Case 004Portfolio constructionWarm up

A fund holds n stocks equally weighted, each with 30% volatility and pairwise correlation 0.25. What is portfolio volatility for n = 1, 10 and 50, and in the limit, and what does that mean for adding more names?

The situation

Neelgagan Pension Trust runs an equity sleeve of equally weighted stocks. For planning, the team treats every stock alike: 30% annual volatility and a correlation of 0.25 between any two of them. The trustees ask whether going from 10 names to 50 would make the sleeve much safer.

Your task

Compute the sleeve's volatility for 1, 10 and 50 stocks and in the limit, and advise on the value of adding names.

Quick check

With a very large number of stocks, what does volatility approach?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

About 30% for one stock, 17.1% for ten, 15.4% for fifty and 15% in the limit. Equal weights split variance into a stock-specific part that shrinks with 1 over n and a shared part fixed at the correlation times one stock's variance. Going from 10 to 50 names cuts volatility by under two points; the remaining risk is the market's and needs a hedge or other assets, not more stocks.

Step 1How does variance split when you add names?

Picture ten shops in one mall. Each has its own bad days, a leaking roof or a sick owner, and those cancel out across the mall. But when footfall to the whole mall drops, every shop suffers together. With equal weights, portfolio variance is one stock's variance times the correlation, plus the rest of that variance divided by n: the first term is the mall, the second is the shops. Only the second term shrinks as you add names.

The relationship

\sigma_p^2 = \sigma^2\left[\rho + \frac{1-\rho}{n}\right] \qquad \sigma_{10} = 30\%\sqrt{0.25 + 0.075} \approx 17.1\%

\sigma	volatility of each stock, 30%
\rho	correlation between any two stocks, 0.25
n	number of equally weighted stocks

What it says in wordsPortfolio variance is a floor set by the correlation plus a stock-specific part that falls in proportion to 1 over n.

Stocks	Specific variance term	Shared term	Volatility
1	675.0	225.0	30.00%
10	67.5	225.0	17.10%
50	13.5	225.0	15.44%
Limit	0.0	225.0	15.00%

Variance in percent squared: the specific term falls from 675 to 67.5 to 13.5 as the sleeve goes from 1 to 10 to 50 stocks, while the shared 225 never moves, so volatility falls from 30% to 17.1% to 15.4% and stops at 15%.

Step 2Is going from 10 names to 50 worth it?

Measure the gain on each step. The first nine additions take volatility from 30% to 17.1%, nearly 13 points; the next forty take it only to 15.4%, about 1.7 points more. At ten names, only 23% of the remaining variance is still diversifiable. The trustees would be paying for forty more names to research and monitor in exchange for a small cut in risk.

With 30% volatility per stock and correlation 0.25, portfolio volatility falls steeply to 17.1% at ten names and then crawls to 15.4% at fifty, flattening towards a floor of 15% that is the market risk every stock shares.

Step 3What would actually reduce the risk that remains?

Attack the shared term directly. The 15% floor is systematic riskThe part of a stock return driven by factors common to all stocks, such as the market, which cannot be removed by holding more stocks., and only something with a lower correlation to these stocks moves it: bonds or other asset classes, stocks from markets with a lower correlation, or a hedge with index futures. Adding a 51st stock with the same 0.25 correlation does nothing to it.

Say the limit of the model. Real stocks are not identical: some have 50% volatility, correlations cluster by sector, and in a sell-off correlations rise, so the floor itself rises exactly when it matters. A sleeve of 50 stocks that are all banks behaves more like a sleeve of five. Count the independent bets, not the names.

Where candidates lose it

The classic error is saying volatility falls towards zero as n grows. That is only true if the stocks are uncorrelated, and with a correlation of 0.25 half of each stock's volatility can never be diversified away.

The second is dividing volatility, not variance, by n, which gives 30% over the square root of 10 for ten stocks, 9.5%, and suggests diversification is far more powerful than it is.

What the interviewer asks next

In a crisis the correlation rises to 0.6. What is the floor now?
How many stocks get you within 1 point of the floor?
Would you rather hold 10 stocks at correlation 0.1 or 50 at correlation 0.25?

Company names and figures are illustrative.