Question

Suppose that the real interest rate in the economy is 4 per cent, while the inflation rate one year from now is known to be 2 per cent. Use the Fisher equation to find the nominal interest rate. Use the nominal interest rate to find the present value of \(£ 100\) one year from now. Now suppose that inflation in one year from now is known to be 4 per cent. How has the present value calculated previously changed? Why?

Short answer

Increasing inflation lowered the present value from £94.27 to £92.45 due to a higher nominal interest rate.

Step-by-step solution

Understand the Fisher Equation

The Fisher equation states that \(1 + i = (1 + r)(1 + \pi)\) where \(i\) is the nominal interest rate, \(r\) is the real interest rate, and \(\pi\) is the inflation rate. In this exercise, \(r = 0.04\) and \(\pi = 0.02\).

Calculate the Nominal Interest Rate

Use the Fisher equation: \(1 + i = (1 + 0.04)(1 + 0.02)\). This gives \(i = 0.061\) or 6.1%.

Find Present Value with Initial Inflation

The present value \(PV\) is calculated using the formula \(PV = \frac{FV}{1 + i}\), where \(FV = £100\) and \(i = 0.061\). Thus, \(PV = \frac{100}{1.061} = £94.27\).

Adjust for New Inflation Rate

With the new inflation rate \(\pi = 0.04\), recalculate the nominal rate: \(1+i = (1+0.04)(1+0.04)\), giving \(i = 0.0816\) or 8.16%.

Recalculate Present Value with New Inflation

Using the new \(i = 8.16\%\), calculate \(PV = \frac{100}{1.0816} = £92.45\).

Compare Present Values and Explain the Change

The present value decreased from £94.27 to £92.45 when inflation increased from 2% to 4%. Higher expected inflation led to a higher nominal interest rate, which in turn lowered the present value.

Key concepts

Nominal Interest Rate

The nominal interest rate is an essential tool in the financial world. It represents the percentage by which the lender expects to be compensated for parting with their money. In simple terms, it's the rate of interest that does not take inflation into account.
To find the nominal interest rate, we often use the Fisher Equation. This equation links real interest rate, nominal interest rate, and inflation rate together. It can be represented as:

• \(1 + i = (1 + r)(1 + \pi)\)
• Where \(i\) is the nominal interest rate, \(r\) is the real interest rate, and \(\pi\) is the inflation rate.

In our example, with a real interest rate of 4% and an inflation rate of 2%, inserting these into the Fisher Equation gives us a nominal interest rate of 6.1%.
It's crucial to understand that the nominal interest rate doesn't indicate how much purchasing power actually increases. For that, we need to consider the inflation rate.

Present Value

Present value is a crucial concept in finance that helps us determine the value today of a sum of money promised in the future. Imagine you have \(£100\) that you'll receive one year from now. But how much is that \(£100\) worth in today's terms?
To calculate the present value, we use the formula:

• \(PV = \frac{FV}{1 + i}\)
• Where \(PV\) is the present value, \(FV\) is the future value, and \(i\) is the nominal interest rate.

For example, with a future value of \(£100\) and a nominal interest rate of 6.1% (or 0.061), the present value comes out to be around \(£94.27\).
This means that \(£100\) one year from now is equivalent to \(£94.27\) today if the nominal interest rate is 6.1%. It's vital for comparing total returns when there are differing timelines of payment or receipt.

Inflation Rate

The inflation rate measures how fast prices for goods and services rise over time, decreasing the purchasing power of money. In simple terms, as inflation rises, each pound can buy fewer goods and services.
The Fisher Equation helps connect the dots between inflation, nominal, and real interest rates. A key insight we can draw is that higher inflation leads to higher nominal interest rates since lenders want to be compensated for lost purchasing power.
In our scenario, increasing the inflation rate from 2% to 4% resulted in the nominal interest rate rising from 6.1% to 8.16%. Consequently, the present value of a \(£100\) future payment decreased from \(£94.27\) to \(£92.45\). The increase in inflation means that future cash flows are worth less in today's terms, which is why we see this decrease in present value.
Understanding inflation's impact lets individuals and businesses make better financial decisions, ensuring the preservation of purchasing power over time.