Case 045Portfolio construction and sizingHard

Oruvel Capital has five ideas with expected alphas of 6%, 8%, 4%, 10% and 5% and volatilities of 25%, 40%, 20%, 50% and 30%. Size each in proportion to alpha divided by variance, with a cap of 8% of NAV. What is the ranking and where does the cap bind?

The situation

Oruvel Capital runs a concentrated long book. The analysts have five ideas, A to E, with expected alphas over the next year of 6%, 8%, 4%, 10% and 5%, and volatilities of 25%, 40%, 20%, 50% and 30%.

The portfolio manager sizes each position as alpha divided by variance, scaled by a risk-aversion factor of 10, so the weight is alpha / (10 x volatility squared). No position may exceed 8% of NAV. Treat the five ideas as uncorrelated for this exercise.

Your task

Rank the ideas by size, find where the cap binds, and explain why the idea with the highest expected alpha ends up among the smallest positions. What does the cap cost, and why have it?

Quick check

Which idea gets the smallest position?

Worked solution

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

30-second answerThe answer to give first

The ranking by alpha over variance is C, A, E, B, D: the cap binds on C and A, at 8% each, while D, with the highest alpha, gets the smallest position at 4%. Variance squares volatility, so doubling volatility quarters the size while doubling alpha only doubles it. The cap trims gross from 34.2% to 30.6% of NAV and expected alpha from 2.05% to 1.88%, a small price for protection against a low-volatility name gapping.

Step 1Why size by alpha over variance?

A driver can take a straight, well-lit road faster than a winding one even if both lead somewhere good; how fast is safe depends on how much the road can surprise you. Sizing in proportion to alpha divided by variance gives each idea a position where the extra return it adds is balanced against the risk it adds, and it gives every idea the same return per unit of the risk it contributes. With a scale of 10, the weight is alpha / (10 x volatility squared). Idea C, 4% alpha at 20% volatility, scores 0.04 / 0.04 = 1.0 and would be 10% of NAV; idea D, 10% alpha at 50% volatility, scores 0.10 / 0.25 = 0.4, a 4% position.

The relationship
wi=αiλ σi2wC=0.0410×0.202=10%wD=0.1010×0.502=4%w_i = \frac{\alpha_i}{\lambda\,\sigma_i^2} \qquad w_C = \frac{0.04}{10 \times 0.20^2} = 10\% \qquad w_D = \frac{0.10}{10 \times 0.50^2} = 4\%
alpha_ithe expected return of idea i above what the market would give
sigma_ithe idea's volatility, so sigma squared is its variance
lambdathe risk-aversion scale, 10 here, which sets the overall size of the book
What it says in wordsEach position is its expected alpha divided by its variance, scaled so the whole book carries the risk the manager wants.
Step 2What are the sizes, and where does the cap bind?
IdeaAlphaVolatilityAlpha / varianceRaw sizeCapped size
C4%20%1.0010.0%8.0%
A6%25%0.969.6%8.0%
E5%30%0.565.6%5.6%
B8%40%0.505.0%5.0%
D10%50%0.404.0%4.0%
Book34.2%30.6%
Ranked by alpha over variance the order is C, A, E, B, D; C and A exceed the 8% cap at 10.0% and 9.6% and are cut to 8%, and D, the highest-alpha idea, is the smallest position at 4.0%.
Alpha over variance: the highest-alpha idea ends up the smallest2%4%6%8%10%cap 8% of NAVraw 9.68.0%Idea Aalpha 6%, vol 25%5.0%Idea Balpha 8%, vol 40%raw 10.08.0%Idea Calpha 4%, vol 20%4.0%Idea Dalpha 10%, vol 50%5.6%Idea Ealpha 5%, vol 30%Expected alphaon NAVraw 2.05%capped 1.88%gross 30.6%of NAV after cap
Raw sizes of 9.6%, 5.0%, 10.0%, 4.0% and 5.6% for A to E become 8.0%, 5.0%, 8.0%, 4.0% and 5.6% under the 8% cap, cutting expected alpha on NAV from 2.05% to 1.88%.

The cap binds on the two lowest-volatility ideas, C and A, because a low variance in the denominator inflates their size. The highest-alpha idea, D, is the smallest position because its volatility is squared: doubling volatility cuts the size to a quarter, while doubling alpha only doubles it. Look at the risk each position carries, size times volatility: A contributes 8% x 25% = 2.0 points, D 4% x 50% = 2.0 points. The method is giving the two ideas the same risk, not the same capital.

Size follows alpha over variance: volatility is squared, alpha is not4%8%12%10%20%30%40%50%size 5%size 8%: the capA: 8.0%B: 5.0%C: 8.0%D: 4.0%E: 5.6%Volatility, a yearExpected alpha, a year
Placed by volatility and alpha, ideas A and C sit above the curve of 8% positions, so the cap binds on them, while D's 10% alpha at 50% volatility sits below the 5% curve, because the size needed for a given alpha rises with the square of volatility.
Step 3What does the cap cost, and why have it?

The cap removes 3.6% of NAV from A and C and cuts expected alpha from 2.05% of NAV to 1.88%, about 0.18 points. That is cheap insurance, because the formula trusts volatility estimates most exactly where they are least reliable: a stock that has moved 20% a year can fall 30% in a day on a fraud allegation or a lost licence. Low-volatility names are also often the most crowded and the hardest to exit in size. The cap also limits what happens if the alpha estimate itself is wrong, which the formula takes as given.

Two limits to say out loud. The ideas were treated as uncorrelated; if A and C are both, say, consumer staples that fall together, the true risk of the capped book is higher than the sizes suggest, and correlation belongs in the calculation. And alpha estimates are noisy: a formula that divides a guess by a small number magnifies the guess. Many managers therefore shrink every alpha towards the average before sizing, which pulls the extreme positions in and makes the cap bind less often.

Where candidates lose it

The instinctive answer ranks by alpha and gives D the biggest position. Alpha over variance does the opposite, and the interviewer wants to see you run the numbers rather than trust the headline return.

The second is dividing by volatility instead of variance. Alpha over volatility gives the risk-adjusted ranking, a different ordering with D level with B and C, and the sizes that follow are wrong by the square of the volatility.

What the interviewer asks next

  • Size by alpha over volatility instead. How does the ranking change?
  • A and C have a correlation of 0.8. What happens to their sizes?
  • Where would you redeploy the 3.6% of NAV the cap frees up, if anywhere?
  • How would you shrink the alpha estimates before sizing?

Company names and figures are illustrative.