no third solution

Blogging about liberty, anarchy, economics and politics

NPV?

August 1st, 2007

If you aren’t very good at math, check out some of my other posts. This won’t interest you. If you are very good at math, I need your assistance, because I’m not. If you’re worried about “academic dishonesty,” don’t be. I don’t have any classes this summer.

I understand that the NPV of a consol bond or perpetuity would be C/(1+r) where C is the stated coupon rate, and r is the discount rate. Let’s assume that C is expected to grow by g% each year, and that this instrument is also a perpetuity. What would be the mathematical expression for such an instrument? For practicality’s sake, assume g < r This is as far as I can get:

NPV = Sum of [C/(1+r)] + [(C*(1+g))/(1+r)^2] + … + [(C*(1+g)^(n-1))/(1+r)^(n)]

Obviously, the sum increases as n increases, but at an ever decreasing rate, the velocity of which is determined by the spread between g and r – distant future growth (in terms of NPV) approaches irrelevance (zero) as n increases, and holding g constant, it approaches zero faster as r increases.

How can I simplify this so that I can plug it into my calculator? If I had taken more calculus, I’d be able to figure this out. I have taken the liberty of putting the formula into an xls spreadsheet and cascading it, which allows me to change the values of C, g, r; with a discount rate of 15% and a growth rate of 1%, the NPV appears to be growing about a penny per year, past 100 years, so it’s still quite a long term deal.

but if you know of a better way, please advise.

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no third solution

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