Private Wealth Management puzzles, solved step by step

Puzzles
100
Traced to a firm
3
Topics
13
Hard
30
Topic
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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Showing 1–3 of 3 · filtered from 100Clear filters
  1. 014One deposit pays 12% a year compounded monthly. Another pays 12.5% a year compounded annually. Which pays more, and by how much on Rs 1 lakh over a year?Compounding and doublingHardIndian wealth management

    Try it first

    Which ends the year ahead?

    Show the worked solution

    The 12% monthly deposit pays more: an effective 12.68% against 12.50%. One per cent a month, compounded twelve times, is 1.01 to the 12th, which is 1.1268. On Rs 1 lakh that is Rs 1,12,683 after a year against Rs 1,12,500, Rs 183 more. The gap is small, but the method is the point: convert every quoted rate to an effective annual rate before comparing.

    Why is 12% monthly not 12%?

    Imagine a savings box where interest is dropped in every month rather than at year end. From the second month on, the interest already in the box also earns. A quoted rate with a compounding frequency is a label, not the return; the return is the effective annual rate, which rises with every extra compounding. For 12% compounded monthly, that effective annual rateThe rate that, compounded once a year, gives the same result as the quoted rate with its compounding frequency. is 12.683%.

    Compare effective annual rates, not the quoted ones1,00,0001,05,0001,10,000Month 0Month 6Month 12Monthly: a step upevery month, 1% ona growing balanceAnnual: nothing until the year endYear-end value, zoomedaxis starts at Rs 1,12,0001,12,68312% monthly1,12,50012.5% annualMonthly wins by Rs 183: 12.68% vs 12.50%
    Rs 1 lakh at 1% a month climbs in twelve steps to Rs 1,12,683, while 12.5% compounded annually jumps once to Rs 1,12,500, so the monthly deposit ends Rs 183 ahead on an effective 12.68%.

    How do you do 1.01 to the 12th in your head?

    Use the binomial shortcut. 1.01 to the 12th is roughly 1 plus 12 times 0.01 plus 66 times 0.0001, the number of pairs of months times the interest on interest. That is 1 plus 0.12 plus 0.0066, about 1.1266, within a hair of the exact 1.1268. The 0.66 point of extra return is the interest on interest, and it is what closes most of the gap to 12.5%.

    The relationship
    EAR=(1+0.1212)12−1=12.68%  >  12.50%\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 12.68\% \;>\; 12.50\%
    0.12/12the monthly rate, 1%
    12the number of compounding periods in a year
    EARthe effective annual rate
    What it says in wordsCompound the periodic rate for a year and subtract one to get the rate you can compare.

    Give the two numbers that frame it. Compounded continuously, 12% would give 12.75%, the ceiling for a 12% quote. And a monthly deposit would need to quote only 11.84% to match 12.5% annual. The limitation for a client: tax, premature withdrawal terms and the credit of the issuer usually matter more than Rs 183.

    Where candidates lose it

    The trap is comparing the quoted numbers, 12% against 12.5%, and picking the annual deposit. The candidate has compared two labels written in different units.

    The opposite loss is overselling the result. Rs 183 on Rs 1 lakh is a small gap; say so, and say that the method, converting to effective rates, is what the interviewer wanted.

    What the interviewer asks next

    • What would 12% compounded quarterly give as an effective annual rate?
    • What monthly-compounded rate would exactly match 12.5% annual?
    • A loan quotes 1.5% a month. What is the effective annual rate?
  2. 040A client can run a Rs 10,000 monthly SIP that rises 10% every year, or a flat Rs 15,000 monthly SIP. Both run 15 years at 12% a year, treated as 1% a month. Which ends bigger, and in which year does the step-up overtake on the monthly instalment?Compounding and doublingHardMutual fund distributionIndian wealth management

    Try it first

    The step-up ends bigger. Where does most of its advantage come from?

    Show the worked solution

    The step-up ends bigger, Rs 86.8 lakh against Rs 75.7 lakh, but it wins late and by saving more. Its instalment first beats Rs 15,000 in year 6, at Rs 16,105, and its corpus overtakes only in year 11. Over 15 years it puts in Rs 38.1 lakh against Rs 27.0 lakh; the growth on each is almost identical, about Rs 48.7 lakh.

    Why does the step-up lag for so long?

    Two students save for a trip: one puts in Rs 150 a week from the start, the other begins at Rs 100 and adds 10% every term. The steady saver is ahead for most of the year because her money arrived first. The step-up pays in less than the flat SIP for the first five years, and money put in early is the money that compounds longest, so its corpus trails until year 11. Its year-5 instalment is still only Rs 14,641.

    The step-up passes on instalment in year 6 and on corpus only in year 1110k20k30k40kYr 1Yr 5Yr 10Yr 15Monthly instalment, Rsflat Rs 15,000step-up +10% a yearyear 6: Rs 16,105Corpus at year 15, Rs lakh75.7in 27.0+48.7Flat86.8in 38.1+48.7Step-upGrowth about equal: extra corpus = extra saving
    The step-up instalment passes the flat Rs 15,000 in year 6 and reaches Rs 37,975 by year 15, but its corpus overtakes only in year 11. At year 15 it holds Rs 86.8 lakh against Rs 75.7 lakh, with almost identical growth of about Rs 48.7 lakh on each.
    Flat Rs 15,000Step-up from Rs 10,000
    Total put inRs 27.0 lakhRs 38.1 lakh
    Corpus at year 5Rs 12.4 lakhRs 9.8 lakh
    Corpus at year 10Rs 34.9 lakhRs 33.7 lakh
    Corpus at year 15Rs 75.7 lakhRs 86.8 lakh
    Growth at year 15Rs 48.7 lakhRs 48.7 lakh
    Both SIPs invested at the start of each month at 1% a month; the step-up rises 10% at the start of each year.

    So is the step-up the better plan?

    It is a different plan, not a smarter one. The step-up ends ahead because it asks the client to save Rs 11.1 lakh more; per rupee saved, the flat SIP compounds better because its money arrives earlier. Where the step-up earns its place is fit: it matches a salary that rises each year, so the client can afford it without strain in year one. The honest comparison for a client is between what each plan asks of his budget in each year, not between two final numbers.

    Limits: 1% every month is smooth, real returns are not, and the step-up puts its biggest instalments in the last years, so a bad market late in the plan hurts it more. The 10% step-up also assumes the client's income actually rises that fast.

    Where candidates lose it

    Most candidates say the step-up wins and credit compounding. The step-up does win, but compounding favours the flat SIP rupee for rupee; the extra corpus is almost exactly the extra saving. That is the insight the interviewer is fishing for.

    The other loss is the crossover year. The instalment crosses in year {P40['cross_contrib']}, when 10,000 x 1.1 to the power 5 first tops 15,000, but the corpus takes until year {P40['cross_corpus']}. Say both, and say why they differ.

    What the interviewer asks next

    • What flat SIP would match the step-up's corpus at year 15?
    • How does the answer change over 25 years instead of 15?
    • The client's salary rises 5% a year, not 10%. Which plan fits him better, and why?
  3. 090Two sisters invest at 10% a year. Sister A puts in Rs 1 lakh a year from age 25 to 34 and then stops. Sister B puts in Rs 1 lakh a year from 35 to 60. Payments are made at the start of each year. Who has more at 60, and by how much?Compounding and doublingHardIndian wealth management

    Try it first

    Who has more at 60?

    Show the worked solution

    Sister A, with about Rs 190 lakh against Rs 109 lakh, ahead by Rs 81 lakh. A pays in only Rs 10 lakh, but her pot is Rs 17.5 lakh at 35 and then compounds untouched for 25 years, about 10.8 times. B pays in Rs 26 lakh, yet her last payments have little time to grow. The early years carry the compounding.

    How do you get the answer without a spreadsheet?

    Work A in two stages. Ten payments of Rs 1 lakh at 10% build to about Rs 15.9 lakh by the last payment at 34, and to Rs 17.5 lakh a year later at 35. From 35 to 60 that pot compounds for 25 years. At 10% money doubles about every 7.2 years, so 25 years is about three and a half doublings, roughly 11 times, which takes Rs 17.5 lakh to about Rs 190 lakh.

    The relationship
    A60=1.126⋅1.110−10.1≈189.9B60=1.126−10.1≈109.2A_{60} = 1.1^{26} \cdot \frac{1.1^{10} - 1}{0.1} \approx 189.9 \qquad B_{60} = \frac{1.1^{26} - 1}{0.1} \approx 109.2
    (1.1^10 - 1) / 0.1ten yearly payments of Rs 1 lakh added up with their growth, about 15.9 lakh
    1.1^26growth from the last payment at 34 to age 60
    B_6026 payments from 35 to 60, the last earning nothing
    What it says in wordsA's small early pot grows for 26 more years; B's larger total has much less time in the market.
    Ten early payments against twenty-six later ones, at 10% a year501001502002535455560AgeRs lakhA paysB paysA stops at 35 with 17.5 lakhA: 189.9B: 109.2A paid in 10 lakhB paid in 26 lakhA ahead by80.8 lakh
    Sister A stops paying at 34 with Rs 17.5 lakh at 35 and still ends at Rs 189.9 lakh at 60, well above Sister B's Rs 109.2 lakh from 26 payments, because A's money has decades longer to compound.

    Why does B never catch up, even with 26 payments?

    Picture a snowball rolled from the top of a long slope against a bigger pile of snow thrown on halfway down. The first has more distance to gather snow. At 10% a rupee invested at 25 is worth about 28 rupees at 60, while a rupee invested at 50 is worth about 2.6, so a decade of early payments outweighs a quarter century of later ones. For B to match A she would need to pay about Rs 1.74 lakh a year from 35.

    Then add the honest limits. A steady 10% is an illustration, not a forecast; a real sequence of returns will differ, and fees and taxes reduce both pots. The ratio between the sisters, though, comes from time in the market, and that holds at any positive rate: the lower the rate, the smaller A's lead.

    Where candidates lose it

    Most candidates answer B because she paid in more than twice as much. They count rupees and forget the time each rupee has to grow, which is exactly the mistake a wealth client makes when he delays starting.

    The second loss is freezing because the sum looks like it needs a spreadsheet. Split A into two stages and use doublings for the second: an estimate within a few lakh is what the interviewer wants.

    What the interviewer asks next

    • At what rate of return would the two sisters end level?
    • How much would B need to invest each year to match A at 60?
    • What does this puzzle say about an employer's retirement contribution that starts at 25?