Question

Describing the economy in England in \(1920,\) the historian Robert Skidelsky wrote the following: "Who would not borrow at 4 percent a year, with prices going up 4 percent a month?" What was the real interest rate paid by borrowers in this situation? (Hint: What is the annual inflation rate, if the monthly inflation rate is 4 percent?)

Short answer

The real interest rate paid by borrowers in this situation was approximately -56.103%

Step-by-step solution

Calculate Annual Inflation Rate

Given that prices are increasing at 4% per month, we can calculate the annual inflation rate by compounding it. Intuitively, this follows the idea that each month's inflation builds on the increased prices from the previous month. Mathematically, we use the formula \( (1 + i_m)^{n} - 1 \), where \( i_m \) is the monthly inflation rate and \( n \) is the number of periods (12 months). So, \( (1 + 0.04)^{12} - 1 = 0.60103 \) or approximately 60.103% annually.

Calculate Real Interest Rates

The real interest rate is determined by subtracting the inflation rate from the nominal interest rate. The nominal interest rate is 4%, or 0.04, and we've found that the annual inflation rate is approximately 0.60103, or 60.103%. So, the real interest rate is \( 0.04 - 0.60103 \), or -0.56103. This means that the real interest rate is approximately -56.103%.

Key concepts

Annual Inflation Rate

To truly understand the value of money over time, it's crucial to grasp the concept of the annual inflation rate. In the context of the exercise, we're dealing with an economy where prices are surging by 4 percent each month, a situation indicative of a high inflationary environment.

Inflation can be thought of as a balloon inflating, hence the term, which reflects the general increase in the price levels of goods and services in an economy over a period of time. Crucially, when we speak of the annual inflation rate, we're interested in how much the cost of living has increased over the span of a year. To clarify, if an economy exhibits a monthly inflation rate of 4 percent, compounding this rate over 12 months provides the annual rate. One might hastily multiply 4 by 12 to get 48 percent, but this straightforward method overlooks the effect of compounding inflation.

Through the formula \( (1 + i_m)^{12} - 1 \) where \( i_m \) is the monthly inflation rate, we can accurately calculate how each month's inflation is compounded over the year. Using this approach as detailed in the step-by-step solution reveals a startling annual inflation rate of approximately 60.103%, far greater than a simple multiple would suggest. This understanding is critical for both investors and borrowers, as it influences the real value of interest rates and returns.

Nominal Interest Rate

When you hear terms like '4 percent interest rate', what is being referred to is the nominal interest rate. It's the rate of interest that banks and other financial institutions cite for various products, like loans, savings accounts, or bonds. It's nominal because it doesn't consider the full picture—specifically, it doesn't factor in inflation, which can drastically alter the value of money over time.

In our exercise, borrowers are offered a nominal interest rate of 4 percent annually. On the surface, this seems straightforward; if you borrow 100 units of currency, a year later you would owe 104 units, right? The reality is, due to inflation, the actual value of money owed could be significantly different. The nominal interest rate is like a facade that hides the true cost or benefit of financial transactions.

By merely looking at the nominal rate, you can't discern the actual purchasing power of your future money – that's where the real interest rate comes into play. We use the nominal interest rate as a starting point, from which we will subtract the annual inflation rate to determine the real interest rate that reflects the true cost of borrowing.

Compounding Inflation

Compounding is a concept that's often associated with the growth of an investment, but it also plays an equally fundamental role in the context of inflation. Compounding inflation means that the increase in prices builds upon itself. Each month's or year's increase is applied to a new, higher base price level.

To detail, if you pay 100 units for a basket of goods this month and there's a 4 percent inflation rate, next month the same basket would cost 104 units. However, compounding means that the following month the prices rise not just by another 4 units, but by 4 percent of 104, which is more than 4. This effect over time leads to exponential, rather than linear, price increases.

Academically and practically, understanding compounding inflation aids in making informed decisions about long-term financial commitments and investments. As vividly shown in the problem from the 1920 England economy, not considering the compounding nature of inflation when evaluating financial options can lead to gross misjudgments about the real cost or return on investments and loans. This concept is fundamental in explaining why an annual inflation rate can be so much higher than 12 times the monthly rate—because each month's inflation is calculated on a price level that has been bumped up by the previous month's inflation.