VaR Mapping of an FX Forward into Discounted Cash Flows
Summary
The document explains how to represent an FX forward as simpler risk-factor positions for value-at-risk analysis. Its example breaks a contract to buy euros for dollars into a currency spot exposure, a euro bill position, and a dollar bill position. The component cash flows are discounted and combined with risk-factor VaRs and correlations to estimate portfolio risk.
The discussion clarifies that the currency exposure should reflect euros needed at maturity, so its value is discounted using the euro rate; the questioner had instead applied the dollar rate. The explanation distinguishes the forward’s maturity cash flow from an immediate spot purchase. It is a conceptual clarification tied to the stated example, and does not provide a complete derivation of VaR mapping or address conventions such as compounding and day counts.
Key ideas
- An FX forward can be mapped to currency and bill exposures to identify its underlying risk factors.
- The euro amount needed at maturity is discounted at the euro rate when valuing the corresponding position today.
- A spot exposure in the mapping represents exchange-rate sensitivity, while discounting reflects the timing of the required cash flow.
- Aggregate VaR combines component risk estimates and their correlations.
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# VaR mapping - Forward Foreign Currency Contract # VaR mapping - Forward Foreign Currency Contract I have a question about VaR mapping for FX forwards. Please bear with me while I outline the problem. Philippe Jorion's book discusses VaR mapping; a means to break down complex instruments into simple risk factors and calculate the amount of risk against these risk factors across the portfolio. The example given for FX forwards is a 1 year contract to buy 100 MM EUR for 138.09 MM USD. The forward rate is $1.3013. The FX contract is broken down into 3 components: a long position in the EUR spot ($1.2877) a long position in a EUR risk free bill (2.281%) a short position in a USD risk free bill (3.330%) A VaR is given for each of the above positions, along with a correlation matrix. The author then calculates the present value of the cash flows from each of the 3 positions above as such: PV cash flow of long position in the EUR spot = $$130.09 * 1/(1+USD risk free bill rate) = 125.89$$ PV of the cash flow of long position in the EUR risk free bill = $$100 * EUR spot * 1/(1+EUR risk free bill rate) = 125.90$$ PV of the cash flow of short position in the USD risk free bill = $$-130.09 * 1/(1+USD risk free bill rate) = - $125.89$$ The PV cash flows are then multiplied by the given VaRs for those risk factors and summed up to produce an aggregate undiversified var. My question: I don't understand why the cash flow for the EUR spot long position is being calculated the way it is. It makes no sense to me. Seeing as it is a spot rate why is it's cash flow not simply spot * 100 MM EUR notional ? Here is a YouTube video on this problem in case my textual description isn't clear. https://www.youtube.com/watch?v=Um8e_teI_dw (This contrived question is a topic on my Financial Risk Manager exam) Thanks ## Answer by ali (score 2) https://quant.stackexchange.com/a/17999 Since Fx Fwd has different underlying risk factors, it decomposes the positions into different cash flows. The spot (currency) position is created to account for the impact of exchange rate fluctuation. ## Answer by mbison (score 2) https://quant.stackexchange.com/a/19507 If I interpret your question below correctly: > I don't understand why the cash flow for the EUR spot long position is being calculated the way it is. It makes no sense to me. Seeing as it is a spot rate why is it's cash flow not simply spot * 100 MM EUR notional ? You want to know why Jorion takes 100MM EUR * spot * 1/(1+eur_rate) instead of 100MM * spot. First of all: Note that you made the mistake of saying 100MM eur * spot * 1/(1+usd_rate) it should be the eur_rate. As in this context euro rate is foreigen. Secondly the reason why he takes 100M EUR * spot * euroDiscount is because you do not need 100MM EUR at t=0; you need 100MM EUR at maturity. Therefore it is sufficient to start off with a smaller position (the discounted one against euro rate).
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