The FRED® Blog

# Constructing forward interest rates in FRED

A forward interest rate is a rate that pertains to a future loan and/or bond purchase. A forward transaction can be arranged in an over-the-counter market with a financial institution, or it can be constructed from existing fixed income instruments. A forward rate contract has at least two elements: the contract start and length. For example, a forward loan contract might commence 2 years in the future and last for 6 months. Such a contract might be termed a 6-month contract, 2 years ahead. Interest rate contracts (bonds or loans) that start immediately (or nearly so) are called spot market contracts.

Under the often-implicit assumption that financial markets price assets in a risk-neutral manner, analysts often use forward rates as market expectations of future interest rates. In other words, analysts often assume that the interest rate that one can lock-in today for a future transaction is the market’s expectation of that interest rate. With this interpretation, many analysts use forward rates to infer information about market expectations of variables such as monetary policy, output and inflation, and currency movements. For example, Hausman and Wongswan (2011) show that the 3-month forward interest rate, 1 year ahead, is closely related to what Gürkaynak, Sack, and Swanson (2005) refer to as the “path” monetary policy shock.

How can we determine the interest rate that should prevail on a forward contract, assuming that markets are working reasonably well? For concreteness, let’s think about the zero-coupon rate that should prevail from 2 years in the future until 10 years in the future. That is, we will consider the 8-year forward rate, 2 years ahead.

To see the relation between the 8-year forward rate, 2 years ahead, and the 2- and 10-year spot rates, let’s think about two ways that one could invest/lend a sum of money for 10 years.

1. Buy a 10-year, zero coupon bond.
2. Buy a 2-year bond and a forward contract to buy an 8-year bond at the end of the 2 years.

If one invests \$1 on the first strategy, one obtains the compounded, 10-year gross yield: (1+i0,10)10, where i0,10 is the annually compounded interest rate paid from now (time 0) until year 10. Likewise, the first leg of strategy 2 yields a payoff of (1+i0,2)2, where i0,2 is the annually compounded interest rate from time 0 to year 2. At year 2, this payoff to the first leg is then invested for 8 years at the forward interest rate to get a total payoff of (1+i0,2)2 (1+i2,10)(10-2), where i2,10 is the annually compounded interest rate from year 2 to year 10.

The payoffs to the two strategies must be approximately equal or arbitrageurs would bid up the price of the cheaper strategy and short the more expensive strategy, driving its price down. The equation below shows the relation between the forward rate (i2,10) and the two spot market rates (i0,10 and i0,2).
(1+i0,10)10=(1+i0,2)2 (1+i2,10)(10-2)

More generally, using time t0 as the base date and annually compounded interest rates, the gross forward interest rate at time t0, from time t1 to t2 can be represented as follows:
(1+i(t0,t2))(t2-t0)=(1+i(t0,t1))(t1-t0) (1+i(t1,t2))(t2-t1)

Solving for the forward rate alone, which is used in constructing the graphs, is as follows:

The above formula is strictly applicable only for zero-coupon bonds—that is, bonds whose only payoff is at maturity. But, depending on the purpose, it might produce a reasonable approximation for bonds that pay coupons.

FRED has 10 forward interest rates that are derived from the Kim-Wright term structure model: 10 instantaneous forward rates, from 1-, 2-, 3- … to 10-years hence. These “instantaneous” forward rates are theoretical constructs for an interest rate that applies to loans of arbitrarily short duration. In practice, these are most reasonably interpreted as approximately overnight rates.

You might wish to construct your own forward rates with different characteristics in FRED. To give you a hand, we construct 2 different forward rates below, an 8-year forward rate, 2 years ahead, and a 3-month forward rate, 3 months ahead.

Example 1: The following directions describe how to construct an 8-year forward rate, 2 years ahead. The graph is at the top of the blog post.

1. In the FRED search box, type in “Fitted yield on a 10 year zero coupon bond” and select the series of that name. Graph the series.
2. Select “Edit Graph.”
3. In the series selection box under “Customize data,” type in “Fitted yield on a 2 year zero coupon bond” and click “Add” to add the series.
4. To construct an 8-year forward rate, 2 years ahead, type the following formula into the formula box: 100*((((1+a/100)^10)/(1+b/100)^2)^(1/8)-1). Then click “Apply.” The 100s in the formula convert the interest rates between percentage and decimal terms.
5. To compare the constructed 8-year forward rate, 2 years ahead, to the 10-2 Treasury spread offered by FRED, click on “Add Line” and type “10-year Treasury Constant Maturity Minus 2-year Treasury Constant Maturity” in the search box, then select that series and click “Add data series.”
6. To compare the 10-year Treasury rate with 8-year forward rate, 2 years ahead, select “Add line” in “Edit Graph” and type in “Market Yield on U.S. Treasury Securities at 10-Year Constant Maturity, Quoted on an Investment Basis,” then click “Add data series.”
7. To see the series over a common sample, set the start of the sample at 1990-01-02.

Perhaps not surprisingly, the graph shows that this forward rate is highly correlated with the 10-year Treasury rate. Coming out of recessions, the 10-2 spread tends to be high, as in 1992, 2003, or 2010, and the forward rate tends to exceed the 10-year rate. But when the 10-2 spread is near or below zero, as in 1998, 2006, or 2019, the yield curve is said to be inverted and the forward rate and the 10-year yield are nearly identical.

Example 2: The following directions describe how to construct a 3-month forward rate, 3 months ahead.

1. In the FRED search box, type in “Market Yield on U.S. Treasury Securities at 6-Month Constant Maturity” and select the series of that name. Graph the series.
2. Select edit graph.
3. In the series selection box under “Customize data,” type in “Market Yield on U.S. Treasury Securities at 3-Month Constant Maturity” and click “Add” to add the series.
4. To construct a 3-month forward rate, 3 months ahead, type the following formula into the formula box: 100*((((1+a/100)^.5)/(1+b/100)^.25)^(1/.25)-1). Then click “Apply.” The 100s in the formula convert the interest rates between percentage and decimal terms.
5. To compare the constructed 3-month forward rate, 3-month ahead, to the federal funds target offered by FRED, click on “Add Line” and type “Federal funds target range, upper limit” in the search box, then select that series and click “Add data series.”
6. To similarly add the lower limit of the federal funds target range, click on “Add Line” and type “Federal funds target range, lower limit” in the search box, then select that series and click “Add data series.”
7. To see the series over a recent sample, set the start of the sample at 2015-01-01.

The 3-month forward rate, 3 months ahead, is interesting because it can illustrate how financial markets anticipate monetary policy movements. The graph shows that it started to rise before the start of federal funds tightening cycles in 2015 and 2022 because markets correctly anticipated that a tightening cycle was about to start. The 2020 fall in the 3-month forward rate, 3 months ahead, was nearly coincident with the decline in the federal funds target rate because the latter was a reaction to the economic implications of the spread of the COVID-19 virus. Thus, the monetary easing was not anticipated very far in advance.

Suggested by Christopher Neely.