Case 005Investment and portfolio riskCore

A family office holds half its money in equities and believes it is balanced. Measure how much of its risk each asset actually carries, and redesign the mix so no asset carries more than 60% of the risk.

The situation

Meruvan Family Office holds Rs 1,000 crore: 50% in Indian equities, 40% in high-grade bonds and 10% in gold. The risk team's assumptions for annual volatility are 18% for equities, 5% for bonds and 15% for gold. Correlations are 0.1 between equities and bonds, minus 0.1 between equities and gold, and 0.2 between bonds and gold.

The family's investment committee has adopted a rule that no single asset should contribute more than 60% of the portfolio's total volatility. These volatilities and correlations are the team's illustrative assumptions, not forecasts.

Your task

What share of the portfolio's risk does each asset contribute today, and what mix would meet the 60% rule?

Quick check

Roughly what share of Meruvan's total risk comes from its 50% equity allocation?

Worked solution

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

30-second answerThe answer to give first

Equities carry about 91% of Meruvan's risk on half its money; bonds carry 7% and gold 2%. Portfolio volatility is 9.45%. To bring equities under 60%, the equity weight has to fall to about a quarter: a 25/65/10 mix gives equities 57.5% of risk at 6.05% volatility. The committee must accept a much lower expected return, or relax the rule.

Step 1Why is a half-equity portfolio not half equity risk?

Picture a household with two earners: one a salaried clerk, the other a commission salesperson whose income swings wildly. They each bring home half the money on average, but almost every bad month comes from the salesperson. Risk share depends on how much an asset moves and how it moves with the rest, not on how much money sits in it. Equities at 18% volatility move more than three times as much as bonds at 5%.

Step 2How do you compute each asset's contribution?

First the portfolio volatility, from the weights, volatilities and correlations: 9.45%. Then each asset's marginal contributionHow much portfolio volatility rises for a small extra holding of one asset: its covariance with the whole portfolio divided by portfolio volatility.: its covariance with the whole portfolio divided by that volatility. For equities that is 17.23%, for bonds 1.69% and for gold 1.59%. Weight times marginal contribution gives each asset's share of risk, and the shares add to exactly 100%. Equities: 0.5 times 17.23% is 8.62 points of the 9.45, about 91%.

The relationship
RCi=wi (Σw)iw⊤Σw∑iRCi=1RC_i = \frac{w_i\,(\Sigma w)_i}{w^{\top}\Sigma w} \qquad \sum_i RC_i = 1
w_ithe asset's weight
(\Sigma w)_ithe asset's covariance with the whole portfolio
w^{\top}\Sigma wthe portfolio's variance
What it says in wordsEach asset's share of risk is its weight times how it moves with the portfolio, divided by the portfolio's total variance, and the shares always sum to one.
AssetWeightVolatilityMarginal contributionRisk share
Equities50%18%17.23%91.2%
Bonds40%5%1.69%7.2%
Gold10%15%1.59%1.7%
Portfolio100%9.45%100.0%
Equities are 50% of Meruvan's capital but 91.2% of its risk; bonds are 40% of the capital and 7.2% of the risk, and gold 10% and 1.7%.
Capital share against risk share: equity dominates until it is cut hardNow: 50 / 40 / 10vol 9.5%Equities50% capital91% riskBonds40% capital7% riskGold10% capital2% risk60% risk capRedesign: 25 / 65 / 10vol 6.0%Equities25% capital57% riskBonds65% capital36% riskGold10% capital7% risk60% risk capPale bar: share of the money. Dark bar: share of the portfolio's volatility.
At 50/40/10 equities carry 91% of Meruvan's risk; only a cut to 25/65/10 brings them to 57%, under the 60% cap, and portfolio volatility falls from 9.5% to 6.0%.
Step 3What mix meets the rule, and what does it cost?

Cut equities and let bonds take the money. At 25% equities, 65% bonds and 10% gold, equities carry 57.5% of risk, bonds 35.5% and gold 7.0%. Volatility falls from 9.45% to 6.05%. Notice how far equity had to move: halving the weight only brings the risk share down by about a third, because equities remain far more volatile than anything else in the portfolio.

Close with the judgement the committee needs. The 60% rule is really a decision to give up much of the portfolio's expected growth, and the family should make that choice knowingly. The limitation matters too: the equity to bond correlation of 0.1 is a calm-market number, and in a sharp sell-off it can rise, pushing equity's real share of risk higher still. Run the same numbers with a stressed correlation before signing off the new mix.

Where candidates lose it

Candidates read the 50/40/10 capital split as a balanced portfolio and never convert it into risk. The whole point of the question is that capital weights hide where the risk sits.

The second miss is computing each asset's standalone volatility times its weight and calling that its contribution. Those numbers do not add to the portfolio volatility, because they ignore diversification; only the covariance-based contributions do.

What the interviewer asks next

  • What mix would give each asset an equal share of risk?
  • Would you use leverage on the bonds to keep the expected return up? What new risk does that add?
  • The equity to bond correlation rises to 0.5 in a crisis. What happens to equity's risk share at 25/65/10?

Company names and figures are illustrative.