Forward Rate Calculator

What this calculator does

You can see today's rate for lending money for two years, and today's rate for lending it for three. What you cannot see anywhere on a screen is the rate for lending money for one year starting two years from now. This calculator works that hidden rate out. It is called the forward rate, and it is already sitting inside the two rates you can see.

You enter two spot rates with their terms, and it returns the rate implied for the stretch between them. Nobody quotes that number directly, and yet it is not a matter of opinion. It is pinned down exactly by the two rates you started with, for a reason worth understanding.

Two routes to the same date must pay the same

Suppose you want your money invested for three years. There are two ways to get there. You can lock in the three-year rate today and leave it. Or you can lock in the two-year rate, then reinvest for the final year at whatever the one-year rate turns out to be by then.

If the market already knew that the second route would pay more, everyone would take it, and the rates would move until it did not. So the two routes have to arrive at the same place. The forward rate is simply the answer to "what would that final year have to pay to make the two routes tie?" That is the no-arbitrage condition, and the formula is nothing more than it rearranged:

Forward rate = [ (1 + longer rate)^{longer term} ÷ (1 + shorter rate)^{shorter term} ]^{1 ÷ (longer term − shorter term)} − 1

Read it as growth divided by growth. The top is what three years of compounding gives you, the bottom is what the first two years already account for, and the division leaves exactly the growth the market is attaching to the remaining stretch.

Getting the two periods the right way round

One practical matter before the numbers. The field labelled time period 1 must be the longer of the two, with time period 2 the shorter. Enter them the other way round and the calculator will stop you, since the maths needs a positive gap between the two dates to describe a forward stretch at all.

Each period has its own months or years selector. Where you can, it is cleanest to express both in the same unit before entering them, since a mixed pair such as two years against eighteen months can trip the check that compares them even though two years is genuinely the longer. Putting both in months, or both in years, avoids the question entirely.

Running the numbers on a real curve

Take a two-year spot rate of 4.00 percent and a three-year spot rate of 4.30 percent. The gap between them is the third year, so what the market is implying for one year of borrowing starting two years out is:

• Three years of growth at 4.30 percent: 1.043³ = 1.134627
• Two years of growth at 4.00 percent: 1.04² = 1.081600
• Forward rate: (1.134627 ÷ 1.081600) − 1 = 4.90 percent

Now check that the two routes really do tie. Two years at 4 percent then one year at 4.90 percent gives 1.0816 × 1.049 = 1.134627, which is exactly what three years at 4.30 percent gives. The tie is perfect, because that is what the number was built to do.

Notice how much more extreme the forward rate is than either spot rate. The three-year rate is only 4.30 percent, yet the implied rate for that third year alone is 4.90 percent. That is not an accident. When the curve slopes upward, the forward rate always sits above the longer spot rate, because the longer rate is an average that includes the cheaper early years, and the forward strips those out.

Implied is not the same as predicted

Here is where people over-read this number, so it is worth being blunt. A forward rate of 4.90 percent does not mean the market is forecasting that one-year rates will be 4.90 percent in two years. It means 4.90 percent is the rate that makes today's two prices consistent with each other. Those are different claims.

There is a long-standing theory, the expectations hypothesis, which says forward rates should be unbiased forecasts of future spot rates. It is a useful starting point and it does not survive contact with the data. The evidence is that forward rates are biased predictors, and the usual explanation is a term premium: investors want extra compensation for tying money up for longer, so the forward rate carries that premium on top of any genuine expectation. In normal times this tends to make forwards overshoot where short rates actually end up. The premium is not fixed either, and has at times been negative.

So use the number for what it is genuinely good for: pricing an agreement that locks in a future rate, checking whether a quoted forward looks fair against the curve, or seeing what return you would need from rolling short-term investments to beat simply locking in the long rate. Treat it as the market's current price for future money, which is solid, rather than as its prophecy about future rates, which it is not.

Questions people ask

What is a spot rate?

The rate today for money committed today, for a stated term. The two-year spot rate is what you get by lending now for two years. Forward rates, by contrast, cover a period that starts in the future.

Why is the forward rate higher than both spot rates?

Because the curve slopes upward in that example. The longer spot rate averages in the cheaper early years, so the rate implied for the later stretch alone has to be higher to pull that average up.

Can a forward rate come out negative or below the spot rates?

Yes, if the longer spot rate is lower than the shorter one, which is an inverted curve over that stretch. The forward then falls below both, and it can be negative if the inversion is steep enough.

Does it tell me where rates are going?

Not reliably. It tells you the rate implied by today's prices. Because forwards include a term premium, they have historically been biased predictors of the rates that actually arrive.

References

The forward rate is derived from the current structure of spot rates by no-arbitrage, so that investing for the longer term matches investing for the shorter term and reinvesting at the forward rate. The interpretation of the implied forward curve as an indicator of expected future short rates, and the evidence that forward rates are biased predictors of future spot rates because of term premiums, are discussed in the central bank and international institution literature below.

1. Bank for International Settlements, Volatility and the Treasury yield curve. https://www.bis.org/publ/confp01m.pdf
2. International Monetary Fund, Estimating and Interpreting Forward Interest Rates, IMF Working Paper. https://www.elibrary.imf.org/view/journals/001/1994/114/article-A001-en.xml