# 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?

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).

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.pdf

• Thank you. Reason I'm wanting to do this is that the exchange I trade on gives all the daily MTM data, but exclude the rate they use to arrive at the MTM data. So they give the spot price, futures price, and dividend assumptions... but they do not supply the discount rate. So it seems that the rate might be inconsistently applied over different stocks. I want to verify and calculate the rates they use for daily MTM. Thanks for the roots answer - appreciated.
– Anon
Jan 1, 2019 at 13:12
• First of all what you call $r$ is not just the interest rate but it includes the repo rate as well. Now the futures price is observed in the market data so i am not sure what you mean by “assumptions” to get the MTM ?
– Ezy
Jan 1, 2019 at 13:17
• Thanks Ezy :) With 'assumptions' I mean the dividend assumptions which are date on which it goes ex div, and also the dividend amount. So we have all the variables except 'r'. 'r' has the risk free part, which is known, but the risk premium is also part of it. And that's what I'm after. What risk premium does the exchange use for their daily mark to market. So I gather from your answer that one can't solve for 'r' algebraically, but one can use an algorithm to search for roots and so approximate it and arrive at the exchange's risk premium. Thanks for the help :)
– Anon
Jan 2, 2019 at 14:02
• @HeinVogel again you are missing one variable which is the repo rate. It seems you are mixing together risk free rate and repo rate.
– Ezy
Jan 2, 2019 at 15:18
• Hello Ezy :) Thanks again for your valuable feedback. Okay so I looked to add the repo rate, and I came up with the formula below. So I used 'Rp' to represent the risk premium, and 'Rr' to represent the repo rate. Then it becomes: F = Se^[(Rr+Rp)*T] + De^[(Rr+Rp)*t]. Is that looking better?
– Anon
Jan 4, 2019 at 12:06