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.

## Comments

## 8 Comments

RSS