Solving for an Implied Interest Rate from a Dividend-Paying Stock Future
Summary
The document asks how to infer an interest rate from a stock futures price when the spot price, expiry, and expected dividend are known. With one dividend, the pricing equation includes the compounded spot value and the dividend discounted according to the time from its ex-date to expiry. The rate appears in multiple exponential terms, so the equation generally does not yield a convenient algebraic rearrangement.
The proposed approach is numerical root search: find the rate that makes the pricing equation balance, provided the given inputs admit a solution. The expression can be extended to include a cost-of-carry term and multiple dividends, each with its own timing. The answer does not specify a particular root-finding algorithm, convergence settings, or how to handle multiple roots and uncertain inputs. It also questions the interpretation of the inferred rate, which may represent effective financing embedded in futures prices rather than a directly observed risk-free rate.
Key ideas
- A dividend-adjusted stock futures equation can place the unknown rate in several exponential terms.
- Numerical root search can find an implied rate when a solution exists.
- Each expected dividend enters the pricing relationship with its own timing to expiry.
- A cost-of-carry term can be incorporated into the futures pricing expression.
- The meaning of an inferred rate depends on the financing quantity being estimated.
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Full text
# Getting rate from a share's given futures price, with known dividend information
# Getting rate from a share's given futures price, with known dividend information
Question was answered by @Ezy - thanks!
This seems to be a basic question, but mysteriously unsolvable as far as I can see.
It concerns calculating the interest rate from a given stock futures price. It seems astonishingly hard to do.
Assume the following are given:
F - the Futures price
S - the Spot price
T - the Time to the futures expiry (days / 365)
D1 - the expected Dividend
t1 - the Time from the dividend ex-date to expiry
R - the risk Rate used
To keep it simple, assume there is only 1 expected dividend. Then the formula for the futures price is:
F = Se^(RT) - D1*e^(R*t1)
Then assume we have all the values except r. We know what F, S, T, t, and D are; and we want to solve for R.
I was unable to solve for R. (Perhaps my algebra is too weak). Wolfram Alpha professional also can't resolve a general 'R' from this equation either. If taken over the real numbers, then Wolfram Alpha can approximate R if all the other values are given. It looks like this is done through some kind of Goal-Seek or numerical analysis.
Why is this simple effort of getting R turning out to be so mysteriously difficult?
Note: The answer from @Ezy shows that the right answer is:
Use roots to get the value of R. One uses roots to get the implied rate. The formula, for one dividend, can be extended by adding cost of carry:
$$F = Se^{(R-q)T} - D1e^{(R-q)t1}$$
Where q is cost of carry.
One can add as many values of D as is necessary (D2, D3 etc.) to represent all the divs due in the period (each will then have a different t to expiry: t2, t3, etc).
## Answer by Ezy (score 2)
https://quant.stackexchange.com/a/43288
You do root search for such an equation. It works perfectly well assuming a solution exists given your parameters.
Aside from this it is not clear to me why you would want to imply the interest rate from this. Are you trying to imply the effective rate of financing from futures investors ?
For a useful reference on the forward price formula you can consult
https://web.ma.utexas.edu/users/mcudina/m339d-lecture-ten-forwards-pricing.pdfShown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.