Case 071Portfolio construction and sizingCore

A pod holds three longs of 4% of NAV each, Kalinda Paints, Rasvi Tyres and Orbel Chemicals, all helped by falling crude, with pairwise correlation of 0.8 and 30% volatility each. How many independent bets is that really, and what is the combined risk?

The situation

A consumer and materials pod on a multi-manager platform holds three long positions, each 4% of NAV: Kalinda Paints, Rasvi Tyres and Orbel Chemicals. The PM pitched them as three separate stock ideas. Each benefits from falling crude oil, which is their largest raw material cost.

Each stock has annualised volatility of 30%, and over the past two years their daily returns have had pairwise correlations of about 0.8. The risk manager asks how diversified the three really are.

Your task

What is the combined risk of the three positions as a share of NAV, how many independent bets is that equivalent to, and what should the PM do about it?

Quick check

Each position carries 1.2% of NAV in risk. What is the combined risk of the three at 0.8 correlation?

Worked solution

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

30-second answerThe answer to give first

The three positions carry about 3.35% of NAV in risk, close to the 3.6% of a single 12% position, and are worth about 1.15 independent bets. Independent, they would carry 2.08%. At 0.8 correlation most of each stock's movement is the same crude factor, so the book is one crude call in three names. Size it as one position, or hedge the crude exposure and keep the stock-specific part, about 0.93% of NAV.

Step 1How do three correlated positions add up?

Not by the square-root rule that works for independent bets. Three friends who each carry an umbrella feel diversified until you notice they all live on the same street: when it rains, it rains on all three. Each position is 4% of NAV at 30% volatility, 1.2% of NAV in risk; with a 0.8 correlation between every pair, combined variance is 3 x 1.2 squared plus six cross terms of 0.8 x 1.2 squared, and combined risk is about 3.35% of NAV. Independent, the same three would carry 2.08%; one 12% position in a single stock would carry 3.6%. The three names are much closer to the second.

The relationship
σp=3(1.2)2+6(0.8)(1.2)2=11.232=3.35%Neff=31+2(0.8)=1.15\sigma_p = \sqrt{3(1.2)^2 + 6(0.8)(1.2)^2} = \sqrt{11.232} = 3.35\% \qquad N_{eff} = \frac{3}{1 + 2(0.8)} = 1.15
1.2each position's risk, % of NAV: 4% x 30%
0.8pairwise correlation
N_effnumber of independent bets the three positions are equivalent to
What it says in wordsThree positions at 0.8 correlation carry 3.35% of NAV risk and behave like 1.15 independent bets.
Three positions, 1.2% of NAV risk each: how they add upIf the three were independentKalinda Paints1.2%Rasvi Tyres1.2%Orbel Chemicals1.2%Combined risk2.08%one 12% bet: 3.6%Equivalent to 3.00 independent betsrisks partly cancelAt 0.8 correlation, all tied to crudeKalinda Paints1.2%Rasvi Tyres1.2%Orbel Chemicals1.2%Combined risk3.35%one 12% bet: 3.6%Equivalent to 1.15 independent betsrisks mostly add
Three 4% positions at 30% volatility would carry 2.08% of NAV risk if independent, but at 0.8 correlation they carry 3.35%, close to a single 12% position's 3.6%, and count as only 1.15 independent bets.
Step 2What does the effective number of bets tell the PM?

How much diversification is real. With three equal positions and an average correlation of 0.8, the effective number of betsHow many independent positions a set of correlated positions is worth, for equal sizes: the number of positions divided by one plus the number less one times the average correlation. is 3 / (1 + 2 x 0.8), about 1.15. The curve falls fast: at 0.3 correlation three names are worth 1.88 bets, at 0.5 only 1.50. The PM has one idea, crude falls and input costs drop, expressed three times, and one idea should be sized as one idea.

Three names, how many real bets? It falls fast as correlation rises11.522.530.00.20.40.60.81.0Average correlation between the three positions0.8: 1.15 betsindependent: 3 bets
Three equal positions count as three independent bets at zero correlation, 1.50 at 0.5 and only 1.15 at 0.8, so the pod's three crude-sensitive names are close to a single bet.
Step 3What should the PM do?

Choose which bet to keep. If the view is on crude, size the three together as one position within the pod's single-name limit, perhaps 6% to 8% of NAV in total rather than 12%. If the view is on the three companies, hedge the crude factor with a short in a business that gains from high crude, such as an invented upstream producer, or with crude futures. With 80% of each stock's variance coming from the shared factor, the stock-specific volatility left is about 13.4%, and the three hedged positions would carry about 0.93% of NAV, genuinely diversified, stock-specific risk.

Close with the limit. Correlations measured over two calm years understate how tightly these names move together in a crude spike, when correlations tend to rise. The risk manager's number is a floor, not a ceiling, which is one more reason to size the trio as a single position.

Where candidates lose it

The frequent loss is applying the square-root rule, which assumes independence, and reporting 2.1% of NAV. At 0.8 correlation that understates the risk by more than a third.

The second is treating three tickers as three ideas. The interviewer is testing whether you see that a shared driver makes the book less diversified than its name count suggests.

What the interviewer asks next

  • Add a fourth crude-sensitive name at 4%. What is the combined risk now, and how many bets?
  • How would you estimate how much of each stock's variance is crude?
  • The platform caps any single factor at 2% of NAV risk. How much can the pod hold?

Company names and figures are illustrative.