© 1999-2020, Douglas A.Ruby (06-11-2020)

A Real Interest Rate represents the real return to lenders measured in terms of the purchasing power of interest paid. For example suppose we have the following:

A one year loan (N = 1) with the following terms:

- Principal 'P' = $1,000, and
**nominal rate of interest**'i' = 5%.

At the time the loan is made, the price of a common commodity 'Gasoline' (P_{gas}) is equal to $1.00/gal.
In *real terms* the lender is providing the borrower with the purchasing power equivalent to 1,000 gallons of gasoline.

At the termination of the loan the borrower repays the principal 'P' of $1,000 plus an interest payment 'I' of
$50 ($1000 x 0.05). If when the loan is repaid one year later, the price of gasoline P_{gas}' has risen to $1.03/gal.
(*a 3% rate of inflation*); the purchasing power of the principal plus interest ($1,050) will be equal to 1,019 gallons of gasoline.
In real terms, the purchasing power of the lender has increased by roughly 2%.

If the price of gasoline had risen to $1.07 (a 7% rate of inflation) then the purchasing power of the repayment would have been equal to 981 ($1,050/$1.07) gallons of gasoline. In this case the lender provided the opportunity for the borrower to acquire 1,000 gallons of gasoline and at the termination of the loan, the borrower repaid to the lender the ability to acquire only 981 gallons. An unexpectedly high rate of inflation had an adverse impact on the lender -- a negative real rate of return.

Given the following:

r = i - π

i = r* + E[π]

where:

i = nominal rate of interest

r* = desired real rate of return

π = the actual rate of inflation

E[π] = the expected rate of inflation

substituting, we have:

r = r* + E[π] - π

If expected inflation 'E[π(_{t})]' is greater than the actual rate of inflation 'π_{t}' then 'r' will exceed 'r*' to the benefit of
lenders (real returns to lending greater than desired and perhaps greater than the rate of real economic growth) as shown in the operation below:

if E[π] > π then r > r^{*}

If the opposite is true, then benefits will accrue to the borrower.

During the 1980's, many economists have felt that the real rate of interest was abnormally high (i.e., in excess of 2.5-3%). This
may be explained in part due to the inflationary expectations that built up in the late 1970's and early 1980's. Nominal interest rates
have taken these expectations into account. The effects of these inflationary expectations differing from the actual rate of inflation
can be seen in the table below where the annualized 6-month**T-bill rate** is used as a measure of the market interest rate:

Year |
T-Bill Rate |
r* [desired] |
E[π_{t}] |
%Δ(CPI) |
r [actual] |

1976 | 5.52% | 5.40% | 0.12% | 4.90% | 0.62% |

1978 | 7.74 | 5.60 | 2.14 | 9.00 | -1.26 |

1979 | 9.75 | 3.10 | 6.65 | 13.30 | -3.55 |

1980 | 10.89 | -0.30 | 11.19 | 12.50 | -1.61 |

1981 | 13.14 | 2.50 | 10.64 | 8.90 | 4.24 |

1982 | 11.07 | -1.90 | 12.97 | 3.80 | 7.27 |

1983 | 8.80 | 4.50 | 4.30 | 3.80 | 5.00 |

1984 | 9.92 | 7.20 | 2.72 | 3.90 | 6.02 |

1988 | 7.13 | 4.10 | 3.03 | 4.40 | 2.73 |

1990 | 7.35 | 1.90 | 5.45 | 6.10 | 1.25 |

1992 | 3.71 | 3.40 | 0.31 | 2.9 | 0.81 |

1996 | 5.21 | 3.70 | 1.51 | 3.30 | 1.91 |

2000 | 5.78 | 4.10 | 1.68 | 3.40 | 2.38 |

2002* | 1.69 | 1.80 | -0.11 | 2.40 | -0.71 |

2004* | 1.57 | 3.50 | -1.93 | 3.30 | -1.73 |

2006* | 4.80 | 2.70 | 2.10 | 2.50 | 2.30 |

2008 | 1.63 | -0.30 | 1.93 | 0.10 | 1.53 |

2010 | 0.23 | 2.40 | -2.10 | 1.50 | -1.20 |

2012 | 0.17 | 2.20 | -2.37 | 1.70 | -1.53 |

2014 | 0.11 | 2.10 | -2.21 | 1.60 | -1.49 |

2016 | 0.60 | 2.30 | -1.70 | 1.30 | -1.70 |

2018 | 1.99 | 2.30 | -0.31 | 2.44 | -0.45 |

Source: St. Louis Federal Reserve (FRED:TB1YR, GDP, GDPC1). ^{*}interpolated 1-yr T-bill rates.

*Note the anticipated real rate of interest (r*) is based on an average of the **actual
rate of real economic growth over the previous three years.*

Over time, changes in market interest rates may be attributed to changes either in the real desired rate 'r* or due to changes in inflationary expectations. Changes in the desired real rate reflects the behavior in the market for loanable funds. If the supply of these funds (public and private savings) exceeds the demand for these funds (public and private borrowing) then the desired rate should fall in reaction to a surplus of these funds. In periods of economic growth the opposite is true. The growing economy is sustained in part by increased borrowing activity for inventory investment and investment in new capital stock to allow for increased production to meet growth in aggregate demand.

**Note:** If we make a statement, like we did above, that the real rate of interest during the 1980's seem abnormally
high; we are implying that there is some *normal* rate (i.e., 2-3% per year). Note that in our examples above, the real rate is
really about the transfer of real goods and services. If a lender earns 5% annually per $1,000 lent, then what is being repaid is the
opportunity to buy $50 worth of goods and services -- $50 worth that must be produced by the real economy in that same time
period. Thus, suggestions of a *normal* real rate are founded on a relationship with the underlying real rate of economic growth.

- Desired Real Rate of Return
- Expected Rate of Inflation
- Nominal Interest Rate
- Purchasing Power
- Rate of Inflation
- Real Interest Rate
- Real Rate of Return