Pure Expectations Theory – Formula, Calculator & Examples!

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Forward contracts are often used as a way to minimize exposure to changes in exchange rates and currency fluctuations. 

These contracts involve locking in rates for a foreign exchange so that you know the price you will pay for buying and selling in the future, helping you manage and predict cash flow more efficiently. This type of trade involves calculating the forward premium with the help of an estimation of spot rate and forward rate. 

Let us discuss the Pure Expectations Theory that plays an important role in determining these rates. We talk in detail about this theory, its applications, calculations, formula, and examples to give you a better idea of the concept. 

Pure Expectations Theory – A Complete Guide

Forward rate models are systems used to analyze and predict the estimated values of economic variables in different financial markets.

Forward rates refer to forward exchange rate or forward interest rate, depending on the asset traded.

Pure expectations theory aims to predict the short-term rates based on current rates.

It asserts that forward rates exclusively represent the expected future rates. 

The pure expectations theory serves as a model to calculate the forward exchange rates and rates of interest.

The forward interest rate refers to the predicted interest rate an instrument or asset offers in the future.

It can be predicted with an analysis of the term structure of interest rates. A forward contract generally has a premium when the foreign exchange rate is quoted higher than the spot exchange rate.

Calculation of the forward exchange rate helps find the forward premium of a currency pair. 

Pure Expectation Theory Formula

To understand the expectation theory formula, consider an example of an N-year bond costing Q(t)N in period t and paying amount X in (t+N) years.

This means the return on the 1-year bond is X/Q(t)1 and the 1-year bond pays X in period t+1.

If an investor buys a 1-year bond now at Q(t)1, he receives amount X at the end of the year and invests the amount on buying 1-year bonds expiring next year. 

The return on such an investment is X/Q(t)1*X/Q(t+1)1

According to the pure expectation theory, the risk-free interest rate on a 2-year bond should give the same return as the expected return on two 1-year bonds.

This means the expected return on N 1-year bonds should equal the return on an N-year bond. The formula is:

Pure Expectation Hypothesis - Formula

The ‘E’ in the formula stands for the expected value as investors are not aware of the future prices of the bonds. 

Pure Expectation Theory Calculator

The pure expectation theory calculator is a tool that helps calculate the future interest rates for instruments to guide investment decisions.

To understand how this calculator works, take an example of a bond market where an investor can decide between 1-year and 2-year bonds based on the results of the expectation theory. 

For example, if a 1-year bond offers 9% interest and the 2-year bond has a 10% interest, this theory should help predict the interest rate for the second year.

Firstly, add 1 to the interest rate of the 2-year bond which gives 110% or 1.1 here. Squaring it gives 1.21. 

The next step is to divide this result by the interest rate of the 1-year bond plus one.

This means 1.21 divided by (9%+1 = 1.09) which gives us 1.11. Finally, subtract 1 to get the predicted 1-year rate for the next year which comes to 11.1% for this example.

This implies that an investor putting in a 1-year bond with a 9% current rate should expect it to give 11.1% in the next year to get an equivalent return of a 2-year bond. 

Expectations Theory Example

To understand how the expectations theory works, consider an example of USD/GBP where £1 = $1.2.

This is the spot exchange rate or the rate faced by a trader who would like to trade in these currencies at present.

The current risk-free rate of interest is 5% in the United States which means a dollar deposited in a bank earns an interest of $0.05 in a year.

On the other hand, this interest rate is 3% in the United Kingdom, meaning a £1 earns £0.03 as interest when kept in a UK bank. 

For example, a trader investing in USD/GBP has £1. He will have £1.03 or $1.26 at the end of the year.

The 1-year forward exchange rate is the rate that rules out the arbitrage probability in the currency pair market.

Thus, the forward exchange rate for USD/GBP is £1 = $1.22 which is more than the spot rate.

Any possibility of arbitrage a trader could enjoy by holding USD with a higher interest rate instead of GBP is neutralized. 

Unbiased Expectations Theory Calculator

One of the most widely used forms of the hypothesis model is the unbiased expectations theory.

This theory assumes that it is possible to predict short-term future interest rates and exchange rates can with the use of current long-term rates.

In other words, the theory suggests that a trader investing in a single 2-year bond should earn the same interest as he would with two consecutive investments in 1-year bonds.

The 1-year bond should have a lower rate of interest as compared to the 2-year bond.

However, the unbiased expectations theory assumes that the net profit should be equal.

Considering the theory to hold true, we can make predictions about the bond profits. 

An investor would prefer to purchase a 1-year bond now and another 1-year bond later instead of buying a 2-year bond.

The unbiased expectations theory holds that he should get the same returns in both cases.

A pure expectations theory calculator uses the formula to calculate the predicted future interest rates for investments.

The formula for calculation remains the same as the expectations theory.

Such a calculator provides an easy way to predict the future interest rates and exchange rates and helps in decision making as the investor can figure out, on the basis of outcomes, whether the future rates are favourable. 

Final Thoughts

Pure expectations theory offers an easy way to predict the future interest and exchange rates in financial markets.

It is a valuable tool investors use when trying to analyze short and long-term investment options across currencies, bonds, and other instruments.

However, this theory is purely based on formula and assumption and should not purely guide investment decisions.

It should be used to analyze the market health and combined with other methods to get reliable insights. 

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