Mutual Fund Mastery puzzles, solved step by step

Puzzles
100
Traced to a firm
29
Topics
13
Hard
30
Topic
All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
Level
AnyWarm upCoreHard
Source
AnyReported at a firmStandard
Showing 1–1 of 1 · filtered from 100Clear filters
  1. 067A fund has a market beta of 1.1 and a size-factor loading of 0.3. Over the year cash paid 6%, the market beat cash by 4%, the size factor (small minus large) returned 4%, and the fund returned 15%. What is its alpha after the factors?Statistics, correlation and diversificationCoreState StreetCambridge · 2019

    Try it first

    What is the fund's alpha after both factors?

    Show the worked solution

    Alpha is 3.4%. The fund's exposures alone would have earned cash of 6%, plus 1.1 x 4% = 4.4% for its market beta, plus 0.3 x 4% = 1.2% for its tilt to small companies: 11.6% in all. It returned 15%, so 3.4% is what the factors do not explain. It beat the market's 10% by 5 points, but 1.6 of those were paid-for risk.

    Why is beating the market by 5 points not 5 points of skill?

    A delivery rider who earns more than others in the monsoon may simply be taking the rainy-day shifts that pay extra. A fund that takes more market risk, or leans towards small companies, is paid for those exposures in years when they do well, and that pay is not skill. A factor modelA way of explaining a fund return as cash plus a set of exposures, each times the return of a common driver such as the market or small companies, with what is left over called alpha. prices each exposure. This fund has a beta of 1.1, so it gets 1.1 times the market's 4% premium over cash: 4.4%. It has a size loadingHow strongly a fund return moves with the gap between small company and large company returns. A positive loading means a tilt towards small companies. of 0.3, so it gets 0.3 times the 4% that small companies beat large ones by: 1.2%.

    Peel off what the fund was paid for, and what is left is alpha0%5%10%15%6.0Cash+4.4Market 1.1 x 4+1.2Size 0.3 x 4+3.4Alpha15.0%Fund returnexpected from risk: 11.6%Three readingsBeat the market+5.015% vs 10%CAPM alpha+4.6after beta onlyFactor alpha+3.4after beta and sizeEach step strips outa reward for risk
    Cash of 6.0%, a market contribution of 4.4% and a size contribution of 1.2% explain 11.6% of the fund's 15% return, which leaves 3.4% of alpha, well below the 5 points by which it beat the market.
    The relationship
    α=R−[rf+βm(Rm−rf)+s⋅SMB]=15−[6+1.1(4)+0.3(4)]=3.4%\alpha = R - \left[r_f + \beta_m (R_m - r_f) + s \cdot SMB\right] = 15 - [6 + 1.1(4) + 0.3(4)] = 3.4\%
    Rthe fund's return, 15%
    r_fthe cash rate, 6%
    beta_mmarket beta, 1.1
    R_m - r_fthe market's return over cash, 4%
    sthe size loading, 0.3
    SMBsmall minus big: small companies' return over large ones, 4%
    What it says in wordsAlpha is what remains after cash and every priced exposure have been paid.

    What changes as you add each factor?

    Each step strips out a reward that anyone could have bought cheaply. Against the market alone, the fund is 5.0 points ahead. After beta, the CAPMThe capital asset pricing model, which explains returns with one factor, the market, scaled by beta. alpha is 4.6%. After the size tilt as well, alpha is 3.4%, so about a third of the apparent outperformance was a small-company bet that a cheap index fund could have delivered. Add a value or momentum factor and the alpha could shrink further, or grow if the fund leaned against a factor that did well.

    The limits are worth stating plainly. One year of data says almost nothing; loadings and alpha are estimated from many periods of returns, and a 3.4% alpha with typical noise needs years before it is distinguishable from luck. The answer also depends on which factors you include and how they are built, so two research teams can report different alphas for the same fund.

    Where candidates lose it

    The first trap is quoting 5%, the gap to the market. That treats a fund with more risk and a small-company tilt as if it were the index, and gives the manager credit for exposures.

    The second is stopping at 4.6% after beta. The question names a size loading because the interviewer wants to see you price every exposure the fund carries before you call anything skill.

    What the interviewer asks next

    • The size factor returns -4% next year. What would the same fund need to return to show the same alpha?
    • Why might a fund with negative alpha still be a reasonable holding?
    • How many years of monthly data would you want before trusting a 3.4% alpha estimate?

    Asked at State Street, Investment Banking, Cambridge, 2019 (Wall Street Oasis): some basic market knowledge, such as factor model (Fama French), portfolio optimization, risk analysis