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The risk-free interest rate is the interest rate that it is assumed can be obtained by investing in financial instruments with no risk.
Though a truly "risk-free" asset exists only in theory, in practice most professionals and academics use short-dated United States Government bonds, often referred to as bills. They are considered to be risk-free because the likelihood of the Government defaulting is extremely low, and because the short maturity of the bill protects the investor from interest-rate risk that is present in all fixed-rate bonds.
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.
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.
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