Risk-free rate: Definition & Examples

The risk-free rate is the return on lending money with no risk of default, over a stated horizon and in a stated currency. No loan is perfectly safe, so practitioners use proxies: yields on government bonds and Treasury bills in the government's own currency, overnight reference rates such as SOFR, SONIA or €STR, and the overnight indexed swap rates built from them. The result is a yield curve, not one number, because each maturity has its own risk-free rate. It is also a nominal rate. Inflation still erodes what the money buys, which is why the real interest rate differs from it. Hull defines

r

as the rate at which money is borrowed or lent when there is no credit risk, so that the money is certain to be repaid.

One bill, six quotes

A fictional 182-day US Treasury bill has a face value $F=100{,}000$ USD and sells for $P=97{,}900$ USD, a dollar discount of 2,100 USD. Dealers quote bills on a bank discount basis, so the first figure on the screen is often the lowest annualised number in the table below.

Convention	Formula	Value
Bank discount yield	$\dfrac{F-P}{F}\cdot\dfrac{360}{t}$	4.1538%
Holding-period return	$\dfrac{F-P}{P}$	2.1450%
Money-market (CD-equivalent)	$\dfrac{F-P}{P}\cdot\dfrac{360}{t}$	4.2429%
Bond-equivalent	$\dfrac{F-P}{P}\cdot\dfrac{365}{t}$	4.3019%
Effective annual rate	$\left(\dfrac{F}{P}\right)^{365/t}-1$	4.3483%
Continuously compounded	$\ln\!\left(\dfrac{F}{P}\right)\cdot\dfrac{365}{t}$	4.2564%

The holding-period return is the only row that is not annualised. It is the actual return on the 97,900 USD invested over 182 days. Every other row annualises the same growth factor, $F/P=1.0214505$ , with a different convention. The continuous rate reproduces it exactly:

\frac{F}{P}=e^{\,0.0425639\times 182/365}=1.0214505

Fabozzi gives two reasons the bank discount yield is not a return measure. It is based on face value rather than the cash invested, and it uses a 360-day year, while Treasury notes and bonds pay interest on a 365-day basis. The money-market (CD-equivalent) yield keeps the 360-day year but divides by the price. Fabozzi's form $360\,Y_d/(360-t\,Y_d)$ returns the same 4.2429%.

The convention matters whenever a formula expects a particular rate. Take a no-dividend index at

S=5{,}000

USD with

T=182/365

years. The forward under continuous compounding is

F=Se^{rT}

. Using

r=4.2564\%

gives

F=5{,}107.25

USD, which equals

5{,}000\times 100{,}000/97{,}900

. Plugging in the 4.1538% discount quote gives 5,104.64 USD, which is 2.61 points low. Even the CD-equivalent 4.2429% is 0.34 points low, because it is not a continuous rate. Black–Scholes and cost-of-carry formulas expect continuously compounded rates, so convert first.

Horizon changes what is risk-free

Priya is the treasurer of a manufacturer with 10,000,000 USD of surplus cash. She must hold it for 182 days and has two options.

Lock the bill. The interest is fixed on the day she buys:

\text{Interest}=10{,}000{,}000\left(\frac{100{,}000}{97{,}900}-1\right)=214{,}504.60\text{ USD}

Roll overnight deposits. She starts at 4.20% and accrues simple ACT/360 interest each day, compounding every calendar day:

W_{182}=10{,}000{,}000\prod_{k=1}^{182}\left(1+\frac{r_k}{360}\right)

Here $r_k$ is the overnight rate on day $k$ , and interest is $W_{182}-10{,}000{,}000$ . Three illustrative paths, which are scenarios rather than forecasts, give:

Path	Overnight rate	Interest, USD	Versus bill, USD
Flat	4.20% on all 182 days	214,590.99	+86.40
Two cuts	4.20% on days 1–60; 3.95% on days 61–120; 3.70% on days 121–182	201,548.80	−12,955.80
One hike	4.20% on days 1–91; 4.45% on days 92–182	221,047.31	+6,542.72

The flat overnight rate that exactly matches the bill is 4.1983%. Two 25 basis point cuts leave her about 13,000 USD short of the bill, while a single 25 basis point hike adds about 6,500 USD. The calendar-day compounding is a stated simplification. SOFR compounds over business days, so a real rolled deposit would differ slightly.

Line chart of cumulative interest earned on 10,000,000 USD over 182 days for three overnight-rate paths. A flat 4.20 percent path ends at 214,590.99 USD, a path with two 25 basis point cuts ends at 201,548.80 USD and a path with one 25 basis point hike ends at 221,047.31 USD. A dashed horizontal line marks the 214,504.60 USD interest locked by the 182-day bill.

Original fictional illustration. Interest accrues on 10,000,000 USD using simple ACT/360 daily accrual with calendar-day compounding. The rate paths are scenario assumptions, not forecasts.

Each overnight loan carries almost no default risk for one night, but over 182 days only the bill fixes the outcome. Rolling carries reinvestment risk, because every future rate decision changes the result. The reverse also holds. If Priya needed the cash tomorrow, selling the bill early would expose her to its market price. A rate is risk-free only relative to a horizon, and the benchmark that fits is the one whose maturity matches the cash need.

Which proxy, and why it matters

Hull notes that a government is usually assumed not to default on debt in its own currency, which makes Treasury rates the natural first candidate. Derivatives desks rarely discount with them. Hull's Business Snapshot 9.1 lists three dealer arguments that these yields are artificially low: regulated institutions must hold them, the capital charge for holding them is small, and their interest is exempt from state tax. The own-currency argument also fails for an individual eurozone member, which cannot control the European Central Bank.

LIBOR was the traditional derivatives benchmark before 2008, but it was not risk-free either. A short-term loan to a AA-rated bank can default, however unlikely that is. The TED spread, the excess of the 3-month Eurodollar deposit rate over the 3-month Treasury bill rate, is normally under 50 basis points. It rarely fell below 100 basis points between October 2007 and May 2009 and peaked above 450 basis points in October 2008.

After the crisis, Hull reports, dealers moved collateralised derivatives to OIS discounting while still discounting uncollateralised trades at LIBOR or higher, reflecting a view that the rate should match the bank's average funding cost. LIBOR itself has since been retired: the UK Financial Conduct Authority's notice on the US dollar panel records that the panel ended on 30 June 2023 and that new use of the remaining synthetic settings was prohibited. Overnight rates such as SOFR are now the anchors. The choice of proxy is not cosmetic: it changes the present value of every derivative cash flow.

SOFR is a secured rate. The New York Fed describes it as a broad measure of the cost of borrowing cash overnight, collateralised by Treasury securities, and publishes it each business day at about 8:00 a.m. ET. €STR is an unsecured rate reflecting the wholesale overnight borrowing costs of euro-area banks, published by the ECB at 08:00 CET on each TARGET2 business day. Because their collateral differs, an OIS curve built from one is not the same curve built from the other.

Proxy	What it measures	Typical role
Government bill or bond yield in own currency	Lending to a government borrowing in its own currency	Textbook benchmark; often seen as artificially low
LIBOR (traditional)	Unsecured interbank borrowing between banks	Pre-2008 derivatives benchmark; carries bank credit risk
SOFR and €STR	Secured (SOFR) or unsecured (€STR) overnight borrowing	Overnight anchor for OIS curves
OIS rate	Fixed rate swapped against compounded overnight rates	Discounting collateralised derivatives

Risk-free never means zero risk. A bill is default-free in its own currency, yet its market value moves before maturity and its real return depends on inflation. A secured overnight rate removes most credit exposure, but not collateral, operational or settlement risk. The Finance Primer's Money Markets lesson (public preview; full lesson requires membership) covers bill quotes, deposits and repo funding calculations, and makes the same point that "risk-free" and "overnight" are not synonyms.

Where the rate enters quant work

Discounting. A payment $X$ due at time $T$ has a present value of $X\,D(0,T)$ , where $D(0,T)$ is the discount factor read from the risk-free curve.
Option pricing. The Black–Scholes model discounts at $e^{-rT}$ . Under the risk-neutral measure $\mathbb{Q}$ the underlying drifts at $r$ , so the rate sets both the forward and the drift:

dS_t = rS_t\,dt + \sigma S_t\,dW_t^{\mathbb{Q}}

Performance. Excess return over the risk-free rate is the input to the Sharpe ratio and to the capital asset pricing model. A fund returning 9% with 12% volatility has a Sharpe ratio of $(0.09-0.043)/0.12\approx 0.39$ at a 4.3% risk-free rate. Using zero instead gives $0.09/0.12=0.75$ , nearly double. The two figures answer different questions, so state the rate and horizon whenever you compare Sharpe ratios.

Sources

Frank J. Fabozzi, Bond Markets, Analysis, and Strategies, 9th ed., Chapter 6: bank discount yield and CD-equivalent yield.
John C. Hull, Options, Futures, and Other Derivatives, 9th ed., Global Edition: Section 4.1 on Treasury rates; Chapter 5 on the definition of $r$ ; Section 9.1 and Business Snapshot 9.1 on proxies, the TED spread and eurozone governments.
UK Financial Conduct Authority, US dollar LIBOR panel – 1 month to go (31 May 2023): end of the US dollar LIBOR panel.
New York Fed, Secured Overnight Financing Rate reference rates.
European Central Bank, euro short-term rate (€STR) overview.

One bill, six quotes

Horizon changes what is risk-free

Which proxy, and why it matters

Where the rate enters quant work

Sources

Related terms

Basis point

Black-Scholes model

Continuous compounding

Discount factor

Inflation

Present value

Real interest rate

Yield curve