Interest Rate Parity

[Biju George]

HI All ,

Can you please help in answering the below

Suppose the spot rate is 0.7102 USD/CHF and
Swiss interest Rate = 7.6%
US interest Rate = 5.2%
If the 1 Year forward rate is 0.72 USD/CHF What's the arbitrage opportunity here ?
--
My Understanding :
As per the Interst Rate parity eqn the Forward Rate should be =0.726 where as the quoted price=0.72 only.Hence future price is under quoted and buy futures ?So buy Forward Contract.
Can you please confirm
Thanks in advance

 

[Dr. Jayanthi Sankaran]

Hi @RiskGuy,

Just hazarding a guess here:

According to the interest rate parity theorem: (1 + r(USD)) = [1/S(t)]*(1 + r(CHF))*F(t)

i.e. Rate on US investment = Hedged return on foreign (CHF) investment

Where 1 + r(USD) = 1 plus interest on US CDs for the foreign investment at time t
S(t) = $/CHF spot exchange rate at time t
1 + r(CHF) = 1 plus the interest rate on CHF CDs at time t
F(t) = $/CHF forward exchange rate at time t

In your example above,
r(USD) = 5.2%
S(t) = $0.7102/CHF
r(CHF) = 7.6%

Substituting in the IRPT equation above:

(1 + .052) = [1/0.7102]*(1 + 0.076)*F(t)
F(t) = (1.052)/(1.515) = 0.694
Since the quoted price for F(t) = 0.72, there is an immediate arbitrage opportunity.
Arbitrageurs will step in to sell F(t) = 0.72 buy the US CD's at (1.052)*.7102 = 0.747 and sell the CHF CDs at 1.076 i.e.
[(1.052)*0.7102]/(1.076) = 0.694. In other words, investing in the foreign exchange CD market is not as lucrative as investing in the US CD market.

Thanks!
Jayanthi

 

[David Harper CFA FRM]

That's such a great answer @Jayanthi Sankaran !

Can I just add: the hard part, for me, is not identifying the existence of an arbitrage. (I will use continuous compounding rather than annual compounding, such that my result is similar to @Jayanthi Sankaran but not identical.) Please note, it's important to clarify which is the base versus quote currency. That is, 0.7102 USD/CHF connotes 0.7102 CHF (ie., the quote currency) per $1.00 (ie, the base currency). The COC says the forward price should be = 0.7102*exp(5.20%-7.60%)*1 = 0.6934; same as Jayanthi's but continuous. And, directionally, this is intuitive: the USD interest rate is higher, so that to be competitive as an investment, the CHF currency needs to appreciate such that the base currency of the forward (the dollar) prices lower. Put another way, in the expected forward currency contract, the dollar has a lower price (depreciates) to offset its higher domestic interest rate.
... btw, the base currency [dollar] is the commodity here, so the base currency's interest rate [7.60%] is subtracted in the exponent because its interest rate is analogous to the dividend yield in commodity COC. While the quote currency's interest rate [5.20%] is analogous to the cash paid/received for the commodity, so its gets added [goes first] in the exponent, as it is analogous to the risk-free rate in the COC.

Okay, so plugging into the COC gives you the expected forward price of CHF 0.6934 which is lower than the traded forward price of CHF 0.720. The hard part for me is identifying the trade that @Jayanthi Sankaran illustrates, what to buy/sell? Well the arbitrage is going to be:

• Borrow CHF (or USD)
• Convert the borrowings at the spot rate; ie, either borrowed CHF to USD (or borrowed USD to CHF)
• Invest the converted USD (or CHF) over the 1.0 year at the USD (or CHF) domestic interest rate
  
• In 1.0 year, sell forward the USD (or CHF)
  
• Payoff the CHF (or USD) loan

The answer is, we want to sell expensive stuff and buy cheap stuff. When we sell forward in one year, which is more expensive (the dollar or the franc) in the 0.720 which we can lock in? CHF 0.720 is a cheaper relative franc and more expensive relative dollar than anticipated by CHF 0.6934. So, we want to be selling the relatively more expensive dollars (and buying the relatively cheaper francs) in that forward trade. Consequently, we want to be invested in dollars over the year and earning the USD rate so we can sell those dollars into the forward. So the arbitrage trade is:

• Borrow CHF
• Convert borrowed CHF to USD
  
• Invest the converted USD over the 1.0 year at the USD interest rate
  
• In 1.0 year, in the forward contract, sell USD in exchange for CHF
  
• Payoff the CHF (or USD) loan

I hope that's helpful,

 

[Biju George]

Hi ,
Wanted to confirm one point related to Interest Rate Parity.
Quote : USDBRL . I assume same as USD/BRL

If the
spot rate for USDBRL=3.5 and
US Interest Rate=1%
BRL Interest Rate=9%

In the IR Parity equation .I am having a confusion which should be foreign currency and which is the local currency. How to identify based on the quote given ?
Also for other quote conventions US/BRL & 3.5BRLUS ...


Also .. As per my understanding for the Currency with higher IR will always depreciate?Please confirm
Thanks in advance

 

[ami44]

It always helps me, to think of the foreign currency as commodity, that is bought or sold with money in the local currency. So if you want to calculate forward rates for USD/BRL, which is the price of 1 USD in BRL, then BRL is the local currency and USD is foreign currency.
So the current price of 1 USD is 3.5 BRL, what will the forward price be:
Ft = 3.5 USD/BRL * exp((IRBRL - IRUSD) * t)
As you can see the interest rate for the foreign currency is treated like a (continuous) dividend for a commodity.

Also, if IRBRL is higher that IRUSD than Ft is higher than the current spot rate. That means the price of 1 USD in BRL will appreciate, or on the other side BRL will depreciate. So your understanding is correct, the currency with the higher interest rate will depreciate.