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FX Forward P&L Requires Discount Curves Consistent with Forward Pricing

Article Quant Q&A · Author: tron
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Summary

The discussion compares two ways to value the P&L of a foreign exchange forward: tracking the change in the forward rate, and valuing the future currency cash flows with discount factors and converting them at spot. The response says these methods agree when the discount curves are consistent with the FX forward market relationship used to derive the forward rate.

A common source of disagreement is using each currency’s risk-free curve without accounting for cross-currency basis. If the curves omit that basis, their implied forward rate can differ from the market-consistent forward, so the two valuation approaches will not match. The explanation is concise and assumes the stated cash-flow and quotation conventions; practitioners still need to use consistent curve inputs and currency units when implementing the calculation.

Key ideas

  • Forward-rate changes and discounted cash-flow valuation should agree under consistent market inputs.
  • The FX forward relationship links spot to the two currency discount factors.
  • Ignoring cross-currency basis can make the chosen curves inconsistent with observed forward pricing.
  • P&L calculations depend on using coherent currency and quotation conventions.

Tags

ForexFixed incomeDerivatives pricing
Full text
# pnl calculation of FX forward


# pnl calculation of FX forward












i am trying to clarify the correct method of computing pnl (in base ccy) on an FX forward. let's assume the following notation:

S(t) = spot rate at time t

df(base,t) = base ccy discount factor (USD in this case) at time t

df(foreign,t) = foreign ccy discount factor (JPY in this case) at time t

FXfwd(t) = FX forward at time t = S(t)*[df(base,t) / df(foreign,t)]

assuming, at time t=0, we buy 100 USDJPY, 1y forward, i would compute the pnl from t=0 to t=1 to be \$100 * [(FXfwd(1) - FXfwd(0)) / FXfwd(1)]

however, an alternate approach seems to be to compute the present value of each of the future cashflows using each ccy's discount factor, sum in USD according to the spot for time t=0 and do the same at time t=1, and taking the difference.

it seems like the first approach i mention is more commonly referenced, but the 2 methods do not produce the same result and there is surprisingly little information found online for more practical matters like this

## Answer by dm63 (score 4)

https://quant.stackexchange.com/a/37593

They are the same, provided your discount curves are consistent with the fx forward market, as defined in your 'Notation' paragraph.

The problem people usually run into is that they use risk free rates in each currency for discount curves , but they forgot to incorporate the currency basis.

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