# difference of carry for zero coupon bonds in Pedersen and Ilmanen

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.

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 $$f_t^{\tau,\tau-1}$$.