Hi,
I will explain.. First this deals with Sharpes Single Index Model which is similar to typical regression line equation with R(m) = Returns from market, being an independent variable, dependent variable R(p) = Returns from portfolio.
R(p) = alpha(p) + Beta(p) * R(m) + error term(p)
We can find the E[R(p) | R(m)] based on the above but your question deals only with the Variance part. So we convert the equation accordingly.
Var(R(p)) = Var(B(p) * R(m)) + Var(e(p))
(Remember Variance of constant variable is zero . Therefore Variance of alpha is zero.)
Var(R(p)) =
B(p)^2 * Var(R(m)) +
Var(e(p))
Total Risk =
Systematic Risk +
Unsystematic Risk/Diversification Risk
B(p)^2*Var(R(m)) is always prevalent irrespective of size because the terms are from Beta of portfolio and the Market Variance.
Var(e(p)) is really firm specific and also Var(e(p)) being independent has a expected mean of zero, the Variance of this term decreases as number of assets increases - law of averages (i.e. Diversification reduces this component)
Now n being total assets in portfolio p. The above assertion can be proved if we have a equally weighted portfolio and therefore each asset will be (
1/n) of total assets and i = 1,2,3.......n
Therefore
summation operator [Var(1/n*e(i))] = summation operator [
(1/n)^2 * Var(e(i))]
Summation operator in this case means "
Sum the values of Var(e(i)), starting at e(1) and ending with e(n)."
We can then say 1/n * 1/n = (1/n)^2 and we can bring 1/n outside the summation term and make the formula more elegant.
=
(1/n) * {summation operator [
(1/n) * Var(e(i))]}
Note that [ summation operator
(1/n) * Var(e(i)] is nothing but
average of sum of variance of error term i.e (var(e(i)) with a dash above: see below) ]
Notice as n increases to infinity the variance of error terms becomes negligible.. In Calculus terms this is a horizontal asymptote and this is how a graph looks..
Hope its clear.
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