William R. Watt (ag384[at]freenet.carleton.ca) writes:
- quote -

um, a change has been made to the inital payment. sorry about that.
the modified formula can be found here: www.ncf.ca/~ag384/RetGrowth.htm
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---------------------------------------------- William R Watt National Capital FreeNet Ottawa's free community network
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I worked out a formula for accumulated value of growing payments (eg
inflation indexed bonds, corporate dividends) because I could not find one
for my retirement planning spreadsheet (under developement). I've appended
a description of the formula below in case it might be of use to anyone.
It's also posted on my website in file www.ncf.ca/~ag384/RetGrowth.htm.

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<preBelow is a formula for accumulated payments which are growing at
some rate. It would be used for interest payments from inflation-
indexed bonds or for dividends from companies which rise their
dividends.

First the formula will be developed, then some comments on its
applications will be made.

Formula
-------

The following simplifying assumptions are made:
- one bond or share
- payments of unity (eg one dollar) are made on the last day of
each year
- payments increase at an annual rate "g"
- payments are reinvested at an annual rate "r"
- no reinvestment restrictions, eg. bank account
- the symbol for exponentiation is "^"

The list below shows the year, the payment, and the accumulated
reinvested payments for a few years and for a final year "n".

Year Payment Accumulated payments (AP)
1 1 1
2 (1+g) (1+r) + (1+g)
3 (1+g)^2 (1+r)^2 + (1+g)(1+r) + (1+g)^2
4 (1+g)^3 (1+r)^3 + (1+g)(1+r)^2 + (1+g)^2(1+r) + g(1+g)^3
. . .
. . .
. . .
n (1+g)^n (1+r)^(n-1) + (1+g)(1+r)^(n-2) + ...+ (1+g)^(n-1)

Multiplying by (1+g)^(n-1)/(1+g)^(n-1) gives accumulated payments
of

AP = (1+g)^(n-1) ( (1+r)^(n-1) (1+r)^(n-2) )
( ----------- + ----------- + ... + 1 )
( (1+g)^(n-1) (1+g)^(n-2) )

We know the sum of a geometric series is sum[a^k] = (1-a^n) / (1-
a), where k= 0, 2, ..., n-1, and this provides the final form of
the equation for the accumulated payments

AP = (1+g)^(n-1) ( (1+r)^n / (1+r) )
( 1 - ------- / 1 - ----- )
( (1+g)^n / (1+g) )


Application
-----------
More than one bond or share: Multiply AP by the number of bonds
or shares.

Non-unity payments: Multiply AP by the initial payment. For
example a 4.5% inflation indexed bond pays $4.50 the first year.
To accumulate payments starting in any later year "j" multiply AP
by the first year's payment and by (1+g)^(j-1)

Non-annual payments: Bonds usually pay interest twice a year and
shares usually pay dividends four times a year. The growth and
reinvestment rates are prorated. Multiply AP by the initial
payment and divide "g" and "r" by the number of payments per
year, and multiply "n" by the number of payments per year. This
turns "years" into "periods", and "annual payments" into
"periodic payments". Income is paid, reinvested, and compounded
at the end of each period.

After tax accumulation: In Canada (2005) bond interest and
foreign dividends are taxed as ordinary income at the full income
tax rate but dividends from Canadian companies get a special
rebate. Let the income tax rate be "t". For after tax interest
and foreign dividends multiply AP by "(1-t)" and replace "r" with
"r(1-t)". For after tax dividends from Canadian corporations
multiply AP by "(1-1.25(t-0.1333)" and replace "r" with "r(1-t)".

--
---------------------------------------------- William R Watt National Capital FreeNet Ottawa's free community network
homepage: www.ncf.ca/~ag384/top.htm
warning: non-FreeNet email must have "notspam" in subject or it's returned