# Forward Rate

Updated July 20, 2021

## What is the Forward Rate?

Usually reserved for discussions about Treasuries, the forward rate (also called the forward yield) is the theoretical, expected yield on a bond several months or years from now.

## Forward Rate Example

The yield curve dictates what today's bond prices are and what today's bond prices should be, but it can also infer what the market believes tomorrow's interest rates will be on Treasuries of varying maturities.

For example, let's assume you receive money that you would like to use for a tuition bill you know will arrive in exactly one year. If you invest the money in Treasuries to keep safe and liquid, you still have two choices: You could either buy a T-Bill that matures in one year, or you could buy a T-Bill that matures in six months, and then buy another six-month T-Bill when the first one matures.

If both options generated the same outcome, you would probably be indifferent and go with whatever was easiest. However, there is the chance that rates will be higher in six months. If so, you'd make more money by buying a six-month T-Bill now and rolling it over into another six-month T-Bill to take advantage of those potentially higher rates. Or maybe rates will be lower, and you'd make more money locking your money up now for the full year. So the real question is, how much will a six-month T-Bill cost six months from now? That is, what is the forward rate on that six-month T-Bill?

The answer isn't clear. After all, by simply looking online you can ascertain how much a one-year T-Bill and a six-month T-Bill yields right now. But there is no way to tell for sure what a six-month T-Bill will yield in six months. However, there is a way to determine what the market is expecting, and that is by calculating forward rates.

Forward Rate Formula

Mathematically, the forward rate is the rate at which you would be indifferent to the two alternatives in our example. In other words, if you just bought the one-year Treasury, which you know from the newspaper is yielding 3% right now, you can easily calculate the price of this T-Bill:

\$100/(1+.015)2 = \$97.09

So you know that if you invest \$97.09 today, you'll have the \$100 you need in a year.

Now, how much do you need to invest if you purchase a six-month T-Bill and then reinvest that after six months in another T-Bill? You don't know for sure unless you know what that second six-month T-Bill is going to earn. If the annual yield on a six-month T-Bill purchased today is 2%, which is 1% semiannually, then the price of purchasing one six-month T-Bill today and then rolling it over into another six-month T-Bill would be:

Investment required today = \$100/((1+.01)(1+f))

Where f is the forward rate -- the rate on a six-month T-Bill six months from now.

In order for you to be indifferent about your two alternatives, you would have to be sure that investing \$97.09 in both scenarios would generate the \$100 you need in a year. Thus, the returns on the two investments have to equal.

That is,
\$100/(1+.015)2 = \$100/((1+.01)(1+f))
or
\$97.06617 = \$100/((1+.01)(1+f))

What is the rate that makes these investments equal? We must solve for f:

f = ((1+.015)2/(1+.01))-1 = 2.00% for six months, or 4.00% for one year.

The forward rate is 4% per year. Thus, we know that the market believes today the six-month T-Bill is going to yield 4% per year in six months. Thus, if you chose to buy a six-month T-bill and reinvest the proceeds in another six-month T-Bill, that second T-Bill would need to have a 4% annual yield to make you indifferent between doing this and just purchasing a one-year T-bill at the going rate. Now the question is, do you think you're really going to get 4%?

## Why the Forward Rate Matters

Forward rates are essentially the market's expectations for future interest rates. If the investor believes that rates will actually be higher or lower than expected, this may present an investment opportunity. Likewise, forward rates serve as economic indicators, telling investors whether the market expects more or less of all the things that correlate to interest rates.

If there is anything to be learned from forward rates, it is that they are prime illustrations of how interest rates tie together across the spectrum. Forward rates can be calculated further into the future than just six months. It's just a matter of doing the math. For example, the investor could calculate the three-year implied forward rate four years from now, the seven-year implied rate two years from now, etc.

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