If you just want to calculate an Annualized return (without all the bumpf
which follows), go 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).
If our annualized Rate of Return is
Adding the current value of all five investments gives the current value of our portfolio,
namely $6,523.33, so we must have: Now, the big question:
In fact, it's 0.09, meaning our annualized return was 9%.
1000(1.09)
>Yeah, but what if we didn't know
We start by staring at Eq.(1) and noting that, if
R)^{n} by 1+nR (for n=5, 4, 3, 2 and 1) and get:
(2) 1000(1+5 which we can rewrite as
(3) 5000+15,000
and solve for >Instead of 9%. That's not bad, but it must depend upon
the numbers you used, like $1000 and years of 5, 4, 3 ... - If the amounts invested were A
_{1}and A_{2}*etc.*(instead of always $1,000) - and the number of years each was invested was T
_{1}and T_{2}*etc.*(instead of 5, 4, 3, 2 and 1) - and the current value of our portfolio is P (instead of 6,523.33),
- and we'd invested for N years (instead of just five years)
(4) A
The
(5) A and, collecting terms as we did in Eq.(3), we get:
(6) (A
and, the
where we're using the notation
Σ u
_{1}, A_{2}, etc.
can also be negative numbers. If we withdrew $1,000 from our portfolio 3 years ago,
then the corresponding term, namely 1,000(1+R)^{3}, will be money we DIDN'T make
over the past three years ... so we stick it in as a negative number (thereby subtracting from
our assets).
Now we can do better by replacing things like (1+.09) >What! >The light grey curve, in Fig. 1! Right?
(7) A which is our replacement for Eq.(5). Collecting terms, we can replace Eq.(6) by:
(8) ΣA
Fortunately, there is a formula for solving Eq.(8) and it gives our
Since the graph corresponding to Eq.(8) is quadratic (like the
> ZZZ...ZZZ...ZZZ In order to deter inevitable tedium, we'll generate some pictures, plotting
_{1}(1+R)^{5} +
A_{2}(1+R)^{4} +
A_{3}(1+R)^{3} +
A_{4}(1+R)^{2} +
A_{5}(1+R)
- P
to see what
Here, the investments A We'll also choose the value of our current portolfio,
The
Uh ... that's my notation for $6K invested 4.0 years ago followed by $3K withdrawn 3.0
years ago (hence the negative value) followed by $1K invested 2.0 year ago followed by a withdrawal of ...>Okay, I get it. ... and the current portfolio, P, is $2020.
>So, if we can't use the quadratic approximation, I take it we're
destined to use that lousy linear approximation. |