# How to come up with this present value in this question? [closed]

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.

## closed as off-topic by LocalVolatility, Bob Jansen♦Feb 25 '17 at 12:45

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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 $n=\infty$:

$$0.8 \sum_{k=1}^{k=\infty} (1.06^{-2})^{k} = 0.8 \cdot 1.06^{-2}\frac{(1.06^{-2})^{\infty}-1}{1.06^{-2}-1} = 0.8 \cdot 1.06^{-2} \frac{0-1}{1.06^{-2}-1} = -\frac{0.8 \cdot 1.06^{-2}}{1.06^{-2}-1} = 6.4729$$