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.