###### Projections Beyond the Short-term Fore ...

Short-dated nominal zero-coupon government bonds have yields closely related to the central bank’s policy rate. Additionally, the uncertainty of inflation over the very short-term investment horizon, typically three months, is negligible. Therefore, the nominal interest rate of a T-bill will only have two elements: the **real risk-free rate** and the **expected inflation**.

As such, the price of a T-bill is given by the following equation:

$$ P_0=\frac{CF_t}{\left(1+R+\pi\right)^t} $$

Where:

- \(P_0\) is the value of the asset (T-bill) today.
- \(CF_t\) is the cashflow in time t.
- \(R\) is the risk-free rate at time t.
- \(\pi\) is the expected inflation rate at time t.

Short-term nominal interest rates depend on short-term real interest rates and short-term expected inflation. As discussed in the previous section, we expect these interest rates to be higher in economies with higher and more volatile GDP growth and higher average inflation levels over time, holding all else constant.

In conclusion, a responsible central bank will refer to inflation and the level of expected economic activity when setting its policy rates.

For longer-term bonds, a **risk premium** is added for uncertainty about inflation. Therefore, the nominal rate of interest has three elements: the real risk-free rate of interest, the expected inflation, and a risk premium.

Taylor’s rule helps rate-setters, such as central banks, measure whether their policy rate is at an appropriate level. The rule is given by:

$$ r_t=R+\pi_t+0.5\left(\pi_t-\pi_t^\ast\right)+0.5(y_t-y_t^\ast) $$

Where:

- \(r_t\) is the policy rate at time \(t\).
- \(R\) is the real short-term interest rate.
- \(\pi_t\) is the rate of inflation.
- \(\pi_t^\ast\) is the target rate of inflation.
- \(y_t\) is the logarithmic level of actual GDP.
- \(y_t^\ast\) is the logarithmic level of potential real GDP.

The **output gap** is defined as the difference between the logarithmic level of the actual GDP and the logarithmic level of the potential real GDP. It is measured in percentage. A positive output gap implies that the economy is producing more than it can sustain. This is associated with a high or rising inflation. A negative output gap implies the converse.

The policy rule has a more significant weight on inflation relative to the weight on output to stabilize inflation over the long term close to the targeted inflation rate. When inflation is close to the targeted rate and the output gap is zero, then the policy rate is called the **neutral policy rate**.

The short-term interest rates and the business cycle have an interdependent relationship. For example, if the short-term interest rates are too low for too long, then there is a risk of creating a credit bubble. On the other hand, if they are set too high for too long, then they could lead to recessionary or even depression-like economic conditions.

Therefore, central banks play a vital role in moderating the business cycle. The banks do this either by modifying the policy rate or exaggerating the cycle by not responding optimally to the changing economic conditions.

In addition to its potential use in forecasting future interest rates, the yield curve is applicable in forecasting future economic activity (i.e., business cycles). The policy rate influences the shape of the yield curve. A steeply sloping yield curve implies the expectation of a sharp increase in interest rates. This is mainly due to high inflation and expected inflation.

On the other hand, steeply inverted curves imply an expectation of sharply falling inflation as well as future interest rates. This may be as a result of the very high nominal policy rates. The inverted curves also imply that investors expected rates to come down once the causes of high current inflation are eliminated.

A recession is mostly preceded by a flattening, or an inversion, in the yield curve. Moreover, the late stages of business expansion are characterized by a peak in inflation and, therefore, relatively high short-term interest rates. If longer maturity yields reflect lower inflation rates and diminished business credit demand, the yield curve tends to flatten or invert. In particular, an inverted yield curve often predicts a recession.

## Question

The central bank of a given country provides you with the following information:

$$ \begin{array}{c|c} \textbf{Variable} & \textbf{Value} \\ \hline \text{The real short-term interest rates} & 3\% \\ \hline \text{The rate of inflation} & 2\% \\ \hline \text{The target rate of inflation after one period} & 3\% \\ \hline \text{The output gap} & 2\% \end{array} $$

The central bank policy rate is

closest to:

- 5%.
- 5.5%.
- 6%.
## Solution

The correct answer is B.Using Taylor’s rule,

$$ r_t=l_t+\pi_t+0.5\left(\pi_t-\pi_t^\ast\right)+0.5(y_t-y_t^\ast) $$

The policy rate is given by:

$$ \begin{align*} {Pr}_t & =3\%+2\%+0.5\left(2\%-3\%\right)+0.5(2\%) \\ &=5\%-0.5\%+1\%=5.5\% \end{align*} $$

Reading 44: Economics and Investment Markets

*LOS 44 (d) Explain how the phase of the business cycle affects policy and short-term interest rates, the slope of the term structure of interest rates, and the relative performance of bonds of different maturities.*