# Present value of a growing perpetuity that grows nonlinearly.

D

#### DarkProtoman

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.

T

#### Tim

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.
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....

D

#### DarkProtoman

Tim said:
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....
But what would that finite value be?

T

#### Tim

But what would that finite value be?
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%
....
!!!!!!!!

D

#### DarkProtoman

Tim said:
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%
...
!!!!!!!!
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?

T

#### 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?
When will it be called?

J

#### john boyle

In message said:
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.

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.

D

#### DarkProtoman

john said:
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.
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

T

#### tim \(back at home\)

DarkProtoman said:
What would be the point of that? It would be tantamount to taking their
money and giving them a worthless piece of paper.
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

J

#### john boyle

In message <[email protected]>,
DarkProtoman said:
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.
So the interest IS paid out then? In that case, on redemption your
company pays back the par value?

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
Quite, I cant see anybody buying it!

D

#### DarkProtoman

john said:
In message <[email protected]>,

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!
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.