diversification benefit: number of stocks and correlation logic

well i thougt the same as you magician, but the curriculum said something like this:
assuming stocks have the same standard deviation of returns:
σ2 =σ2((1−ρ)/n+ρ)
If the portfolio contains one stock, the portfolio variance is σ2. As n increases, port-
folio variance drops rapidly. In our example, if the portfolio contains 15 stocks, the
portfolio variance is 0.347σ2, or only 34.7 percent of the variance of a portfolio with
one stock. With 30 stocks, the portfolio variance is 32.3 percent of the variance of a
single-stock portfolio. The smallest possible portfolio variance in this case is 30 percent
(6)
of the variance of a single stock, because σ2 = 0.30 σ2 when n is extremely large. With p
only 30 stocks, for example, the portfolio variance is only approximately 8 percent larger than minimum possible value (0.323σ2/0.30σ2 − 1 = 0.077), and the variance is 67.7 percent smaller than the variance of a portfolio that contains only one stock.
For a reasonable assumed value of correlation, the previous example shows that a portfolio composed of many stocks has far less total risk than a portfolio composed of only one stock. In this example, we can diversify away 70 percent of an individual stock’s risk by holding many stocks. Furthermore, we may be able to obtain a large part of the risk reduction benefits of diversification with a surprisingly small number of securities.
What if the correlation among stocks is higher than 0.30? Suppose an investor wanted to be sure that his portfolio variance was only 110 percent of the minimum possible portfolio variance of a diversified portfolio. How many stocks would the investor need? If the average correlation among stocks were 0.5, he would need only 10 stocks for the portfolio to have 110 percent of the minimum possible portfolio variance. With a higher correlation, the investor would need fewer stocks to obtain the same percentage of minimum possible portfolio variance. What if the correlation is lower than 0.30? If the correlation among stocks were 0.1, the investor would need 90 stocks in the portfolio to obtain 110 percent of the minimum possible portfolio variance”
 
You are confusing two different things : Speed of convergence and diversification benefits.
Smaller correlation means that we could potentially achieve better diversification but it also means it takes more stocks to get there. A crude way to think of this maybe is that, since there is more risk reduction to be gained, there is more way to go there so more stocks needed. If the maximum diversification benefit is small to begin with then we can get close with less stocks (we have less “distance” to cover).
 
panos.kollias

thanks for your comments…maybe i didn’t understand that paragraph above correctly. Your comments kind of lighten it up for me.. but really words by words, i understood it as the higher correlation, the more stocks needed to achieve better diversification.
 
jaychou wrote:
panos.kollias

thanks for your comments…maybe i didn’t understand that paragraph above correctly. Your comments kind of lighten it up for me.. but really words by words, i understood it as the higher correlation, the more stocks needed to achieve better diversification.
Click to expand...
Glad to be of help!
 
I think the idea is this:
If you have a low average correlation, you get huge diversification benefits, but it takes more stocks to get close to the maximum.
If you have a high average correlation, you get modest diversification benefits, but it takes fewer stocks to get close to the maximum.
So, for example, with a low average correlation, your diversification benefit might be a 50% drop in volatility (compared to having all correlations at +1.0), but you’ll need 75 stocks to get most of that 50% drop. With a high average correlation, your diversificaation benefit might be only a 10% drop in volatility, but you can most of that 10% with only 20 stocks.
 
S2000magician wrote:
I think the idea is this:
If you have a low average correlation, you get huge diversification benefits, but it takes more stocks to get close to the maximum.
If you have a high average correlation, you get modest diversification benefits, but it takes fewer stocks to get close to the maximum.
So, for example, with a low average correlation, your diversification benefit might be a 50% drop in volatility (compared to having all correlations at +1.0), but you’ll need 75 stocks to get most of that 50% drop. With a high average correlation, your diversificaation benefit might be only a 10% drop in volatility, but you can most of that 10% with only 20 stocks.
Click to expand...
mate , i think this is exactly what the bloody curriculum is trying to explain in that long paragraph that got me confused so much…but u summed it up nicely! Ok now my logic and the text have clicked togather. Thank you!!
 
jaychou wrote:
S2000magician wrote:I think the idea is this:
If you have a low average correlation, you get huge diversification benefits, but it takes more stocks to get close to the maximum.
If you have a high average correlation, you get modest diversification benefits, but it takes fewer stocks to get close to the maximum.
So, for example, with a low average correlation, your diversification benefit might be a 50% drop in volatility (compared to having all correlations at +1.0), but you’ll need 75 stocks to get most of that 50% drop. With a high average correlation, your diversificaation benefit might be only a 10% drop in volatility, but you can most of that 10% with only 20 stocks.
Click to expand...
mate , i think this is exactly what the bloody curriculum is trying to explain in that long paragraph that got me confused so much…but u summed it up nicely! Ok now my logic and the text have clicked togather. Thank you!!
Click to expand...
My pleasure.
Now, go back and reread your comments about stocks going up and going down, and my rant about correlations of prices vs correlations of returns. That’s something that’s really important (aside from this curriculum) but that people in finance get wrong all the time.
 
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