# Risk-free interest rate

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The risk-free interest rate is the interest rate that one can be obtained by investing in financial instruments bearing no risk. It is usually noted as ${\displaystyle R_{F}}$ in finance textbooks.

## Conditions

Though a truly "risk-free" asset exists only in theory, in practice most professionals and academics use short-maturity government bonds of the currency in question. The rate used as Risk-Free rate has to met two conditions:

• Very low probability of default

For U.S. dollar investments, US Treasury bills are usually used, while a common choice for investments denominated in Euro are German government bills or Euribor rates. Those securities are considered to be risk-free because the likelihood of a government defaulting is extremely low, but not equal to zero. Damodaran (2002) argues that the reason behind the very low probability of default of a sovereign government is the control that they have on the printing of money. However, many governments have defaulted on their sovereign debt throughout history.

• No reinvestment risk

Because of the short maturity of the bill choosen, the investor will be protected from interest-rate risk that is present in all fixed rate bonds (if interest rates go up soon after the bill is purchased, the investor will miss out on a fairly small amount of interest before the bill matures and can be reinvested at the new interest rate).

## Logic behind the Risk-free rate

Since this interest rate can be obtained with no risk, it is implied that any additional risk taken by an investor should be rewarded with an interest rate higher than the risk-free rate (or with preferential tax treatment; some local government US bonds give below the risk-free rate).

The risk-free interest rate is thus of significant importance to modern portfolio theory in general, and is an important assumption for rational pricing. It is also a required input in financial calculations, such as the Black-Scholes formula for pricing stock options.

An alternative interpretation would be that, while no investment is truly free of risk, scenarios in which a major government with a long track record of stability defaults on its obligations are so far outside what is known that one cannot make quantitative statements about their chances of happening, and therefore it is simply not feasible to include them in financial planning. A German circa 1904 deciding whether to purchase long-term bonds issued by the German government could scarcely have been able to anticipate a World War followed by hyperinflation.

## References

Capinski M. and Zastawniak T. (2003), "Mathematics for Finance-An Introduction to Financial Engineering", Springer-Verlag

Damodaran A., 2002, "Investment Valuation", Second Edition, Wiley and Sons.