Private Wealth Management puzzles, solved step by step

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All topicsCompounding and doubling8Returns arithmetic9Fee and cost drag8Inflation and real return7Tax arithmetic7Probability and risk of loss9Retirement and withdrawal8Fixed income numeracy8Behavioural traps8Estimation and sizing8Options and structured products6Leverage and borrowing6Wealth business economics8
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  1. 026A client's equity fund returned 14% last year while its benchmark index returned 12%. The fund's beta is 1.3 and the risk-free rate was 6%. Did the manager add value, and how much?Returns arithmeticHardWealth management

    Try it first

    Before you work it: how much of the 2-point beat was the manager's skill?

    Show the worked solution

    Barely: the risk-adjusted excess, or alpha, is about 0.2%. The market premium was 12 minus 6, or 6 points. A beta of 1.3 earns 1.3 times that, 7.8 points, on top of the 6% risk-free rate, so the fund should have made 13.8% just by holding more market risk. It made 14%. Of the 2-point beat, 1.8 points were paid for by extra risk and 0.2 came from the manager.

    Why is beating the index not the same as adding value?

    Picture two drivers who both arrive early. One drove carefully and knew a shortcut; the other simply drove 30% faster on the highway. You would not call the second one a better navigator. A fund with a beta of 1.3 is the fast driver: in a rising market it is expected to beat the index simply because it carries more of the market's risk. The client paid for that extra speed with extra drawdown risk, and would have got it from any higher-beta fund.

    So the fair yardstick is not the index but what the index would have paid at 1.3 times the risk. That is the capital asset pricing modelA model that says a portfolio should earn the risk-free rate plus its beta times the market premium, the market return less the risk-free rate. expectation, and whatever sits above it is called Jensen's alpha.

    The fund's 14%, rebuilt: most of the beat was extra market riskindex 12%6.0Risk-freethe floor+7.8Beta x premium1.3 x (12 - 6)+0.2Alphathe skill14.0Fund returnreportedThe 2-point beat over the index1.8 from extra risk0.2 of skill0.3 extra beta x 6 premiumWhat a 1.3 beta should have earned6 + 1.3 x 6 = 13.8%Fund 14.0% beats that by 0.2
    The fund's 14% rebuilds as 6% risk-free, plus 7.8 points from a 1.3 beta on a 6-point market premium, plus only 0.2 points of alpha. Of the 2-point beat over the index, 1.8 points were paid for by carrying extra market risk.
    The relationship
    α=Rp−[Rf+β (Rm−Rf)]=14−[6+1.3×6]=0.2%\alpha = R_p - \left[R_f + \beta\,(R_m - R_f)\right] = 14 - [6 + 1.3 \times 6] = 0.2\%
    R_pthe fund's return, 14%
    R_fthe risk-free rate, 6%
    R_mthe benchmark's return, 12%
    \betathe fund's sensitivity to the market, 1.3
    What it says in wordsAlpha is what the fund earned beyond what its level of market risk alone should have paid.

    What would you tell the client, and what can one year not tell you?

    Tell the client the fund did roughly what a higher-risk version of the index would have done, with a sliver on top. One year of 0.2 points of alpha is indistinguishable from noise; it takes several years and a stable beta before anyone can call it skill. Also say the mirror image: in a year the index falls 10%, the same 1.3 beta implies a fall of about 6 + 1.3 x (minus 16), or minus 14.8%, before any skill at all. The client should expect to feel that.

    The limitation is the beta itself. It is estimated from past returns, it moves, and a different benchmark gives a different number. Say that you are treating 1.3 as given for the puzzle.

    Where candidates lose it

    The fast answer is 2 points of value added, because 14 beats 12. It ignores that the fund took 30% more market risk than the index, and in a rising year extra risk is rewarded whether or not anyone is skilful.

    The second loss is doing the sum wrong: multiplying the whole 12% by 1.3 to get 15.6% and concluding the manager destroyed value. Beta scales the premium over the risk-free rate, not the total return. Say 12 minus 6 first, then multiply.

    What the interviewer asks next

    • The same fund had a beta of 0.8. What is its alpha now?
    • Next year the index falls 10%. What return would you expect from this fund before any skill?
    • Why might a Sharpe ratio tell a different story from alpha?
    • How many years of data would you want before calling this skill?
  2. 066A manager makes plus 20% in year one on a client's Rs 1 crore. Impressed, the client adds Rs 4 crore at the start of year two, which returns minus 10%. The factsheet shows a two-year time-weighted return of plus 8%. What did the client actually earn on his money?Returns arithmeticHardPrivate banking

    Try it first

    Is the client up or down in rupees?

    Show the worked solution

    He lost Rs 32 lakh, about -5.4% a year, while the manager reports plus 8%. Year one made Rs 20 lakh on Rs 1 crore. The client then had Rs 5.2 crore invested when the 10% fall came, costing Rs 52 lakh. He put in Rs 5 crore and holds Rs 4.68 crore. Both numbers are honest: 8% measures the manager's skill, the money-weighted loss measures the client's outcome.

    Why can both numbers be right?

    A bus that averages 60 km an hour tells you about the driver, not about a passenger who boarded only for the slow stretch through traffic. A time-weighted returnA return that chains each period growth rate together, so it ignores when money was added or withdrawn. Used to judge a manager. chains the period returns and ignores the size of the balance, so it judges the manager; a money-weighted returnThe internal rate of return on the actual rupees the client put in and took out, so it depends on the timing and size of his flows. weighs each period by the rupees actually at work, so it measures the client. The manager did not choose when the Rs 4 crore arrived; the client did.

    Same account, two honest numbers that disagreeThe manager's record (time-weighted)The client's rupees (money-weighted)+20%-10%Year 1Year 21.2 x 0.9+8%over twoyearsEvery rupee weighted the same, whatever the timingMoney at workRs 1 crore: +Rs 20 lakhRs 5.2 croreyear 2 on Rs 5.2 crore: -52 lakhPut in: Rs 1 crore + Rs 4 crore = Rs 5.00 croreHolds: 5.2 x 0.9 = Rs 4.68 crore-32 lakh, about -5.4% a year
    The manager's record is plus 20% then minus 10%, a time-weighted plus 8% over two years, but the client had Rs 1 crore at work in the good year and Rs 5.2 crore in the bad year, so he put in Rs 5 crore and holds Rs 4.68 crore, a loss of Rs 32 lakh.

    How do you get the client's yearly rate?

    Find the rate that makes his flows add up. Rs 1 crore invested for two years plus Rs 4 crore invested for one year must grow to Rs 4.68 crore. Solving gives a money-weighted return of about -5.4% a year, the number that describes what actually happened to his money. The quadratic is quick: with x as one plus the rate, x squared plus 4x equals 4.68, so x is about 0.9462.

    The relationship
    1⋅x2+4⋅x=4.68  ⇒  x=−4+16+4×4.682≈0.94621\cdot x^2 + 4\cdot x = 4.68 \;\Rightarrow\; x = \frac{-4 + \sqrt{16 + 4\times4.68}}{2} \approx 0.9462
    xone plus the client's yearly money-weighted return
    1, 4the rupees in crore added at the start of year one and year two
    4.68the ending value, Rs crore
    What it says in wordsThe money-weighted return is the single yearly rate that grows the client's actual deposits into his actual ending value.

    The wealth lesson is behavioural. Clients tend to add money after strong years and pull it after weak ones, so their money-weighted results often trail the funds they hold. Showing a client both numbers, and why they differ, is one of the most useful conversations an adviser can have.

    Where candidates lose it

    The trap is quoting the factsheet 8% as the client's return. It answers a different question, how good the manager was, and a client who has lost Rs 32 lakh will not accept it as his result.

    The opposite error is calling the 8% misleading. It is the right measure for the manager, who did not control the flows. The strong answer gives both numbers and says what each is for.

    What the interviewer asks next

    • If the client had withdrawn Rs 50 lakh after year one instead of adding, which way would the gap run?
    • Which return should a fund's factsheet show, and which should a client's statement show?
    • How would you explain this gap to a client who is angry about the factsheet?
  3. 091A portfolio has an arithmetic average annual return of 10% and a volatility of 20%. Roughly what compound annual return should the client expect to earn over the long run?Returns arithmeticHardPrivate banking

    Try it first

    Your estimate of the long-run compound return.

    Show the worked solution

    About 8% a year. The compound, or geometric, return is roughly the arithmetic mean less half the variance. Volatility of 20% is a variance of 0.2 squared, 0.04, and half of that is 0.02, or 2 points. So an average year of 10% compounds at about 8%. The higher the volatility, the wider the gap.

    Why does volatility pull the compound return below the average?

    Walk up a hill 30 steps and back down 10, then repeat: you are fine. But returns multiply: a rise of 30% followed by a fall of 10% leaves 1.3 x 0.9, which is 1.17 over two years, an average of 10% that compounds at only 8.17%. A loss is taken from a larger base than the gain that preceded it, so the more returns swing, the further compound growth falls below the average return.

    The relationship
    g≈μ−σ22=0.10−0.2022=0.08g \approx \mu - \frac{\sigma^2}{2} = 0.10 - \frac{0.20^2}{2} = 0.08
    gthe geometric, or compound, annual return the client actually earns
    \muthe arithmetic mean of yearly returns, 10%
    \sigmathe volatility, the standard deviation of yearly returns, 20%
    What it says in wordsThe compound return is roughly the average return less half the square of the volatility.
    The average year and the compound return are two different numbers-40%-10%10%40%70%At full scale, 10% and 8%sit almost on top of each otherOne year's return, volatility 20%Magnified: 6% to 11%6%7%8%9%10%11%average10%compound8%2 ptsgap = half the variance0.5 x 0.20 x 0.20 = 0.02
    The average year sits at 10% but the compound return the client earns sits at about 8%, two points lower, and the gap is half the variance, 0.5 x 0.20 x 0.20, which is too small to see on the full bell and plain on the magnified ruler.

    How far can you trust the rule?

    Check it on a case you can work exactly. The fund that alternates plus 30% and minus 10% has an average of 10% and a volatility of exactly 20%, and its true compound rate is the square root of 1.17 less one, 8.17%. The half-variance rule gives 8% against an exact 8.17%, close enough to say out loud, and it gets less accurate as volatility rises. At 40% volatility the rule takes off 8 points, and the exact answer depends heavily on the shape of the returns.

    Then say why a client should care. Two portfolios with the same 10% average but volatility of 10% and 20% compound at about 9.5% and 8%. Over 20 years on Rs 1 crore that is roughly Rs 6.1 crore against Rs 4.7 crore. Lower volatility at the same average is worth real money, which is part of the case for diversification.

    Where candidates lose it

    The weak answer is 10%, treating the average year as the rate the client compounds at. It is the same mistake as reading a fund's average return as the growth in the client's statement.

    The second loss is knowing the rule but not the reason. Give the plus 30, minus 10 example in one line: it shows the interviewer you understand why the gap exists, not only its formula.

    What the interviewer asks next

    • What compound return does a 10% average with 30% volatility give, roughly?
    • Two funds have the same compound return but different volatility. Which has the higher average?
    • Why do some fund factsheets show the average return rather than the compound one?