Case 004Factor investing and quantWarm up

A low volatility index has a beta of 0.7. In a year the market returns 25% with cash at 6%, what does CAPM expect from it, and why do low volatility investors accept that?

The situation

The Sthir Low Volatility Index holds the least volatile half of a broad Indian equity universe. Its beta to the market is 0.7, its volatility is 12% a year against the market's 16%, and cash earns 6%.

A client of your fund has just watched the market return 25% in a year and wants to know why the low volatility fund he owns returned so much less, and whether he should be worried.

Your task

What does CAPM expect the index to return in that year, what would it expect in a year the market falls 15%, and why do investors hold it anyway?

Quick check

The market returns 25% and cash earns 6%. What does CAPM expect from a beta of 0.7?

Worked solution

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

30-second answerThe answer to give first

CAPM expects about 19.3%, a lag of 5.7 points, and that lag is the design working. Beta scales only the market's return above cash: 6 plus 0.7 times 19. In a year the market falls 15%, the same arithmetic gives about -8.7%, a cushion of 6.3 points. Over those two years together, the index compounds to 8.9% against the market's 6.2%.

Step 1How does CAPM turn a market return into an expected return for the index?

CAPMThe capital asset pricing model: expected return equals the cash rate plus beta times the market return above cash. says the reward for market risk is paid on the return above cash, and beta says how much of that risk you carry. So you scale the excess return, 25 minus 6, by 0.7, and add cash back: 6 plus 13.3 is 19.3%. Think of a bus fare that has a fixed base charge plus a per-kilometre rate: beta changes the per-kilometre part, not the base charge everyone pays.

The relationship
E[R]=Rf+β (Rm−Rf)=6+0.7×(25−6)=19.3%E[R] = R_f + \beta\,(R_m - R_f) = 6 + 0.7 \times (25 - 6) = 19.3\%
R_fcash rate, 6%
\betasensitivity to the market, 0.7
R_mmarket return in the year, 25%
What it says in wordsThe index is expected to earn cash plus seven tenths of whatever the market earns above cash.
Step 2What does the same line say in a falling year?

Run it with the market at minus 15%. The excess return is minus 21 points, 0.7 of that is minus 14.7, plus 6 is -8.7%, a loss 6.3 points smaller than the market's. The lag in the good year and the cushion in the bad year are the same property seen from two sides, and a client who wants one has to accept the other.

Low volatility lags the rally by design and earns its keep in the fallUp year: market +25%0+25.0%Market+19.3%Sthir indexlags the marketby 5.7 pointsDown year: market -15%0-15.0%Market-8.7%Sthir indexfalls lessby 6.3 pointsBoth years together:market +6.25% against Sthir index +8.92%smaller falls need smaller recoveries
CAPM expects the Sthir index to return 19.3% when the market returns 25% and -8.7% when it returns minus 15%, so across the two years the index compounds to 8.92% against the market's 6.25%.
Step 3Why would anyone accept the lag?

Three reasons, in order of strength. First, compounding: a smaller fall needs a smaller recovery, so across a 25% year and a minus 15% year the index ends 8.9% up against the market's 6.2%, despite lagging the rally by almost 6 points. Second, the low volatility anomalyThe finding, across many markets and decades of academic study, that low-beta stocks have earned more than CAPM predicts for their beta and high-beta stocks less.: studies across many markets have found low-beta stocks earning more than CAPM predicts. Frazzini and Pedersen's betting-against-beta argument, 2014, explains it by investors who cannot borrow buying high-beta stocks for extra return instead, bidding them up. Third, behaviour: clients who fall less are less likely to sell at the bottom.

Step 4What are the limits you should say out loud?

The numbers are consistent: 0.7 times 16% is 11.2% of market-driven volatility, and a total of 12% leaves about 4.3% that is specific to the index's holdings. The limit is that a low volatility index is a bundle of sector bets, often heavy in staples, utilities and healthcare, and those sectors behave like long-dated bonds. In a year when rates rise sharply they can fall with the market and lose the cushion. The anomaly has also been widely published, so a crowded and expensive low volatility basket can earn less than its history.

Where candidates lose it

The classic slip is multiplying the whole market return by beta, 0.7 times 25, and answering 17.5%. That treats the cash rate as if it were a reward for market risk.

The second is calling the lag a failure. Interviewers want to hear that the index behaved as designed: less of the market's excess return in both directions, and that the value shows up over a full cycle, not in one rally.

What the interviewer asks next

  • The index returns 22% in the up year. What is its alpha against CAPM, and what might explain it?
  • Why might a low volatility index fall more than beta suggests in a year rates rise sharply?
  • How would you combine this index with a momentum strategy, and what would you expect the correlation to be?

Company names and figures are illustrative.