Private Wealth Management puzzles, solved step by step

Puzzles: 100
Traced to a firm: 3
Topics: 13
Hard: 30

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017A client's salary rose from Rs 12 lakh to Rs 18 lakh over six years, while inflation ran at 6% a year. How big was his raise in real terms?Inflation and real returnCoreWealth management
Try it first
Pick the real raise over the six years.
Show the worked solution
About 5.7% in total, not 50%. The salary rose 1.5 times. At 6% a year, prices rose 1.06 to the 6th, 1.419 times, or 41.9%. The real raise is 1.5 divided by 1.419, which is 1.0574, about 0.9% a year. Rs 18 lakh today buys what Rs 12.69 lakh bought six years ago.
Why is the real raise so much smaller than it looks?
Think of a family whose grocery bill has quietly climbed from Rs 10,000 to Rs 14,000 a month over six years. If their income rose by half in the same years, most of the extra rupees simply cover the same groceries. A raise is measured against what the money buys, so inflation has to be divided out of it before it means anything. Six years at 6% raise prices by 41.9%, which eats almost all of a 50% raise.
The salary rose 50% over six years, but prices rose 41.9% at 6% a year, so the real raise is only 5.7%, and Rs 18 lakh now buys what Rs 12.69 lakh bought six years ago.
Why divide instead of subtracting inflation?
Because both numbers are growth multiples. Subtracting works only for small rates over one year; over several years it misstates the answer, first by ignoring compounding and then by subtracting growth rates that should be divided. Six times 6% is 36%, which understates inflation; 50% less 41.9% is 8.1%, which overstates the real raise. The correct real growthGrowth after removing the effect of rising prices, found by dividing the nominal growth multiple by the price growth multiple. is 1.5 over 1.419.
The relationship
$1 + r_{\text{real}} = \frac{1 + r_{\text{nominal}}}{1 + \pi} = \frac{1.50}{1.06^6} = \frac{1.50}{1.4185} = 1.0574$
r_nominal the salary growth over the period, 50%
pi price growth over the same period, 1.06 to the 6th minus one
r_real growth in what the salary buys
What it says in wordsReal growth is the nominal growth multiple divided by the price growth multiple, minus one.
Bring it back to the client. A family whose income rose 50% may feel richer and spend accordingly, when their buying power has grown by less than a tenth of that. Planning savings on the nominal figure is how lifestyle creep starts, and the 6% here is an illustration, not a forecast of future prices.
Where candidates lose it
Three wrong answers compete: 50%, which ignores inflation; 14%, which subtracts six years of simple inflation; and 8.1%, which subtracts compounded inflation. Each sounds like a method, which is what makes the trap work.
Give the ratio, 1.5 over 1.4185, and then the plain words: his buying power rose by under 6% in six years.
What the interviewer asks next
What salary would he have needed to keep his buying power exactly flat?
If inflation had been 4%, what would the real raise have been?
How would you use this number in a conversation about his savings rate?
030A client's monthly spending is 40% on food and household items rising at 4% a year, 30% on rent and services rising at 8%, and 30% on school fees and healthcare rising at 12%. What is his personal inflation rate, and why might it differ from the headline index?Inflation and real returnCoreWealth management
Try it first
Before you calculate: what is his personal inflation rate?
Show the worked solution
His personal inflation rate is 7.6%. Weight each category's price rise by its share of his spending: 40% of 4 is 1.6, 30% of 8 is 2.4 and 30% of 12 is 3.6, which add to 7.6%. It differs from the headline index because the index weights a national average basket, heavy in food, while this household spends much more on schooling and healthcare, whose prices rise faster.
How is a personal inflation rate built?
Think of a monthly bill as a thali. If dal gets 4% dearer but dal is only a small part of the plate, it barely moves the bill; if the costliest item on the plate jumps, the whole bill jumps. An inflation rate is a weighted average: each item's price rise counts in proportion to its share of what you actually spend. The same rule builds the national index, only with national weights.
The client's 40, 30 and 30 split contributes 1.6, 2.4 and 3.6 points of inflation, adding to 7.6%, well above an illustrative 5% headline print. Over 20 years that gap turns a Rs 1 lakh monthly spend into Rs 4.33 lakh rather than Rs 2.65 lakh.
The relationship
$\pi_{\text{personal}} = \sum_i w_i\,\pi_i = 0.4(4) + 0.3(8) + 0.3(12) = 7.6\%$
w_i the category's share of monthly spending
\pi_i the category's own yearly price rise
What it says in wordsPersonal inflation is each category's price rise weighted by how much of the budget it takes.
Why does the gap matter for a financial plan?
Because the plan's target corpus is built from future spending. At 7.6% a year, today's Rs 1 lakh of monthly spending needs Rs 4.33 lakh in 20 years; at an illustrative 5% headline rate it would need only Rs 2.65 lakh. A retirement plan built on the headline number would come up about 39% short on the spending side. Affluent households tend to spend more on education, healthcare and services, which is why advisers often assume a higher inflation rate than the index for them.
The limitations are real. Weights shift as life changes: school fees end, healthcare grows in retirement. The category rates here are illustrative, and the current headline figure should be read from the official release, not assumed.
Where candidates lose it
The common slip is averaging 4, 8 and 12 to get 8%, ignoring that the categories are not equal shares of spending. A smaller group then fails the second half by saying the headline index is simply wrong.
The index is not wrong; it measures a different basket. Say that the difference comes from weights, then show what the gap does to a 20-year spending target.
What the interviewer asks next
When the children finish school, his basket becomes 55% food, 35% services and 10% healthcare. What is his inflation rate now?
What real return does his portfolio need to hold its purchasing power if it earns 10% nominal?
Which categories would you expect to dominate a retired couple's basket?
056A fixed deposit pays 7% a year. The client pays tax on the interest at an illustrative 30% slab, and inflation runs at 5.5%. What is his post-tax real return?Inflation and real returnCoreIndian wealth management
Try it first
Is the client getting richer in real terms?
Show the worked solution
About minus 0.6% a year: the deposit loses buying power. Tax at 30% takes 2.1 of the 7 points, leaving 4.9% after tax. Inflation of 5.5% is larger than that, so the real, after-tax return is roughly 4.9 minus 5.5, or minus 0.6%; exactly, 1.049 over 1.055 less one is -0.57%. The deposit is safe in rupees and still shrinks in what the rupees buy.
In what order do tax and inflation come out?
Tax first, because the tax office charges on the rupee interest, not on the real gain. The client keeps 70% of the 7%, which is 4.9%, and only then does inflation get its turn. Think of a salary rise of 7% in a year when your rent, school fees and groceries go up 5.5%: after tax you are behind. On Rs 10 lakh, the deposit pays Rs 70,000, tax takes Rs 21,000, and the client holds Rs 10,49,000 when the same basket of goods now costs Rs 10,55,000.
The deposit's 7.0% becomes 4.9% after an illustrative 30% tax, and inflation of 5.5% takes more than that, so the post-tax real return ends just below zero at about minus 0.6%, a loss of about Rs 5,700 of buying power a year on Rs 10 lakh.
The relationship
$r_{\text{real}} = \frac{1 + r(1-t)}{1 + \pi} - 1 = \frac{1.049}{1.055} - 1 \approx -0.57\%$
r the deposit rate, 7%
t the illustrative tax rate, 30%
pi inflation, 5.5%
What it says in wordsTake tax off the interest, then divide by the rise in prices, and what is left is the growth in buying power.
What rate would the deposit need to break even?
Work backwards. After tax the deposit must at least match inflation, so the pre-tax rate must be 5.5% divided by 0.7, which is 7.86%. At a 30% slab, a deposit has to pay about 7.9% just to stand still in real terms. That is the sentence a client remembers, and it frames every later conversation about where safety money sits. Slabs, surcharge and cess change; the 30% here is illustrative and you should confirm the current rates before using them with a client.
Say the limit too. The deposit still does its job if the job is to keep money stable in rupees for a known bill next year. The point is only that over long periods, a return below inflation after tax slowly shrinks what the money can buy.
Where candidates lose it
The common answer is 7 minus 5.5, plus 1.5% a year. It forgets tax entirely, and it is exactly the arithmetic a client does in his head, so the interviewer is checking whether you correct it.
The quieter error is taking tax off the real return, 30% of 1.5%. Tax is charged on the whole rupee interest, including the part that only keeps up with prices.
What the interviewer asks next
At what tax rate does this deposit exactly break even in real terms?
Inflation drops to 4%. What is the real post-tax return now?
Why does the same 7% deposit look very different to a retired client in a low tax slab?
080A client wants to fund her daughter's MBA in 15 years. The course costs Rs 25 lakh today and education costs rise about 10% a year. Roughly what will it cost when the daughter enrols?Inflation and real returnCoreIndian wealth management
Try it first
Pick the closest figure.
Show the worked solution
About Rs 1.04 crore. At 10% a year costs double roughly every 7.2 years (72 / 10), so 15 years is just over two doublings: Rs 25 lakh becomes 50 and then 100 lakh, plus a little. Exactly, 25 x 1.1 to the power 15 is Rs 104.4 lakh. The goal to fund is that figure, not Rs 25 lakh.
Why is the answer four times today's fee and not two and a half?
A school that raises fees 10% every year raises them on last year's fee, not on the fee from when your child started. The rises pile on each other. Cost inflation compounds exactly like an investment return, so a goal's cost has to be grown at its own inflation rate before anyone asks how much to save. Adding 10% of today's fee each year gives Rs 62.5 lakh, which is simple interest and leaves the plan Rs 42 lakh short.
At 10% a year the Rs 25 lakh fee reaches Rs 50 lakh in about 7.3 years, Rs 1 crore in about 14.5 years and Rs 104.4 lakh by year 15, while a plan that forgets inflation stays stuck at Rs 25 lakh.
The relationship
$FV = 25 \times 1.10^{15} = 25 \times 4.177 = 104.4 \text{ lakh}$
25 today's cost in Rs lakh
1.10 one plus the 10% annual rise in education costs
15 years until the fee is paid
What it says in wordsThe future fee is today's fee grown at education inflation for the years until it is paid.
What do you tell the client after the number?
Two things. First, the inflation rate is an assumption, not a fact: education costs may rise faster or slower than 10%, and a two point change moves the answer by tens of lakhs, since 25 x 1.08 to the 15th is about Rs 79 lakh and 25 x 1.12 to the 15th is about Rs 1.37 crore. The inflation assumption matters as much as the return assumption, so state it and show the range.
Second, the saving plan must beat this cost growth, not general inflation. If the portfolio earns 10% and fees rise 10%, the money only keeps pace; it does not get ahead. That is why goal planning uses the goal's own inflation rate.
Where candidates lose it
The common answer is about Rs 62 lakh, from adding Rs 2.5 lakh a year for 15 years. It treats inflation as simple interest and quietly underfunds the goal by about 40%.
The second slip is using general consumer inflation for an education goal. Say which inflation rate you used and why, and give the range for two points either side.
What the interviewer asks next
How much must the client invest today at 12% to meet this cost?
If the course is abroad and the rupee weakens 3% a year, what changes?
Why might education inflation differ from consumer price inflation?
093A client's portfolio returns 12% a year and inflation is 6%. How long before his purchasing power doubles?Inflation and real returnCoreIndian wealth management
Try it first
How long until what his money buys has doubled?
Show the worked solution
About 12.6 years, not 12. The rupees double in about 6.1 years at 12%, but purchasing power grows at the real rate. The real rate is 1.12 divided by 1.06, less 1, which is 5.66%, not 6%. At 5.66% money doubles in about 12.6 years; 72 / 6 gives 12, a few months short.
Why is the real rate 5.66% and not 6%?
If your money grows 12% while a thali's price rises 6%, you can afford 1.12 / 1.06 = 1.0566 times as many thalis, not 1.06 times. Real return is found by dividing out inflation, not subtracting it, because inflation reduces what each rupee of the gain buys as well as the principal. Subtraction overstates the real rate slightly, and the error grows with inflation.
The relationship
$r_{real} = \frac{1.12}{1.06} - 1 = 5.66\% \qquad n = \frac{\ln 2}{\ln 1.0566} = 12.6 \text{ years}$
r_real the real return, growth in what the money buys
1.12 one plus the nominal return
1.06 one plus inflation
n years for purchasing power to double
What it says in wordsDivide out inflation to get the real rate, then find how many years at that rate make two.
At 12% the rupees in the account double in about 6.1 years, but at the 5.66% real rate what they buy takes about 12.6 years to double, a little longer than the 12 years that 72 / 6 suggests.
Is the difference between 12 and 12.6 years worth making?
In the room, yes, briefly: 12 years from the shortcut, 12.6 done properly, and the reason is division against subtraction. The bigger point is the gap between 6 years and 12.6: a client who hears his money doubles every six years believes he will be twice as well off, when in spending power he will need more than twice as long. That gap is what goal planning is built around.
Note the limits: inflation is not steady, the client's own inflation, school fees or medical costs, may run above the headline rate, and tax on the nominal return lowers the real rate further. Each of those pushes the doubling time out, not in.
Where candidates lose it
The fast wrong answer is six years, the rule of 72 on the nominal 12%. It answers when the rupees double, and the question asked about purchasing power.
The second slip is presenting exactly 12 years as the answer. It is a sound shortcut, but say that the real rate is 5.66% by division and that the true figure is about 12.6 years.
What the interviewer asks next
With inflation at 8% and the same 12% return, how long does purchasing power take to double?
How does a 20% tax on the nominal return change the answer?
Why might a retired client's personal inflation differ from the headline figure?

r_nominal	the salary growth over the period, 50%
pi	price growth over the same period, 1.06 to the 6th minus one
r_real	growth in what the salary buys

w_i	the category's share of monthly spending
\pi_i	the category's own yearly price rise

r	the deposit rate, 7%
t	the illustrative tax rate, 30%
pi	inflation, 5.5%

25	today's cost in Rs lakh
1.10	one plus the 10% annual rise in education costs
15	years until the fee is paid

Try it first

Why is the real raise so much smaller than it looks?

Why divide instead of subtracting inflation?

Where candidates lose it

What the interviewer asks next

Try it first

How is a personal inflation rate built?

Why does the gap matter for a financial plan?

Where candidates lose it

What the interviewer asks next

Try it first

In what order do tax and inflation come out?

What rate would the deposit need to break even?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer four times today's fee and not two and a half?

What do you tell the client after the number?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the real rate 5.66% and not 6%?

Is the difference between 12 and 12.6 years worth making?

Where candidates lose it

What the interviewer asks next