Hi

@Hesham_87 To me, that formula is merely the covered interest rate parity (CIP; aka, interest rate parity, IRP) formula. In FRM P1.T3, we reviewed why IRP is an application of cost of carry. In this (discrete) annual notation, IRP say it should be the case that the forward price is a function of the spot price and the net cost of carry, F(0) = S(0)*(1+c)^T where c is the net cost of carry. The most basic carry is the risk free rate such that F(0) = S(0)*(1+Rf)^T but if the asset pays a dividend (or provides income) then F(0) = S(0)*[(1+Rf)/(1+q)]^T which is the discrete analog to (for some of us the more familiar) continuous version F(0) = S(0)*exp[(Rf-q)*T].

The CIP/IRP is merely this COC but where the commodity is another (foreign) currency. Importantly, just as the S&P Index (an investment commodity) pays a dividend, cash invested in the foreign currency pays a "dividend" in the form of its own riskfree rate. Consequently, when the commodity is a currency, the same COC becomes the CIP/IRP formula: F(0) = S(0)*[(1+Rf)/(1+Rf_f)]^T, or the single-period version which you show: F(0) = S(0)*[(1+Rf)/(1+Rf_f)]. Subtracting the spot rate, S(0), retrieves the "typically quoted" (according to the BIS) forward points given by F(0) - S(0) = S(0)*[(1+Rf)/(1+Rf_f)] - S(0).

So from my perspective the CIP/IRP formula doesn't "need" the forward rate. Rather, it is simply a formula that gives us a theoretical price of the forward FX rate which, under assumptions, should hold because we can easily demonstrate an arbitrage profit (is in my learning XLS) if the actual F(0,T) varies from this theoretical F(0,T).

At the onset, the rational vanilla currency swap has a value of zero to both parties (if it has initial positive value for one counterparty, whey would the other enter the contract?). They will exchange principal at the spot rate. CIP/IRP tells us that the currency paying the higher (lower) riskfree rate should, in terms of the forward rate, depreciate (appreciate) relative to the other currency; intuitively, if we swap currencies and you enjoy the higher coupon (risk-free) rate, then you should expect a less favorable future exchange (aka, depreciation). At the initial swap, if we assume that the forward FX rate is an unbiased predictor of the future spot rate, then it doesn't theoretically really matter if our agreement, at the end, is to swap the principal back at the forward rate or the future spot rate; i.e., the expected future spot rate is predicted by the forward rate. Of course, the E[S(t)] is unknown while the F(0,T) can be locked in immediately, so really

*either *can be the determinant (i.e., future spot FX versus forward FX) of the final principal exchange; e.g, we tend to assume the vanilla fixed-for-fixed currency swap entails a swap of the principal amounts (which implicitly assumes the future spot FX rate is the determinant) but the counterparties can agree on a predetermined forward rate per the FX swap agreement in the BIS link (which really just implies the FX price risk is shifted from one to the other).

In any case, the point of the Borio reading is to ask,

*why does CIP/IRP appear to fail?* See below, please--where the emphasis is mine--just to draw attention to the focus of the paper, which is to illustrate that these instruments do tend to have a non-zero basis (which here means that the difference between the FX spot rate and the contractual forward FX rate is not fully explained by CIP/IRP):

"Yet since the onset of the Global Financial Crisis (GFC), CIP has failed to hold. This is visible in the persistence of a cross-currency basis since 2007. The cross-currency basis indicates the amount by which the interest paid to borrow one currency by swapping it against another differs from the cost of directly borrowing this currency in the cash market. **Thus, a non-zero cross-currency basis indicates a violation of CIP. **Since 2007, the basis for lending US dollars against most currencies, notably the euro and yen, has been negative: borrowing dollars through the FX swap market became more expensive than direct funding in the dollar cash market.

...

In an FX swap, one party borrows one currency from, and simultaneously lends another currency to, a second party (see also Baba et al (2008)). The borrowed amounts are exchanged at the spot rate, S, and then repaid at the pre-agreed forward rate, F, at maturity. The implicit rate of return in an FX swap is determined by the difference between F and S, and the contract is typically quoted in forward points (F – S). **If the party lending a currency via FX swaps makes a higher or lower return than implied by the interest rate differential in the two currencies, then CIP fails to hold**. Typically, the US dollar has tended to command a premium in FX swaps. In this case, rearranging the CIP equation yields the following relationship between (F – S), r and r*:

F-S > S*[(1+r)/(1+r*) - 1]

I hope that's helpful,

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