net present value perpetuity question

[pompas]

I have a homework assignment that I don't understand. A person has a book deal to receive $10 million. It will take three years to complete. After the first year that the book is completed, he will receive $3 million as royalty. After that, the royalty decreases 30% in perpetuity. The person, who is extremely famous, could also make $8 million per year (paid at the end of the year) giving speeches instead of taking this book deal.

If he takes the book deal, what is the net present value of this deal? The person's cost of capital is 10%.

If I understand it correctly, this person's cost is $24 million because he lost the 3 years that he could have made $8 million per year. Is that correct?

I don't understand how to calculate the PV of this perpetuity of uneven cash flows that decrease 30% in perpetuity.

 

[tkhunny]

Can you say, "Basic Principles"?

i = 0.10 -- Annual Interest Rate
v = 1/(1+i) -- Annual Discount Factor

Lacking the Book Deal, we have:

8,000,000(v + v^2 + v^3 + ...) = 8,000,000(v/(1-v)) = 80,000,000

This is an 8,000,000 / year perpetuity.

With the Book Deal, we have:

Decreasing 30% leaves 70% after the decrease.

10,000,000 + v^4(3,000,000 + 3,000,000*0.7v + 3,000,000*(0.7v)^2 + 3,000,000*(0.7v)^3 + ...) =
10,000,000 + v^4(3,000,000)(1 + 0.7v + (0.7v)^2 + (0.7v)^3 + ...) =
10,000,000 + v^4(3,000,000)(1/(1 - 0.7v)) = 15,634,861

Comparing 80 to 16 isn't very compelling. Are you sure the 8,000,000 wasn't for a limited time rather than into perpetuity? For example, if it is for only the 3 years writing the book, it is 19,894,816 - quite a bit closer to $15.6MM

There is a lot in there. Give it some thought and let's see where we land.