Investment Banking puzzles, solved step by step

Puzzles: 100
Traced to a firm: 36
Topics: 12
Hard: 29

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021A company writes down Rs 10 of inventory and the write-down is tax deductible at 25%. Walk it through the three statements.Accounting riddlesWarm upBulge bracket IBMiddle market IB
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What happens to the company's cash?
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Net income falls Rs 7.5, cash rises Rs 2.5, inventory falls Rs 10 and retained earnings fall Rs 7.5. The write-down is an expense, so pre-tax income falls 10 and, after the Rs 2.5 tax saving, net income falls 7.5. No cash left the business, so the cash flow statement adds the 10 back and cash ends Rs 2.5 higher. Assets fall 7.5 and equity falls 7.5, so it balances.
Why does a loss leave the company with more cash?
A shopkeeper finds a carton of biscuits past their date. They cost Rs 10 and are now worth nothing. No money changes hands today; the cash went out when the biscuits were bought. What changes is that the loss lowers this year's taxable profit. A write-down is a non-cash loss, so its only cash effect is the tax it saves, and cash rises by 25% of Rs 10. Inventory is carried at the lower of cost and net realisable valueWhat the stock can be sold for, less the costs of selling it., which is why the carrying amount is cut when goods lose value.
The Rs 10 write-down cuts net income by Rs 7.5 after tax, the cash flow statement adds the non-cash Rs 10 back so cash rises Rs 2.5, and on the balance sheet inventory down 10 and cash up 2.5 match retained earnings down 7.5.
How does each statement move?
Income statement: the write-down usually sits inside cost of goods sold, so pre-tax income falls 10, tax falls 2.5 at 25%, and net income falls 7.5. Cash flow statement: start from net income of minus 7.5, add back the 10 because no cash left, and operating cash flow is plus 2.5. Balance sheet: inventory is down 10 and cash is up 2.5, so total assets are down 7.5, and retained earnings are down 7.5. The check is minus 10 plus 2.5 on the asset side equalling minus 7.5 in equity.
The relationship
$\Delta \text{Cash} = \underbrace{-10\times(1-0.25)}_{\text{net income}} + \underbrace{10}_{\text{add-back}} = -7.5 + 10 = +2.5$
-10 x (1 - 0.25) the write-down after its 25% tax saving, which is the fall in net income
10 the write-down added back because no cash left the business
What it says in wordsCash moves by net income plus the non-cash charge, which leaves only the tax saving.
What assumption should you say out loud?
That the write-down is deductible for tax now, as the question states. Whether a tax system allows the deduction when the stock is written down or only when it is sold depends on its rules, which you would confirm. If the deduction comes later, cash does not move this year and the company records a deferred tax asset of Rs 2.5 instead, with the balance sheet still balancing. Offering that variant in one sentence shows you understand why the cash moved in the first place.
Where candidates lose it
The usual slip is to say cash falls by 10, as if the write-down were a payment. The cash went out when the inventory was bought; today's entry only recognises that the asset is worth less.
The second slip is forgetting the tax. Without it, net income falls 10, the add-back is 10, cash is unchanged and the balance sheet still balances, so the error hides itself. The tax rate is in the question precisely so that cash moves by 2.5.
What the interviewer asks next
What changes if the write-down is not deductible until the goods are sold?
How is an impairment of goodwill treated differently for tax?
If the written-down stock is later sold for Rs 4, walk that through the statements.
023At 9% a year, roughly how long does money take to double? Check the rule of 72 against the exact answer and say where the rule breaks down.Growth and compoundingWarm upMiddle market IBPrivate equity
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Answer inside five seconds.
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About 8 years: 72 divided by 9 is 8.0, and the exact answer is 8.04 years. The exact doubling time is the log of 2 divided by the log of 1.09. The rule is a shortcut built for moderate rates: it is almost exact around 8%, slightly long at low rates and increasingly short at high ones. At 40% it says 1.8 years against an exact 2.06.
Why does 72 work at all?
Think of a sapling that grows 9% taller each year; the question is how many of those steps multiply up to 2. The exact answer uses logarithms: years equal ln 2 divided by ln(1 + r). For small r, ln(1 + r) is close to r, and ln 2 is 0.693, so the exact rule is close to 69.3 divided by the rate in per cent. 72 replaces 69.3 because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because ln(1 + r) sits below r at the rates people actually meet, which pushes the true constant up. At 9%, ln 1.09 is 0.0862, and 0.693 / 0.0862 is 8.04.
The rule of 72 and the exact doubling time almost coincide at moderate rates, 8.0 against 8.04 years at 9%, but the rule runs about 3% long at a 2% rate and 12.6% short at 40%.
The relationship
$t = \frac{\ln 2}{\ln(1.09)} = \frac{0.6931}{0.0862} = 8.04 \qquad t_{72} = \frac{72}{9} = 8.0$
ln 2 the natural log of 2, about 0.693, because the money must double
ln(1.09) the log of one year's growth factor at 9%
72 / 9 the rule of 72 with the rate in per cent
What it says in wordsThe exact doubling time is the log of 2 over the log of one year's growth; the rule of 72 approximates that ratio for moderate rates.
Where does the rule break down?
At high rates. ln(1 + r) falls further below r as r grows, so the true doubling time is longer than 72 divided by r: at 20% the rule says 3.6 years against 3.80, and at 40% it says 1.8 against 2.06, an error of 12.6%. At very low rates it errs the other way: 36 years against 35.0 at 2%. The rule is within about 1% of the exact answer only between roughly 6% and 10%, and should be adjusted outside that band.
A common adjustment for high rates adds one to the 72 for every three points of rate above 8%. At 20% that gives 76 divided by 20, 3.80 years, and at 40% about 82.7 divided by 40, 2.07 years, both within a hundredth or two of the exact figures. For deal work, where target returns of 20% to 30% are common, that adjustment is worth knowing.
Where candidates lose it
Candidates either answer 8 and stop, or try to compute logarithms in their head and stall. Give 8 at once, then say the exact figure is a touch above, about 8.04, because the rule is tuned for rates near 8%.
The loss that costs more is not knowing where the rule fails, when the question asks. Say that at high rates it understates the time, give the 40% example, and offer the adjustment of one extra point on the 72 for every three points above 8.
What the interviewer asks next
How long does money take to triple at 9%?
Why is 69.3 the exact constant under continuous compounding?
An investment doubles in 5 years. What annual return is that, roughly and exactly?
025Mental maths round: what is 17% of 340 plus 34% of 170?Mental maths and countingWarm upBulge bracket IBMiddle market IB
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Your answer?
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115.6. Notice that 34% of 170 is the same as 17% of 340: halving one number and doubling the other leaves a product unchanged. So the sum is 17% of 340 twice, which is 34% of 340. That is 30% of 340, which is 102, plus 4% of 340, which is 13.6, giving 115.6.
What is the trick hiding in the numbers?
Two plots of land, one 34 metres by 17 and one 17 metres by 34, have the same area; turning a rectangle on its side does not change it. Percentages behave the same way, because x% of y is x times y divided by 100. The percentage and the base can trade places, so 34% of 170 equals 17% of 340: each is 17 x 340 / 100, which is 57.8. Once you see that, the question is one product, not two.
Drawn to scale, a rectangle 340 wide and 17 tall has the same area as one 170 wide and 34 tall, so 17% of 340 and 34% of 170 are both 57.8 and together make 34% of 340, which is 115.6.
The relationship
$17\% \times 340 + 34\% \times 170 = 2 \times \frac{17 \times 340}{100} = 34\% \times 340 = 115.6$
17% x 340 the first term, 57.8
34% x 170 the second term, the same product with the factor of 2 moved across
34% x 340 the two equal terms combined
What it says in wordsHalving the base and doubling the percentage leaves a percentage unchanged, so the two terms are equal and add to one simple product.
How do you do 34% of 340 in your head?
Split it into easy pieces: 30% of 340 is 102 and 4% of 340 is 13.6, so the total is 115.6. Or notice that 34% of 340 is 34 x 3.4, and 34 x 34 is 1,156, so the answer is 115.6. Spotting a structure first and calculating second is the habit the interviewer is checking, and it usually turns two awkward products into one easy one.
How do you check it before you say it?
Estimate first: 17% is a little more than a sixth, and a sixth of 340 is about 57, so each term is near 57 and the sum near 115. Then check each term directly: 10% of 340 is 34 and 7% is 23.8, so 17% is 57.8; 30% of 170 is 51 and 4% is 6.8, so 34% of 170 is 57.8. The two terms match, which confirms the swap.
Where candidates lose it
The common loss is grinding out both products separately and dropping a decimal in one of them; 17 x 3.4 and 34 x 1.7 are easy to mangle under pressure, and the interviewer watches the hesitation.
The other loss is missing the point of the question. A mental maths round with suspiciously related numbers is inviting you to look for a shortcut; saying out loud that 34% of 170 is the same as 17% of 340 earns more credit than fast arithmetic.
What the interviewer asks next
What is 8% of 25?
What is 12.5% of 64 plus 25% of 32?
What is 15% of 60 plus 30% of 30 plus 45% of 20?
045Twelve bankers meet at an offsite and every pair shakes hands exactly once. How many handshakes are there?Mental maths and countingWarm upBulge bracket IBMiddle market IB
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Answer fast.
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66 handshakes. Each of the 12 bankers shakes hands with the other 11, which gives 12 x 11 = 132, but every handshake has two people in it and so has been counted twice. Half of 132 is 66. The same number falls out of adding 11 + 10 + 9 and so on down to 1, or from 12 choose 2.
Why do you halve the count?
Think of counting the phone calls in a group where everyone rang everyone else once. If each person reports the calls they took part in, every call shows up in two reports. Each handshake involves two people, so counting from each person's side counts every handshake exactly twice, and the true number is half. Twelve people times eleven others is 132 ends of handshakes, which is 66 handshakes.
Twelve people on a circle with one line for every pair make 66 lines: each person sits at the end of 11 lines, giving 132 line ends, and every line has two ends, so the count halves to 66.
How do you check 66 another way?
Let people arrive one at a time. The second person shakes 1 hand, the third 2, and so on to the twelfth, who shakes 11, and adding 1 through 11 gives 66. Pair the terms to add them quickly: 1 and 11, 2 and 10, and so on make five pairs of 12 plus a 6 in the middle, 66. Two routes to the same number is the check interviewers like to hear.
The relationship
$\binom{12}{2} = \frac{12 \times 11}{2} = 66 = 1 + 2 + \cdots + 11$
12 x 11 each person times the others they meet, counting every handshake from both sides
2 the two people in every handshake
What it says in wordsPairs from a group of n are n times (n minus 1), halved.
Where does this count show up on a desk?
Anywhere pairs matter. The number of pairs grows roughly with the square of the group: double the group to 24 and the handshakes rise to 276, more than four times as many. That is why a 12-stock portfolio has 66 pairwise correlationsMeasures of how closely two things move together, from minus 1 to plus 1. to estimate, and why a deal with many parties needs far more conversations than its headcount suggests.
Where candidates lose it
The fast wrong answer is 132, from 12 times 11 without halving. It treats one handshake between two people as two separate events.
The other slip is 144 or 78, from letting people shake their own hand. Say each person shakes hands with the other 11, and both errors disappear.
What the interviewer asks next
How many people must be in the room for there to be 105 handshakes?
At a dinner of six couples, everyone clinks glasses with everyone except their own partner. How many clinks?
How many pairwise correlations does a 30-stock portfolio have?
046Fund A returns +30% and then -10%. Fund B returns +10% and then +10%. Starting with Rs 100 in each, which ends higher?Growth and compoundingWarm upMiddle market IBPrivate equity
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Which ends higher?
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Fund B ends higher, at Rs 121 against Rs 117. Fund A goes to 130 and then loses 10% of 130, ending at 117. Fund B compounds 10% twice to 121. Both average 10% a year, but A's compound growth rate is only about 8.2% because its returns swing. Volatility drags on compound returns: roughly half the variance comes off the average each year.
Why does the same average give different endings?
Think of a salary that rises 30% one year and is cut 10% the next, against one that rises 10% twice. The cut lands on the higher salary, so it takes away more rupees than the same percentage would have earlier. Returns multiply rather than add, so a loss after a gain is taken from a bigger base, and an uneven path ends below a smooth one with the same average. Rs 100 grows to 130 and then gives back Rs 13; the steady fund never gives anything back.
Fund A rises to 130 and falls to 117 while Fund B climbs to 110 and then 121; both average 10% a year, but A compounds at only 8.2% because of its swing, so B ends Rs 4 higher.
How big is the drag, and can you estimate it in your head?
The growth rate that matters is the geometric averageThe constant yearly return that would turn the starting amount into the ending amount over the same period.. Fund A's is the square root of 1.17 minus 1, about 8.2%, against B's 10%, even though both arithmetic averages are 10%. A quick estimate: subtract half the variance. A's returns sit 20 points either side of 10%, so half of 0.2 squared is 2 points, and 10% minus 2% is about 8%, close to the exact 8.2%.
The relationship
$g_A = \sqrt{1.30 \times 0.90} - 1 \approx 8.2\% \qquad g_A \approx \bar r - \tfrac{\sigma^2}{2} = 10\% - 2\% = 8\%$
r-bar the arithmetic average return, 10%
sigma how far returns swing around the average, 20 points for Fund A
What it says in wordsCompound growth is roughly the average return minus half the variance.
Why would an interviewer care?
Because fund reports often quote average returns, and investors live on compound ones. Two funds with the same average return can leave an investor with very different money, and the more volatile one leaves less. The same arithmetic explains why a fund that rises 50% and then falls 50% is down 25%, and why leverage that doubles volatility can lower long-run growth even while it raises the average.
Where candidates lose it
The trap is answering that they end level, because both funds average 10% a year. Averaging percentages assumes they add, and returns multiply.
The second loss is getting 117 and 121 without saying why. Name volatility drag and give the half-the-variance estimate; that turns a calculation into an insight.
What the interviewer asks next
A fund rises 50% and then falls 50%. Where does it end?
What steady yearly return matches Fund A over the two years?
Fund C returns +40% and then -20%. How does it compare with A and B?

054On the last day of the financial year, a company buys a Rs 100 crore machine on 60-day credit from the supplier. What changes on each of the three statements at year end?Accounting riddlesWarm upBulge bracket IBMiddle market IB

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Which statement moves at year end?

Show the worked solution

Only the balance sheet changes. Property, plant and equipment rises by Rs 100 crore and accounts payable rises by Rs 100 crore, so both sides grow by the same amount. No cash has moved, so the cash flow statement is untouched, and no time has passed for depreciation, so the income statement is untouched too. Cash and capex appear in 60 days, when the supplier is paid.

Why does a purchase on credit touch neither cash nor profit?

Think of buying a refrigerator on a shop's 60-day credit. The fridge is in your kitchen today and you owe the shop, but your bank balance has not moved and nothing has come out of this month's budget. A credit purchase adds an asset and a debt of the same size, so the balance sheet grows on both sides and nothing else moves. The machine will be used for years, so its cost is not an expense on the day it arrives. It reaches the income statement slowly, through depreciation, over the years it is used.

At year end the Rs 100 crore machine raises property, plant and equipment by 100 and accounts payable by 100, while the income statement and the cash flow statement show nothing. Cash and capex move only on day 60, when the supplier is paid.

What happens over the next 60 days and beyond?

Day 60 is when the cash flow statement wakes up. The company pays the supplier: cash falls by 100 and payables fall by 100. The Rs 100 crore appears as capital expenditure in investing cash flow in the year it is paid, not the year the machine arrived. The notes to the accounts usually flag the year-end purchase as a non-cash investing item, so a reader is not surprised. From the following year, depreciation starts: on a 10-year straight line, Rs 10 crore a year comes off pre-tax profit, is added back in operating cash flow, and lowers the machine's book value.

Moment	Balance sheet	Income statement	Cash flow statement
Year end, machine arrives	PP&E +100, payables +100	No change	No change
Day 60, supplier paid	Cash -100, payables -100	No change	Investing outflow -100
Each later year, 10-year life	PP&E -10	Depreciation -10 before tax	Depreciation added back

Rs crore. The same machine touches the balance sheet on day one, the cash flow statement on day 60 and the income statement only from the following year, through depreciation of Rs 10 crore a year.

Close by saying the balance check out loud. Assets up 100, liabilities up 100: the sheet balances, and that one sentence tells the interviewer you walk the statements in a fixed order rather than guessing. Interviewers use small timing questions like this one to see whether you separate when something is owned, when it is paid for and when it is expensed.

Where candidates lose it

The common slip is putting Rs 100 crore of capex on the cash flow statement at year end, because buying a machine feels like capex. The cash flow statement records cash paid, and the company has paid nothing yet.

The second slip is expensing the machine, or charging a full year of depreciation on day one. Say the timing out loud: asset and debt today, cash in 60 days, depreciation from next year.

What the interviewer asks next

Now the company pays cash on day one instead. Walk me through the three statements.
At the end of next year, with a 10-year life and a 25% tax rate, what has changed on each statement?
The machine turns out to be faulty and is returned before the invoice is paid. What reverses?

056A company is funded 60% by equity at a 14% cost and 40% by debt at 10% before tax, and the tax rate is 25%. What is its weighted average cost of capital?DCF and cost of capitalWarm upBulge bracket IBMiddle market IB
Try it first
Pick the WACC.
Show the worked solution
The WACC is 11.4%. Equity is 60% of the funding at 14%, contributing 8.4 points. Debt is 40% at 10% before tax, but interest is tax-deductible, so its after-tax cost is 10% x (1 minus 25%) = 7.5%, contributing 3.0 points. Add them: 8.4 + 3.0 = 11.4%. Leaving out the tax shield gives 12.4%.
Why is it a weighted average and not a plain one?
A family buying a flat with 60% from savings and 40% from a home loan pays a blended cost that leans towards the bigger source. Each source of money counts in proportion to how much of the funding it provides. Here equity provides 60 rupees in every 100, so its 14% carries more weight than debt's rate. A plain average of 14% and 10% would be 12.0%, which pretends the two sources are the same size.
Equity is 60% of funding at 14% and contributes 8.4 points; debt is 40% at 7.5% after tax and contributes 3.0 points, so the WACC is 11.4%. Using the 10% pre-tax rate adds a wrong extra point and gives 12.4%.
Why does debt go in after tax?
Interest is deducted before profit is taxed, so every 10 rupees of interest cuts the tax bill by 2.50 rupees at a 25% rate. The tax saved pays a quarter of the interest, so the company's true cost of debt is 7.5%, not 10%. That saving is the tax shieldThe tax a company avoids because interest is deducted from profit before tax is worked out.. Equity has no such shield, because dividends are paid out of profit after tax, which is why the adjustment sits on the debt term only.
The relationship
$\text{WACC} = \tfrac{E}{V}\,r_e + \tfrac{D}{V}\,r_d(1-t) = 0.6(14\%) + 0.4(10\%)(0.75) = 11.4\%$
E/V equity's share of total funding, 60%
D/V debt's share of total funding, 40%
r_e cost of equity, 14%
r_d pre-tax cost of debt, 10%
t tax rate, 25%
What it says in wordsWeight each source's cost by its share of the funding, and cut the cost of debt by the tax it saves.
One more sentence wins the point. The weights should be market values, not the book values on the balance sheet, because investors demand a return on what their stake is worth today. Then say what the number is for: 11.4% is the rate at which you would discount the company's unlevered free cash flows in a DCF.
Where candidates lose it
The usual miss is forgetting the tax shield and answering 12.4%. It is the most common one-point error in a first-round cost of capital question, and interviewers ask it because it is so easy to skip.
The second is applying the tax adjustment to equity as well, or to the whole WACC. The shield belongs to interest alone, so say which term it sits on.
What the interviewer asks next
The company moves to 60% debt at the same rates. What happens to WACC, and why would the rates not stay the same?
Why is the cost of equity higher than the cost of debt?
If the company makes losses and pays no tax, what is its WACC?
057Logical test style: what number comes next in 2, 6, 12, 20, 30, and why?Logic and brainteasersWarm upConsulting style brainteasersSales and trading
Try it first
What comes next?
Show the worked solution
42. The gaps between terms are 4, 6, 8 and 10, rising by 2 each time, so the next gap is 12 and 30 + 12 = 42. The rule underneath is that the nth term is n x (n + 1): 1 x 2, 2 x 3, 3 x 4, 4 x 5, 5 x 6, and next 6 x 7, which is 42.
What should you do first with any number series?
Imagine a taxi meter that charges a little more for each extra kilometre than it did for the one before. The fares look irregular, but the jumps between them tell the story. Write the differences between terms underneath the series before guessing, and if they are not constant, take differences again. Here the first differences are 4, 6, 8 and 10, and the second differences are a steady 2, which tells you the rule is a quadratic: something times something.
The differences between 2, 6, 12, 20 and 30 are 4, 6, 8 and 10, rising by a steady 2, so the next difference is 12 and the next term is 42, which is also 6 x 7 under the rule n x (n + 1).
How do you check the answer, not just find it?
A second route that lands on the same number is your proof. Each term splits into two neighbouring whole numbers: 2 = 1 x 2, 6 = 2 x 3, 12 = 3 x 4, 20 = 4 x 5, 30 = 5 x 6. Two methods agreeing, differences and a formula, turn a guess into an answer. The formula also lets you jump ahead: the 10th term is 10 x 11 = 110 without writing out the terms in between.
The relationship
$a_n = n(n+1) \qquad a_6 = 6 \times 7 = 42$
a_n the nth term of the series
n its position, starting at 1
What it says in wordsEach term is its position multiplied by the next position.
Why does a bank put this in an online test?
Numerical and logical screens are timed tightly and filter large numbers of applicants before anyone reads a CV. The skill being tested is spotting structure fast, the same skill you use when a line in a model grows by a changing amount each year. A revenue line whose yearly increase itself rises by a fixed amount has exactly this shape, and seeing it saves you from projecting the last increase forward as if it were fixed.
Where candidates lose it
The fast wrong answer is 40, from assuming the last gap of 10 repeats. Under time pressure candidates spot the first differences and stop before noticing that the differences are themselves growing.
The other loss is spending a minute hunting for an exotic rule. Differences first, second differences next, then neighbouring products: those three checks crack most test series inside the time allowed.
What the interviewer asks next
What is the 20th term of the series?
What comes next in 1, 3, 6, 10, 15, and how is that series related to this one?
What is the sum of the first five terms, and is there a shortcut?
061A bond has a modified duration of 7. Yields rise by 50 basis points. Roughly how much does its price change?Rates, risk and optionsWarm upDebt capital marketsSales and trading
Try it first
Pick the move.
Show the worked solution
The price falls by about 3.5%. Modified duration says how many per cent the price moves for a one-percentage-point move in yield, in the opposite direction. A 50 basis point rise is half a point, so 7 x 0.5 = 3.5% down. On a Rs 100 crore holding that is a loss of about Rs 3.5 crore. The bend in the price curve makes the true fall slightly smaller.
Why do bond prices fall when yields rise?
A bond pays fixed cheques. If new bonds start paying more, nobody will pay the old price for the old, smaller cheques, so the old bond's price drops until its yield matches the market. Think of a flat let at a fixed rent when rents nearby jump: a buyer now pays less for it. Price and yield move in opposite directions, and duration measures how hard the price reacts. A longer bond reacts more, because more of its cash arrives far in the future, where a change in the discount rate bites hardest.
For a bond with modified duration 7, the straight tangent predicts a 3.5% fall for a 50 basis point rise and the true curve gives 3.43%, almost identical, but for a 300 basis point rise the line says 21.0% while the curve falls only 18.7%.
How accurate is the duration answer?
Duration draws a straight line, the tangent, along a price curve that actually bends. For small moves the line and the curve are almost the same, and for large moves the curve sits above the line, so the true loss is smaller than duration says. Take a zero-coupon bond yielding 6% with 7.42 years to run, which has a modified duration of exactly 7. A 50 basis point rise cuts its price by 3.43%, against 3.5% from the shortcut. A 300 basis point rise cuts it by 18.7%, against 21.0% from the line.
The relationship
$\frac{\Delta P}{P} \approx -D_{mod} \times \Delta y = -7 \times 0.005 = -3.5\%$
D_mod modified duration, here 7
Delta y the change in yield, 50 basis points or 0.005
Delta P / P the percentage change in price
What it says in wordsThe percentage price change is roughly minus duration times the change in yield.
Add one sentence that marks you out. The bend is convexityHow much the price curve of a bond bends, which is the change in duration as yields move., and for an ordinary bond it works in the holder's favour both ways: the price falls less than duration predicts when yields rise and gains more than predicted when they fall. For a 50 basis point move the effect is a few hundredths of a per cent, so 3.5% is the answer to give.
Where candidates lose it
The usual slip is an order-of-magnitude error, 0.35% or 35%, from mixing up basis points and percentage points. Say that 50 basis points is half a percentage point before you multiply.
The second is getting the sign wrong. Yields up means prices down; say the direction first, then the number.
What the interviewer asks next
What is the modified duration of a 5-year zero-coupon bond yielding 6%?
Two bonds have the same duration but different convexity. Which would you rather hold, and when does it matter?
How would you hedge the rate risk on Rs 100 crore of this bond?
062Target shareholders are offered 0.5 acquirer shares for each target share. The acquirer trades at Rs 200 and the target at an undisturbed Rs 80. What premium is being offered?Deal mathsWarm upElite boutique IBPrivate equity
Try it first
What premium are target holders being offered?
Show the worked solution
A 25% premium. Each target share is exchanged for 0.5 acquirer shares worth Rs 200 each, so the offer is worth Rs 100 a share. The target trades at Rs 80 undisturbed, so holders get Rs 20 more, and 20 over 80 is 25%. The exchange ratio is not itself the premium: a straight swap at market prices would be 80 over 200, or 0.4.
How do you turn an exchange ratio into rupees?
Picture a swap where someone offers you half a gold coin for your silver one. To judge the offer you price both coins first. Multiply the exchange ratio by the acquirer's share price to get what each target share is being offered: 0.5 x Rs 200 = Rs 100. Then compare that with the undisturbed priceThe target share price before news or rumours of the deal moved it., Rs 80. The premium is the gap over the undisturbed price, Rs 20 on Rs 80, which is 25%.
A target holder gives up a share worth Rs 80 and receives half an acquirer share worth Rs 100, so the offer carries Rs 20 of premium, 25% of the undisturbed price, the same as offering 0.5 shares where 0.4 would be a straight swap.
How do you check it a second way?
Work out the ratio that would be a straight swap at today's prices: Rs 80 over Rs 200, or 0.4 acquirer shares per target share. An offer of 0.5 is 25% more shares than 0.4, the same premium seen through share counts instead of rupees. Two routes agreeing is what an interviewer wants to hear before you commit to a number.
The relationship
$\text{Premium} = \frac{0.5 \times 200 - 80}{80} = \frac{100 - 80}{80} = 25\%$
0.5 the exchange ratio, acquirer shares per target share
200 the acquirer's share price
80 the target's undisturbed share price
What it says in wordsThe premium is the offer value per share over the undisturbed price, minus one.
What can happen to the premium before the deal closes?
In a fixed exchange ratio deal, the target holders' payout moves with the acquirer's share price. If the acquirer falls 10% to Rs 180, the offer is worth Rs 90 and the premium shrinks to 12.5%; if the acquirer rises, the premium grows. That is why target boards push for collars or cash, and why the premium quoted on announcement day is only a snapshot.
Where candidates lose it
The fast wrong answer is 20%, from measuring the Rs 20 gap against the Rs 100 offer rather than the Rs 80 undisturbed price. A premium is always quoted on what the target was worth before the offer.
The other miss is reading 0.5 as a 50% premium. The ratio only means something once both share prices turn it into rupees.
What the interviewer asks next
The acquirer's stock falls 10% before closing. What premium do target holders now receive?
If the target has 50 crore shares and the acquirer 100 crore, what share of the combined company do target holders own?
Why might a target board prefer a fixed value offer to a fixed exchange ratio?

ln 2	the natural log of 2, about 0.693, because the money must double
ln(1.09)	the log of one year's growth factor at 9%
72 / 9	the rule of 72 with the rate in per cent

17% x 340	the first term, 57.8
34% x 170	the second term, the same product with the factor of 2 moved across
34% x 340	the two equal terms combined

12 x 11	each person times the others they meet, counting every handshake from both sides
2	the two people in every handshake

r-bar	the arithmetic average return, 10%
sigma	how far returns swing around the average, 20 points for Fund A

Try it first

Why does a loss leave the company with more cash?

How does each statement move?

What assumption should you say out loud?

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What the interviewer asks next

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Why does 72 work at all?

Where does the rule break down?

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What the interviewer asks next

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What is the trick hiding in the numbers?

How do you do 34% of 340 in your head?

How do you check it before you say it?

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Why do you halve the count?

How do you check 66 another way?

Where does this count show up on a desk?

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Why does the same average give different endings?

How big is the drag, and can you estimate it in your head?

Why would an interviewer care?

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Why does a purchase on credit touch neither cash nor profit?

What happens over the next 60 days and beyond?

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Why is it a weighted average and not a plain one?

Why does debt go in after tax?

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What should you do first with any number series?

How do you check the answer, not just find it?

Why does a bank put this in an online test?

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Why do bond prices fall when yields rise?

How accurate is the duration answer?

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How do you turn an exchange ratio into rupees?

How do you check it a second way?

What can happen to the premium before the deal closes?

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