The risk free rate is not calculated from a formula. It is selected: you read the yield on a government security whose maturity matches the horizon of the cash flows you are discounting, and that observed yield is the input. The work is in choosing which yield, and in saying which one you chose.
That choice is the reason two analysts using the same model on the same business get different answers. Nothing about it is arithmetic.
Step One: Name the Horizon
Every yield on a government curve is a different number, so the phrase the risk free rate means nothing until a maturity is attached to it. A three-year projection and a going-concern valuation do not use the same input, and neither one is more correct in general.
The convention is a matching one: the maturity of the instrument you read should correspond to the period over which the cash flows you are valuing arrive. Pick the maturity first, then look up the yield. Doing it the other way round is how a habitual number gets applied to a horizon it was never chosen for.
Step Two: Read the Published Curve
For US dollar cash flows the observable source is the Treasury’s own daily curve, and how it is built decides what the number actually is.
The Treasury’s official yield curve is a par yield curve derived using a monotone convex method. Its inputs are indicative, bid-side market price quotations, not actual transactions, for the most recently auctioned securities, obtained by the Federal Reserve Bank of New York at or near 3:30 PM each trading day (US Department of the Treasury, Treasury Yield Curve Methodology).
Three things in that sentence matter to anyone quoting the output. The quotations are indicative rather than traded, so the curve is a fitted estimate and not a print. The securities are the most recently auctioned ones at each point, so the curve reads on-the-run issues. And the observation is taken near a fixed afternoon time, so a rate carries a date and a time rather than a general market level.
The maturity points the curve is fitted to are published as well. The current inputs are indicative bid-side market price quotations for the most recently auctioned 4, 6, 8, 13, 17, 26 and 52 week bills, the most recently auctioned 2, 3, 5, 7 and 10 year notes, and the most recently auctioned 20 and 30 year bonds (US Department of the Treasury, Treasury Yield Curve Methodology).
So the yields between those points are interpolated rather than observed. A rate quoted at a maturity that is not an input point is the output of the fitting method, which the same page describes as a monotone convex interpolation performed on forward rates midway between the input points.
Step Three: Match Nominal to Nominal, Real to Real
Treasury publishes a nominal par curve and a real one, and the choice between them has to follow the cash flows.
The nominal series is the Daily Treasury Par Yield Curve Rates, which relates the par yield on a security to its time to maturity and is based on the closing market bid prices on the most recently auctioned Treasury securities in the over-the-counter market. The real series is the Daily Treasury Par Real Yield Curve Rates, and its par real curve relates the par real yield on a Treasury Inflation Protected Security, or TIPS, to its time to maturity, based on the closing market bid prices on the most recently auctioned TIPS (US Department of the Treasury, Interest Rate Statistics).
Nominal cash flows, the kind almost every forecast contains, are discounted at a nominal rate. A model built in constant purchasing power reads the TIPS-derived series instead.
Mixing them is the quiet error. A real rate against nominal cash flows understates the discount and inflates the value, and nothing in the arithmetic complains.
Where the Number Goes
The rate is a model input rather than an answer on its own. In a single-factor pricing model it is the base the equity risk premium is added to, and both terms are worked through in the CAPM equation. In a multi-factor model it plays the same role under a longer sum, set out in arbitrage pricing theory. Where a required return is being assembled for a business rather than for a security, the wider construction is in cost of capital.
What the Risk Free Rate Is Not
- Free of risk. A government security carries reinvestment risk over any horizon shorter than the holding period, and inflation risk on the nominal series. Risk free is a modelling label for an asset assumed not to default, not a description of everything that can happen to the holder.
- The credit-adjusted risk-free rate. That is a different input with a different job. It measures an entity’s own obligation over time and is fixed at initial measurement, and it belongs to the accretion machinery covered in asset retirement obligation and accretion.
- A single number for the year. The curve moves daily and the published series is dated. A valuation states the date its rate was read.
- The rate on any government’s debt. The convention holds for a government borrowing in its own currency and treated as free of default risk. Reading a foreign yield as risk free imports a credit view along with it.
Common Questions
What Is the Formula for the Risk Free Rate?
There is no derivation formula. The nominal rate is looked up on the published curve at the maturity matching the horizon. The only calculation involved is choosing between the nominal and the real series, and Treasury publishes both.
Which Treasury Maturity Should I Use?
The one matching the period the cash flows cover. The curve is fitted to bill maturities from 4 weeks to 52 weeks, note maturities from 2 to 10 years, and bond maturities at 20 and 30 years, so most horizons sit at or between published points.
Why Do Two Valuations of the Same Business Use Different Risk Free Rates?
Because they made different choices, and both may be defensible. A different horizon, a different observation date, or the real series instead of the nominal one each moves the input. This is why the chosen maturity and the observation date belong in the workpaper next to the rate.
Is the Risk Free Rate the Same as the Discount Rate?
No. It is one component of a discount rate. The rest of the discount rate is the compensation demanded for bearing risk the government security does not carry.
Accountably places trained offshore accountants inside US CPA and EA firms, and a firm can grade a real block of its own valuation support work through a trial task before any client file moves.
