A Treasury note with a maturity of four years carries a nominal
rate of interest of 10 percent. In contrast, an eight-year Treasury
bond has a yield of 8 percent.
a. If inflation is expected to average 7 percent over the first four
years, what is the expected real rate of interest?
b. If the inflation rate is expected to be 5 percent for the first
year, calculate the average annual rate of inflation for years
2 through 4.
a. To calculate the expected real rate of interest, we need to adjust the nominal interest rate for inflation. The formula to calculate the real interest rate is:
Real Interest Rate = Nominal Interest Rate - Inflation Rate
In this case, the nominal interest rate is 10 percent, and inflation is expected to average 7 percent over the first four years. Therefore, the expected real interest rate can be calculated as:
Real Interest Rate = 10% - 7% = 3%
So, the expected real rate of interest is 3 percent.
b. To calculate the average annual rate of inflation for years 2 through 4, we need to consider the inflation rate for each of those years and then calculate the average. Given that the inflation rate is expected to be 5 percent for the first year, we can assume the following rates for years 2 to 4:
Year 1: 5%
Year 2: X% (unknown)
Year 3: X% (unknown)
Year 4: X% (unknown)
To find the average, we need to solve for X. Since the average is the sum of all the rates divided by the number of years, we have the following equation:
(5% + X% + X% + X%) / 4 = Average Annual Inflation Rate
Simplifying the equation, we get:
(5% + 3X%) / 4 = Average Annual Inflation Rate
To find X, we can rearrange the equation:
5% + 3X% = 4 * Average Annual Inflation Rate
Now, substitute the average annual inflation rate given in the question (5%) into the equation:
5% + 3X% = 4 * 5%
Simplify further:
5% + 3X% = 20%
Subtracting 5% from both sides of the equation:
3X% = 15%
Finally, divide both sides of the equation by 3:
X% = 15% / 3 = 5%
So, the average annual rate of inflation for years 2 through 4 is 5 percent.
a. To calculate the expected real rate of interest, we need to subtract the expected inflation rate from the nominal rate of interest.
Expected real rate of interest = Nominal rate of interest - Expected inflation rate
Nominal rate of interest = 10%
Expected inflation rate = 7%
Expected real rate of interest = 10% - 7% = 3%
Therefore, the expected real rate of interest is 3%.
b. To calculate the average annual rate of inflation for years 2 through 4, we need to find the average of the inflation rates for each year.
Inflation rate for year 1 = 5%
Inflation rate for year 2 = ?
Inflation rate for year 3 = ?
Inflation rate for year 4 = ?
Average annual rate of inflation = (Inflation rate for year 2 + Inflation rate for year 3 + Inflation rate for year 4) / 3
We know that the average annual rate of inflation for years 1 through 4 is 7%.
From this information, we can solve for the inflation rates for years 2 through 4.
(5% + Inflation rate for year 2 + Inflation rate for year 3 + Inflation rate for year 4) / 4 = 7%
5% + Inflation rate for year 2 + Inflation rate for year 3 + Inflation rate for year 4 = 28%
Now we need to solve for the average annual rate of inflation for years 2 through 4.
Average annual rate of inflation = (Inflation rate for year 2 + Inflation rate for year 3 + Inflation rate for year 4) / 3
(28% - 5%) / 3 = Average annual rate of inflation
23% / 3 = Average annual rate of inflation
Therefore, the average annual rate of inflation for years 2 through 4 is approximately 7.67%.