Understanding Forward Rates Models: Analysis & Prediction

Forward rates models are theoretical frameworks used to analyze and predict the expected valueExpected ValueExpected value (also known as EV, expectation, average, or mean value) is a long-run average value of random variables. The expected value also indicates of economic variables in the future. Forward ratesForward RateThe forward rate, in simple terms, is the calculated expectation of the yield on a bond that, theoretically, will occur in the immediate future, usually a few months (or even a few years) from the time of calculation. The consideration of the forward rate is almost exclusively used when talking about the purchase of Treasury bills usually refer to either the forward interest rate or the forward exchange rate.

Expectations Hypothesis

Consider the following example: An N-year government bond costs Q(t)_N in period t and pays an amount X in period t+N years. Therefore, the return on a 1-year bond is X/Q(t)₁. The 1-year bond pays X in period t+1.

Consider an investor who does the following:

1. Buys a 1-year bond now by paying Q(t)₁
2. At the end of the year receives X
3. At the end of the year, spends all of X on buying 1-year bonds that will expire at the end of next year

The return on a sequence of two 1-year investments is X/Q(t)₁ * X/Q(t+1)₁.

If investors are risk neutral, then the return on a 2-year bond should be equal to the expected return on a sequence of two 1-year bonds. In general, the expected return on N 1-year bonds should be equal to the return on one N-year bond.

The ‘E’ denoted around future bond prices in the denominator represents the expected values since investors don’t know the future bond prices.

Forward Interest Rate

The expectations hypothesisLocal Expectations TheoryIn finance and economics, the Local Expectations Theory is a theory that suggests that the returns of bonds with different maturities should be the same over the short-term investment horizon. Essentially, the local expectations theory is one of the variations of the pure expectations theory can be used as a model to derive forward interest rates and exchange rates. The forward interest rate is the expected rate of interest offered by a security in the future. The forward interest rate can be inferred by analyzing the term structure of interest ratesInterest RateAn interest rate refers to the amount charged by a lender to a borrower for any form of debt given, generally expressed as a percentage of the principal..

Consider the following example:

• A $1 2-year zero coupon bond gives a return of 12% per year.
• A $1 1-year zero coupon bond gives a return of 9% per year.

An investor investing in a $1 2-year zero coupon bond will have $1.2544 at the end of two years. An investor investing in a $1 1-year zero coupon bond will have $1.09 at the end of one year. If the investor proceeded to invest in a $1 1-year zero coupon bond at the end of the first year, the return would depend on the future interest rate offered by a $1 1-year zero coupon bond. The implied 1-year forward rate is that rate of interest that rules out the possibility of arbitrage. Since there is no possibility of arbitrage, the expectations hypothesis says that the product of the two 1-year rate should equal the 2-year rate. Therefore, the answer is 1.09(1 + r_forward) = 1.2544, implying a 1-year forward rate of 15.08%.

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Forward Exchange Rate

The forward exchange rate between two currencies is the exchange rate between two currencies when the actual exchange takes place in the future. Consider the following example: The current USD/GBP exchange rate is £1 = $1.2. It is known as the spot exchange rate or the exchange rate faced by a currency trader willing to deliver or take delivery of either USD or GBP right now.

The current 1-year risk-free rate of interest in the US is 5%. Therefore, $1 deposited in a bank in the US earns interest worth $0.05. The current 1-year risk-free rate of interest in the UK is 3%. Therefore, a £1 deposited in a bank in the UK earns an interest of £0.03.

Consider a currency trader in the USD/GBP market who initially has £1. At the end of one year, the trader can either have £1.03 or $1.26. The efficient 1-year forward exchange rate is the exchange rate that rules out the possibility of arbitrage in the USD/GBP market.  Therefore, the 1-year forward USD/GBP exchange rate is £1 = $1.22, which is higher than the spot rate. This forward rate neutralizes any possible arbitrage that an investor could have if they held USD – which has a higher interest rate – than GBP.

Additional Resources

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To keep learning and developing your knowledge of financial analysis, we highly recommend the additional resources below:

• Coupon RateCoupon RateA coupon rate is the amount of annual interest income paid to a bondholder, based on the face value of the bond.
• Interest Rate SwapInterest Rate SwapAn interest rate swap is a derivative contract through which two counterparties agree to exchange one stream of future interest payments for another
• Swap Rate CurveSwap Rate CurveThe swap rate curve is a chart that depicts the relationship between swap rates and all available corresponding maturities.
• USD/CAD Currency CrossUSD/CAD Currency CrossThe USD/CAD currency pair represents the quoted rate for exchanging US to CAD, or, how many Canadian dollars one receives per US dollar.  For example, a USD/CAD rate of 1.25 means 1 US dollar is equivalent to 1.25 Canadian dollars. The USD/CAD exchange rate is affected by economic and political forces on both