2 - Nominal and Real interest rates

Quantity of goods and services  unit of money can purchase is defined as purchasing power of money. With 100$ people could purchase more items 20 years back than they can do now.(Ref inflation calculator)

Inflation is defined as general rise of prices of goods and services over a period of time.In normal times, inflation occurs year after year and money loses its purchasing power.

If you keep 100 $ in your shelf for 10 years, at 10% yearly inflation (Still some emerging economies have high inflation.Analysis will be simple to understand at 10%) it is worth only 35$ (100*(.9^10)).Inflation and purchasing power loss due to inflation is said to be biggest risk by investors.

Say bank is providing 10% interest on deposit and the actual yearly inflationary expectation is 7%. What is the real returns we get?

This 10% interest is called Nominal interest, as its not adjusted for inflation.If you need to adjust for inflation you will get real interest rate.


Ni - Inflation = Ri
10%-7% = 3% is the real interest rate... right?

Not exactly. It is approximately 3% , not exactly 3%.How?


Assume today you can have your favourite family dinner at 100$ and 1 Kg apple costs 3 $.
But you have only 100$ but want to have both. Your purchasing power is less compared to your desire to consume.

You skip having the dinner now, go to bank and deposit your 100 $ @10%.

After a year you have got 110 $. Assume all things will vary exactly as inflation figure for analysis simplicity.

One year from now, you can get the dinner at 107 $. You have 3 $ more. Can you buy 1 KG apple? It's price next year will be 3.21 $. So 3 $ you will have next year is not 3% return with respect to base year. (All returns ,inflation calculated with respect to base year..in our example current year).

3 $ next year is 3/1.07 $ this year.That is 2.8$ (rounded)or 2.8 percent as we keep principal as 100 $.
This is real return and using this return, purchasing power increase you get if you save at 10 percent interest.
(Here also income tax may come into picture,which would be on nominal interest amount.Do your math.)

You could figure out same calculation holds true when one gets x% increment every year when a nation's economy runs at y% inflation rate.

How we arrive at the approximation?

Say P is principle , Ni is nominal interest , y is inflation rate,Ri is the real interest rate.
      P*(1+Ni) = P(1+Ri)(1+y)

       1+Ri = (1+Ni)/(1+y)

Use Taylor series for 1/(1+x) = 1-x+x^2.... take only first two terms,leave higher order terms to get approximation
      1+Ri approx = (1+Ni) *(1-y)
                  = 1 +Ni -y - Ni * y
last figure since both will normally fraction multiplication of both will be even less. ignore the last term to arrive at

Real interest rate(Ri) approx = Nominal interest rate(Ni) - inflation(y)


When nominal interest rate offered by banks is less than annual inflation rate , real interest rate is negative.You cannot keep your purchasing power even when you save in this situation.

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