I know that carry was discussed broadly on this forum but I can't get my head around the following difference.
If we talk about carry / rolldown I have trouble to see the connection / differences between two very well known paper. These are
- Market Rate Expectations and Forward Rates by Antti Ilmanen, link
- Carry by Koijen, Moskowitz, Pedersen and Vrugt, link
In the latter carry for a fixed maturity instrument, e.g. zero coupon bond with maturity $n$, is derived as
$$ C^p = \left(f(t,n -1, n) -r^f_t\right)\frac{1}{1+r^f_t}$$
where $f(t,n-1,n)$ is the forward rate at time $t$ between $n-1$ and $n$ and $r^f_t$ the riskfree rate at time $t$. Note the scaling factor $\frac{1}{1+r^f_t}$ is not that important here and could be ignored.
On the other hand Ilmanen defines break-even rates and says on page 7 (I quote):
"The break-even yield change $f(t,1,3)-s(t,3)$ shows how much the three >year zero's yield can rise before its carry advantage is offset".
so it seems Ilmanen defines carry as
$$ C^I = f(t,1,n)-s(t,n)$$
with $s(t,n)$ the spot rate at time $t$ for maturity $n$. Ilmanen then continues to add roll to the picture and ends up with a total cushion,$C^I_2$, against adverse price movements of
$$C^I_2 =f(t,1,n)-s(t,n)+(s(t,n)-s(t,n-1))=f(t,1,n)-s(t,n-1) $$
see equation $(6)$ in his paper for $n=3$.
I'm puzzeled how these two things go together. That is why I tried to calculate with the term structure provided in Ilmanen both quantities, i.e. for $n=3$
$$ f(t,1,3)=0.0864$$ $$ f(t,2,3)=0.0927$$ $$ s(t,3) = 0.0775$$ $$ s(t,2) = 0.07$$ $$ s(t,1) = 0.06$$
leads to $C^p = 0.0327$ while $C^I = 0.0089$ and $C^I_2 = 0.0164$. These numbers seem completely different. I've noted that in this example it seems to hold $C^p = (n-2)*C^I_2$. But I didn't verify if this is the case in general.
I would like to know what is the connection between $C^p, C^I$ and $C^I_2$? Are there different underlying assumption or why do they all talk about carry in some way or the other.
