Portfolio Management puzzles, solved step by step

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  1. 008A fund manager has a true information ratio of 0.5: genuine skill, with annual active returns averaging half their volatility. How many years of returns do you need before the track record is statistically significant at the 5% level?Performance measurementCoreFund selectionPerformance analysis

    Try it first

    Roughly how long must you wait?

    Show the worked solution

    About 15 years. The t-statistic of a track record is the information ratio times the square root of the number of years. Set 0.5 x root T equal to 1.96: the root of T is 3.92, so T is about 15.4 years. A manager twice as good, with an information ratio of 1.0, would still need nearly four years, and most careers and fund lives are shorter than this test demands.

    Why does proof take so long even for a good manager?

    Think of a cricketer whose true average is a little above the team's. In one season, luck swamps that small edge; only after many seasons does the average clearly separate. Skill adds up in proportion to time, but noise adds up in proportion to the square root of time, so the ratio between them grows only with the square root. An information ratio of 0.5 means one year's excess return is half a standard deviation of noise. It takes four years to reach a t-statistic of 1.0 and about 15.4 years to reach 1.96.

    A t-statistic grows only with the square root of the years you wait1230510152025Years of track recordt-statistic = IR x root of years1.96: significant at 5%IR 1.0IR 0.53.8 years15.4 years
    A manager with an information ratio of 0.5 sees the t-statistic of the track record reach the 1.96 significance line only after about 15.4 years, and even a manager with an information ratio of 1.0 needs about 3.8 years.
    The relationship
    t=IRT≥1.96  ⇒  T≥(1.960.5)2≈15.4t=IR\sqrt{T}\ge 1.96 \;\Rightarrow\; T\ge\left(\frac{1.96}{0.5}\right)^2\approx 15.4
    IRthe information ratio: mean active return over tracking error, per year
    Tyears of track record
    1.96the two-sided 5% critical value
    What it says in wordsThe years needed are the critical value divided by the information ratio, squared.

    What should a fund selector do with that number?

    Accept that statistics alone will not settle the question within a useful time. A fund selector who waits for significance hires managers after their best years are behind them; one who does not wait must lean on evidence other than the return series. That means the process, the people, turnover, whether the returns came from the bets the manager says they make, and how much of the record is explained by factors that could be bought cheaply. State the limitation: the calculation assumes independent years and a constant information ratio, and real skill decays as assets grow.

    Where candidates lose it

    The common answer is three to five years, because that is how long most reviews look back. The interviewer wants to see you compute rather than guess, and then notice how uncomfortable the answer is.

    The other slip is squaring the wrong thing. Write t equals IR times root T, solve for root T first, then square.

    What the interviewer asks next

    • Using monthly data, does the answer change?
    • How many years for a 90% confidence level instead?
    • If 1,000 managers have no skill, how many will look significant after 15 years?
  2. 042A fund category had 100 funds five years ago. Twenty of them were closed or merged away after averaging minus 4% a year; the 80 survivors averaged 11% a year. What was the true average return of the category, and how big is the survivorship bias in the survivors' figure?Performance measurementCoreFund selectionMutual funds

    Try it first

    What was the category's true average return?

    Show the worked solution

    About 8% a year, so the survivor average overstates the category by 3 points. Weight each group by its share of the 100 funds: 0.8 x 11% plus 0.2 x minus 4% is 8.8% minus 0.8%, or 8%. The survivors' 11% leaves out exactly the funds that did worst, so it describes the winners, not what the average investor in the category earned.

    Why does a survivors-only average flatter the category?

    Look at a school's alumni wall of fame and you would think every student became a success. The wall shows who is on it, not who was in the class. Funds that do badly are closed or merged into better ones, so a list of funds that exist today is a list of funds that did well enough to survive, and its average is biased upward. The bias is not random: the missing funds are exactly the ones with the worst numbers.

    The average you can see leaves out the funds that are gone80 survivors20 closedeach closed fund averaged -4% a year11%Survivors onlywhat the table shows8%All 100 fundswhat investors earned-3 pts
    Of 100 funds, the 80 survivors averaged 11% while the 20 closed funds averaged minus 4%, so the full-category average was 8%. A table built only from survivors shows 11% and overstates what investors earned by 3 points a year.

    How do you compute the true figure and the bias?

    Take a weighted average by number of funds. The true category average is 80% of 11 plus 20% of minus 4, which is 8%, and the bias is the gap: 3 points a year. A quick way to see it: the closed funds were 15 points behind the survivors, and they were a fifth of the category, so they pull the average down by a fifth of 15, which is 3.

    The relationship
    Rˉ=80×11+20×(−4)100=8%bias=11−8=3 points\bar{R} = \frac{80 \times 11 + 20 \times (-4)}{100} = 8\% \qquad \text{bias} = 11 - 8 = 3 \text{ points}
    80, 20the survivors and the closed funds
    11, -4each group's average annual return, per cent
    What it says in wordsThe honest average counts every fund that existed at the start, including the ones that disappeared.

    State the simplifications. This is an equal-weighted average across funds; an asset-weighted figure could differ, since closed funds are often small. The closed funds also ran for less than five years, so a precise study would compound each fund's returns over the time it existed. Neither changes the direction of the bias, and in real data sets the bias is larger over longer windows because more funds disappear.

    Where candidates lose it

    Most candidates either quote 11% or take the simple midpoint of 11 and minus 4, landing on 3.5% or 7.5%. The first ignores the dead funds; the second ignores that there are four survivors for each dead fund.

    Weight by count, give 8% and 3 points, then say why it matters: a fund selector comparing a manager with a survivor-only peer average is holding the manager to a bar that no real investor earned.

    What the interviewer asks next

    • How would you build a peer group for a manager review that avoids this bias?
    • If the closed funds were mostly small, how would an asset-weighted average differ?
    • Where else in investing does survivorship bias show up?
  3. 065A fund returned 18% in a year when its benchmark index returned 14% and cash returned 6%. The fund's beta to the index is 1.3. What is its Jensen's alpha?Performance measurementCorePerformance analysisAsset management

    Try it first

    What is the fund's alpha for the year?

    Show the worked solution

    1.6%, not 4%. A fund with a beta of 1.3 should earn cash plus 1.3 times the market's return over cash: 6% plus 1.3 times 8%, which is 16.4%. It earned 18%, so the return its market exposure does not explain is 1.6 points. Most of the 4-point lead over the index came from simply taking more market risk in a rising year.

    Why is beating the index by 4 points not the answer?

    Imagine two drivers on a downhill road. One coasts, the other presses the accelerator. The second arrives first, but that says nothing about who drives better. A fund with a beta above 1 is expected to beat a rising market, so raw outperformance mixes skill with extra market exposure. Jensen's alpha strips out the part of the return that the fund's beta alone would have delivered, and asks what is left.

    The relationship
    α=Rp−[Rf+β(Rm−Rf)]=18−[6+1.3×8]=1.6%\alpha = R_p - [R_f + \beta(R_m - R_f)] = 18 - [6 + 1.3 \times 8] = 1.6\%
    R_pthe fund's return, 18%
    R_fthe cash, or risk-free, rate, 6%
    R_mthe index return, 14%
    betathe fund's sensitivity to the index, 1.3
    What it says in wordsAlpha is the fund's return less what its market exposure alone would have earned.
    Security market line: judge the fund against its own beta, not the index0%5%10%15%20%cash 6%index: beta 1, 14%required at beta 1.3: 16.4%fund: 18%alpha = 1.6beat index by 40.00.51.01.31.6Beta to the index
    The security market line gives a 1.3-beta fund a required return of 16.4% in a year when the index made 14% and cash 6%, so the fund's 18% is only 1.6 points of alpha, well short of its 4-point lead over the index.

    What would you warn about the 1.6%?

    Three things. Beta is estimated from past returns, and an error of 0.1 in beta moves alpha by 0.8 points here, half the answer. A single year's alpha is mostly noise, so it describes what happened rather than proving skill. And the answer flips in a falling market: in a year when the index loses, a 1.3-beta fund is expected to lose more, and a fund that merely matches the index would show positive alpha. Always read alpha next to the market's direction and the number of years behind it.

    Where candidates lose it

    The trap is answering 4%, the gap to the index, which ignores beta completely. The interviewer gave you the beta precisely so you would use it.

    The second trap is multiplying the whole 14% by 1.3 to get 18.2% and calling alpha negative. Beta scales the market's return over cash, not the cash part. Write the formula first and put the numbers in.

    What the interviewer asks next

    • In a year when the index falls 10% and cash earns 6%, what return gives this fund zero alpha?
    • How many years of 1.6% alpha, with 4% tracking error, would you need before trusting it?
    • Why might a manager prefer to be measured on information ratio rather than Jensen's alpha?
  4. 085An investor puts Rs 10 lakh into a fund that gains 20% in year one. Just before year two, the investor adds Rs 40 lakh, and the fund then falls 10%. What is the fund's time-weighted return, and how did the investor actually do?Performance measurementCorePerformance analysisWealth management

    Try it first

    Over the two years, how did the investor do in rupees?

    Show the worked solution

    The fund's time-weighted return is plus 8% over two years, but the investor lost Rs 3.2 lakh. Time-weighted return chains the yearly returns, 1.20 x 0.90, and ignores cash flows. The investor had Rs 10 lakh in for the gain and Rs 52 lakh in for the fall, so Rs 50 lakh became Rs 46.8 lakh, a money-weighted return of about minus 5.4% a year.

    How can both numbers be right?

    A cricket team's run rate says how well it batted; a spectator who arrived only for the collapse saw a bad match. Both are true. Time-weighted return measures what the fund did with each rupee over each period, so it judges the manager; money-weighted return weights each period by how much money was in, so it judges the investor's own timing. Here the timing was poor: most of the money arrived just before the fall.

    Same fund, same two years: the manager is up, the investor is down10.0Start12.0End of year 152.0After adding 4046.8End of year 2+40 new+20% on 10-10% on 52Rs lakhTime-weighted+8.0%1.20 x 0.90, over two yearsjudges the managerMoney-weighted-5.4% a yearput in 50, left with 46.8judges the investor's timing
    Rs 10 lakh grew to Rs 12 lakh, the investor added Rs 40 lakh, and the Rs 52 lakh then fell 10% to Rs 46.8 lakh, so the fund shows plus 8% time-weighted while the investor lost Rs 3.2 lakh, about minus 5.4% a year money-weighted.

    How do you get the money-weighted return?

    It is the internal rate of return on the investor's own cash flows. Find the single yearly rate at which Rs 10 lakh put in two years ago and Rs 40 lakh put in one year ago grow to exactly Rs 46.8 lakh today. That is a quadratic, and its root is a growth factor of about 0.9462, or minus 5.4% a year.

    The relationship
    10(1+m)2+40(1+m)=46.8  ⇒  m≈−5.4%10(1+m)^2 + 40(1+m) = 46.8 \;\Rightarrow\; m \approx -5.4\%
    mthe money-weighted return, a yearly rate
    10, 40the investor's contributions, Rs lakh, two years and one year before the end
    46.8the ending value, Rs lakh
    What it says in wordsThe money-weighted return is the one rate that grows the investor's contributions into the ending value.

    This is why fund factsheets quote time-weighted returns: the manager does not control when investors add money. It is also why many investors earn less than the funds they own, because inflows tend to follow good years. Annualised, the fund's figure is 3.9% a year against the investor's minus 5.4%.

    Where candidates lose it

    The trap is reporting plus 8% as the investor's experience. The interviewer wants to hear that the fund's return and the investor's return are different quantities, and why.

    The second slip is computing the investor's figure as a simple return on the total, Rs 3.2 lakh over Rs 50 lakh. That ignores that Rs 10 lakh was invested for two years and Rs 40 lakh for one. Name the internal rate of return as the correct tool.

    What the interviewer asks next

    • Reverse it: the investor adds Rs 40 lakh after the fall instead. What changes?
    • Which return should appear on a fund factsheet, and which on a client statement?
    • Why do investors in aggregate often earn less than the funds they hold?