# 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.

## 1 Answer

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

• I agree. Ilmanen carry is the annualized carry, while the pedersen number is the carry of the bond price. To get from one to the other you need to multiply by the remaining duration (2 in your example). By the way shouldn’t your last equation read (n-1) rather than (n-2) ?
– dm63
Commented Aug 27, 2022 at 10:04