Case 021Alternatives and private marketsHard

Add 10% of a 60/40 portfolio to a long-short fund returning 12% with beta 0.4, or to more long-only equity returning 14%. With the market at 12% volatility, which improves the portfolio's return for its risk more?

The situation

The Gokhale family's portfolio is 60% equity and 40% debt. Their adviser proposes moving 10% of it, taken proportionally from both sleeves, into one of two places. Option A is an invented long-short equity fund expected to return 12% a year with a beta of 0.4 to the market and 5% of volatility of its own that is unrelated to anything else. Option B is more long-only equity, expected to return 14% with a beta of 1.

Use illustrative assumptions: equity volatility 12%, debt return 7% with volatility 3%, a correlation of 0.2 between equity and debt, and a risk-free rate of 6.5%. Option A leaves the family at 54% equity, 36% debt and 10% long-short; Option B at 64% equity and 36% debt.

Your task

Work out return, volatility and return per unit of risk for today's portfolio and each option, and say which addition improves the portfolio more and why.

Quick check

Which addition improves the portfolio's return for its risk more?

Worked solution

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

30-second answerThe answer to give first

The long-short fund improves the portfolio more: return per unit of risk rises from 0.624 to 0.657, against 0.625 for more equity. Its beta of 0.4 means most of its 12% is not market return: about 2.5 points is return the market does not explain. Adding it lowers volatility from 7.53% to 7.27% and still lifts return slightly. More equity adds return and risk in the same proportion.

Step 1Why can a lower-return asset improve the whole?

Think of a household with one earner in a cyclical job. A second earner on a lower but steadier salary can make the household far more secure than a raise for the first earner. What an addition does to a portfolio depends on how it moves with what is already there, not only on its own return. The long-short fund's 12% comes with a beta of 0.4: only 0.4 of the market's moves, plus 5% of its own unrelated noise, about 6.9% volatility in all. More equity at 14% brings a full unit of the same market risk the portfolio already has.

Step 2How much of the long-short fund's return is really new?

Strip out what its market exposure alone would earn. A beta of 0.4 to a market returning 14% against a 6.5% risk-free rate should earn about 6.5 + 0.4 x 7.5 = 9.5%. The fund's expected 12% is about 2.5 points above that, the alphaReturn above what an investment should earn for the market risk it takes, as measured by its beta. the family would actually be paying for. That number is an assumption about the manager, and it is the one to question hardest.

The relationship
σp=(βp σm)2+(wd σd)2+2ρ βpσmwdσd+(wℓ σε)2,βp=we+0.4 wℓ\sigma_p = \sqrt{(\beta_p\,\sigma_m)^2 + (w_d\,\sigma_d)^2 + 2\rho\,\beta_p\sigma_m w_d\sigma_d + (w_\ell\,\sigma_\varepsilon)^2}, \quad \beta_p = w_e + 0.4\,w_\ell
βpportfolio beta to the market: equity weight plus 0.4 times the long-short weight
σmmarket volatility, 12%
wd σddebt weight times debt volatility
wℓ σεlong-short weight times its own unrelated volatility, 5%
What it says in wordsPortfolio risk is the market risk from its beta, the debt risk and their overlap, plus the long-short fund's own noise, which mostly diversifies away at a 10% weight.
Each addition against today's 60/40: what changesAdd 10% long-short fundReturn, % a yeartoday 11.20%11.28%Volatility, % a yeartoday 7.53%7.27%Return per unit of risk (Sharpe)today 0.624with it 0.657better return for the riskAdd 10% more long-only equityReturn, % a yeartoday 11.20%11.48%Volatility, % a yeartoday 7.53%7.97%Return per unit of risk (Sharpe)today 0.624with it 0.625more return, more risk, same ratio
Adding the long-short fund moves the Gokhale portfolio to 11.28% return and 7.27% volatility, lifting return per unit of risk to 0.657; adding equity moves it to 11.48% and 7.97%, leaving the ratio at 0.625.
Step 3What do the numbers show?

Today: 11.20% return, 7.53% volatility, a Sharpe ratioReturn above the risk-free rate divided by volatility: how much extra return each unit of risk earns. of 0.624. With the long-short fund: 11.28%, 7.27%, 0.657. With more equity: 11.48%, 7.97%, 0.625. More equity moves the portfolio up its existing line, more return for proportionally more risk; the long-short fund moves it up and to the left, onto a steeper line. If the family wanted more return at today's risk, they could hold the long-short version with a little more equity and still come out ahead.

Moving left beats moving up: the long-short fund steepens the line7%8%9%10%11%12%0%2%4%6%8%Volatility, % a yearrisk-free 6.5%With long-short: 11.28%, 7.27% volToday 60/40: 11.20%, 7.53% volWith more equity: 11.48%, 7.97% volslope 0.657: with long-shortslope 0.624: today; 0.625 with more equity
On a return against volatility chart, the long-short addition moves the portfolio left and slightly up onto a steeper line from the risk-free rate, a slope of 0.657, while more equity moves it up along today's line at about 0.625.
Step 4What would make you doubt the answer?

Three assumptions carry it. The alpha: long-short funds charge high fees and many deliver little above their beta, so ask for a long record, net of fees, through a falling market. The low correlation: in a crash, long-short funds often turn out more correlated with equities than their calm-period beta suggests, just when the diversification is needed. And liquidity: many such funds have lock-ups or notice periods. The case for the fund is only as good as the evidence that its beta stays low and its alpha survives fees.

Where candidates lose it

The standard slip is comparing 12% with 14% and choosing equity. The question asks about return for risk, and a candidate who never computes portfolio volatility has not answered it.

The second is treating the long-short fund's volatility on its own, about 7%, as the whole story. What matters is how much of it the portfolio already has: 0.4 of it is market risk, and the rest largely diversifies away at a 10% weight.

What the interviewer asks next

  • The long-short fund's beta turns out to be 0.8 in a crash. Redo the comparison for that year.
  • The fund charges 2% and 20%. How much of the 2.5 points of alpha survives?
  • How much extra equity would the family need alongside the long-short fund to match today's volatility with more return?

Company names and figures are illustrative.