An upward-sloping yield curve, also known as a normal yield curve, occurs when longer-term interest rates are higher than shorter-term interest rates. This is the most commonly observed yield curve shape in healthy economic conditions. An upward-sloping yield curve suggests that market participants expect interest rates to rise in the future.
The expectations hypothesis theory postulates that long-term interest rates are determined by the average of expected future short-term rates. If the yield curve is upward-sloping, it implies that the market participants expect the future short-term rates to be higher on average, compared to the current short-term rate. Mathematically, the expectations hypothesis can be represented as follows:
$$R_{t}^{n} = \frac{1}{n}\sum_{i=0}^{n-1} E_t(R_{t+i}^1)$$
Here, Rtn is the observed yield on an n-period bond at time t, and Et(Rt + i1) is the expected yield on a 1-period bond at the end of the period i.
Let’s take an example:
Consider a 2-year bond and a 1-year bond. Suppose the yield on the 1-year bond is 2%, and the yield on the 2-year bond is 4%. The upward-sloping yield curve suggests that the market participants believe that the 1-year interest rate, one year from now (i.e., at the end of the first year), should be around 6%. This is because, according to the expectations hypothesis:
$$R_{t}^{2} = \frac{1}{2}\left(E_t(R_{t}^1) + E_{t+1}(R_{t+1}^1)\right)$$
Substituting the given yields,
$$4\% = \frac{1}{2}\left(2\% + E_{t+1}(R_{t+1}^1)\right)$$
Solving for the expected 1-year rate, one year from now,
Et + 1(Rt + 11) = 6%
Thus, in this situation, the market expects future short-term interest rates to increase, which is consistent with an upward-sloping yield curve. However, it is crucial to note that the expectations hypothesis theory is not universally accepted, and there could be other factors such as the liquidity preference theory, risk premia, and supply-demand dynamics that influence the shape of the yield curve.