Excel
02 Dec 2008
Jul
02
Interest rate parity is applied cost of carry model
by David Harper, CFA, FRM, CIPM
FRM |
A classic, tough exchange rate question was posted here on the forum. Here is the question:
The two year risk free rate in the United Kingdom and France are 8% and 5% per annum continuously compounded, respectively. The current French Franc (FF) to the GBP currency exchange rate is that one unit of GBP currency cost is 0.75 units of FF. What is the two year Forward Price of one unit of the GBP in terms of the FF (now Euros, of course) so that no arbitrage opportunity exists? If the observed two year forward price of one unit of the GBP is 0.850 of the FF what is your strategy to make an arbitrage profit?
I input the answers into the EditGrid spreadsheet below. I hope it helps with the intuition. The short answer is given by (spot FX)*EXP[(5%-8%)(2)] because interest rate parity is a variation on the cost of carry model. (Note the continuous compounding. Saunders uses annual compounding for a slightly different result). Recall the universal cost of carry model:
Forward (F0) = (spot)*EXP[(costs of ownership – benefits of ownership)(T)
Where costs are the interest rate (financing costs) and storage costs. Benefits are dividends and their intangible equivalent, convenience yield. Cost/benefit handles all fundamentals (including, in the case of oil futures, the tendency of a convenience yield to drive long-run backwardation. But note the simple COC model does not impound ‘technical' factors, so it really doesn't incorporate supply/demand. It is tempting to conflate technical supply/demand with fundamental cost of carry. But supply/demand overwhelms the model.)
Application of cost of carry to interest rate parity (IRP)
To apply the cost of carry to this FX problem, we just have to take Hull to heart: "a foreign currency can be regarded as an investment paying a known yield. The yield is the risk-free rate of interest in the foreign country." In the practice question, then, the forward (FF/GBP) is
Forward (F0) = (0.75 FF/GBP)*EXP[(5% – 8%)(T) = 0.7063 FF/GBP
In this way, we treat the forward as an investment asset paying a dividend yield (i.e., a dividend yield lowers the forward price because it is a benefit of holding the asset; the long forward does not need to compensate the owner for it). And the dividend yield is here really the 8% rate in the U.K.
But that's not how I solve the problem because I can't intuitively place the 5% and the 8%. Instead, you only need to imagine that no arbitrage says that two alternative investment scenarios must produce the same terminal value:
- Stay home: Pick either country and call it home. For example, U.K. is home (but it doesn't matter which). Your first scenario is to invest at home. In this case, "staying home" earns you (domestically) 8%.
- Round trip abroad: Your second scenario is to invest abroad (in this case, France) by taking a "round trip." If U.K. is home, the round trip equivalent is intuitive: convert at spot rate (from GBP to FF), invest in Francs (now Euros of course) for the two year period, then convert them back to Sterlings with the forward contract.
The forward exchange rate is set by the no-arbitrage rule. I illustrated that below. After that is illustrated the second part of the question: if the forward is higher or lower, what is the arbitrage. In orange are shown covered arbitrage. The trick is to see which country is the better investment. For example, in the first case, the forward is mis-priced at 0.85 FF/GBP. That is a depreciated (weaker) Franc vis a vis the correct forward rate of 0.7063. Therefore, we don't want to be invested during the period in Francs, but rather pound sterlings. We borrow Francs in order to invest in sterlings.
EditGrid:
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