Portfolio Management puzzles, solved step by step

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100
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All topicsStatistics and forecasting9Portfolio risk maths10Logic brainteasers7Behavioural and decision traps7Probability and expected value8Bond maths10Valuation riddles8Performance measurement8Private and real asset maths8Funds, ETFs and implementation7Compounding and fee drag7Market sizing and estimation6Currency and global returns5
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Showing 11–14 of 14 · filtered from 100Clear filters
  1. 077Money doubles every six years. How long does it take to grow to eight times, and roughly what annual return does that imply?Compounding and fee dragWarm upWealth managementAsset management

    Try it first

    How many years to reach eight times?

    Show the worked solution

    Eighteen years, at about 12.2% a year. Eight is two times two times two, so reaching eight times takes three doubling periods of six years each. The return that doubles money in six years is 2 to the power one sixth, less one, which is 12.2%; the rule of 72 gives 12% as a quick check.

    Why count doublings instead of dividing?

    A rumour passed on by each listener to two new people reaches 2, then 4, then 8. Nobody counts that as adding 2 each round; it doubles each round. A compounding balance works the same way, so any multiple that is a power of two is just a count of doubling periods. Eight is two cubed: three doublings, eighteen years. Sixteen times would be four doublings, twenty four years.

    Eight times is three doublings of six years each1x2x4x8xYear 0Year 62xYear 124xYear 188xx 2x 2x 2Doubling every 6 years means2^(1/6) - 1 = 12.2% a yearRule of 72 check: 72 / 6 = 12%
    Money that doubles every six years is worth 2 times at year 6, 4 times at year 12 and 8 times at year 18, and the smooth path between those points is a steady 12.2% compounded each year.

    How do you get the annual return without a calculator?

    Use the rule of 72: the doubling time multiplied by the rate is roughly 72, so 72 divided by 6 is 12%. The exact figure is 12.25%, and the rule of 72 is within a quarter point anywhere from about 6% to 12%. Give 12% first, then say it is a shade over, which shows you know the rule is an approximation.

    The relationship
    (1+r)6=2  ⇒  r=21/6−1≈12.2%(1+r)^6 = 2 \;\Rightarrow\; r = 2^{1/6} - 1 \approx 12.2\%
    rthe annual return
    6years to double
    What it says in wordsThe annual return is the sixth root of two, less one.

    Portfolio managers ask this because clients think in multiples and managers think in annual rates. Being able to move between the two in your head is the whole skill.

    Where candidates lose it

    The fast wrong answer is 48 years, eight times six, which treats the multiple as a count of periods. It is the same straight-line habit that makes people underestimate what compounding does over a career.

    The quieter loss is saying 12% and stopping. Give 12% from the rule of 72, then correct it to about 12.2%, so the interviewer sees you know where the shortcut comes from.

    What the interviewer asks next

    • How long to reach 10 times at the same rate?
    • If fees take 1.5% a year off that return, how long does one doubling take?
    • Why does the rule of 72 get worse at very high rates?
  2. 078You bought a stock at Rs 800. It now trades at Rs 500, and your updated estimate of fair value is Rs 450. Do you hold it until it gets back to Rs 800?Behavioural and decision trapsWarm upAsset managementWealth management

    Try it first

    Which numbers belong in the decision?

    Show the worked solution

    No. The Rs 800 you paid does not enter the decision; on your own numbers the stock is worth less than it trades for. Getting back to Rs 800 needs a 60% rise. Your own value of Rs 450 sits 10% below the Rs 500 price. Holding it is a fresh decision to own an overpriced stock.

    Why does the purchase price feel like it matters?

    Someone who paid Rs 2,000 for a concert ticket will go out in a storm with a fever rather than waste it, although the money is gone either way. The only question left is whether the evening is worth it now. Money already spent is sunk: it is the same whatever you do next, so it cannot help choose what to do next. The purchase price of a stock is exactly that kind of number.

    The purchase price is not on the decision; only today's two numbers are400450500550600650700750800850Rs 800: what you paidsunk, not a live inputPrice today Rs 500Your value Rs 450+60% needed just to get back: irrelevant to the choiceThe live gap: value is 10% below today's priceTest: would you buy this stock today at Rs 500 if you had never owned it? If not, holding it is the same bet.
    The Rs 800 purchase price sits outside the decision, and the 60% rise needed to reach it is irrelevant; the live comparison is today's price of Rs 500 against your updated value of Rs 450, which says the stock is overpriced.

    How do you say it so it sounds like judgement rather than a slogan?

    Turn the question round. If you had Rs 500 in cash today and no history with this stock, would you buy it at a price above your own value? If the answer is no, holding it is the same bet in disguise. The anchoringLeaning on a reference number, here the purchase price, when judging something that does not depend on it. pull comes from the Rs 800, and the disposition effect, holding losers to avoid booking the loss, is a well documented habit among professional managers as well as individuals.

    Then give the honest limits. Your Rs 450 is an estimate, so ask how confident you are and whether anything has changed that the price already reflects. A booked loss can also offset gains for tax, which is a reason to act, not to wait. Replacement matters too: the money should go to whatever has the best expected return per unit of risk, which may or may not be this stock.

    Where candidates lose it

    The trap is answering the question as asked, with a view on how long the stock might take to get back to Rs 800. That accepts the anchor, and the interviewer is testing whether you reject it.

    The opposite slip is a flat sell with no reasoning. Say the sunk cost point, give the would-I-buy-it-today test, then note that your Rs 450 value is an estimate that deserves a second look.

    What the interviewer asks next

    • Your value was Rs 900 instead of Rs 450. What changes?
    • Why do managers hold losers longer than winners, and how would you guard against it in your own process?
    • How would you explain this to a client who refuses to sell below cost?
  3. 079An Indian investor holds a US stock that rises 8% in dollars over a year, while the rupee weakens 5% against the dollar. What is the investor's return in rupees?Currency and global returnsWarm upGlobal investingIndian wealth management

    Try it first

    Pick the rupee return.

    Show the worked solution

    About 13.4% in rupees. The stock turns each dollar into 1.08 dollars, and each dollar now buys 5% more rupees, so each rupee invested becomes 1.08 x 1.05, or 1.134 rupees. The extra 0.4% over simple addition is the currency gain earned on the stock gain.

    Does a weaker rupee help or hurt this investor?

    A student in India whose parents send a fixed dollar allowance from abroad is better off when the rupee weakens: the same dollars convert into more rupees. An Indian investor holding dollar assets is in the student's position, so a weaker rupee adds to the return. Say the direction first, because half the candidates who get this wrong get the sign wrong, not the arithmetic.

    Currency moves compound with the local return; they do not just add+8.0%Stock, in dollars+5.0%Rupee weakens 5%+0.4%Cross term13.4%Rupee return1.08 x 1.05 = 1.134The cross termis the 5% currencygain earned on the8% stock gain:0.08 x 0.05= 0.4%
    The stock's 8% dollar gain and the rupee's 5% fall add to 13%, and the extra 0.4% comes from the currency gain applying to the grown dollar amount, taking the rupee return to 13.4%.

    Where does the extra 0.4% come from?

    Follow Rs 100. At the start it buys a dollar amount; a year later that amount has grown 8%. The 5% currency gain is earned on the grown amount, not the original one, so the currency also earns 5% on the 8% profit, which is 0.4%. For small moves the cross term hardly matters; for a 30% stock gain with a 10% currency move it is 3 points.

    The relationship
    1+rRs=(1+r$)(1+rFX)=1.08×1.05=1.1341 + r_{Rs} = (1 + r_{\$})(1 + r_{FX}) = 1.08 \times 1.05 = 1.134
    r_$the stock's return in dollars, 8%
    r_FXthe change in rupees per dollar, 5%
    r_Rsthe return measured in rupees
    What it says in wordsThe home currency return is the local return compounded with the currency return.

    Run it the other way as a check. If the rupee had strengthened 5% instead, the return would be 1.08 x 0.95, less one, which is 2.6%: a good stock year mostly erased by currency. That is why global allocations report returns in both currencies.

    Where candidates lose it

    The costly slip is the sign: treating a weaker rupee as a loss and answering 3%. It shows the candidate is thinking about the rupee's health rather than about which currency the investor owns.

    The smaller slip is 13% from simple addition. Say 13.4%, and name the cross term, so the interviewer hears that you know returns multiply.

    What the interviewer asks next

    • What if the rupee strengthens 5% instead?
    • How would the investor hedge the currency, and what would that cost or earn?
    • Over ten years, why might currency matter more than the one-year cross term suggests?
  4. 080A fund turns over 120% of its portfolio a year, and each round trip, selling a holding and buying its replacement, costs 40 basis points. What is the annual drag on returns from trading?Funds, ETFs and implementationWarm upPortfolio implementationMutual funds

    Try it first

    What is the annual trading drag?

    Show the worked solution

    About 0.48% a year. Turnover of 120% means the fund sells and replaces the equivalent of its whole portfolio 1.2 times a year. At 40 basis points per round trip, the drag is 1.2 x 40, or 48 basis points. On a Rs 1,000 crore fund that is Rs 4.8 crore a year, taken from returns rather than charged as a fee.

    What exactly is a round trip, and why count it that way?

    Trading in a car costs you twice: the dealer pays less than it is worth when you sell, and charges more than it is worth when you buy the next one. A fund switching one stock for another pays the same two-sided cost, so the natural unit is the round trip: one sale and one purchase together. Reported turnover is usually the lesser of purchases and sales over average assets, which counts each switch once, so 120% maps to 1.2 round trips.

    Turnover becomes round trips, and round trips become basis points of returnTurnover a year1 full round trip+0.2 = 1.2 round tripsCost per round trip40 bp: spread, impact, brokerage and taxes on one sale plus one purchaseDrag a year40 bp+8 bp = 48 bp, or 0.48% a yearWhat the investor sees against what the investor pays, basis points a yearExpense ratio, say100 bp: on the factsheetTotal cost of ownership+48148 bp
    Turnover of 120% is 1.2 round trips a year at 40 basis points each, a 48 basis point drag; beside an assumed 1.00% expense ratio, the investor's true cost is 148 basis points, and the trading part never appears on the factsheet.
    The relationship
    drag=turnover×cost per round trip=1.2×40=48 bp\text{drag} = \text{turnover} \times \text{cost per round trip} = 1.2 \times 40 = 48 \text{ bp}
    turnoverportfolio replaced per year, 120%
    cost per round tripspread, market impact, brokerage and taxes on a sale and a purchase, 40 bp
    What it says in wordsMultiply how many times the portfolio is replaced by what one replacement costs.

    Why does this matter if the expense ratio looks fine?

    The expense ratio covers the manager's fee and running costs. Trading costs are paid inside the portfolio, through worse prices and brokerage, so they reduce the return without ever appearing in the expense ratio. A fund with a modest fee and high turnover can cost more in total than a pricier fund that trades little. For a Rs 1,000 crore fund, 48 basis points is Rs 4.8 crore a year.

    Say the limitation too. The 40 basis points is an average; impact rises with trade size, so a fund that grows while keeping the same turnover usually pays more per round trip, not less.

    Where candidates lose it

    The common slip is doubling the answer to 96 basis points on the grounds that turnover counts both the buy and the sell. The usual definition already counts each switch once, and the cost per round trip already includes both legs.

    The second slip is saying the cost is already in the expense ratio. It is not, and the interviewer asks precisely to see if you know where trading costs hide.

    What the interviewer asks next

    • The fund doubles in size and keeps the same turnover. What happens to cost per round trip?
    • How would you estimate a fund's trading costs from its published numbers?
    • Why do index funds usually have far lower turnover than active funds?