Case 028Portfolio construction and optimisationWarm up

A risk parity fund holds equities with 18% volatility and bonds with 5%, uncorrelated. What weights equalise their risk, what volatility results, and how much leverage reaches a 10% target?

The situation

Trivikram Risk Parity Fund holds two assets: an equity index with 18% annual volatility and a government bond portfolio with 5%. For the case, their returns are uncorrelated. The fund's rule is that each asset must contribute the same share of total portfolio risk, and its mandate targets 10% volatility, close to a conventional 60/40 fund.

The investment committee also wants a rough check on whether the result earns its keep. Assume, for illustration, expected returns of 12% for equities and 7.5% for bonds, and that the fund can borrow at 6.5%.

Your task

What weights equalise the risk contributions, what volatility does that give, and how much leverage reaches 10%? Is the leverage worth taking?

Quick check

Roughly what equity weight gives equal risk from the two assets?

Worked solution

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

30-second answerThe answer to give first

About 21.7% equities and 78.3% bonds, giving 5.53% volatility, so the fund needs about 1.81 times leverage to reach 10%. With no correlation, equal risk means weight times volatility is equal, so weights run inversely to volatility. The unlevered mix is far too quiet for the mandate, which is why every risk parity fund borrows or uses futures.

Step 1What does equal risk mean when the assets are uncorrelated?

Imagine two people carrying a plank, one strong and one weak. To share the load equally, the weaker one holds nearer the middle. Risk parity sizes each asset so that it contributes the same share of the portfolio's variance, and with no correlation each asset's contribution is simply its weight times its volatility, squared. Setting weight times volatility equal for both gives weights proportional to one over volatility.

The relationship
wE=1/181/18+1/5=21.7%σP=(0.217×18)2+(0.783×5)2=5.53%w_E = \frac{1/18}{1/18 + 1/5} = 21.7\% \qquad \sigma_P = \sqrt{(0.217 \times 18)^2 + (0.783 \times 5)^2} = 5.53\%
w_Eequity weight
18, 5volatility of equities and bonds, % a year
\sigma_Pportfolio volatility with no correlation
What it says in wordsEach asset contributes 3.91 points of weight-times-volatility, and two equal uncorrelated pieces combine to 5.53%.
Step 2Why does a 60/40 fund fail the same test?

Run the same arithmetic on 60/40. Equities contribute 0.6 times 18, 10.8 points, and bonds 0.4 times 5, 2 points. Squared, that is 116.6 against 4. A 60/40 fund is 60% equities by money but about 97% equities by risk, so its results are almost entirely an equity bet. Risk parity was built to fix exactly that, and the fix has a price: at 5.53% volatility, the balanced mix is too quiet for a client expecting a normal fund.

Equal risk needs a small equity weight, so risk parity must borrow60.0%40.0%capital96.7%3.3%risk21.7%78.3%capital50.0%50.0%risk60/40Risk parityequitiesbonds5.5%Risk parityunlevered10.0%Levered1.81x11.0%60/4010% targetPortfolio volatility, % a year
A 60/40 fund takes about 97% of its risk from equities, while the risk parity mix of 21.7% equities splits risk 50/50 but runs at only 5.53% volatility, so it must be levered about 1.81 times to reach the 10% target.
Step 3How much leverage, and does it pay?

Leverage scales risk in a straight line: 10 divided by 5.53 is 1.81 times, so for every Rs 100 of capital the fund holds about Rs 181 of assets. On the assumed returns, the unlevered mix earns 8.48%. Levered 1.81 times and paying 6.5% on the Rs 81 borrowed, it earns about 10.1% at 10% volatility, against 10.2% for 60/40 at 11.0%. Return per unit of risk above the borrowing rate is 0.36 for risk parity against 0.34 for 60/40: a real edge, but a small one.

PortfolioEquitiesBondsVolatilityExpected return
Risk parity, unlevered21.7%78.3%5.53%8.48%
Risk parity, 1.81x, borrowing at 6.5%39.3%141.4%10.00%10.07%
Same, borrowing at 7.5%39.3%141.4%10.00%9.27%
60/4060.0%40.0%10.98%10.20%
Levered to 10% volatility, risk parity earns about 10.1% if it borrows at 6.5% but only 9.3% if borrowing costs as much as the bonds yield, against 10.2% for 60/40 at 11.0% volatility.

Now the limit. The edge lives in the gap between what the bonds earn and what the borrowing costs. If borrowing rose to 7.5%, the levered return would fall to 9.3%, below 60/40. The design also leans on the zero correlation: in a year when inflation surprises upwards, stocks and bonds can fall together, and a fund holding about 141% of capital in bonds through leverage feels it twice. A strong answer gives the weights, then names borrowing cost and correlation as the two things to watch.

Where candidates lose it

The common error is answering 50/50 because the question says equal. Equal capital is not equal risk, and the whole idea of risk parity is that it weights by risk.

The second is stopping at the weights. A 5.5% volatility portfolio cannot meet a 10% mandate, and interviewers want to hear that leverage is not an add-on but built into the design, with its own cost and failure mode.

What the interviewer asks next

  • How do the weights change if equities and bonds have a correlation of 0.3?
  • Add gold at 15% volatility, uncorrelated with both. What are the three weights?
  • Why might risk parity funds sell equities after a volatility spike, and what does that do to the market?

Company names and figures are illustrative.