The **expectations hypothesis** of the term structure stands as a foundational concept for understanding the relationship between short-term and long-term interest rates. It provides valuable insights into bond market dynamics, yield curve behavior, and interest rate forecasting.

In this lesson, we begin by introducing the key terms and, then, explain how this theory works with practical examples. You can also find a video tutorial at the end, which goes over these examples in greater detail.

#### Contents

## Definitions: spot rates, short rates, forward rates

Before moving on to details, lets begin with a couple of definitions that will be helpful later on:

**Spot rate**: The current yield to maturity of a zero coupon bond that matures at some point in future. For example, in Figure 1, the spot rate of bond A, which matures in one year, is 7%/yr and that of bond B, which matures in two years, is 5%/yr.

**Short rate**: The interest rate that will prevail during a particular period in future. In Figure 1, we see that Bond A’ is a 1-year bond that will be issued next year. However, we don’t know at the moment what the short rate will be for that bond.

**Forward rate**: This is the future interest rate implied by the current spot rates. Forward rates may or may not be equal to expected short rates. In Figure 1, the forward rate for year 2, which we denote as *f _{2}*, can be obtained as follows. Imagine that your investment horizon is two years. If you invest in Bond B, your holding period return is:

(1 + 5%)^{2}

Or, if you invest in Bond A in year 1 and at the forward rate in year 2, your holding period return is:

(1 + 7%) * (1 + *f _{2}*)

If we set these equal to each other such that:

(1 + 5%)^{2} = (1 + 7%) * (1 + *f _{2}*)

we can solve for the forward rate to get *f _{2}* = 3.04%.

While we can observe the spot rates and forward rates now, short rates are unknown. However, we can have *expectations* of short rates. And, the expectations hypothesis is all about the link between forward rates and expected short rates as we explain next.

## Understanding the expectations hypothesis

Let’s imagine a simple bond market with two zero coupon bonds as follows:

Bond | Maturity | Yield to maturity |
---|---|---|

A | 1 year | 9% |

B | 2 years | 10% |

Let’s first calculate the** forward rate** implied by these two bonds (see also Figure 2). If I invested in Bond B at 10% for two years, my holding period return would be:

(1 + 10%)^{2}

If I invested in Bond A at 9% in the first year and then a new 1-year bond at *f _{2}* in the second year, I’d get:

(1 + 9%) * (1 + *f _{2}*)

Setting these returns equal to each other:

(1 + 10%)^{2} = (1 + 9%) * (1 + *f _{2}*)

And, solving for the forward rate yields:

*f _{2}* = 11%

### Short-term investors vs long-term investors

How is this forward rate of *f _{2}* = 11% determined in the bond market? What does it reflect? This is where the expectations hypothesis comes into play. To understand this, let’s consider two investors as follows:

Investor | Investment horizon |
---|---|

Jake | Short term: 1 year |

Susan | Long term: 2 years |

Both investors can buy either bond A or bond B to meet their investment goals. Let’s begin with Jake. If he purchases bond A, he knows that he will get exactly 9% return when this bond matures in one year’s time. So, there is no uncertainty involved in this investment. However, if he purchases bond B, his return will depend on the price he sells the bond in one year’s time as the bond would still have one year maturity remaining. This means that his return with bond B is uncertain. Therefore, he would consider Bond B only if it offered a **liquidity premium** to compensate him for the risk he is taking. This would imply that the forward rate in year 2 would be equal to:

where *E[r _{2}]* is the expected short rate in year 2 and

*λ*

_{2}*> 0*is the liquidity premium, or Jake’s reward for choosing bond B over bond A.

In other words, for Jake to invest in the long-term bond, the forward rate should be larger than the expected short rate:

*f _{2}* >

*E[r*

_{2}]How about Susan? She faces the opposite scenario. If she invests in bond B, she guarantees herself a yield to maturity of 10% per annum and her holding period return over two years is:

(1 + 10%)^{2}

She can also invest in bond A, earn 9% when the bond matures next year, and invest in a new 1-year bond then. But, at the moment, she can’t know the exact return she’d earn from the latter bond… Her expected return in this scenario would be:

(1 + 9%) * (1 + *E[r _{2}]*)

So, for Susan to choose bond A over bond B, we should have:

(1 + 9%) * (1 + *E[r _{2}]*) > (1 + 10%)

^{2}

And, recall that we derived the forward rate as: (1 + 10%)^{2} = (1 + 9%) * (1 + *f _{2}*). Then, the inequality above becomes:

(1 + 9%) * (1 + *E[r _{2}]*) > (1 + 9%) * (1 +

*f*)

_{2}This simplifies into:

*f _{2}* <

*E[r*

_{2}]This means that the short-term bond is attractive for Susan when the forward rate is less than the expected short rate.

### Forward rates reflect expected short rates

In summary, short-term investors like Jake would invest in long-term bonds only if forward rates exceed expected short rates: *f _{2}* >

*E[r*. Conversely, long-term investors like Susan would invest in short-term bonds only if forward rates are below expected short rates:

_{2}]*f*<

_{2}*E[r*. Overall, both groups of investors demand a premium to hold bonds whose maturities differ from their investment horizons. So, how does the market balance that out?

_{2}]The expectations hypothesis argues that when both short-term investors and long-term investors are present in the market, the forward rate *f _{2}* will be equal to the expected short rate

*E[r*:

_{2}]*f _{2}* =

*E[r*

_{2}]This implies that there won’t be a liquidity premium (i.e., *λ _{2}* = 0) due to varying maturities of bonds. And, it means that the forward rates we obtain from the yield curve serve as market expectations of future short rates.

### Explaining the shape of yield curve

According to the expectations hypothesis, if investors anticipate increasing short rates in the future, this will lead to an upward sloping yield curve.

Conversely, if the expectation is a fall in short rates, we will observe a downward sloping yield curve. And, a flat yield curve would imply no expected changes in short rates in the future.

## Video tutorial

##### What is next?

If you enjoyed this lesson, we recommend you check our other courses and tutorials as well. And, for further reading on the expectations hypothesis, we suggest the following resources.

Bekaert and Hodrick (2002), ‘Expectations Hypotheses Tests‘, *The Journal of Finance*, Vol. 56(4), pp. 1357-1394.

In the next lesson, we’ll be talking about the liquidity preference theory.