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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.

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I believe Ilmanen is talking about an annualized carry, while your second link seems to talk about $\tau-$period carry. That might be the difference. Also just before equation (12) in Pedersen the authors seem to have forgotten a $-1$ in the definition of $f_t^{\tau,\tau-1}$.

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