Present value of a growing perpetuity

"Inquisitive" wrote

You mean "at the end of Year 1"; (not "each period")

"Inquisitive" wrote

You mean 150? (see below)

"Inquisitive" wrote

Agreed.

"Inquisitive" wrote

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.

You mean £150 here, surely.

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.

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.

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

2. 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.

"Inquisitive" wrote

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??!]

"Inquisitive" wrote

Ah, a four-year "annuity-certain".

"Inquisitive" wrote

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

"Inquisitive" wrote

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.

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.

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.