Suppose we observe the following rates: IR 1 = 8\%, IR 2 = 10\%. If the unbiased expectations theory of the term structure of interest rates holds, what is the one year interest rate expected one…

Solution to the Term Structure of Interest Rates Problem

Part 1: Expected One-Year Interest Rate One Year from Now (6 marks)

The unbiased expectations theory (UET) of the term structure of interest rates states that the long-term interest rate is the geometric average of current and expected future short-term rates. Mathematically, for a two-year bond, the relationship is given by:

$$(1+I{R}_2{)}^2=(1+I{R}_1)\times (1+E[I{R}_1])$$

Where:

• $I{R}_1=8\%=0.08$ (1-year interest rate)
• $I{R}_2=10\%=0.10$ (2-year interest rate)
• $E[I{R}_1]$ = Expected 1-year interest rate one year from now (unknown)

Step 1: Substitute the given values into the equation

$$(1+0.10{)}^2=(1+0.08)\times (1+E[I{R}_1])$$

Step 2: Simplify the left-hand side

$$(1.10{)}^2=1.21$$

So,

$$1.21=(1.08)\times (1+E[I{R}_1])$$

Step 3: Solve for $E[I{R}_1]$

$$1+E[I{R}_1]=\frac{1.21}{1.08}=1.1204$$

$$E[I{R}_1]=1.1204-1=0.1204=12.04\%$$

Conclusion: The expected one-year interest rate one year from now is 12.04%.

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Part 2: Implied Rate of Inflation in Year Two (4 marks)

The Fisher equation relates nominal interest rates, real interest rates, and expected inflation:

$$1+IR=(1+r)\times (1+\pi )$$

Where:

• $IR$ = Nominal interest rate
• $r$ = Real risk-free rate (given as 3% = 0.03)
• $\pi$ = Expected inflation rate

We are asked to find the inflation rate in year two (${\pi }_2$), given that the real risk-free rate is 3% and the nominal interest rate in year two is the expected one-year rate from Part 1 (12.04%).

Step 1: Apply the Fisher equation for year two

$$1+I{R}_2=(1+r)\times (1+{\pi }_2)$$

But $I{R}_2$ here is the expected nominal rate in year two, which we found to be 12.04%. However, we must clarify whether the question refers to the second-year inflation rate or the average inflation over two years.

Since the question asks for the implied rate of inflation in year two, we assume it refers to the inflation rate in the second year, which is the same as the expected inflation rate in year two.

Step 2: Rearrange the equation to solve for ${\pi }_2$

$$1+{\pi }_2=\frac{1+I{R}_2}{1+r}$$

$$1+{\pi }_2=\frac{1+0.1204}{1+0.03}=\frac{1.1204}{1.03}=1.0878$$

$${\pi }_2=1.0878-1=0.0878=8.78\%$$

Conclusion: The implied rate of inflation in year two is 8.78%.

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Final Answer Summary

1. Expected one-year interest rate one year from now: 12.04%
2. Implied rate of inflation in year two: 8.78%