Equity Research puzzles, solved step by step

The market is pricing about a 23.5% chance of approval; at 30% the stock is worth about Rs 201. Set the price equal to the probability-weighted outcomes: 190 = p x 320 + (1 - p) x 150, so p = 40/170 = 23.5%. At 30%, the value is 150 + 0.3 x 170 = Rs 201, about 5.8% above the price, ignoring the time value of money and any premium for bearing a binary risk.

How does a price turn into a probability?

Think of a ticket that pays Rs 320 if it rains tomorrow and Rs 150 if it does not. Nobody would sell it below Rs 150 or buy it above Rs 320, and where it trades in between tells you how likely the crowd thinks rain is. Where the price sits between the two outcomes is the market's probability: Rs 190 is 40 of the 170 rupees from the downside to the upside, 23.5% of the way.

What does the gap between 23.5% and 30% actually buy you?

The expected edge is the difference in probability times the spread: 6.5 points x Rs 170, about Rs 11 a share, or 5.8% of the price. But the outcome is binary: the stock either falls 21% to Rs 150 or rises 68% to Rs 320, and on a 30% view it falls seven times in ten. That is why desks size binary positions small and spend their effort asking how confident the 30% itself is, not just whether it beats 23.5%.

What does the simple formula leave out?

Three things worth naming. The decision may be months away, so the outcomes should be discounted. Investors want paying for binary risk, so part of the gap between your view and the price is a premium, not a mistake. And the implied probability is only as good as the two end values: if the rejection value is Rs 160 instead of Rs 150, the implied probability drops from 23.5% to 18.8%. Most of the real work is in the end values, not the division.

About 6 years to double and about 18 years to grow eightfold. The rule of 72 says the doubling time is roughly 72 divided by the percentage rate, so 72 / 12 = 6 years; the exact figure is 6.1 years. Eightfold is three doublings, 2 x 2 x 2, so it takes three doubling times, about 18 years, and exactly 18.3. It is not eight times six.

Why does 72 divided by the rate work?

Think of a savings pot that earns 12% a year and never has anything taken out. Each year's interest earns interest the next year, so the pot grows faster in rupees every year while growing at the same percentage. The time to double depends only on the rate, and for everyday rates it is close to 72 divided by the rate in percent. The exact answer is the log of 2 over the log of 1.12, 6.12 years. The number 72 is used because it sits near the exact constant and divides neatly by 2, 3, 4, 6, 8, 9 and 12.

Why is eightfold 18 years and not 48?

Because growth multiplies. Eight is 2 x 2 x 2, so eightfold is three doublings, and each doubling takes the same six years whatever the size of the pot. The same counting gives the other useful anchors: fourfold is two doublings, 12 years; a thousandfold is about ten doublings, since 2 to the power 10 is 1,024, so about 60 years. Tenfold is a bit more than three doublings, exactly 20.3 years.

Where does an analyst use this?

Anywhere a growth rate needs to be turned into a sense of scale, fast. A company promising 24% growth is promising to double every three years; a fund charging 2% a year in fees gives up about a third of the pot over twenty years; inflation at 6% halves the value of money in about 12 years. Turning rates into doubling times is the quickest way to check whether a claim is plausible before any spreadsheet is opened.

About 15.2x. Convert the numerator and the denominator separately. Enterprise value is 10 x 100 = 1,000; take off net debt of 200 and equity value is 800. EBITDA of 100 less D&A of 20 and interest of 10 is pre-tax profit of 70; after 25% tax, net income is 52.5. The P/E is 800 / 52.5 = 15.2x.

Why can't you read the P/E straight off the EV multiple?

A flat worth Rs 1 crore with a Rs 20 lakh loan on it is worth Rs 80 lakh to its owner, and the rent the owner keeps is what is left after the loan interest and the tax. The flat's value and the owner's stake are different numbers, and so are the rent and the owner's income. EV and EBITDA belong to lenders and shareholders together; equity value and net income belong to shareholders alone, so a multiple must pair one with the other of the same kind.

Why is the P/E higher than the EV multiple here?

Because the two sides shrink by different amounts. The price side loses 20% on the way from EV to equity, but the earnings side loses nearly half on the way from EBITDA to net income, so the P/E ends above the EV multiple. Leverage changes the balance: with net debt of 400 and interest of 20, equity value is 600, net income is 45 and the P/E is 13.3x. Heavy D&A or a high tax rate pushes the P/E up; debt that costs less than the earnings yield pulls it down.

When would an analyst do this conversion for real?

Whenever two sources quote different multiples for the same company, or when a sector trades on EV/EBITDA but a client thinks in P/E. Saying out loud which claims each number belongs to, all funders or shareholders only, is the habit that stops the conversion going wrong. Assume here that D&A is tax deductible in full and that the company has no minority interests or associates, and say so.

On day one nothing touches the income statement: equipment and debt each rise by Rs 100 crore, and cash flow shows minus 100 in investing and plus 100 in financing. A year later, depreciation of Rs 10 crore and interest of Rs 9 crore cut pre-tax profit by Rs 19 crore and net income by Rs 14.25 crore. Cash falls Rs 4.25 crore, equipment is Rs 90 crore, debt is still Rs 100 crore and equity is down Rs 14.25 crore.

Why does nothing hit profit on the day of purchase?

Buy a delivery van on the last evening of March with a bank loan and you are no poorer that night: you swapped borrowed money for a van. The costs come later, as the van wears out and as the loan collects interest. Buying an asset with borrowed money swaps one balance sheet item for another; costs reach the income statement only as the asset is used up and as time passes on the loan. So on day one the only movements are equipment up 100, debt up 100, and the cash flow statement showing the money coming in through financing and going out through investing.

How do the numbers flow through at the end of year one?

Start on the income statement. Depreciation is 100 / 10 = Rs 10 crore and interest is 9% of 100 = Rs 9 crore, so pre-tax profit falls Rs 19 crore. Tax falls by 25% of that, Rs 4.75 crore, so net income falls Rs 14.25 crore. On the cash flow statement, add back the Rs 10 crore of depreciation, which never left the bank: cash falls Rs 4.25 crore. The cash cost of the year is the interest less the tax it saves, 9 - 2.25 = 4.25; depreciation costs profit but not cash.

What assumptions should you say out loud?

Four of them. Interest is paid in cash at the end of the year. The tax saving is real because the company has other profits to set it against. Tax depreciation matches book depreciation. No principal is repaid in year one. Stating the assumptions turns a walkthrough into an argument the interviewer can follow, and each one is a lever for the next question. Change the second, and a loss-making company saves no tax this year, so net income falls the full Rs 19 crore.

About 11.3% a year, and about 1.71 times after five years. EPS is net income over shares. Net income is multiplied by 1.08 each year and the share count by 0.97, so EPS is multiplied by 1.08 / 0.97 = 1.113. Over five years that compounds to about 1.71x, against 1.47x for net income alone. Adding 8% and 3% to get 11% is close, but slightly understates it.

Why divide by 0.97 rather than add 3%?

Picture a pot of profit shared among a group of friends. If the pot grows 8% and one friend in every 33 leaves each year, each remaining friend's share grows by more than 8%, and a little more than 3% on top. EPS growth is (1 + net income growth) divided by (1 - buyback rate), minus one, because the share count sits in the denominator. Dividing by 0.97 is multiplying by 1.0309, a 3.09% boost, and that compounds with the 8%: 1.08 x 1.0309 = 1.1134.

Is buyback-driven EPS growth as good as profit growth?

Not automatically. EPS growth from a shrinking share count is paid for with cash that could have been reinvested or paid out, and it adds value for the remaining holders only when the shares are bought below what they are worth. A company buying back stock at a high multiple can show fast EPS growth while transferring value to the shareholders who sell. The question to ask is what the cash would have earned elsewhere.

What would an analyst check before trusting the 11.3%?

Two things. First, whether the 8% net income growth already carries the cost of funding the buyback: interest lost on cash, or interest paid on new debt. Second, whether staff share awards are quietly adding shares back. What matters is the net fall in the share count: if awards add 2% a year, EPS growth drops to about 9.2%. The share count line in the notes to the accounts settles both.

The real rate is about 5.7%, and discounting real cash flows at the nominal rate understates value. Real and nominal rates are linked by multiplying: 1.11 / 1.05 = 1.0571, so the real rate is 5.71%, a little below the 6% you get by subtracting. A forecast in today's prices has no inflation in it, so an 11% rate removes inflation twice. Rs 100 in year 10 is worth Rs 57.4 at the matched rate but Rs 35.2 at the mismatched one.

How are real and nominal rates linked?

Your salary rises 11% in a year when prices rise 5%. You can buy more, but not 6% more: you have 1.11 rupees for every 1.05 rupees of goods you used to buy, which is 5.71% more stuff. A real rate is the nominal rate with inflation divided out, not subtracted, so 1 plus the real rate equals 1.11 over 1.05. At low rates the two methods differ by a fraction of a point; at high inflation the gap matters.

Why does the mismatch understate value?

A nominal discount rate assumes the cash flows it meets include inflation, and discounting removes it. If the forecast was built in today's prices, the inflation was never added, so the nominal rate removes it anyway and the value falls by about 39% on a year-10 cash flow. The reverse mistake, inflated cash flows at a real rate, overstates value by just as much in the other direction. Both errors grow with the length of the forecast, which is why they hurt long-lived assets most.

Which approach does an analyst use?

Either, as long as it is consistent. Most equity models are nominal, because reported accounts, debt costs and tax are all in current rupees. Real models are common for long-lived assets such as infrastructure, where prices are linked to inflation. One trap sits inside real models: tax depreciation is fixed in rupees at the purchase price, so in a real model its value must be deflated or it is overstated. Check the terminal growth rate too: a 6% nominal growth rate is only about 1% in real terms at 5% inflation.

RASK rises about 2%. RASK is yield times load factor: revenue per passenger kilometre times the share of seat kilometres that carry a passenger. The load factor rises by 85/80 = 1.0625, a 6.25% gain, not 5%, and yield is multiplied by 0.96. Together 1.0625 x 0.96 = 1.02, so RASK is up 2.0%.

Why is RASK yield times load factor?

Think of a 50-seat bus. Revenue per seat on a trip is the fare each rider pays times the share of seats with a rider in them. Fill more seats at a lower fare and revenue per seat can go either way, depending on the sizes. Revenue per available seat kilometre splits exactly into what each flown passenger kilometre earns, the yield, and the share of available seat kilometres that are flown, the load factor, so percentage changes in the two multiply.

Why is the load factor gain 6.25% and not 5%?

Because 5 is the change in percentage points, and the question needs the percentage change. A move from 80% to 85% is 5 points, but 5 on a base of 80 is a 6.25% increase in seats filled, and it is the percentage change that multiplies through to revenue. Get that wrong and you add +5 and -4 to say +1%, which halves the true answer. The break-even is useful too: with yield down 4%, RASK stays flat at a load factor of 83.3%.

What should an analyst ask next?

Whether the two changes are linked, and what happened to cost. Airlines often fill seats by cutting fares, so a higher load factor and a lower yield can be one decision, not two pieces of news. RASK only matters against CASK, the cost per available seat kilometre: a 2% gain in RASK is a margin gain only if cost per seat kilometre rose by less. A change in the average trip length also moves yield, since longer flights earn less per kilometre, so check the route mix before reading the fall in yield as weaker pricing.

About Rs 4.4 crore a year, on stated assumptions. Five shows a day on 200 seats at 30% occupancy is 300 visitors a day, about 1,09,500 a year. At Rs 250 a ticket that is Rs 2.74 crore; at Rs 120 of food per head, Rs 1.31 crore; screen advertising adds about Rs 0.30 crore. Food is under a third of revenue, but because a distributor takes about half of each ticket, it is well over a third of what the screen keeps.

How do you build the visitor count?

Count it the way you would count lunches at an office canteen: seats, times sittings, times how full each sitting is, times days. Visitors are shows x seats x occupancy x days, and occupancy is the number to defend, because most screens are far from full outside weekend evenings. Five shows on 200 seats at 30% gives 300 visitors a day and 1,09,500 a year. Every figure here is an assumption for the exercise, and prices are taken net of GST, whose rates should be confirmed.

Why does food matter more than its price suggests?

Because the two rupees are not shared the same way. A ticket rupee is split with the film's distributor, while a food rupee mostly stays with the screen, so Rs 120 of popcorn and a drink keeps about Rs 90, close to the Rs 125 the screen keeps from a Rs 250 ticket. That is why chains price food high and push it hard, and why an analyst tracks spend per head as closely as ticket prices. The distributor's share and the food cost here are assumptions to check against a company's disclosures.

Which assumption would the interviewer push on?

Occupancy first. It moves both big lines at once: at 35% instead of 30%, revenue rises to about Rs 5.03 crore. A screen's economics turn on filling empty seats, because rent, staff and power are the same whether the hall is a third full or packed. Then the average ticket price, which blends weekday discounts with weekend premiums, and the number of shows, which depends on film lengths. Name all three and you have shown how the business works, not just its size.

A P/E of 20x, an earnings yield of 5%, and a price of Rs 500. P/E is price over EPS, 400 / 20 = 20. The earnings yield is the inverse, EPS over price, 20 / 400 = 5%. If EPS rises 25% to Rs 25 and the market still pays 20 times, the price is 20 x 25 = Rs 500, also 25% higher: at a constant multiple, price moves one for one with earnings.

What does a P/E of 20 actually say?

Buy a small shop for Rs 20 lakh that earns Rs 1 lakh a year and you have paid twenty years of today's profit, a 5% return on your price before any growth. A P/E is the number of years of today's earnings you pay for, and its inverse, the earnings yield, is those earnings as a return on the price. Neither says whether the stock is cheap; that depends on how fast the earnings grow and how risky they are.

When does the price not follow earnings?

When the multiple moves. If EPS rises 25% but the P/E falls from 20x to 16x, the price does not move at all: 25 x 16 = Rs 400. The price change is the earnings change and the multiple change compounded: 1.25 x 0.8 = 1.0. That is how a company can grow profit for years while its share price stands still, and why analysts separate earnings growth from rerating when they explain a move.

Why is the earnings yield worth quoting?

It puts the price on the same scale as other returns. A 5% earnings yield can be set beside a deposit rate or a bond yield, as long as you say what differs: earnings are not all paid out, they can grow, and they are not promised. When bond yields rise above a stock's earnings yield, the case for the stock rests more heavily on growth, which is one reason multiples tend to fall when rates rise.

Net income falls Rs 15 crore and cash rises Rs 5 crore; inventory falls Rs 20 crore while equity falls Rs 15 crore. The write-down is a Rs 20 crore expense, cut to Rs 15 crore after the Rs 5 crore tax saving. It costs no cash today, so the cash flow statement adds the Rs 20 crore back, leaving cash up by the tax saved. Assets fall Rs 15 crore, and so does equity.

Why does a Rs 20 crore loss raise cash?

A shopkeeper who bought stock last year and finds this year that it has spoiled is not paying for it again; the money went out when the goods came in. Admitting the loss costs nothing new, and if the tax rules let her deduct it, she pays less tax this year. The cash for inventory leaves when it is bought; the write-down only records the loss, and the tax deduction is the one real cash effect today. So profit falls by the loss after tax, while cash rises by the tax saved.

How do you walk it through the statements in order?

Income statement first: a Rs 20 crore expense, usually inside cost of goods sold, lowers pre-tax profit by 20; tax falls by 25% of that, Rs 5 crore; net income falls Rs 15 crore. Cash flow statement next: start from net income of minus 15 and add back the Rs 20 crore, because no cash left; cash is up Rs 5 crore. Balance sheet last: cash up 5 and inventory down 20 take assets down 15, and retained earnings fall by the same 15, so the two sides move together.

What if the tax rules are different?

Tax treatment of write-downs varies by jurisdiction, and some rules allow the deduction only when goods are actually scrapped or sold, so confirm it for the case at hand. If the deduction comes later, book net income still falls Rs 15 crore, but cash tax does not change this year, so cash is flat and a Rs 5 crore deferred tax asset appears on the balance sheet instead. Saying this without being asked shows you know where the Rs 5 crore of cash came from.

What does an analyst read into a write-down?

Two things. The charge is usually treated as one-off, so analysts look at profit before it, but repeated write-downs say something about how the company buys and forecasts demand. A write-down also moves cost out of future periods: goods written off now cannot be expensed again when sold, so next year's gross margin can look better than the business really is.