line (par-yield shift) is just the 10y key-rate shift, where we take the standard assumption that a key-rate shift at a given term only affects the adjacent terms (in this case, 5y and 30y), and is thus zero beyond these points.

I'm trying to understand the shape of the spot-rate shifts relative to the key-rate shift, namely why it increases faster.

In a book I read I saw roughly the following:

Par yields in 0 to 5y are unchanged (by definition of key-rate shifts). This implies that spot rates in 0 to 5y are unchanged (because if the par yields don't change, then the spot rates can't). Hence, the par-yield changes for terms greater than 5y can only be reflected in spot-rate changes for terms greater than 5y.

This all makes sense to me. Then the author says:

Hence, spot rates between 5y and 10y have to increase faster than par rates in the same time interval.

Can someone please explain the logic behind this final statement? It is invariably necessary to understand this in order to understand why it decreases faster beyond 10y as well.

Answer 1

Intuitively, this is the "coupon effect" at work – when the yield curve is upward sloping, lower coupon bonds have higher yield and their yields move up more when the overall curve shifts up (all else equal). The opposite is true when the yield curve is downward sloping. We'll focus on when the curve is upward sloping below.

I think it's probably best to think through what happens in a parallel shift in the par yield curve and go from there.

Imagine that the par curve is upward sloping and we shock the par curve by 100 basis points. You'll notice that the corresponding shift in the spot curve is not parallel; the shock size increases with maturity.

Let's work this out mathematically. Let's say that our par curve is spaced out evenly at 1-year intervals; it's also upward sloping, with 1- and 2-year par rates at 5% and 10%, respectively. Through simple bootstrapping, we can work out that the 1-year spot rate is 5%, and the 2-year spot rate is 10.263%. This is easily verifiable – a 2-year coupon bond with a coupon rate of 10% would indeed be priced at par:

$$ \frac{10}{1 + 5\%} + \frac{100 + 10}{1 + 10.263\%} = 100. $$

Notice that the 2-year spot rate is higher than the 2-year par rate. Intuitively, the 2-year par rate is a complex weighted average of 1- and 2-year spot rates. Since the curve is upward sloping and since 1-year spot rate is lower than 10%, for the 2-year par rate to be 10%, the 2-year spot rate must be higher than 10%. This is the intuition behind the "coupon effect."

Now let's shock the par curve by 100 basis points. The 1-year par rate becomes 6% and the 2-year par rate is now 11%. It's easy to verify that the 1-year spot rate is now 6%, and the 2-year spot rate is now 11.289%:

$$ \frac{11}{1 + 6\%} + \frac{100 + 11}{1 + 11.289\%} = 100. $$

As you can see, the 2-year spot rate has increased by 129 basis points!

The part that can be confusing is that when you shock the par curve, you're not shocking the yield to maturity of the original par bonds. Instead, you're creating brand new par bonds. In our example, you're not shocking the yield of the 2-year 10% par bond from 10% to 11%. You've created a new 2-year 11% par bond, at which point the old 10% bond is no longer a par bond. Because it now has a lower coupon than the par bond, and because lower coupon bonds have higher yields, its yield will increase by more than 100 basis points if the yield curve is upward sloping. Spot rates increase even more because their corresponding bonds have even lower coupons (0%!).

So let's turn back to the key rate shift. The discussion above is directly applicable to the first segment (the 5- to 10-year spot rates rise more for the same reason). But since the 30-year par rate is unchanged, the 30-year spot rate has to decline a bit to compensate for the fact that all the intermediate zero rates increased too much.

Answer 2

5Y is 1% and 10Y is 1%.
Now 10Y rises to 2% and 5Y stays at 1%.
What is now the rate on the 5Y measured 5Y forward in time, i.e. 5Y5Y?
Well it is 3% (not precisely due to discounting), since... $$ 10Y \approx \textbf{Average} \left ( 5Y + 5Y5Y \right )$$ The actual equations are only slightly more complicated but this highlights the effect of leverage which is what you have observed.