# Present value of a growing perpetuity

I

#### Inquisitive

I'm currently taking a finance course and hope someone can help me with
a query regarding growing perpetuities.

I know the formula for calculating the present value of a growing
perpetuity is

C/(r-g)

where C is the cash flow each period, r is the interest rate, and g is
the rate of growth. But this formula assumes that the first payment in
the perpetuity occurs in one year's time. What happens if the first
payment won't occur for a few years.

For example, say that the first payment at the end of year 1 will be
£100, the annual rate of interest is 10%, and the perpetuity needs to
grow at an annual rate of 6%. In this case the present value will be
150/(0.10-0.06)=£3750 (I think).

However, what would be the case if the first payment didn't happen
until the end of year 5? Do you simply work out the present value of
150 and then plug that into the equation?

Hope you can help

T

#### Tim

I know the formula for calculating the
present value of a growing perpetuity is

C/(r-g)

where C is the cash flow each period, ...
You mean "at the end of Year 1";
(not "each period")

... r is the interest rate, and g is the rate of growth.
But this formula assumes that the first payment
in the perpetuity occurs in one year's time.

For example, say that the first payment
at the end of year 1 will be £100, ...
You mean £150?
(see below)

... the annual rate of interest is 10%, and the perpetuity
needs to grow at an annual rate of 6%. In this case the
present value will be 150/(0.10-0.06)=£3750 (I think).
Agreed.

However, what would be the case if the first
payment didn't happen until the end of year 5?
Simply discount your above result for the
four years (between 'end of Year1' and
'end of Year5') at the interest rate used :-

£3750 / (1.10^4) = £2561.30.

Assuming, of course, that you mean that the payment at end of Year5 is to be
£150 and not £189.37 (as it would be after 4 increases at 6%).
If the latter is the case, then the answer is £3233.58.

R

#### Ronald Raygun

Inquisitive said:
I know the formula for calculating the present value of a growing
perpetuity is

C/(r-g)

where C is the cash flow each period, r is the interest rate, and g is
the rate of growth. But this formula assumes that the first payment in
the perpetuity occurs in one year's time. What happens if the first
payment won't occur for a few years.

For example, say that the first payment at the end of year 1 will be
£100,
You mean £150 here, surely.
the annual rate of interest is 10%, and the perpetuity needs to
grow at an annual rate of 6%. In this case the present value will be
150/(0.10-0.06)=£3750 (I think).
Correct, because the 10% interest will grow your £3750 to £4125, and
after paying out the £150, there'll be £3975 left in the kitty, which
is 6% more than you started with. Similarly, a year later, you can
harvest 6% more than £150 and still leave the kitty 6% fuller, and so
on forever.
However, what would be the case if the first payment didn't happen
until the end of year 5? Do you simply work out the present value of
150 and then plug that into the equation?
I'd have thought you'd need to work out the "present" value at the
end of year 4 using the above formula, and then work back to the
real present from there by applying the interest rate in reverse for
4 years.

So if the year 4 value is £3750, and the interest rate 10%, then the year
0 value should be £2561 (i.e. £3750 divided by the 4th power of 1.10).
I'm assuming here that the first payment at year 5 is still to be £150
as originally, not £150 scaled up by 4 lots of 6%. Otherwise amend as
appropriate.

I

#### Inquisitive

Many thanks to both you and Tim for your replies.

I have an associated question, based upon the answer of my query above:

I need to calculate the present value today of a cash flow stream that
starts at 150 in one year's time and grows at an annual rate of 6% (the
corresponding interest rate is 10%) up to and including the end of year
4.

I believe that I can do this by working out the present value of two
growing perpetuities, which both have the same interest rate (10%) and
growth rate (6%) but the first cash flow starting at different times
(at the end of year 1 for the former perpetuity and at the end of year
5 for the latter), then subtracting the latter from the former. The
details of these are obviously from my previous posting, but my query
is this - does this further question assume that the 150 payment has
also grown with the rate of interest (i.e. the PV is 3233.58 as you
have indicated above rather than 2561.30)?

What it boils down to is whether the present value is equal to:
(1) 3750 - 2561.30
or
(2) 3750 - 3233.57
Any further insight you can provide would be most appreciated.

T

#### Tim

Many thanks to both you and Tim for your replies.
You're welcome for my contribution, but I don't know who you are replying
to, nor what they said, as their post has not appeared on my newsgroup
server ...
[I'd take a guess at it having been Ronald??!]

I need to calculate the present value today of a cash
flow stream that starts at 150 in one year's time and
grows at an annual rate of 6% (the corresponding interest
rate is 10%) up to and including the end of year 4.
Ah, a four-year "annuity-certain".

I believe that I can do this by working out the present value of two
growing perpetuities, which both have the same interest rate (10%)
and growth rate (6%) but the first cash flow starting at different times
....

That's one way. Another way is to simply value the four payments.

... does this further question assume that the 150 payment
has also grown with the rate of interest (i.e. the PV is
3233.58 as you have indicated above rather than 2561.30)?

What it boils down to is whether the present value is equal to:
(1) 3750 - 2561.30
or
(2) 3750 - 3233.57
Neither - it's actually:
(2a) 3750 - 3233.58 = 516.42.

Think about it - you want the second perpetuity to cancel-out the first one
for subsequent years, hence the payment at the end of year 5 needs to be the
same as that for the first perpetuity - which is 150 x 1.06^4 = 189.37.

R

#### Ronald Raygun

Inquisitive said:
Many thanks to both you and Tim for your replies.

I have an associated question, based upon the answer of my query above:

I need to calculate the present value today of a cash flow stream that
starts at 150 in one year's time and grows at an annual rate of 6% (the
corresponding interest rate is 10%) up to and including the end of year
4.
Why not just add up the Present Values of each of the four payments?
This is going back to basics and is a bit boring, but it gives a good
way of checking whether you get the same result by any other method.

The PV of the first £150 is £150/1.1,
that of the 2nd payment is £150x1.06/1.1^2,
the 3rd £150x1.06^2/1.1^3 etc, so the four add up to:

£150 x (1/1.1 + 1.06/1.1^2 + 1.06^2/1.1^3 + 1.06^3/1.1^4)

which looks pretty messy but can be simplified to:

(£150/1.1) x (1 + 1.06/1.1 + (1.06/1.1)^2 + (1.06/1.1)^3)

There's a formula for working out (1 + k + k^2 + k^3), i.e. taking
k to be 1.06/1.1, namely (k^4-1)/(k-1).

So here k=.96364, (k^4-1)/(k-1)=3.78706, and the answer is £516.42.
I believe that I can do this by working out the present value of two
growing perpetuities, which both have the same interest rate (10%) and
growth rate (6%) but the first cash flow starting at different times
Cunning! You think this might be simpler than using the boring method?
Makes sense. The value now (year 0) of paying £150 in year 1, and
forever annually thereafter index linked at 6% on capital growing at
10%pa, is, as you say, £3750.

The value at year 4 of paying £150 index-linked forever from year 5
is also £3750, but from year 5 you'd be paying not £150 index linked
but 1.06^4 times as much. So the Y4 value of the tail end is
1.2625x£3750 or £4734.29, and the Y0 equivalent of *that* is obtained
by dividing it by 1.1^4, which gives £3233.58.

Subtract this from £3750 and you get the same £516.42 as the boring
method.