# Interest rate

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An interest rate is the "rental" price of money. When a resource or asset is borrowed, the borrower pays interest to the lender for the use of it. The interest rate is the price paid for the use of money for a period of time. One type of interest rate is the yield on a bond.

When money is loaned the lender defers consumption (or other use of the money) for a specific period of time. The lender does this in exchange for an expected increase in future income. The expected increase in interest payments (relative to the amount loaned) is the nominal interest rate, and can be defined as the face value of money received by the lender, or paid by the borrower. For example, suppose a household deposits \$100 with a bank for 1 year and receives interest payments totalling \$10. In this case, the nominal interest rate is 10% per annum. The real interest rate, which measures the purchasing power of interest receipts, is calculated by adjusting the actual rate charged (known as the money or nominal interest rate) to take inflation into account. (See real vs. nominal in economics.) A first approximation for the real interest rate for a one-year loan is:

ir = inpe

where:

in = nominal interest rate
ir = real interest rate
pe = expected or projected inflation over the year.

After the fact, there is the realized or ex post real interest rate:

ir = inp

where p = the actual inflation rate over the year.

Thus, if the (expected) inflation rate is 5% and the nominal interest rate is 7%, the (expected) real interest rate is 2%.

If financial markets have adjusted for the effects of expected inflation and the real interest rate is given, then the nominal rate approximately equals:

ir + pe

Thus, if the real interest rate is 3% and the inflation rate equals 5%, the nominal interest rate = 8%. The theory of rational expectations is sometimes applied to say that this equation applies in most cases. Most economists would agree that it applies over several years, as financial markets adjust: higher inflation leads to higher nominal rates, all else being equal.

Irving Fisher proposed a better approximation of the relationship between nominal interest rate, inflation and real interest rate. For a one-year bond, the expected real rate equals

ir = [(1 + in)/(1 + pe)] — 1

Using the first numerical example above, the expected real rate equals [1.07/1.05]-1 = 0.19 or 1.9%, which is similar to (but not the same as) the 2% calculated above.

When comparing different interest rates on different kinds of loans, a different kind of formula is used. For the nominal rate on a single type of asset,

in = i*n + d + mrp + lp

where

i*n = the nominal interest rate on a short-term risk-free liquid bond (such as U.S. Treasury Bills).
d = default premium (reflecting the likelihood of default by the borrower)
mrp = maturity risk premium (risk factor for length of borrowing period)
lp = liquidity premium (reflecting the perceived difficulty of converting the asset into money and thus into goods).

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