Private Wealth Management puzzles, solved step by step

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

Showing 11–20 of 100

011A client holds a stock bought at Rs 100 and sells a one-month call option with a Rs 110 strike for a premium of Rs 3. What is his maximum gain, and what does he give up if the stock ends the month at Rs 130?Options and structured productsWarm upPrivate banking
Try it first
The stock ends at Rs 130. What is the covered position worth, per share?
Show the worked solution
His maximum gain is Rs 13 a share, and at Rs 130 he gives up Rs 17. Above Rs 110 the call is exercised and the share goes at Rs 110, so the position is capped at Rs 110 plus the Rs 3 premium, Rs 113. At Rs 130 the stock alone would be worth Rs 130. The premium cushions a fall by only Rs 3: below Rs 97 he is losing money.
What exactly has the client sold?
Think of a landlord who takes a small non-refundable deposit from someone for the right to buy his flat at a fixed price within a month. If flat prices jump, the buyer exercises and the landlord gets only the fixed price plus the deposit. A covered call sells the upside above the strike in exchange for a small, certain fee today. The client keeps all the downside of owning the stock, less the premiumThe price the option buyer pays the seller up front, kept by the seller whatever happens next. received.
The covered call is worth Rs 3 more than the stock below the Rs 110 strike, but it is capped at Rs 113 above it, so at a price of Rs 130 the client gives up Rs 17 against simply holding the stock.
When does the trade help and when does it hurt?
Walk the three regions out loud. Below Rs 110 the covered call beats the plain stock by exactly the Rs 3 premium; above Rs 113 it falls behind by every rupee the stock rises. Between Rs 110 and Rs 113 the stock alone catches up. So the trade suits a client who expects the stock to drift sideways and wants some income, and it hurts the client who is secretly hoping for a sharp rally.
The relationship
$V_T = \min(S_T,\,110) + 3 \qquad V_{130} = 110 + 3 = 113$
S_T the stock price at expiry
110 the strike price of the call sold
3 the premium received up front
V_T the value of the stock plus the short call, per share
What it says in wordsThe covered position is worth the lower of the stock price and the strike, plus the premium already banked.
Say the risk plainly, because clients hear the word income and relax. If the stock falls to Rs 70, the position is worth Rs 73: the premium barely registers. A covered call is not protection; it is a trade of upside for a small, steady fee.
Where candidates lose it
The trap is adding the premium on top of the full stock price and saying Rs 133. It forgets that the share is called away at the strike, which is the whole point of the option sold.
The second loss is describing a covered call as a safe income strategy. The downside is almost entirely intact, and a wealth interviewer is listening for whether you tell a client that.
What the interviewer asks next
Where is the breakeven, and what is the position worth if the stock ends at Rs 90?
What changes if he sells a Rs 105 call for Rs 5 instead?
How would you explain this trade to a client who calls it free money?
012A product gives twice the daily return of an index, resetting every day. The index rises 10% one day and falls 10% the next, ending down 1%. Where does the 2x product end?Leverage and borrowingCorePrivate banking
Try it first
Pick the 2x product's result over the two days.
Show the worked solution
Down 4%. The index goes 100, 110, 99, a 1% loss. The 2x product goes up 20% to 120, then down 20% to 96. It resets its leverage each day, so over several days it compounds twice the daily moves rather than delivering twice the period's return. In a choppy market that difference always works against the holder.
Why is it not simply twice the index's 1% loss?
Imagine a shop that runs a 20% sale one day and a 20% price rise the next. The price does not come back to where it started: 100 becomes 80 and then 96. Equal percentage moves up and down do not cancel, and leverage doubles the size of each move, so it more than doubles the leak. The index loses 1% from the up-and-down, the 2x product loses 4%.
The index goes from 100 to 110 to 99, down 1%, while the 2x daily product goes from 100 to 120 to 96, down 4%, twice the naive expectation of 2%.
Where does the extra loss come from?
From volatility dragThe shortfall of compounded growth below the average return, caused by returns bouncing up and down. It grows roughly with the square of the size of the moves.. A move of plus x then minus x leaves you with 1 minus x squared. For the index that is 1 minus 0.01; for the 2x product it is 1 minus 0.04, four times the leak, because the drag grows with the square of the move. Double the leverage and you quadruple the drag.
The relationship
$(1 + Lx)(1 - Lx) = 1 - L^2x^2 \qquad L=2,\; x=0.10:\; 1 - 0.04 = 0.96$
L the leverage multiple, here 2
x the size of each daily move, 10%
What it says in wordsAn up move and an equal down move leave the position short by the square of the leveraged move.
Say what it means for a client. A daily reset product can be close to its promise over a day and far from it over a year, even if the index ends flat. It is built for short holding periods; held for months in a choppy market it can lose money while the index goes nowhere.
Where candidates lose it
The trap is answering minus 2%, doubling the period's result. That reading treats a daily product as if it gave twice the return over any horizon, which is exactly the misunderstanding that hurts clients who hold these products for months.
The second loss is getting 96 and not explaining why. The one-line reason is that leverage doubles each move and the leak grows with the square of the move.
What the interviewer asks next
The index goes up 10% and then up 10% again. How does the 2x product do against twice the index?
What happens to a 3x daily product over the same two days?
Why is a daily reset product a poor fit for a client planning to hold for a year?
013A client with Rs 5 crore can pay an adviser a flat 1% advisory fee on all his assets, or use a distributor who earns a 1.2% trail commission on the 70% of his money held in mutual funds. Which costs him less each year?Wealth business economicsWarm upIndian wealth management
Try it first
Which annual bill is smaller, in rupees?
Show the worked solution
The commission route costs less on these numbers: Rs 4.2 lakh a year against Rs 5.0 lakh. The advisory fee is 1% of the whole Rs 5 crore. The trail is 1.2% of only the Rs 3.5 crore held in funds. The two bills would match if 83.3% of his assets earned trail, and the cheaper bill says nothing yet about the quality or independence of the advice.
Why does the higher rate give the smaller bill?
A phone plan charging Rs 2 a minute on calls only can cost less than one charging Rs 1.50 a minute on calls and data together, if you rarely use data. A fee rate means nothing until it is multiplied by the base it is charged on, so fee models are compared in rupees on the client's actual asset mix. Here 1.2% applies to Rs 3.5 crore and 1% applies to Rs 5 crore.
On a Rs 5 crore portfolio the 1% advisory fee is a Rs 5.0 lakh bill, while a 1.2% trail on the Rs 3.5 crore in funds is Rs 4.2 lakh, and the two only match when 83.3% of assets pay trail.
What else should the comparison include?
Visibility and incentives. A trail is paid out of the fund's expense ratio, so the client never sees it as a bill, while an advisory fee arrives as an invoice he has to approve. The trail also pays more when more money sits in trail-paying funds, which is a pull away from direct equity, bonds or direct plansVersions of a mutual fund scheme bought without a distributor, with a lower expense ratio because no commission is paid out of them.. If the distributor moved another Rs 1 crore into funds, the trail bill would rise to Rs 5.4 lakh.
The relationship
$\text{Fee} = 0.01 \times 5 = \text{Rs } 5.0\text{ lakh} \qquad \text{Trail} = 0.012 \times 0.70 \times 5 = \text{Rs } 4.2\text{ lakh}$
5 the client's assets, Rs crore; 0.01 of a crore is Rs 1 lakh
0.70 the share of assets held in trail-paying funds
0.012 the trail rate a year
What it says in wordsMultiply each rate by the assets it is charged on, then compare rupees.
Add the regulatory point as a framework, not a fact from memory: Indian rules separate registered advisers who charge fees from distributors who earn commissions, and limit doing both for the same client. The current regulations need checking before any of this reaches a client.
Where candidates lose it
The trap is comparing 1% with 1.2% and declaring the fee cheaper. The rates apply to different bases, and the rupee bill is the only honest comparison.
The second loss is stopping at the cheaper bill. The interviewer wants to hear that the trail is invisible and that it rewards keeping money in funds, because that is how the two models shape advice differently.
What the interviewer asks next
At what share of assets in funds do the two models cost the same?
The distributor proposes moving the other Rs 1.5 crore into funds. What happens to his income and the client's bill?
How would you explain the invisible trail to a client in one sentence?
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%.
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
$\text{EAR} = \left(1 + \frac{0.12}{12}\right)^{12} - 1 = 12.68\% \;>\; 12.50\%$
0.12/12 the monthly rate, 1%
12 the number of compounding periods in a year
EAR the 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?
015An equity index rises 8% a year on price for 20 years and also pays a 1.5% dividend yield, reinvested every year. How much more wealth does the total return version end with than the price index?Returns arithmeticCoreMutual fund distributionIndian wealth management
Try it first
Pick the extra ending wealth from reinvesting the dividends.
Show the worked solution
About 32% more wealth. Price growth alone turns Rs 1 into 1.08 to the 20th, Rs 4.66. Reinvesting a 1.5% dividend each year makes the total return 9.5% a year, and 1.095 to the 20th is Rs 6.14. The ratio is 1.32. A 1.5 point yearly difference compounds into a gap of about a third of the price-only result.
Why does a small yield make such a large gap?
Think of a fruit tree whose fruit you plant every season instead of eating. Each season's seeds become trees that also bear fruit. Reinvested dividends buy more units, and those units then earn both the price growth and their own dividends for the rest of the period. So the 1.5% is not added twenty times to the starting amount; it raises the compounding rate on the whole holding from 8% to 9.5%.
Over 20 years Rs 1 grows to 4.66 on price alone at 8% a year but to 6.14 with a 1.5% dividend reinvested, so the total return version ends about 32% richer, with most of the gap opening late.
What if the client takes the dividends as cash?
Then the dividends stop compounding. Each year's dividend is 1.5% of the price-only holding, and across 20 years they add up to about 0.69 of the starting rupee. Price plus cash dividends ends at about 5.35 times, well short of the 6.14 times from reinvesting. This is why comparing a fund's return with a price indexAn index that tracks only the prices of its stocks and ignores the dividends they pay. flatters the fund: the fair benchmark is the total return index.
The relationship
$\frac{(1.095)^{20}}{(1.08)^{20}} = \frac{6.14}{4.66} \approx 1.32$
1.095 total return a year: 8% price growth plus a 1.5% dividend yield on the opening price
1.08 price return a year
20 years
What it says in wordsThe extra wealth from reinvesting is the ratio of the two compounded growth factors.
State the simplification. Real dividend yields move from year to year and are taxed in the client's hands, which lowers what is reinvested. The 32% is the gap under the question's clean assumptions, not a promise about any index.
Where candidates lose it
The trap is adding the yield in a straight line: 1.5% times 20 years is 30 points, so the total return index ends at about 4.96 times. That treats the dividends as cash put in a drawer, not reinvested.
The mirror error is thinking the dividend makes little difference because 1.5% sounds small. Show the 6.14 against 4.66 and the point makes itself.
What the interviewer asks next
What would the gap be over 30 years instead of 20?
A fund reports beating the price index by 1% a year. What does that say about its skill?
How does tax on dividends change the answer for a client in a high bracket?
016A portfolio management service charges a 2% fixed fee plus 20% of returns above an 8% hurdle. In a year when the gross return is 15% on Rs 1 crore, what is the client's net return, and what share of the gross gain went in fees?Fee and cost dragHardIndian wealth management
Try it first
Pick the client's net return, with the performance fee charged on the return after the fixed fee.
Show the worked solution
The client nets 12%, and fees take 20% of the gross gain. The 2% fixed fee brings 15% down to 13%. That is 5 points above the 8% hurdle, and 20% of 5 points is a 1% performance fee. Net return is 12%, Rs 12 lakh on Rs 1 crore. Fees are Rs 3 lakh of the Rs 15 lakh gross gain. The exact order of fees is set by the agreement.
Why do the two fees stack rather than sit side by side?
Think of a restaurant bill with a fixed cover charge and then a service charge worked out on what is left. Each charge is modest on its own, but together they take a real share of the meal. The fixed fee is paid in every year, good or bad, and the performance fee then takes a slice of whatever is left above the hurdle, so in a good year the two stack. Here 2 points and 1 point together take a fifth of a 15% year.
On Rs 1 crore a 15% gross year loses 2 points to the fixed fee and 1 point to the performance fee on the 5 points above the 8% hurdle, leaving the client 12%, so fees take Rs 3 lakh of the Rs 15 lakh gain.
What changes if the agreement charges the performance fee on the gross return?
Then the hurdle is measured before the fixed fee. The performance fee becomes 20% of 7 points, 1.4%, the net return falls to 11.6%, and fees take 22.7% of the gain instead of 20%. The same headline terms give two answers, which is why the order of calculation, the high-water markA rule that performance fees are only paid on gains above the highest value the account has previously reached, so the client does not pay twice for recovering a loss. and the basis for the fixed fee all have to be read in the agreement.
On Rs 1 crore Fee after fixed fee Fee on gross
Gross gain, Rs lakh 15.0 15.0
Fixed fee, 2% (2.0) (2.0)
Performance fee, 20% above 8% (1.0) (1.4)
Net gain, Rs lakh 12.0 11.6
Share of gross gain paid in fees 20.0% 22.7%
Charging the performance fee after the fixed fee leaves the client Rs 12.0 lakh; charging it on the gross return leaves Rs 11.6 lakh, so the same headline terms differ by Rs 40,000 on Rs 1 crore.
One more layer is worth a sentence: fees usually attract GST in India, charged on top, which widens the gap further. Confirm the current rate and treatment before quoting a client a net figure.
Where candidates lose it
The trap is taking the performance fee as 20% of the whole 15% gross return, or forgetting the fixed fee comes out first. Both give a wrong net figure with total confidence.
The quieter loss is quoting 12% without the convention. Say that you have assumed the performance fee is charged after the fixed fee, and give the other answer, 11.6%, in the same breath.
What the interviewer asks next
What is the net return in a year when the gross return is 8%?
At what gross return do fees take exactly a quarter of the gain?
How does a high-water mark change the fee in the year after a loss?
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?
018One fund in ten is truly skilled and beats its benchmark in 70% of years; the rest are unskilled and beat it in 50% of years. A fund has just beaten its benchmark three years running. What is the chance it is skilled?Probability and risk of lossHardWealth management
Try it first
After three straight wins, how likely is the fund to be skilled?
Show the worked solution
About 23%. Take 1,000 funds: 100 skilled and 900 unskilled. A skilled fund wins three years running with chance 0.7 cubed, 34.3%, so 34.3 funds. An unskilled fund does it with chance 0.5 cubed, 12.5%, so 112.5 funds. Of the 146.8 funds with a streak, 34.3 are skilled: 23.4%. The streak moves the odds from 10% to 23%, but most streak funds are still unskilled.
Why is the answer not close to 70%?
Think of a medical test that is good but not perfect, used for a rare condition. Most positive results come from the many healthy people, simply because there are so many more of them. How common skill is to begin with matters as much as how well skill shows up in results. Skilled funds are one in ten, so even though they streak more often, unskilled funds produce most of the streaks by sheer numbers.
Of 1,000 funds, 34.3 skilled and 112.5 unskilled funds post a three-year winning streak, so a fund with a streak is skilled only 23.4% of the time, up from a starting 10%.
How do you set it up without a formula sheet?
Use natural frequencies. Pick a round population, 1,000 funds, and count how many land in each branch; the answer is one count over the total count. That is Bayes ruleA way of updating a starting probability with new evidence, by weighing how likely the evidence is under each possible explanation. without the notation, and it is far harder to get wrong out loud. The formula version gives the same 23.4%.
The relationship
$P(S\mid W^3) = \frac{0.1 \times 0.7^3}{0.1 \times 0.7^3 + 0.9 \times 0.5^3} = \frac{0.0343}{0.1468} \approx 23\%$
S the fund is skilled
W^3 three wins in a row
0.1, 0.9 the share of skilled and unskilled funds before any results
0.7^3, 0.5^3 the chance of a three-year streak for each type
What it says in wordsThe chance of skill given a streak is the skilled streaks divided by all streaks.
The client version is one sentence: a three-year record is weak evidence on its own. It is also worth naming the limitation of the model: real skill is not a clean 70% and fund returns are not independent year to year, but the direction of the answer survives both.
Where candidates lose it
The trap is answering 70% or 34%, the numbers attached to skilled funds. Both describe how a skilled fund behaves, not how many streak funds are skilled, and confusing the two is the most common error in probability questions.
The other loss is fumbling the formula. Counting 1,000 funds through the tree is faster, is easier to say, and checks itself.
What the interviewer asks next
What if the fund has won five years running?
If one fund in four were skilled, what would three wins imply?
How would you use this when a client wants to buy last year's top fund?
019An annuity pays 7% of the purchase price every year for life and returns nothing on death. Is the client earning 7%? What is the return if he lives for 20 years?Retirement and withdrawalHardPrivate banking
Try it first
For a client who lives exactly 20 years after buying, what is the internal rate of return?
Show the worked solution
No: for a 20-year life the return is about 3.4% a year. He pays 100 and receives 7 a year for 20 years, 140 in total, of which 100 is his own capital coming back and only 40 is return. The rate that makes 20 payments of 7 worth 100 today is 3.44%. The payout rate is not a yield, because the capital is never returned.
Why is a 7% payout not a 7% return?
Imagine lending a friend Rs 1 lakh and being repaid Rs 7,000 a year until one of you dies, with nothing more after that. The Rs 7,000 is partly interest and partly your own Rs 1 lakh coming back in slices. An annuity payout rate includes the return of the buyer's own capital, because nothing is paid back at death; only the interest part is return. A bank deposit paying 7% hands back the Rs 1 lakh at the end as well; the annuity does not.
Over a 20-year life each Rs 7 payment splits into interest at 3.44% and the client's own capital coming back, so of the Rs 140 received only Rs 40 is return and Rs 100 is his own money.
How does the return change with how long he lives?
Lifespan is the whole trade. If he dies after 10 years the return is about -6.0%, negative, because he gets back only 70; at 20 years it is 3.4%; at 30 years it reaches 5.7%. An annuity is insurance against living long, priced so that the insurer wins on clients who die early. That is its purpose, and the client should buy it for longevity protectionIncome that continues however long the client lives, so the risk of outliving savings passes to the insurer., not for yield.
The relationship
$100 = \sum_{t=1}^{20} \frac{7}{(1+r)^t} \;\Rightarrow\; r \approx 3.44\%$
100 the purchase price
7 the yearly payment
20 the number of years the client lives
r the internal rate of return
What it says in wordsThe return is the one rate that makes the stream of payments worth exactly what the client paid.
Add what a private banker would. Annuity income is usually taxed as income in the year received, and payouts are fixed in rupees, so inflation erodes them. The 7% here is an illustration; real annuity rates depend on age, the option chosen and the current rate environment, and have to be taken from a current quote.
Where candidates lose it
The trap is calling the 7% a yield and comparing it with a 7% deposit. The deposit returns the capital at the end; the annuity never does, so the two numbers measure different things.
The opposite loss is dismissing the annuity as a bad return. It is insurance against a long life, and the interviewer wants to hear that the return depends on lifespan, not a verdict.
What the interviewer asks next
What if the annuity returned the purchase price to his heirs at death? How would the payout rate change?
At what lifespan does the return reach 5%?
How would inflation of 5% a year change the real value of the last payment?
020A perpetual bond with a 9% coupon trades at 104 per 100 of face value. The issuer can call it at par in two years and is expected to. What yield is the client actually buying?Fixed income numeracyHardPrivate banking
Try it first
If the bond is called at par in two years, what is the yield?
Show the worked solution
About 6.8%, not 9%. The client pays 104 and, if the call is used, receives a coupon of 9 after one year and 9 plus 100 after two. The Rs 4 of premium above par is lost at the call. The yield that discounts 9 and 109 back to 104 is 6.79%. For a bond above par that is likely to be called, the yield to call is the honest figure.
Why does the call matter so much?
Think of paying extra for a flat on a lease you expect to run for ever, then learning the landlord can end it after two years and refund only the base price. The rent was fine; the premium you paid is gone. A bond bought above par loses the premium if it is called at par, so its yield must be measured to the call date, not as if the coupon ran for ever. Here 4 of premium is lost over just two years.
The client pays 104, receives 9 and then 109 at the call, losing the 4 of premium above par, so the yield to call is 6.79% against a coupon of 9% and a current yield of 8.65%.
How do you estimate it in your head?
Spread the premium loss over the years to the call. Coupons of 9 a year less 4 of premium over two years is about 7 a year, on an average price of about 102, which is roughly 6.9%, close to the exact 6.79%. The shorter the time to the call, the bigger the drag per year, because the same premium is lost over fewer coupons.
The relationship
$104 = \frac{9}{1+y} + \frac{109}{(1+y)^2} \;\Rightarrow\; y \approx 6.79\%$
104 the price paid
9 the annual coupon
109 the final coupon plus the 100 repaid at the call
y the yield to call
What it says in wordsThe yield to call is the rate that makes the coupons up to the call and the call price worth the price paid.
Name the second risk. A perpetual bond is callable, not must-call: if rates rise or the issuer weakens, it may not be called, and the client then holds a bond with no maturity at all. Desks quote the yield to worstThe lowest yield across all the dates on which the bond could be called or mature, the conservative figure for a callable bond. for exactly this reason, and a client needs both scenarios before buying.
Where candidates lose it
The trap is quoting the 9% coupon, or the 8.65% current yield, as the return. Both ignore that the client paid 104 for something that will most likely be repaid at 100 in two years.
The second loss is treating the call as certain. Say what happens if it is not called: a perpetual with no maturity, whose price can fall a long way.
What the interviewer asks next
What is the yield to call if the call is in one year instead of two?
The bond is not called and trades at 90. What is its current yield?
Why do issuers usually call a bond like this when rates fall?

S_T	the stock price at expiry
110	the strike price of the call sold
3	the premium received up front
V_T	the value of the stock plus the short call, per share

L	the leverage multiple, here 2
x	the size of each daily move, 10%

5	the client's assets, Rs crore; 0.01 of a crore is Rs 1 lakh
0.70	the share of assets held in trail-paying funds
0.012	the trail rate a year

0.12/12	the monthly rate, 1%
12	the number of compounding periods in a year
EAR	the effective annual rate

Try it first

What exactly has the client sold?

When does the trade help and when does it hurt?

Where candidates lose it

What the interviewer asks next

Try it first

Why is it not simply twice the index's 1% loss?

Where does the extra loss come from?

Where candidates lose it

What the interviewer asks next

Try it first

Why does the higher rate give the smaller bill?

What else should the comparison include?

Where candidates lose it

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Why is 12% monthly not 12%?

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

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Why does a small yield make such a large gap?

What if the client takes the dividends as cash?

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Why do the two fees stack rather than sit side by side?

What changes if the agreement charges the performance fee on the gross return?

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

Why divide instead of subtracting inflation?

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Why is the answer not close to 70%?

How do you set it up without a formula sheet?

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Why is a 7% payout not a 7% return?

How does the return change with how long he lives?

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Why does the call matter so much?

How do you estimate it in your head?

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