Where does the formula for terminal value come from?

Question

I am a beginner investor and I had this doubt in DCF models of valuation. Where does the formula for terminal value even come from? Like I have seen this video: https://www.youtube.com/watch?v=hCGn1ejYs1I, and I don't seem to get the intuitive part (the first few minutes) of the video.

The formula is: T.V.=FCF(n) {1+r%}/(R%-r%), where n is the last projection year, R%=return wanted (WACC) in 1 yr and r%=FCF growth rate of 1 year

1. Why do we grow the FCF of the last year only once and not compound it for years like make the formula FCF(n) {1+r%)^t for t years after the last projection and then discount each back? Is a possible answer that the time will be indefinite and so we will have to grow the FCF for an indefinite amount of time, so we will be saying that the company's FCF is growing at the same rate for an infinite amount of time, so we can't really add an infinite number of FCFs up? If not, then why?

Basically, I am asking why isn't there a (^t) in the numerator and sigma to sum them up

Please give an intuitive answer to this and not maths cause I am sure that the math will turn out to be correct.

2. Where does the R%-r% come from, and where is the (^t) here? don't we want to get that back too to the https://www.youtube.com/watch?v=fd_emLLzJnk at 24:08? And why is the r% being subtracted? (Please: again an intuitive answer required with logic)

This is my interpretation of inflation: A $100 is worth the same as a $107 the next year if inflation is 7%. SO, even if we grow our money by 7%, the real value of that money is the same, so the return is 0% or R%-r(inf)%.

What if WACC=r%? then WACC-r%=0, so T.V. is undefined. Why are we subtracting r% from WACC? Isn't r% also supposed to increase our money's value nevertheless? (Assume inflation=0, so is r=7, then $100 today will grow to $107 tomorrow, and will DEFINITELY give a return, until and unless r(inf)%>=r%)

3. If I take the terminal value after 5yrs of projection vs 4yrs of projection, won't I be missing out on 1 yr's free cash flow in the 2nd case, thus impacting intrinsic value? Does the equal nature of intrinsic value then come from the "t" in discounting factor of the terminal value to the present, as in the 1st case it would be 5, and in the second case it would be 4?

(EDIT 2: I tried this out with an initial cash flow of $100, R=10, r=1, and for case 1, I took the cash flow projections for 2 yrs and then T.V. and got an intrinsic valuation of almost $268, and in the 2nd case, I took only for the first year the cash flow projection and then T.V, so I got a valuation of about $193 (r=growth rate perpetual, R=expected return) on DCF, so my assumption is probably wrong. )

Again, please give all your answers with logic and examples if possible. That would be very well appreciated. Thanks in advance.

EDIT 1: I have seen and understood the math derivation easily. I am looking for the intuition behind it sort of to build this formula on our own without the use of geometric series.

Answer 1

It's derived from the formula for a growing perpetual annuity. The present value of a perpetual annuity is the initial cash flow (the payment after one period) divided by the discount factor minus the growth rate (C1/(r-g)) - you can look up the derivation of that formula offline.

So if we treat all of the cash flows after year N as a growing perpetual annuity, the initial cash flow (the first year after N) will be the cash flow from year n after one year of growth, or FCF(n) * (1+r)

So the value of the company at year N is then [FCF(N) * (1+r)]/(R-r)

Note that the derivation of the annuity formula is exactly a closed-form formula of the infinite series

C*(1+r)^1     C*(1+r)^2    C*(1+r)^3    C*(1+r)^4
---------  +  --------- +  --------- +  --------- ...
  (1+R)^1       (1+R)^2      (1+R)^3      (1+R)^4 

So you're not losing any cash flows after the terminal year, you're just summarizing all of the remaining cash flows into one closed-form term.

What if WACC=r%?

Imagine you could borrow money at r% indefinitely with no payments (meaning the interest just compounds) and use it to buy a house whose rent increased by g% indefinitely with no risk - if you calculated the present value of each of those rent payments, the rent payments would grow faster than the accrued interest and you'd (in theory) have an infinite value. So the terminal value formula requires that the constant, perpetual growth rate is less than the "cost of capital", or inflation, or risk-0free interest rate, or whatever metric you use for discounting.

You can have higher growth for a period of time, and certainly growth companies often do grow faster then their WACC, but not forever. Eventually they have to grow more slowly and become more of a "cash cow".

Answer 2

its me, the asker of the question

I think we should go by this model as :

To understand the concept of DCF Terminal Value, we must know that after a period of growth, a company reaches a “steady state” which is when all sources of competitive advantage are exhausted and its profitability and efficiency ratios are stabilized. The steady-state period typically coincides with the end of the explicit forecast of the DCF analysis. The value of the future steady-state cash flows can be summarized in a single number called the DCF terminal value.

And then we derive that formula already knowing that IT HAS ACCOUNTED FOR ALL FUTURE CASH FLOWS AND HAS BROUGHT THEM TO THE START OF THE TERMINAL PERIOD OR THE END OF THE FINAL CASH FLOW PROJECTION PERIOD, which so happens to come out after using the rigorous theory that T.V.=FCF(n)*(1+r%)/(R%-r%), and there may/ may not be intuition to develop the formula from scratch.

This is the website: https://www.fe.training/free-resources/financial-modeling/dcf-terminal-value-formula/#:~:text=The%20terminal%20value%20in%20year%20n%20%28for%20example%2C,LTM%20EBITDAn%20%2A%20Multiple%20N%20%3D%20year%205