## Interest rate parity theorem

Let’s see how it works. Assume that the nominal interest rate for a certain instrument with a specified validity period in camp X is higher than the nominal interest rate for an instrument with the same validity period and similar risk level in country Y.

Also, suppose that spot (cash) and futures rate (at a future point in time) are equal at the moment. Then, investors with available funds in country Y will convert their currency “y” into currency “x” and invest it in country X at the higher rates adopted in country X.

- E (D) XY – forward or futures exchange rate, after the number of days D;
- EXY – exchange cash (spot) rate;
- R (D) X and R (D) Y – interest rates for instruments with a validity period D and similar risk;

Based on the formula of the Law on Interest Rate Parity, it is possible, using simple mathematical calculations, to determine the dependence of the cash rate on the difference in interest rates.

We will analyze the situation using an ideal example using US and UK treasury bills and bonds with a maturity of 1 year, assuming that the cash and futures GBP / USD in this situation are equal. As of May 22, UK annual bonds have a yield of 0.729%, and US treasury bills with a yield of 2.362%. The yield on US Treasury bills is 1.633% higher than the yield on UK bonds. Both bonds have one maturity and the same credit rating, that is, the same lowest risk level. These are the so-called “risk-free” assets with the highest degree of reliability.

It is easy to guess that having the opportunity to receive financing in pounds at lower interest rates, an international investor will exchange pounds for dollars and place them in the United States at a higher percentage. The yield on this operation for one million pounds will be 1.633%. According to the law of supply and demand, when converting pounds to US dollars, the investor will lower the pound and increase the dollar.

It was smooth on paper but forgot about the ravines. This situation is ideal, and cannot be realized in practice due to the difference between cash and futures. The fact is that when receiving financing in pounds and buying US dollars, the investor acquires an undesirable short position in pounds, i.e., has an uninsured currency risk, which can lead to significant losses exceeding the investor’s income.

In order to avoid the risk of currency fluctuations, the investor must purchase British pounds in the forward or futures market with a delivery period of one year, i.e., make the reverse conversion, thus insuring his short cash position. Then, in the futures market, the investor will buy British pounds, increasing their rate in the future and selling American dollars, lowering their rate at the time of delivery. This is a very important circumstance that allows you to determine the direction of capital flows. If the currency of the future exchange rate is higher than cash, the cash flow goes in the direction of the currency in the denominator of the currency pair. If the futures rate is lower than cash, then the cash flow goes in the direction of the currency in the numerator. The difference between the cash and the futures rate is called the forward point.

We will analyze our example taking into account the difference in cash and futures rates. As of May 22, 2019, the GBP / USD cash rate is 1.2670, while the futures rate for delivery in June 2020 is 1.2924. We convert one million British pounds to US dollars at the cash rate and get $ 1,267,000. The return on investment of 2.362% is $ 29.927. The total amount received as a result of the investment in dollars, in a year will be equal to $ 1,296,927.

Convert it back to pounds at the futures rate and get 982.662 pounds. As you can see, the loss from this operation on investing in US assets amounted to 17.738 pounds and taking into account the fact that investing money in UK bonds could have earned 7.290 pounds without any risk, the lost profit from such a transaction would be 24.628 pounds or 2.463% of the investment. As we see, risk-free arbitrage at exchange rates using treasury bonds is really impossible. However, for other instruments, for example, corporate or municipal bonds, as well as shares with dividend payments and rising in price, such arbitration becomes quite real, the main thing is that the return on investment covers possible risks. Such arbitration in the slang of traders is called “carry trade”.