What is the negative compound annual growth rate for a starting point
of  12,022 and an ending point of -613 if there are 104 compounding
periods and how do you calculate it?  The formula:
(Futureval/Presentval)^(1/Periods)-1 doesn't yield an answer.  The
series is a straight regression line, actually, and the equation is y=
-121.487x+12022.045.  Also, is there a general relationship between
the slope of the line and the compound growth rate?

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Request for Question Clarification by leapinglizard-ga on 23 Mar 2005 17:34 PST

If the value plotted over time is a straight line, then the interest
is not calculated on a compounding basis. Compound interest must, by
definition, result in exponential change. Another clue is the negative
end value, which is also impossible in the case of compound interest.
Linear change implies that the value changes by a constant amount in
each time interval. Does that solve your problem?

leapinglizard

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Clarification of Question by mike1269-ga on 23 Mar 2005 18:02 PST

This problem is not about compound interest.  I have a series of
average annual Aluminum prices, in $/tonne, starting in the year 1900
and through 2004 (104 periods).  While the nominal growth of the price
of aluminum is positive, the real prices (which I have calculated from
the CPI index for each of these years) yeilds a data series with a
downward trend.  The regression line for this period yields a line
with the data I gave in the original question.  Using this line of
best fit, I am trying to determine the real compound annual growth (in
this case negative) of aluminum prices over the period.  The problem
is that the trendline (line of best fit from the regression) at the
end of the series is a negative number, which makes calculating CAGR
using the normal formula impossible.  So how to do it?  I can see
graphically that the value is dropping, there must be a negative CAGR
associated with the line.

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Clarification of Question by mike1269-ga on 23 Mar 2005 21:05 PST

I am using the regression just to get the trend over the period, to
get a starting and ending point to figure out the equivalent
compounding rate.  If there is a way to use the equation of the line
it would be helpful.  It sounds like there isn't.  I thought there
might be some higher manipulation of the equation (integration
perhaps?) using the starting and ending data points that might get me
there, but it sounds like there isn't.  Still, I would sure like to
figure this out without having to chop-off older data (which is much
higher on a real basis) just to get a line that stays (y-variable)
positive over the series.

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Request for Question Clarification by leapinglizard-ga on 23 Mar 2005 22:05 PST

Fine, it's not interest, but compounding is still compounding.
Compound growth cannot possibly result in a straight line. Let me
answer your question another way: the compound annual growth rate for
this straight line of yours is zero. There is, however, a fixed annual
growth of -(12022+613)/104.

leapinglizard

If you talk about a negative compound annual growth rate you're
telling me that, every period,you need to take of a <<<constant
percentage>>>> of last period price to get today price

      price today = last period price x (1+decrease)
      or 
      price today = original price x (1+decrease)^(number of period)

this is a exponential you cannot get the "decrease" number with a
regression, you need to use a exponential fit ( y=a*e^(bx) ) in this
case "decrease" = e^b-1

Now if you want to take a <<<<<<<constant amount>>>>>>> out of last period price
you get

      price today = last period price + decrease
      or 
      price today = original price + decrease x number of period

there you have your regression 

      y= 12022.045-121.487x 

      original price = 12022.045 and the decrease (slope) =-121.487

The equation y = -121.487n + 12022.045 gives the average annual
aluminium price in period n. For example in period 104, the equation
gives a value of -612.6 which is an approximation for the price in
period 104 (2004).

In order to get a relation between the compound annual growth rate and
the slope of the line, a simple way will be to equate the two
relations. That is price in any period n can also be given by the
relation y = 12022*(1+r)^n where r is the annual growth rate and n is
the period. However as mentioned in other comments this will work only
is y is positive.

In such a scenario when y is positive, we get 12022*(1+r)^n =
-121.487n + 12022.045. For any given n we can get a value of r as long
as the price is positive. In its general form, the equation can be
written as a(1+r)^n = mn + b where a is the starting value, r is the
annual compounding rate, n is the period, m is the slope of the
regression line and b is the intercept. For small r (<<1), we can use
the approximation (1+r)^n = 1+rn to solve for r. r = (mn+b-1)/n.

However, this will work only if y is positive. And this too will only
be an approximation for r in any given period. The highest n that
keeps y positive is 98 and using that to solve for r, we get an annual
growth rate of -4.62%. As is obvious, this excludes the last 6
periods.

A minor correction in the relation between slope and annual growth rate r 

r = ([(mn+b)/a]-1)/n.