Why is there no carry if interest rates follow projected forward rates?

Question

"Carry is a function of the shape of the interest rate curve. When the curve is upwardly sloping (ie, longer dated rates are higher than shorter rates, as they are currently), then the market is implying that interest rates are expected to rise in the future. If interest rates follow projected forward rates (and these expected rises materialise), then carry will be zero."

I'm struggling to understand why this would be the case?

Source: redington.co.uk/base/redington/publications/download/id/12 [Note: It auto downloads the PDF]

Answer 1

Carry tends to have different meanings in the derivatives world and in the asset management world. The paper you are referring to comes from the asset management world.

In the derivatives world people tend to think that if the random part is zero the variables "age" according to their forward value: tomorrow's interest rates will be today's 1 day forward interest rates, tomorrow's implied vol will be today's 1 day forward implied vol, etc.

In the asset management world people tend to think that if the random part is zero the variables remain the same: tomorrow's interest rates are the same as today's. So they have introduced a concept of carry that can be thought of as the impact of the difference between rates not changing and rates changing according to their forwards.

In essence it's the same difference than between theta and modified theta in the option world: theta is just the derivative to time, but modified theta includes the various funding effects, volatility aging, etc.