| Rate of Return a la Newton a continuation of Part I |
If you just want to calculate an Annualized return (without all the bumpf which follows), go here
Suppose we invest $1,000 each year, for five years, and our portfolio is now worth $6,523.33 (the last $1K investment having been made one year ago).
We're making investments of A1, A2, A3, ... AN
for time periods (in years*) of
T1, T2, T3, ... TN and our current portolio is $P.
* If the times were in MONTHS, then we'd get the
MONTHLY return instead of the ANNUALized return.
The annualized return, R, must satisfy:
(1) f(R) = A1(1+R)T1 + A2(1+R)T2 + A3(1+R)T3 + ... + AN(1+R)TN - P = 0
We have a linear approximation for R, namely:
| RLinear = { P - Σ An} / ΣAnTn |
which is the solution to the Linear equation: fL(R) = Σ An + { ΣAnTn } R - P = 0
We also have a Quadratic approximation ... explained in Part I. We'll ignore it - for now.
A notation such as Σ un is shorthand for u1+u2+u3+ ... +uN.
We consider a particular problem where our current portfolio is P = $2020, and
 |  Fig. 1 |
| f(R) | = 6000(1+R)4 - 3000(1+R)3 + 1000(1+R)2 - 1000(1+R) + 2000(1+R)0.5 - 2020 >= 0 at RE = -0.275 |
| fL(R) | = 6000(1+4R) - 3000(1+3R) + 1000(1+2R) - 1000(1+R) + 2000(1+0.5R) - 2020 |
| = 2980 + 17000R >= 0 at RL = -0.175 | |
| fQ(R) | = 6000(1+4R+6R2) - 3000(1+3R+3R2) + 1000(1+2R+R2) - 1000(1+R) + 2000(1+0.5R-0.125R2) - 2020 |
| = 2980 + 17000R + 27480R2 >is NEVER 0 !! |
The Exact annualized return is RE = -0.275 (or -27.5%)
... which can be verified by
substituting this R-value into f(R) and getting
f(-0.275) = $0
although, to get $0 to within 2-bits, you'll
have to subtitute the more precise value of R=-0.2747 :)
And, alas, the green quadratic fails
to provide an estimate 
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Here we'll use the Linear approximation to get a better approximation. First, we construct a tangent line to the f(R) curve at the point where R = the Linear approximation, namely RL = -0.175 (or -17.5%). That means:
|  Fig. 2 |
 Fig. 2a |
>That's pretty good! The exact value is -27.5% so ... Wait. We're not finished. We can repeat this procedure, moving along the tangent line starting at this IMPROVED approximation (namely R1) and that'll give us an even BETTER approximation (which we'll call R2), so we can start from there and construct yet another tangent line and move to where it cuts the horizontal axis yielding and even BETTER, BETTER approximation (which we'll call R3) so we can repeat with this BETTER, BETTER ...
>I get it! I get it! So when do you get to the exact value, RE? |
Note, too, that we'll call RL = R0 for sanitary reasons.
>And for our example? What are the values of ...?
Here they are:
R0 = -0.17500
R1 = -0.25709
R2 = -0.27403
R3 = -0.27467
R4 = -0.27467
and from there on, you get the same five digits, so we conclude that the
exact annualized return is, to five digits, -0.27467
(or -27.467%) although we could get more decimals if we wanted. Do you want more decimals?
>Definitely not - but do I have to do all this graphically? Do I draw tangent lines and ...?
Definitely not. As you might expect, there's a magic formula.
To get the magic formula, we first rewrite Eq. 1, replacing (1+R) by x, getting:
(2) F(x) = A1xT1 + A2xT2 + A3xT3 + ... + ANxTN - P = ΣAnxTn - P
and we wish to determine the value of x which makes F(x) = 0.
If we let
(1+R0)=x0,
(1+R1)=x1,
(1+R2)=x2, etc., then the prescription for computing
the various approximations is:
| x1 = x0 - F(x0)/F'(x0) = x0 - { ΣAnx0Tn - P }/{ΣAnTnx0Tn-1} |
| x2 = x1 - F(x1)/F'(x1) = x1 - { ΣAnx1Tn - P }/{ΣAnTnx1Tn-1} |
| x3 = x2 - F(x2)/F'(x2) = x2 - { ΣAnx2Tn - P }/{ΣAnTnx2Tn-1} |
etc. etc.
If we carry five digits and our approximations don't hardly change after, say, "k" iterations, then we stop and get our annualized return as RE = xk - 1.
>The formulas got F'(x0) and F'(x1) and ...
Them's derivatives. Don't worry about 'em.
>Mamma mia! Did you figure all this out?
Well ... uh, it was actually Sir Isaac Newton.
>The guy who took a blow from the falling apple?
Yeah, that's him.
>That's he.
The method is called Newton's Method and we'll call the various approximations x1, x2, etc. Newton Iterations. Fortunately we have a .ZIPd spreadsheet to do all this. We just enter the CASH FLOWS A1, A2, etc. and the DATEs when they occurred - the spreadsheet computes the time periods in YEARS: T1, T2, etc. - and, finally, we type in the current value of the portfolio, $P.
Here's an example with CASH FLOWS and DATEs; it looks like so:
to download the .ZIPd spreadsheet
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>Doesn't Excel already have a command to do all this? Yes, the XIRR command. It goes like this:
where the range A1:A50 contains the cash flows and the range B1:B50
contains the dates and ...
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until it's zero ... or close enough ... to within 0.000001 percent.
>Then why are you describing what Newton does? Why not just use XIRR?
That's so you understand what's going on. Aren't you interested?
>Not really.
Besides, XIRR may not be able to find an annualized return and you should know why. For example, if your investments are very volatile ...
>Like investing in dot.coms?
Exactly. Suppose, for example, you invest $10K and one year later withdraw $10K and a year later withdraw another $10K then, a year later, invest $10K and, a year later you find that your portfolio is worth just $1K, then there are three possible annualized returns.
>How's that again??
It goes like this:
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The pertinent equation is: f(R) = 10,000(1+R)4 - 10,000(1+R)3 - 10,000(1+R)2 + 10,000(1+R) - 1,000 = 0
where R is the annualized rate of Return >Picture? |
**
It's a quartic so it's got four-count-'em-four solutions 
...as pointed out by R.R.R. 
|
 Fig. 5b |
>Are those numbers, I mean +$10K and - $10K and so on,
are they reasonable?? >I invested in one of those stocks recently. It was called ...
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>So Newton always gives an answer, right?
Well ... uh ... not always. It depends upon that first guess:
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>I take it all these numbers are your inventions? I mean, they're not actual ...
We invest $100 each month, for ten years (Jan 1/90 to Dec 31/99), in the S&P 500.
Our portfolio grows to P = $31,077.26 ... like so
The annualized gain for the S&P, over this ten-year period, was 15.5% but we've been investing
$100/month so didn't have much invested in those earlier years when the S&P wasn't growing
dramatically, but in the later years ... aaah ...
|  Fig. 4 |
where ... uh ... we're going to find the MONTHLY, not the ANNUALIZED, gain so our time periods are in MONTHS..
>What's the MONTHLY version of ANNUALIZED? Is it MONTHLY-IZED or maybe ...
Pay attention. We need to compute:
- ΣAn = 100 + 100 + ... + 100 = 120*100 = 12,000
- ΣAnTn = 100(120) + 100(119) + ... + 100(1) = 100 (120+119+118+...+2+1) = 100(120)(121)/2 = 726,000
using a magic formula: 1+2+3+...+n = n(n+1) / 2 - RLinear = {
P - Σ An}
/ ΣAnTn = (31,077.26 - 12,000)/726,000 = 0.02628 or 2.628% MONTHLY.
which makes an ANNUALIZED return of 1.0262812 - 1 = 0.365 or 36.5% and that's our Linear Approximation - to begin our Newton Iterations. - From Eq. 2 we get: F(x) = A1xT1 +
A2xT2 +
A3xT3 + ...
+ ANxTN
- P = 100(x120+x119+ ...
+x2+x) - 31,077.26
which makes exactly 120 monthly contributions, the last $100 being invested for a single month. - Use another magic formula, namely:
x120+x119+ ...
+x2+x = x(x120-1)/(x-1) so
F(x) = 100x(x120-1)/(x-1) - 31,077.26 - Further:
F'(x) = d/dx{ 100(x120+x119+ ... +x2+x) - 31,077.26} = 100(120x119+119x118+ ... +2x+1)
and there's a magic formula for this - but you won't like it. - Finally, our Newton Iteration formulas are:
xn+1 = xn - F(xn)/F'(xn) with n = 0, 1, 2, ...
x0 = 1.02628 x1 = 1.01890 x2 = 1.01491 x3 = 1.01400 x4 = 1.01400 x5 = 1.01400 etc. etc.
The corresponding ANNUALIZED return is then: 1.0140012 - 1 = 0.181 or 18.1%
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>Dollar Cost Averaging works, right? We got a larger return than the 15.5% annualized S&P return, right? For example, here's our portfolio if the monthly S&P returns occurred in the reverse order, with the Dec/99 return first, Nov/99 second ... and the Jan/90 last. With the better five years coming first, the annualized return is now only 13.6%, for our investments. |  Fig. 5 |
The annualized return for the S&P500 would remain the same, 15.5%, since we'd multiply the monthly gain factors in the reverse order, but get the same final value for the product.
 Fig. 6 |
>How about a picture.
Here are the three curves: f(R),
fQ(R)
and fL(R)
which we considered earlier The places where each crosses the horizontal R-axis are:
18.1% (Exact) that's the RED dot
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Sorry. Just remember that we were first working on the MONTHLY returns (because our investments were monthly), and we put
x=1+R for convenience, then, finally, switched to ANNUALIZED - that's 12-months worth of MONTHLY returns - so we use
(1+Rmonthly)12=1+Rannual or (xmonthly)12=xannual to switch back and forth
... and, of course, we multiply by 100 anytime we want a percentage. Got it?
>ZZZZZzzzz
for Part III.
