## Wednesday, November 18, 2015

### Interest Rate Parity and Exchange Rates

[If you want to skip all the math and theoretical stuff, go ahead and jump down to the conclusion]

Take a bond, any bond.

Actually take two. Make them zero coupon as well, with a one year maturity (for ease of exposition). Bond A is issued in Country A, while Bond B is issued in Country B, both at a discount of 5% to face value i.e. both bonds yield 5% in a single year.

Let’s define the spot exchange rate (S) as the ratio of currency B to currency A, say for example, \$4 of country B currency is exchangeable to \$1 of country A currency.

The uncovered interest rate parity condition for these variables are:

iA = iB + E(St+k)/St

where:

E is the symbol for market expectations
iA is the interest rate in country A
iB is the interest rate in country B
St+k is the spot exchange rate k periods in the future
St is the current spot exchange rate

All that gobbledygook says is that if the interest rate in country A is higher than in country B, the exchange rate would increase by the same magnitude. Taking a numerical example, if iA = 6% and iB = 5%, you would expect the exchange rate to appreciate (from A’s point of view; depreciate from B’s point of view) by 1% per period.

Clear? Sure? Let’s proceed.

Let’s take the numbers I laid out earlier and put them in the model:

iA = iB + E(St+k)/St

5% = 5% + 0%

I’m totally ignoring the risk premium here but you get the gist – assuming interest rates are the same, there should be no change in the current or future expected exchange rate. For a 1 percent increase in Country A:

6% = 5% + 1%

So a higher interest rate in Country A results in a positive change of 1% in the expected future value of S. But both the examples above make the crucial and unspoken assumption that the only interest bearing instruments are cash deposits.

What happens if the majority of investors are in bonds instead?

The peculiarity of bonds is that prices are inverse to the yield. So to take my example above, a 5% 1-year zero coupon bond with a face value of \$100 would have a value at issuance of 100/105 = \$95.24. If you raise the yield to 10%, the value would be 100/110 = \$90.90. Increasing the yield (interest rate) results in a drop in the principal value of the bond. An investor holding a bond yielding 5%, would suffer a 4.5% loss on his capital should yields rise to 10% (90.90/95.24). In other words, the increase in yield (+5.0%) is almost completely offset by the loss of principal (-4.5%).

I’m ignoring here (again for the sake of exposition) the possibility of inter-period trading, and the fact that the issuer would still pay off the bond at the 5% yield, not the market rate of 10% (providing an incentive for bond holders to sit on their investments). Nevertheless, the point I’m trying to make here is that bond holders would suffer a capital loss when interest rates rise. This has some interesting implications for the interest rate parity view of exchange rate determination.

Let’s try it out:

iA + dA = iB + dB + E(St+k)/St

…where I’ve inserted a term for a change in the capital value of the bond instruments (dx). I’d write out the math for that, but this is getting complicated as it is. Suffice to say that an increase in ix would result in an (almost) equal and opposite value for dx.

Let’s plug the numbers in and contrast what the standard and modified interest rate parity conditions say about the reaction of the exchange rate to an interest rate increase. Let’s assume country A raises interest rates by 1%, and focus on the outcome for the exchange rate (E(St+k)/St):

Standard:
Expected change in exchange rate = 6% – 5% = 1%

Modified:
Expected change in exchange rate = (6%-0.9%) – (5%-0%) = 0.1%

Big difference, no?

Conclusion

If investors betting on exchange rate movements are holding their funds in bonds instead of cash, the argument that a change in the interest rate differential would cause an appreciation of the exchange rate is much weaker than normally recognised. The fact that a rise in interest rates results in capital losses for bond holders means that the incentive to pile into a currency where interest rates are rising (as opposed to expected to rise) doesn’t really materialise.

To a lesser degree, a similar argument could be made for holdings in equities, as interest rate increases raise borrowing costs, and thus earnings valuations. Exchange rate movements based on interest rate parity only makes sense if everybody keeps their money in bank or interbank deposits. But most investors have mandates that strictly regulate their ability to hold cash. Also, the yield pickup from holding bonds instead of cash are attractive enough that it’s worthwhile doing so, even if you’re expecting interest rates to rise and therefore losses on principal.

The point of the foregoing discussion is this:

Everybody seems to be expecting the US Dollar to strengthen further when (if) the Fed starts raising interest rates. But historically, that hasn’t really happened – the last couple of times the Fed has embarked on a tightening stance, the US Dollar has generally weakened instead, having appreciated well beforehand.

I’m presenting here a stronger argument for that empirical oddity than the tired old “buy on rumour, sell on fact” explanation. There is here a mechanism involved beyond market psychology, and interest rate parity isn’t as strong a force as people think it is. My idea isn’t completely fool-proof – some people would indeed have their money in bank deposits or money market funds, while changes in the Fed Funds Rate don’t always translate into one-for-one yield increases for securities at the longer end of the yield curve. Also, this tightening cycle is likely to turn out to be the slowest and longest drawn out such affair in history, on top of the fact that we continue to have other major central banks heading in the opposite direction.

Nevertheless, what I’m expecting to happen is that when the Fed does finally get around to raising the FFR, it will mark the beginning of the end of the USD’s bull run.

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