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Risk Free Interest Rate Calculation
Risk Free Interest Rate Calculation
The Black Scholes model used to value an option requires a risk free continuously compounded interest
rate as one of its inputs. Since U.S. Treasury securities are considered to be risk free, the interest rate at
the time of grant on the Treasury security with a maturity equal to the expected term of the grant is a valid
input to the Black Scholes model. The calculations used by Fidelity to derive this rate are explained below.
The Federal Reserve publishes the current fixed maturity yield on Treasury securities of varying maturity on
a weekly basis. Since the expected term of a grant is rarely equal to the maturity of one of these securities,
the first step is to calculate the equivalent yield on a security with maturity equal to the expected term.
For example, assume the expected term is 5.5 years and the fixed maturity yields on a 5 year bond and 7
year bond are 3.12% and 3.43%, respectively. To calculate the equivalent 5.5 year rate, a standard linear
function is created out of the known maturities and rates to find y, where x = expected term:
(Interest Rate) = (Slope) x (Months) + (Intercept)
Months Rate
84 3.43
60 3.12
66 3.1975
Yield Curve
3.45
3.4
Rate
3.35
3.3
Rate
3.25
3.2
3.15
3.1
55
60
65
70
75
Months
80
85
90
On the line created from the 5 year and 7 year rates, where months = 66, rate = 3.1975.
According to the logic above, the fixed maturity yield for the equivalent of a 5.5 year maturity is 3.1975%.
The fixed maturity yields published by the Federal Reserve are annual interest rates, so the second step is
to find a continuously compounded interest rate equivalent to the calculated annual rate. To accomplish
this, first a standard formula is used to compound the interest:
A = P (1 – r/n)^nt
Where:
A = Future Amount
P = Principal Invested
r = Compounded Interest Rate
n = Number of Times Interest is Compounded
t = Time in Years
Since an annual interest rate is returned, the formula will look at one year, so let t = 1. Also, let m = n/r.
Continuous compounding means that the interest will be compounded infinitely many times, so taking the
above into account, the formula can be restated as:
A / P = Lim m –>∞ (1 – 1/m)^mr
Lim m –>∞ (1 – r/m)^m is a commonly occurring limit in mathematics that has been proven to be equal
to the Euler constant e (this proof can be provided if necessary). The ending amount divided by the
principal originally invested for one year will always be equal to one plus the interest accrued, so let
(A / P) = (1 + Tr) where Tr is the annual rate on a Treasury security, calculated above. Taking this into
account, the formula can be restated as:
(1 + Tr) = e^r
Solving this equation for the continuously compounded interest rate r, we have:
ln(1 + Tr) = ln(e^r)
r = ln(1 + Tr)
For this example of Tr = 3.1975%, there is an equivalent continuously compounded interest rate
of 3.147%.
For more information, please contact a Fidelity
Stock Plan Services Customer Service Manager.
FOR PLAN SPONSOR USE ONLY.
Please note that tax information is general in nature and should not be considered tax or legal advice.
Fidelity does not provide legal or tax advice. Consult with an attorney or tax professional regarding any
specific legal or tax situation.
Stock plan recordkeeping and administrative services are provided by Fidelity Stock Plan Services, LLC.
Fidelity Brokerage Services LLC, Member NYSE, SIPC, 900 Salem Street, Smithfield, RI 02917
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