Present value of a growing perpetuity that grows nonlinearly.

My company has decided to issue perpetual bonds. They can ONLY be redeemed a) if the company goes out of business b) is liquidated, or c) after 50yrs, either by being called away or put to issuer (what's the term for activating a bond's put feature?). They have a par of $1000. The interest rate is 10%, which grows at rate equal to it's size; ie after 1 year it would be 11%, having been increased by 10%, year after that it would be 12.21%, having been increased by 11%, and so on. My question is, if the conditions for the bond to be redeemed were in place, how much would my company have to pay? How do I calculate this; I know that a perpetuity's present value is given by coupon payment divided by the interest rate, and a linearly growing perpetuity's value is given by c/(i-g), but how do I calculate the value of a perpetuity that grows nonlinearly? Could you help me? Thanks!!!!! I'm in the US, btw.

Reply to
DarkProtoman
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"DarkProtoman" wrote

If it were truely a perpetuity (as in your other formulae), ie never redeemed -- and growing as you suggest -- then the present value is infinite at any finite discount rate. It only gains a finite value by being redeemed....

Reply to
Tim

But what would that finite value be?

Reply to
DarkProtoman

"DarkProtoman" wrote

Depends when redemption occurs - eg if in 50 years' time, then it's the value of a

50-year annuity with the quoted growth rate.

But if it's anything over about 15 years, then it is a very large value indeed:-

Year Coupon 1 10.0% 2 11.0% 3 12.2% 4 13.7% 5 15.6% 6 18.0% 7 21.2% 8 25.8% 9 32.4%

10 42.9% 11 61.3% 12 98.9% 13 196.6% 14 583.0% 15 3,981.5% 16 162,504.9% ... !!!!!!!!
Reply to
Tim

My poor company will be bankrupt!!!!!! Ok, how about changing the terms so the rate will grow by 10% every year. What will be the rate at the time of call?

Reply to
DarkProtoman

"DarkProtoman" wrote

When will it be called?

Reply to
Tim

In message , DarkProtoman writes

Perhaps I am missing something simple here, but I will press on anyway.

Are you saying that the coupon isnt paid out? Usually, bonds pay out the interest and only the par value is payable on redemption.

Reply to
john boyle

What would be the point of that? It would be tantamount to taking their money and giving them a worthless piece of paper. Obviously, you don't understand perpetuities. A perpetuity, you pay the company the par value, and it pays out a steady stream of interest forever. I know a great and extremely cheap way for companies to issue debt: a zero coupon perpetuity; use what you know about a perpetuity and a zero coupon bond to figure out the joke :-)

Reply to
DarkProtoman

I presume that this comment isn't in reply to John?

I must say that I was wondering the same thing.

Why would someone invest in something which only had value if the company went bankrupt (so it would probably be a lost investment) or so far in the future that no-one can reasonably use it in any form of income planning?

tim

Reply to
tim (back at home)

In message , DarkProtoman writes

So the interest IS paid out then? In that case, on redemption your company pays back the par value?

Quite, I cant see anybody buying it!

Reply to
john boyle

I can see the prospectus now: "This offers value at an extremely low price. Since there is no interest to be paid out, it is impervious to interest rate fluctuations, and since it accrues interest forever, the redemption value will be infinite." Really, what would a prospectus for that sort of "zero coupon perpetuity" read like? I really want to know.

Reply to
DarkProtoman

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