Closed-Form Valuation of Biennial Royalties and a Perpetuity
Summary
The discussion values a film’s screening rights when they generate a fixed payment every two years, beginning after the most recent screening. It expresses the finite stream as a geometric series of discounted payments, using the discount factor for a two-year interval. Summing the series gives a closed-form formula for a finite number of screenings instead of discounting each cash flow separately.
For an ongoing stream, the same series extends to infinity and reduces to the perpetuity value. The example assumes payments remain fixed and arrive regularly, with a constant discount rate. It demonstrates the algebraic method but does not address uncertainty about whether screenings continue, changing royalty amounts, or variation in discount rates. The answer includes the stated finite-period and perpetuity valuations as illustrations.
Key ideas
- Payments every two years use a two-year discount factor in the present-value series.
- A finite stream of equal payments can be summed with the geometric-series formula.
- The perpetuity value follows by taking the infinite-series limit when the discount factor is below one.
- The calculation assumes fixed payments, regular timing, and a constant discount rate.
- The formula does not account for uncertainty in future screenings or royalties.
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Full text
# How to come up with this present value in this question?
# How to come up with this present value in this question?
I'm starting to learn corporate finance on my own and have read about this question:
You sell the rights to screen a film on TV once every two years for €0.8m. The film has just been screened. You make the assumption that screenings will be possible for 30 years or in perpetuity. The discount rate is 6%. What is the value of your asset?
The answers to this questions are: €5.34m for the 30-year period €6.47m if it's a perpetuity
For the 30-year period, I have used an excel sheet to discount each cashflow to find 5.34m. I would like to know if there's a formula that I could have used for both situations.
Thank you.
## Answer by will (score 1)
https://quant.stackexchange.com/a/32677
So, you have this:
$$ \sum_{k=1}^{k=15} 0.8 \cdot 1.06^{2k} = 5.3456\ldots $$
And you want to know if there's a formula, or closed form. Yes there is.
$$ \sum_{k=1}^{k=n} x^{k} = x \frac{x^{n+1}-1}{x-1} $$
Where, we're going to set $x=1.06^{-2}$, and sum from 1 to 15 (since you're not including the first period in the valuations).
$$ 0.8 \sum_{k=1}^{k=15} (1.06^{-2})^{k} = 0.8 \cdot 1.06^{-2}\frac{(1.06^{-2})^{n+1}-1}{(1.06^{-2})-1} $$
You can see it here.
For the perpetuity, just set $n=\infty$:
$$ 0.8 \sum_{k=1}^{k=\infty} (1.06^{-2})^{k} = 0.8 \cdot 1.06^{-2}\frac{(1.06^{-2})^{\infty}-1}{1.06^{-2}-1} = 0.8 \cdot 1.06^{-2} \frac{0-1}{1.06^{-2}-1} = -\frac{0.8 \cdot 1.06^{-2}}{1.06^{-2}-1} = 6.4729$$
again, with a link.
To be honest though, this question probably doesn't belong here, it's fairly basic.Shown 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.