The Expectations Hypothesis and Term Premium

Look at almost any government bond market on almost any day and the same thing is true: longer maturities yield more. A one-year bill might pay 4.00% while a ten-year note pays 4.70%. Why should the market pay you more simply for waiting longer?

There are two candidate answers, and telling them apart is the whole subject of this page. Either the market genuinely expects short rates to be higher in future, so the long bond is just averaging in those higher rates — or the market is paying you a fee to take duration risk. Both are happening at once. The question is in what proportion, and if you get it wrong you will read the bond market as a forecast when it is really a price for risk.

The hotel-booking analogy

You need a room for two nights. The hotel offers a two-night rate booked today, or you can book tonight only and take your chances on tomorrow's walk-up price. Suppose the two-night deal works out dearer per night than tonight's rate. Two explanations, and they are the same two:

1. Everyone thinks tomorrow's walk-up price will be higher — a conference is in town. The two-night rate is just the average of a cheap night and an expensive one.
2. Tomorrow's price is genuinely unknown, and you would rather not find out. You pay a little extra for the certainty of a locked room.

Explanation one is expectations. Explanation two is the term premium. The hotel bill does not come itemised, and neither does a bond yield.

Writing it down

Start with two numbers you can read off a screen. Let $y_1$ be today's one-year zero rate and $y_2$ today's two-year zero rate. Now define the forward rate $f$: the one-year rate, one year from now, that today's prices already imply. It is pinned down by insisting that two ways of investing money for two years must cost the same:

$$(1 + y_2)^{2} = (1 + y_1)\,(1 + f) .$$

In plain English: locking in the two-year rate must give you the same ending wealth as taking the one-year rate and then rolling into the forward rate. If it didn't, one route would dominate the other and you could arbitrage the difference.

The forward rate is a fact. What the expectations hypothesis adds is an opinion: that this forward equals the market's genuine expectation of next year's one-year rate,

$$f = E[\,y_1'\,] ,$$

where $y_1'$ is the one-year rate that will actually be quoted a year from today. Read that as: the curve is a pure forecast, and no one is being paid anything for taking the longer route. Extended across all maturities, it says the $n$-year yield is simply the average of expected future short rates.

Reality inserts a wedge. The honest version is

$$y_n = \underbrace{\tfrac{1}{n}\bigl(r_1 + E[r_2] + \cdots + E[r_n]\bigr)}_{\text{expected path}} \;+\; \underbrace{\text{TP}_n}_{\text{term premium}} .$$

Said plainly: the ten-year yield is what you would get from rolling short bills for ten years, if you knew the future, plus whatever extra the market demands for not having to roll. $\text{TP}_n$ is that extra. It grows with maturity because uncertainty grows with maturity, and it moves around over time with the appetite for duration.

Worked example: reading a forward out of two yields

The one-year zero rate is 4.00% and the two-year zero rate is 4.30%. Invest 100 for two years at the two-year rate:

$$100 \times (1.043)^{2} = 108.7849 .$$

Now take the other route. One year at 4.00% turns 100 into 104.00. To finish at the same 108.7849 you need

$$1 + f = \frac{108.7849}{104.00} = 1.04601 ,$$

so the implied one-year forward rate is $f = 4.60\%$. In words: today's prices are consistent with the one-year rate being 4.60% a year from now.

Under the expectations hypothesis, you would stop there and report that the market forecasts a 60 basis point rise. But suppose survey data says investors actually expect next year's one-year rate to be 4.20%, not 4.60%. Then the expected two-year average is $(4.00 + 4.20)/2 = 4.10\%$, while the market yield is 4.30%. The gap,

$$\text{TP}_2 = 4.30\% - 4.10\% = 0.20\% ,$$

is the term premium: 20 basis points a year of compensation, not prophecy. Check it in cash. Rolling gives $104.00 \times 1.042 = 108.368$; locking gives 108.785. The two-year bond is expected to beat the roll by 0.417 per 100 over two years — about 0.20% a year, exactly the premium.

Worked example: decomposing a ten-year yield

The ten-year note yields 4.70%. A survey of primary dealers puts the average expected overnight rate over the next decade at 3.50%. Then

$$\text{TP}_{10} = 4.70\% - 3.50\% = 1.20\% .$$

Over a quarter of the yield is risk compensation rather than forecast. Now the case that breaks people's intuition. The central bank has hiked hard, the one-year bill yields 5.20%, and the ten-year yields 3.80% — an inverted curve. Dealers expect the average short rate over ten years to be 4.30%, because rates are high now and expected to fall later. Then

$$\text{TP}_{10} = 3.80\% - 4.30\% = -0.50\% .$$

A negative term premium. Investors are accepting half a point a year less than they expect from rolling bills, in exchange for locking in duration. That is not irrational: pension funds have long-dated liabilities to match, and in a downturn long bonds pay off exactly when everything else is losing. Duration can be a hedge, and hedges command a negative premium.

Move the level, slope and curvature sliders below and watch what an inversion actually looks like. A slope flip is the market's expectation and its term premium moving in opposite directions:

What this means in practice

• Do not read the curve as a forecast. Forward rates are poor predictors of future short rates. Fama–Bliss and Campbell–Shiller regressions consistently reject the pure expectations hypothesis: when the curve is steep, long bonds tend to earn excess returns rather than short rates rising as implied.
• Carry and rolldown are the term premium in disguise. A steep curve means a long bond earns more than its funding cost and rolls down to lower yields as it ages. That trade works precisely because a premium exists, and it loses when the premium unwinds. See Bond Carry and Rolldown.
• Policy signal versus risk price. Quantitative easing works mainly by compressing the term premium, not by changing expectations. Attribute a falling long yield to expected cuts when it was really QE and you mis-time everything downstream.
• The premium is a model output, not data. ACM, Kim–Wright and survey-based estimates of the same ten-year term premium can differ by 50 basis points. Quote which model you used.

Key terms

• Forward rate — the future rate implied by today's curve, pure arithmetic from two yields.
• Expectations hypothesis — the claim that forwards equal expected future spot rates, i.e. zero term premium.
• Term premium — the extra yield demanded for holding a long bond instead of rolling short ones; can be negative.
• Expected path — the average short rate investors think will prevail over the bond's life.
• Inversion — long yields below short yields, from expected cuts, a negative term premium, or both.