Risk-free rate

Risk-free rate is the theoretical rate of return on an investment that is assumed to have no risk of financial loss over a specified period. It is commonly used as a reference rate in financial economics, portfolio management, asset pricing, and risk-adjusted performance measurement.

In practice, a truly risk-free investment does not exist. Financial analysts therefore use the yield or return on a highly creditworthy government security as a proxy for the risk-free rate.

Concept

The risk-free rate represents the return an investor could theoretically earn without taking investment risk.

A simplified relationship between a risky investment's expected return and the risk-free rate is:

$Risk Premium = E (R_{i}) - R_{f}$

where:

$E (R_{i})$ = expected return of investment $i$ ;
$R_{f}$ = risk-free rate;
$Risk Premium$ = additional expected return associated with taking investment risk.

Risk-free asset

A risk-free asset is a theoretical investment whose future return is known with certainty and whose issuer is assumed to have no probability of default.

Government securities issued by highly creditworthy sovereigns are commonly used as practical proxies for the risk-free asset.

The appropriate security depends on:

the currency of the investment;
the investment horizon;
the jurisdiction;
the purpose of the analysis.

Government securities

Government securities are commonly used as proxies for the risk-free rate because governments with strong creditworthiness are generally considered to have a very low probability of default in their own currency.

Examples can include:

U.S. Treasury securities for U.S. dollar-denominated analysis;
UK government securities for pound sterling-denominated analysis;
German government securities for euro-denominated analysis;
government securities of other highly creditworthy sovereign issuers.

The appropriate proxy depends on the currency and maturity relevant to the analysis.

Maturity

The risk-free rate should generally correspond to the relevant investment horizon.

For example, a short-term analysis may use a short-term government security yield, while a long-term valuation model may use a longer-term government bond yield.

A simplified matching principle is:

$Investment Horizon \approx Risk-Free Rate Maturity$

The appropriate maturity depends on the specific financial model and its assumptions.

Risk premium

The risk-free rate is a fundamental component of the risk premium.

For an investment with expected return $E (R_{i})$ :

$Expected Risk Premium = E (R_{i}) - R_{f}$

The higher the risk-free rate, holding the expected investment return constant, the smaller the resulting expected risk premium.

Sharpe ratio

The Sharpe ratio uses the risk-free rate to measure risk-adjusted investment performance.

The Sharpe ratio is:

$Sharpe Ratio = \frac{R_{p} - R_{f}}{σ (R_{p})}$

where:

$R_{p}$ = portfolio return;
$R_{f}$ = risk-free rate;
$σ (R_{p})$ = standard deviation of portfolio returns.

The numerator represents the portfolio's return above the risk-free rate.

Capital Asset Pricing Model

The risk-free rate is also a fundamental component of the Capital asset pricing model (CAPM).

The CAPM expresses the expected return of an asset as:

$E (R_{i}) = R_{f} + β_{i} [E (R_{m}) - R_{f}]$

where:

$E (R_{i})$ = expected return of asset $i$ ;
$R_{f}$ = risk-free rate;
$β_{i}$ = beta of asset $i$ ;
$E (R_{m})$ = expected market return;
$E (R_{m}) - R_{f}$ = expected market risk premium.

Discounted cash flow valuation

The risk-free rate can be used as a component of discount rates in discounted cash flow valuation and other financial models.

For a single future cash flow:

$P V = \frac{C F_{t}}{(1 + r)^{t}}$

where:

$P V$ = present value;
$C F_{t}$ = cash flow received at time $t$ ;
$r$ = discount rate;
$t$ = time period.

The discount rate may incorporate the risk-free rate together with appropriate risk premiums.

Currency considerations

The risk-free rate is generally expressed in the same currency as the investment or cash flows being analyzed.

For example, a U.S. dollar cash-flow valuation would generally use a U.S. dollar risk-free rate rather than a rate denominated in another currency.

This is important because interest rates and inflation expectations differ between currencies.

Nominal and real risk-free rates

A nominal risk-free rate includes the effects of expected inflation.

A real risk-free rate represents a return adjusted for inflation.

A simplified relationship between nominal and real rates is:

$1 + r_{nominal} = (1 + r_{real}) (1 + π)$

where:

$r_{nominal}$ = nominal interest rate;
$r_{real}$ = real interest rate;
$π$ = inflation rate.

For relatively small rates, this can be approximated as:

$r_{nominal} \approx r_{real} + π$

Risk-free rate and interest rates

Changes in the risk-free rate can affect the valuation of financial assets.

A simplified relationship for a fixed future cash flow is:

$P V = \frac{C F}{(1 + r)^{t}}$

Holding the future cash flow constant, an increase in the discount rate generally reduces its present value.

The effect on different assets depends on their cash-flow characteristics, duration, and other risk factors.

Importance

The risk-free rate is a fundamental reference point in finance. It is used in calculating risk premiums, valuing investments, measuring risk-adjusted returns, estimating the cost of capital, and constructing financial models such as the Capital Asset Pricing Model. Because the appropriate proxy depends on factors such as currency, maturity, and jurisdiction, the risk-free rate is not necessarily represented by a single universal interest rate.

Risk-free rate

Contents

Concept

Risk-free asset

Government securities

Maturity

Risk premium

Sharpe ratio

Capital Asset Pricing Model

Discounted cash flow valuation

Currency considerations

Nominal and real risk-free rates

Risk-free rate and interest rates

Importance

See also