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Calculating and Understanding Real Interest Rates

Calculating and Understanding Real Interest Rates

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[Q:] I have a couple of questions about the real interest rate I was hoping you could answer:
  1. I have the following consumer price index (CPI) and nominal interest rate data:

    CPI Data
    Year 1: 100
    Year 2: 110
    Year 3: 120
    Year 4: 115

    Nominal Interest Rate Data
    Year 1: --
    Year 2: 15%
    Year 3: 13%
    Year 4: 8%

    How can I figure out what the real interest rate is for years 2, 3, and 4?

  2. I’m offered the following deal: I lend $200 to a friend at the beginning of year 2 and charge him the 15% nominal interest rate and he pays me back the $230 at the end of year 2. If I agree to this deal will I be made better or worse off?
[A:] Thanks for your great questions! First we’ll look at what the real interest rate is and then we’ll see how to calculate real interest rates.

What is the Real Interest Rate?

In the article “What’s the Difference Between Nominal and Real?” we learned that “a real variable, such as the real interest rate, is one where the effects of inflation have been factored in. A nominal variable is one where the effects of inflation have not been accounted for.” An example shows how real interest rates work:

    Suppose we buy a 1 year bond for face value that pays 6% at the end of the year. We pay $100 at the beginning of the year and get $106 at the end of the year. Thus the bond pays an interest rate of 6%. This 6% is the nominal interest rate, as we have not accounted for inflation. Whenever people speak of the interest rate they're talking about the nominal interest rate, unless they state otherwise.

    Now suppose the inflation rate is 3% for that year. We can buy a basket of goods today and it will cost $100, or we can buy that basket next year and it will cost $103. If we buy the bond with a 6% nominal interest rate for $100, sell it after a year and get $106, buy a basket of goods for $103, we will have $3 left over.

Now we know what the real interest rate is we’ll look at how you can calculate it.

How Do I Calculate the Real Interest Rate?

Before we start making the calculations we need to introduce some notation:

Notation
i: is the Inflation Rate
n: is the Nominal Interest Rate
r: is the Real Interest Rate

To calculate the real interest rate, we need to know the inflation rate (or expected inflation rate, if we’re making a prediction about the future). From the data given we don’t have the inflation rate, but we can calculate it from the CPI data:

Calculating the Inflation Rate

We need to use the following formula:

i = [CPI(this year) – CPI(last year)] / CPI(last year).

So the inflation rate in year 2 is [110 – 100]/100 = .1 = 10%. We do this for all three years and get the following:

Inflation Rate Data
Year 1: --
Year 2: 10.0%
Year 3: 9.1%
Year 4: -4.2%

Now we can calculate the real interest rate. The relationship between the inflation rate and the nominal and real interest rates is given by the expression: (1+r)=(1+n)/(1+i). However for low levels of inflation we can use the much simpler Fisher Equation to calculate the real interest rate:

FISHER EQUATION: r = n – i

Using this simple formula, we can calculate the real interest rate for years 2 through 4:

Real Interest Rate (r = n – i)
Year 1: --
Year 2: 15% - 10.0% = 5.0%
Year 3: 13% - 9.1% = 3.9%
Year 4: 8% - (-4.2%) = 12.2%

So the real interest rate is 5% in year 2, 3.9% in year 3, and a whopping 12.2% in year 4.

Your Second Question

You wanted to know if you should lend $200 to your friend at the beginning of year 2. If you do make that loan, you will earn a real interest rate of 5%. Since 5% of $200 is $10, you will be financially ahead by making the deal. It doesn’t necessarily mean that you should make the deal. Which is more important to you: Getting $200 worth of goods (at year 2 prices) at the beginning of year 2, or getting $210 worth of goods (also at year 2 prices) at the beginning of year 3. There is no right choice to this problem: it depends on how much you value consumption (or happiness) today relative to consumption one year from now (Economists refer to this as a person’s discount factor ).

If you know what the inflation rate is going to be, real interest rates can be a powerful tool in judging the value of investment as they take into account how inflation erodes purchasing power.

If you'd like to ask a question about nominal interest rates, real interest rates, or any other topic or comment on this story, please use the feedback form. If you're interested in winning cash for your economics term paper or article, be sure to check out "The 2004 Moffatt Prize in Economic Writing"

Final Interest Rate Data

Year CPI Nominal Interest Rate Inflation Rate Real Interest Rate
1 100 -- -- --
2 110 15% 10% 5%
3 120 13% 9.1% 3.9%
4 115 8% -4.2% 12.2%

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