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 start our investment portfolio with $P₀ and

• make investments of A₁, A₂, A₃, ... A_N
• for time periods, in MONTHS, of T₁, T₂, T₃, ... T_N and
• our current portolio is $P and
• we wish to calculate a PERSONAL Monthly Rate of Return at the end of T months.

>Personal?
A monthly return that reflects my own strategies, market timing, my investments and withdrawals, my ...

>Okay, I get the idea.

This MONTHLY return, R, must satisfy:

(1)       f(R) = P₀(1+R)^T + A₁(1+R)^T₁ + A₂(1+R)^T₂ + A₃(1+R)^T₃ + ... + A_N(1+R)^T_N - P = 0

where the original amount, P₀, has been invested for T months and the first investment, A₁, for T₁ months and the next investment, A₂, for T₂ months and the next investment ...

>Okay, I get the idea ... and I assume that an "A" is negative if that amount is withdrawn.
Yes.

We have a linear approximation for R, namely:

(2)       R_Linear = { P - P₀ -Σ A_n} / {P₀ + ΣA_nT_n }

obtained from (1) by replacing things like (1+R)^T by the linear approximation 1+TR.

To keep things neat, we'll relabel (1+R), calling it x, so we must solve, in place of (1):

(3)       F(x) = P₀x^T + A₁x^T₁ + A₂x^T₂ + A₃x^T₃ + ... + A_Nx^T_N - P = P₀x^T + ΣA_nx^T_n - P = 0

where our linear approximation, a la Equation (2), now looks like:

(4)       x₀ = 1 + R_Linear = 1 + { P - P₀ -Σ A_n} / {P₀ + ΣA_nT_n }

Now we do a single Newton Iteration, starting with x₀ and we get - voila!:

(5)

>What! I prefer Modified Dietz!
Wait just a minute ... we'll make it better.

>What about pictures? I like ...
Patience.
First, we notice that the starting value, our first approximation, namely x₀, can be Linear or Dietz formula or anything else.

>It looks real bad!
We need a little ritual. Here's an example; a picture of a spreadsheet which does the calculations:

This Exact and Newton annualized returns are independent of the values of the portfolio after each transaction. However, if we invent portfolio values - and that's what the spreadsheet does - we can compute a Return rate for each of the ten transactions and get an annualized Modified-Dietz-type Return. Would you like to play with the spreadsheet? The numbers are generated randomly and every F9 re-calculation gives a new set and it's great fun and ...

>zzz ZZZ

Of course, you could just use XIRR, right?

>zzz ZZZ