Hello everyone,
the book states: The contribution of each individual asset’s variance to the portfolio’s volatility decreases as the number of assets in the equally weighted portfolio increases,
the book kinda explains, but i do not fully get it. Personally i feel the concept a little abstract
Does anyone know why?
Thanks in advance
Suppose you have a portfolio of stocks with returns R_1, R_2, …, R_n, each with weights x_1, x_2, …, x_n respectively. The portfolio is the sum of the returns multiplied by their respective weights.
R_{portfolio} = x_1R_1 + x_2R_2 + ... + x_nR_n
From probability theory, we know that
Var(R_{portfolio}) = Var(x_1R_1 + x_2R_2 + ... + x_nR_n) =
Var(x_1R_1) + Var(x_2R_2) + ... + Var(x_nR_n) +
Cov(x_1R_1, x_2R_2) + ... + Cov(x_1R_1, x_nR_n) +
… + Cov(x_2R_2, x_1R_1) + ... + Cov(x_2R_2, x_nR_n) +
… + Cov(x_{n-1}R_{n-1}, x_nR_n)
=
x_1^2Var(R_1) + x_2^2Var(R_2) + ... + x_n^2Var(R_n) +
x_1x_2Cov(R_1, R_2) + ... + x_1x_nCov(R_1, R_n) +
… + x_2x_1Cov (R_2, R_1) + ... + x_2x_nCov(R_2, R_n) +
… + x_{n-1}x_nCov(R_{n-1}, R_n)
For an equally-weighted portfolio, x_1 = x_2 = ... = x_n = 1/n, and so above becomes
(1/n)^2Var(R_1) + (1/n)^2Var(R_2) + ... + (1/n)^2Var(R_n) +
(1/n)^2Cov(R_1, R_2) + ... + (1/n)^2Cov(R_1, R_n) + ... +
(1/n)^2Cov(R_2, R_1) + ... + (1/n)^2Cov(R_2, R_n) +
… + (1/n)^2Cov(R_{n-1}, R_n)
= (1/n)^2(Var(R_1) + Var(R_2) + ... + Var(R_n)) +
(1/n)^2(Cov(R_1, R_2) + ... + Cov(R_{n-1}, R_n))
Now here is the trick to the formula (according to a Corporate Finance book, i wish i was clever enough to find this on my own)
There are a total of n^2 terms, n of which are variance terms and the rest (n^2 - n) are covariance terms. We therefore have:
n(1/n)^2((Var(R_1) + Var(R_2) + ... + Var(R_n))/n) +
(n^2 - n)(1/n)^2((Cov(R_1, R_2) + ... + Cov(R_{n-1}, R_n))/(n^2 -n)
(we are dividing the first term by n because there are n variance terms, and the second term by n^2 - n because there are these many covariance terms. also, don’t forget to multiply by the same amount so that the number doesn’t change)
Because ((Var(R_1) + Var(R_2) + ... + Var(R_n))/n is just the average variance of individual stocks, and ((Cov(R_1, R_2) + ... + Cov(R_{n-1}, R_n))/(n^2 -n) is just the average covariance between any pair of different stocks, the formula becomes
n(1/n)^2(Average variance of individual stocks) + (n^2 - n)(1/n)^2(Average covariance of any pair of different stocks)
n(1/n)^2 = 1/n, and (n^2 - n)(1/n)^2 = 1 - 1/n
so we finally arrive at the famous formula for an equally-weighted portfolio:
Var(R_{equallyweightedportfolio}) = (1/n)(Average variance of individual stocks) + (1 - 1/n)(Average covariance of any pair of different stocks)
If n increases to infinity, the first term goes to zero and the second term approaches 1. That is, as n increases, the contribution of individual stock variance decreases.
edit: fixing Mathjax
edit: for some reason the mathjax cuts off when the formula is too long…
edit: i’m trying to make it show all the formula cutoff by mathjax, but mathjax is turning it into bullet points and i don’t know why.