The Scenario:

We invest $1000 on April 1
>April Fool's Day?
... and another $1000 half-way through April,
then we fall asleep until next April
when we discover that our $2000 investment is worth: $2000.

While we slept, the market carried our April investments to $2500 by April's end.

In a month where there are no cash flows, meaning that every A_n = 0 for that month, then the
Dietz gain is given by the magic formula

and, if we end the year with P₁₂ = $2000 and end the first month with P₁ = $2500, we get just 2000/2500 = 4/5.

In that first month (with cash flows: $1000 at the start of the month and another $1000 half-way through the month) we get
r = (2500 - 0 -A)/(0+B),     the initial portfolio amount, before investments, was $0
where A is the sum of cash flows, namely A = $1000+$1000 = $2000, and
B is the sum of time-weighted cash flows, namely B = $1000(1) +$1000(1/2) = $1000+$500 = $1500.
(Note that the first $1000 was in for (1) month and the second was in for (1/2) month.)

We now have the Dietz gain for that first month, namely r = (2500 -2000)/(1500) = 1/3   so the gain factor is 1+1/3 = 4/3 .
Finally, multiplying all the gain factors together: (4/3)(4/5) = 16/15
and, annualized, we Dietz gets: (16/15) - 1 = 1/15 = .0667 or 6.67%

>So?
So you can stick in any portfolio values you like after that first month, as long as you end up with $2000 you'll get 6.67% Dietz.

>That's 6.67% for the whole year, right?
And Newton says 0% for the whole year, right?
Aah, but what does Newton say about that first month where Dietz screws up?
Good question. The Linear approximation for that first month is the same as Dietz, namely 1/3 = .333333 or a gain factor of 1.333333 and, continuing from there, Newton iterations are:

>But Dietz ain't so bad for that first month. So how come ...?
Actually, Dietz does a Linear Approximation to get his monthly gain, and since the monthly gain, R, satisfies an equation like

which we can write as:

which we can write as

which we can write as

which we can write as

so the monthly Dietz gain and the exact monthly gain (namely R) differ by a quantity, namely {f ''(c)/f '(0)}R²/2, which depends upon the second derivative of the function f, hence whether f is concave UP or concave DOWN ... and that depends upon ...

>zzz ZZZZ

Here's another interesting point:

Even if Dietz got exactly the right value for that first month, say gain factor = 1.341688, the Dietz methodology asks that we multiply together the monthly gain factors and that'd give: (1.341688)(4/5) = 1.0734 meaning a percentage gain of 7.34% so ...

>What! You said the exact annual gain was 0.0% and now you say ...
What I'm saying is that multiplying together the twelve monthly gain factors - that's called geometric linking - doesn't give the annualized gain ... except, of course, in the case where there are no ins and outs, no cash flows, no transaction activity, no ...

>Yeah, I get the idea, but that's wierd, eh? If my exact gain is 2% each month, so the gain factor each month is 1.02, you'd think my annualized gain factor would be (1.02)(1.02) ... (1.02) = 1.02¹² = 1.2682 and that means a 26.82% annualized gain. You said that before, in an earlier example! You even got that very same value, before!

Okay, here's the deal.

• Suppose we have exact gain factors of g₁, g₂, ... g₁₂ for each of twelve months.
• Suppose we invest $1.00 at the beginning of each month, for twelve months.
• The first $1.00 grows to g₁g₂...g₁₂ after twelve months.
• The second $1.00 grows to g₂g₃...g₁₂ after eleven months.
• The third $1.00 grows to g₃g₄...g₁₂ after ten months.
• ...
• The twelfth and last $1.00 grows to g₁₂ after one month.
• Our final portfolio is worth the sum of all these dollar values, namely: P = g₁g₂...g₁₂ + g₂g₃...g₁₂ + g₃g₄...g₁₂ + ... + g₁₂
• Let G_n be the gain factor over the first n months: G_n = g₁g₂...g_n
• We can now rewrite the expression for our portfolio value, P, namely:

  (6)         P = G12(1 + 1/G1 + 1/G2 + 1/G3 + ... + 1/G11)

• We note that the Dietz annual gain factor is G₁₂ and the Dietz annual gain is G₁₂ - 1 or, as a percentage, it's 100(G₁₂ - 1).

• Let the monthly gain be R (where, by R = 0.123, we'd mean 12.3%)
• Each $1.00 grows by a factor (1+R) each month.
• Let x = 1+R so each $1.00 grows by a factor x each month.
• The final portfolio value, after twelve months of investing ($1.00 each month) is: x¹² + x¹¹ + x¹⁰ + ... + x = P where the first $1.00 grows by a facor x¹², the second $1.00 grows by x¹¹, etc.
• We can rewrite this equation like so:

  (7)         x12(1 + 1/x + 1/x2 + 1/x3 + ... + 1/x11) = P

• We note that Newton's annual gain factor is x¹² and his annual gain is x¹² - 1 or, as a percentage, it's 100(x¹² - 1).

>Can you prove that?
Sure! Wanna see?
>No.

Alas, if the monthly gains were quite different (a Volatile market), then Newton and Dietz would NOT agree.

>Never?
Well, hardly ever.

Consider our earlier example, but now we invest $A at the start of each month (instead of just $1.00). If the end-of-month portfolios are P₁, P₂, etc., then, since the cash flow each month is A (which is in for the entire month) and the time-weighted cash flow is also A (because it's in for the entire month!), then the Dietz gain for month n (according to our magic formula) would be:

If x is the solution to F(x) = Ax12 + Ax11 + Ax10 + ... + Ax - P = 0 where the final portfolio value is fixed at some value P = P12 (and we start with a portfolio of P0 = 0), then is the following true regardless of the intermediate, end-of-month portfolio values: P1, P2, ... P11 ?

So, is the above equality true, for every choice of intermediate end-of-month portfolio values?

>I'd guess NO.
Is that your final answer?

>You haven't had nearly enough charts. A picture is worth a thousand ... Okay, here's a beautiful chart: I invest $1.00 at the start of each month, for twelve months. I assume an annualized return, R, from -20% to +20%. For each R, I compute the corresponding monthly gain factor, x=(1+R)1/12 and the final portfolio value: P = P12 using Equation (7). I then invent eleven random monthly gains, rn, which differ from the monthly gain, x-1, by no more than plus-or-minus 1%. I compute eleven corresponding, intermediate, end-of-month portfolio values: P1, P2, ...P11 I stare at the Dietz annualized gain (1+r1)(1+r2)...(1+r12) - 1 and the Newton annualized gain, R. I compute their difference ... using a spreadsheet (where each F9 re-calculation generates another set of random, intermediate portfolio values). I do this seven times and get seven sets of differences ... which I plot. Here's a typical chart:

>Why seven times?
Isn't seven a lucky number?

>Fine. Let me see if I understand this. In one example ... I'm looking at the yellow curve ... when you selected a 20% annualized return, so R = 0.20, then a random selection of eleven intermediate portfolio values, you got a Dietz annual return which was greater by more than 2.5%, meaning the Dietz return was more than 22.5%, right?
Right.

>Now I'm looking at the pinkish curve. You selected a 10% annualized return (so R = 0.10) and those random intermediate portfolio values and got a Dietz annual return which was smaller by more than 2%, meaning the Dietz return was less than 8%. Am I right?
zzz ZZZ

for Part IV.