Case 009Portfolio construction and sizingCore

A Nesavu Capital pod has three ideas with volatilities of 20%, 35% and 50%. The PM wants each to contribute the same risk, assuming low correlation between them. How should capital be split, and how does that differ from equal weights?

The situation

A portfolio manager at Nesavu Capital, a multi-manager platform, runs a small pod with three positions she likes equally. Idea A, a staples pair trade, has annualised volatility of 20%. Idea B, a mid cap industrial, has 35%. Idea C, a small cap biotech, has 50%. The three have little to do with each other, so treat their correlations as zero.

She has been running them with equal capital. The platform's risk team says her book's risk is dominated by one name and asks her to size so that each idea contributes the same risk.

Your task

How should the capital be split so each idea contributes equal risk, and what was wrong with equal weights?

Quick check

With equal capital in each, roughly what share of the book's risk comes from the 50% volatility idea?

Worked solution

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

30-second answerThe answer to give first

Weight each idea by one over its volatility: about 51%, 29% and 20% of capital. Each position then carries the same risk, weight times volatility of about 10.1%. With equal weights, the 50% volatility idea carried about 61% of the book's risk and the calm one only 10%, so one bet decided the pod's month.

Step 1Why is equal money not equal risk?

Picture three friends sharing a taxi bill equally when one lives across the street and another at the airport. The money is equal; the distance each one travels is not. A position's risk is its capital times its volatility, so Rs 1 in a 50% volatility biotech is two and a half times as risky as Rs 1 in a 20% volatility staples pair. With zero correlation the variances add, and the squares make the gap wider still: at equal weights idea C carries 61% of the risk, idea B 30% and idea A only 10%.

Step 2How do you set the weights so each idea carries the same risk?

Make weight times volatility the same for every idea, which means weight proportional to one over volatility. One over 0.20, 0.35 and 0.50 gives 5.0, 2.86 and 2.0; divided by their total of 9.86, the weights are 50.7%, 29.0% and 20.3%. Each idea now carries about 10.1 points of volatility and a third of the risk. This is inverse-volatility weightingSizing each position in proportion to one over its volatility, so that with low correlations each contributes roughly the same risk. The simplest form of risk parity., and the calm idea gets the most capital.

The relationship
wi=1/σi∑j1/σj5.09.86=50.7%, 2.869.86=29.0%, 2.09.86=20.3%w_i = \frac{1/\sigma_i}{\sum_j 1/\sigma_j} \qquad \frac{5.0}{9.86} = 50.7\%,\ \frac{2.86}{9.86} = 29.0\%,\ \frac{2.0}{9.86} = 20.3\%
w_ithe share of capital in idea i
sigma_ithe annualised volatility of idea i: 20%, 35% and 50%
What it says in wordsEach weight is one over that idea's volatility, scaled so the weights sum to one.
Equal money is not equal risk: the wild idea dominates an equal-weight bookCapitalIdea Avol 20%33%51%Idea Bvol 35%33%29%Idea Cvol 50%33%20%equal weightsinverse volatilityShare of portfolio riskIdea Avol 20%10%33%Idea Bvol 35%30%33%Idea Cvol 50%61%33%equal weightsinverse volatility
Equal capital puts a third in each idea but leaves the 50% volatility idea carrying 61% of the risk; inverse-volatility weights of 51%, 29% and 20% give each idea one third of the risk.
Step 3What does the book look like after the change, and where does the method stop working?

At equal weights the three ideas together had volatility of about 21.4%; at inverse-volatility weights it falls to about 17.6% for the same capital. The PM can now scale the whole book up to her risk budget, and every idea will earn its keep in proportion to how often it is right rather than how wild it is. The method has two limits worth saying. It assumes low correlation; if A and B both depend on the same factor, their combined risk is understated and a full covariance model is needed. And it ignores conviction: if the biotech has twice the expected return per unit of risk, it deserves more than a third of the risk budget.

Where candidates lose it

Most candidates say equal weights spread the risk evenly, which is the error the question is built to expose. Capital and risk are different currencies, and the conversion rate between them is volatility.

The second trap is weighting by one over variance instead of one over volatility. That over-corrects: it gives the calm idea far more capital than equal risk requires, because variance already squares the volatility.

What the interviewer asks next

  • Ideas A and B turn out to have a correlation of 0.6. Which way do the weights move?
  • The pod's risk budget is 8% annual volatility. How much gross capital can she run?
  • How would you combine inverse-volatility sizing with a conviction score?

Company names and figures are illustrative.