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Carry and Mark-to-Market PnL in Forward Interest Rate Swaps

Article Quant Q&A · Author: Eric Aldrin
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Summary

The document asks how to decompose the one-day mark-to-market PnL of a position receiving a forward six-month interest rate into changes in rates and carry. It gives initial and next-day quotes for a six-month rate and a six-month rate starting six months forward, then proposes valuing the rate move as the initial forward rate minus its next-day quote. It approximates daily carry as the forward rate’s premium over the current six-month rate divided by trading days in a year.

The question highlights an apparent puzzle: if the market’s expected evolution occurs, why should the position earn carry? However, it provides no answer or validation of the proposed arithmetic. A rate difference is not itself a complete swap PnL calculation; valuation depends on the contract’s cash flows, discounting, accrual conventions, notional, and sensitivities. Carry and rate changes also need consistent units and a defined holding-period valuation. The setup is therefore useful as a decomposition question, but not as a reliable worked method for quantifying swap PnL.

Key ideas

  • The proposed decomposition separates a forward-rate mark change from an estimated carry component.
  • The example estimates daily carry from the spread between a forward rate and a shorter current rate.
  • A rate difference alone does not determine the mark-to-market PnL of a swap position.
  • Swap valuation depends on contract cash flows, discounting, accrual conventions, notional, and rate sensitivity.
  • The document poses the free-lunch intuition but does not resolve it or establish that its calculation is correct.

Tags

Fixed incomeCarryRisk management
Full text
# Realized Term Structure: Forward Interest Rate Swaps


# Realized Term Structure: Forward Interest Rate Swaps












Is an realized term structure a PnL generating event? I'm going through Tuckman's book and am trying to split up PnL into carry and change in rates.

Say that the forward rates look like this:

| Term | Rate_t0 | Rate_t1 |
| 6mo rate | .1013 | .1100 |
| 6mo rate, 6m forward | .1746 | .1700 |

Say that I want to estimate my PnL from receiving the 6mo rate, 6mo forward. On day t0, I choose to receive the 6m6m rate at .1746. One day later, the 2 components of my PnL are change in rates and the carry:

Change in rate: the 6mo rate 6m forward rate went from .1746 to .17. Since I am receiving, rates down is good for me. Therefore, I make .1746-.17 = .0046.

Carry: As an approximation, the carry per trading day would be (.1746 - .1013)/252 = .00029.

Therefore, after 1 day, my mark to market PnL is .0046+.0029.

Is this correct? If so, intuitively why am I making money from a situation where what is expected from markets is what occurred. If the market expectation occurred and I made money, doesn't this feel like a "free lunch" situation?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.