Financial Analysis puzzles, solved step by step

X carries more debt and owns a stake in an associate. Debt of Rs 750 crore at 8% costs 5.6% after tax, less than the 6.7% its operations earn on their value, so levering lowers the P/E. The associate adds Rs 22 crore to net income but nothing to EBITDA, while its Rs 300 crore value sits inside EV. Strip it out and X is 7.0x, like its peers.

Why can two multiples on the same company disagree?

Picture a house with a mortgage. Its price compared with the rent it earns is one ratio; your equity in it compared with the rent left after the mortgage payment is another. The two only agree if there is no loan. EV/EBITDA looks at the whole business before financing, while P/E looks at the shareholders' slice after interest, tax and anything non-operating. Every gap between them is explained by something that sits between EBITDA and net income, or between EV and equity value.

What numbers make both facts true?

Give both companies the same operations: EBITDA Rs 150 crore, depreciation Rs 50 crore, operating profit Rs 100 crore, tax 30%. The peer has no debt, earns Rs 70 crore and is worth Rs 1,050 crore: 15.0x earnings and 7.0x EBITDA. X borrows Rs 750 crore at 8%, so interest of Rs 60 crore leaves Rs 28 crore from operations, and it books Rs 22 crore as its share of an associateA company in which the group holds a significant minority stake, usually 20% to 50%. The group books its share of that company profit below operating profit, never in revenue or EBITDA.'s profit. Net income of Rs 50 crore at 12x is equity of Rs 600 crore, and with the debt that is an EV of Rs 1,350 crore, 9.0x EBITDA.

Which reason moves which multiple?

Separate them, because the follow-up always asks. Debt lowers the P/E when its after-tax cost is below the earnings yield of the operations. Here debt costs 8% times 0.7, which is 5.6%, while the operations earn 70 on 1,050, or 6.7%. Swapping expensive equity for cheaper debt leaves the operating slice at about 10.7x earnings. The associate does the other job: its Rs 300 crore of value sits inside EV while its profit sits below EBITDA, which lifts EV/EBITDA from 7.0x to 9.0x.

Name a third candidate if you have time: a lower tax rate than peers raises net income without touching EBITDA, so it also lowers P/E alone. Say what you would check to choose between them: the notes on debt, associates and the effective tax rate.

About Rs 157.5 crore. Start from potential rent of Rs 20 crore and take off 10% vacancy to reach effective gross income of Rs 18 crore. Operating expenses at 30% of that are Rs 5.4 crore, leaving net operating income of Rs 12.6 crore. Divide NOI by the 8% cap rate: Rs 12.6 crore over 0.08 is Rs 157.5 crore, which is 12.5 times NOI.

Why does a cap rate apply to net income and not to rent?

Think of a flat you rent out for Rs 30,000 a month. Some months it sits empty, and the society charges, repairs and property tax come out of your pocket. What you would pay for the flat depends on what is left, not on the rent written in the agreement. A cap rate is the yield a buyer wants on net operating income, the cash the building throws off after vacancy and running costs but before any loan payments. Applying it to gross rent values money the owner never receives.

How do you walk from potential rent to value?

Three steps, in order. Gross potential rent is what the building would earn fully let: Rs 20 crore. Take off vacancy first, because operating expenses here are a share of the income actually collected, not of the potential. 10% vacancy leaves effective gross incomeRent the building actually collects after vacancy and bad debts, before operating expenses. of Rs 18 crore. Expenses at 30% of 18 are Rs 5.4 crore, which leaves NOI of Rs 12.6 crore. Then divide by the cap rate.

What makes the exit cap rate the number to argue about?

Value is very sensitive to the cap rate: one point lower, at 7%, the building is worth Rs 180 crore; one point higher, at 9%, Rs 140 crore. That 1 point swing moves value by about Rs 40 crore on a Rs 157.5 crore building. Analysts usually set the exit cap rate a little above today's rate, because the building will be older when it is sold. Say the limit as well: a single cap rate assumes NOI is stable, so a building with large leases expiring soon needs a cash flow model, not one division.

The implied tax rate is 37.5%. Market cap of 300 at 15x earnings gives net income of 20. Adding net debt of 180 gives EV of 480, so EBITDA at 6x is 80. Less D&A of 30 is EBIT of 50; less interest of 18 is pre-tax profit of 32. Tax is 32 minus 20, which is 12, and 12 over 32 is 37.5%, high enough to ask why.

Where do you start when six numbers arrive at once?

Think of working out a friend's take-home pay from what they spend and what they save. You do not start with their job title; you start with the two numbers that pin the answer. A tax rate is tax divided by pre-tax profit, so every other number in the question is a road to net income or to pre-tax profit. Say that first. Net income is one step: 300 over 15 is 20. Pre-tax profit needs EBITDA, which needs EV, which needs the net debt you were handed.

What does a 37.5% rate tell you, and what have you assumed?

You assumed there are no minority interests, no associates and no interest income on cash, so EV is just market cap plus net debt and every rupee of pre-tax profit belongs to shareholders. Say that. Then read the number. An effective rate well above the statutory rate usually means some costs are not tax deductible, some losses sit in units that cannot use them, or there are one-off tax charges. Compare it with the statutory rate the company actually pays, which you should confirm for the year in question. A second check: interest of 18 on net debt of 180 is 10%, which is plausible, so the inputs hang together.

Market capitalisation is Rs 2,000 crore. With no other liabilities, assets equal debt plus equity. A D/E of 0.5 means debt is half of equity, so assets are 1.5 times equity and book equity is Rs 1,500 crore over 1.5, which is Rs 1,000 crore, with Rs 500 crore of debt. At 2.0x book, the market values that equity at Rs 2,000 crore.

How does a debt-to-equity ratio split the balance sheet?

Think of a Rs 15 lakh car bought with a loan that is half the size of your down payment. If the loan is half the down payment, the car cost one and a half down payments, so the down payment was Rs 10 lakh and the loan Rs 5 lakh. A D/E of 0.5 does not mean half the assets are debt; it means debt is half of equity, so assets are 1.5 times equity. Rs 1,500 crore over 1.5 gives book equity of Rs 1,000 crore and debt of Rs 500 crore.

What assumptions sit under the answer, and what could you add?

The question rules out other liabilities, such as payables and provisions, and that matters: a real balance sheet has plenty, and then assets are not just debt plus equity. Say the identity you are using, assets equal liabilities plus equity, before you lean on it. You can also go one step further than asked. With Rs 500 crore of debt and no cash given, enterprise value is about Rs 2,500 crore. And a P/B of 2.0 says the market thinks the assets earn more than their cost of capital; read it next to return on equity rather than on its own.

Unlevered free cash flow is Rs 1,00,000 a year and the NPV is about Rs 1,43,308. Sales are Rs 4,50,000, goods Rs 2,70,000 and rent Rs 60,000, so EBITDA is Rs 1,20,000. Less Rs 40,000 of depreciation is EBIT of Rs 80,000; after 25% tax, NOPAT is Rs 60,000. Adding back depreciation gives Rs 1,00,000. Five years of that at 14% is worth Rs 3,43,308, less the Rs 2 lakh machine.

How do you get from sales to unlevered free cash flow?

Think of a tea stall owner counting what she can take home each year: takings, less tea and milk, less rent, less the tax man's share. The stall's old kettle wearing out is a cost on paper, but no cash leaves her purse for it each year. Unlevered free cash flow is the cash the asset throws off before any financing: operating profit after tax, plus non-cash charges, less the capital spending and working capital it needs. Here: sales of Rs 4,50,000 less goods of Rs 2,70,000 and rent of Rs 60,000 is EBITDA of Rs 1,20,000. Depreciation of Rs 40,000 gives EBIT of Rs 80,000, tax takes Rs 20,000, and NOPAT is Rs 60,000.

Why does depreciation matter only through tax?

Depreciation is subtracted to reach EBIT and added back to reach cash flow, so on its own it washes out. Its only cash effect is the tax it saves: Rs 40,000 of depreciation at 25% cuts tax by Rs 10,000 a year. That is why taxing EBITDA directly is wrong: it would give Rs 90,000 a year and lose the shield. The answer leans on assumptions you should say out loud: no working capital for stock in the machine, no repairs, no salvage value, a steady 50 sales a day, and a 14% rate that reflects the risk of the location. The NPV is positive by a wide margin, so the decision is most sensitive to daily volume, the one number nobody can check in advance.

About Rs 156 crore on cash flow, but a buyer will not pay much above Rs 150 crore, the cost of building the same plant. Revenue is Rs 100 crore and cash costs Rs 70 crore. Taking depreciation equal to the Rs 5 crore of capex, tax is 25% of Rs 25 crore, Rs 6.25 crore, leaving Rs 18.75 crore a year: over 12% that is Rs 156.25 crore. Replacement cost caps it.

How do you get from units to a value?

Valuing a factory is the same sum as valuing a flat you rent out: the rent left after upkeep and tax, divided by the return you need. A business with no growth is worth its steady yearly free cash flow divided by the discount rate. All the work is in getting the yearly cash right, and the two lines people forget are tax and the capex needed just to stand still.

Why does the rebuild cost set the ceiling?

Suppose someone asks Rs 25 lakh for a used car when the same model costs Rs 20 lakh new at the showroom. You walk to the showroom. No sensible buyer pays much more for an asset than it would cost to build an identical one, so replacement cost caps the price even when the cash flows say more. Here the gap is small, Rs 156.25 crore against Rs 150 crore, so a defensible answer sits close to Rs 150 crore.

The cap is loose in three ways, and naming them is what separates a good answer. Building takes time, perhaps two years with no cash coming in, so a working factory earns a premium for the lost years. Land, permits and trained staff may be hard to copy. And if cash flow value stays well above rebuild cost, rivals build plants, supply rises and prices fall, which pulls the cash flows back down. That last force is why the two numbers tend to converge over time.

What if the numbers pointed the other way?

If the cash flow value were Rs 100 crore against a Rs 150 crore rebuild cost, the factory would be worth about Rs 100 crore. Replacement cost is a ceiling, not a floor: nobody pays Rs 150 crore for a plant whose cash flows justify only Rs 100 crore. The floor is what the land and machinery would fetch if sold off, which is a third number worth asking for before you commit.

On earnings yield alone equities look dear, 4% against 7%, but that ignores growth; on expected return they offer about 12% against 7%, a premium of about 5 points. The dividend yield is half of 1/25, 2%, and growing it at 10% gives roughly 12%. Whether 5 points is enough to pay for equity risk is the judgement the question is really after.

Why is 4% against 7% the wrong comparison?

A flat that rents for 3% of its price looks poor next to a 7% fixed deposit, yet people still buy flats, because rents rise over time and the deposit's interest never does. A bond's yield is close to its whole return if held to maturity, while an earnings yield is only the first year of a stream that grows, so setting one against the other treats a growing payment as a fixed one. The 4% earnings yield is 1 / 25; it says nothing yet about the 10% growth in the question.

How do you turn a multiple into an expected return?

Use the dividend growth relation: the return on a share held for the long run is the dividend yield plus the growth rate of the dividend. The payout is 50%, so the dividend yield is 0.5 / 25 = 2%. Growth is 10%, so the expected return is about 12%. Using next year's dividend, 2% grown by 10%, gives 12.2%; the difference does not change the verdict.

What does the growth assumption have to survive?

Growth carries the whole verdict, so test it. Retaining half the earnings and growing 10% forever requires a 20% return on every rupee reinvested, which is a demanding assumption for an entire market. If growth is 7% instead, the expected return is 9% and the premium shrinks to 2 points; at 5% growth equities offer no more than the bond. The answer to 'which is cheaper' is a statement about growth: at 25 times earnings, equities beat bonds by a healthy margin only if 10% growth is believable. Say that, and resist a one-word verdict.

Enterprise value stays at Rs 1,000 crore at every step; only the split between lenders and shareholders moves. Raising Rs 200 crore and holding it leaves net debt at 300 and equity at 700. Paying it out takes net debt to 500 and equity to 500, with shareholders holding the Rs 200 crore in cash. As PIK interest accrues at 10%, the notes grow to 242 after two years and equity falls to 458, value moving to the lenders.

Why does borrowing not change enterprise value?

Suppose your house is worth Rs 1 crore and you take a Rs 20 lakh loan against it, keeping the money in the bank. The house is still worth Rs 1 crore; you owe 20 lakh more and hold 20 lakh more, and your net position is unchanged. Enterprise value is the value of the operating business and does not depend on how it is funded; a financing step only changes who has a claim on that value. Here the PIK notes add 200 of debt and 200 of cash, so net debt stays 300, equity stays 700, and EV stays 1,000.

What does the dividend do?

The cash leaves the company and lands in shareholders' pockets. Net debt rises from 300 to 500 because the cash that offset the notes is gone, and equity value falls from 700 to 500. Shareholders are no richer and no poorer: they hold Rs 500 crore of shares plus Rs 200 crore of cash, 700 in all, exactly the 700 they started with. What changed is that the lenders now have a 500 claim ahead of them on the same Rs 1,000 crore of operations, so the equity is riskier and the dividend was financed by that risk.

What happens as the PIK interest accrues?

A PIK noteA loan whose interest is paid in kind: added to the principal each period instead of paid in cash, so the amount owed compounds. charges no cash interest; the coupon is added to the principal. At 10%, the Rs 200 crore becomes 220 after a year and 242 after two. With operations unchanged, EV is still 1,000, so equity is 1,000 - 542 = 458: the 42 of accrued interest is value transferred from shareholders to lenders without a rupee moving. In practice EV does move, but for operating reasons: earnings grow or shrink. Two second-order effects can touch EV through financing: interest that is deductible lowers tax and adds a shield, and heavy leverage raises the chance of distress, which costs value; confirm whether accrued but unpaid interest is deductible under the current tax rules before counting the shield.

The trap in the question is the word increase. A candidate who hears PIK and reaches for a bigger EV is adding debt to a number that already includes it. The market capitalisation does fall, from 700 to 500 and then 458, and adding net debt of 500 back to a market cap of 500 gives 1000, the same 1,000. The limit: this holds the operating business fixed to isolate the financing; a real company's EV changes every day for other reasons.

In the year it happens, the capex cut: it adds the full Rs 10 crore to free cash flow, while more revenue or lower cost of goods sold adds Rs 7.5 crore after 25% tax. Capex is a direct cash item. If the change repeats every year, lower capex means lower depreciation and more tax, so all three converge on Rs 7.5 crore a year, and an exit multiple on EBITDA favours revenue and cost.

Why does capex hit cash harder than revenue or cost?

If you earn Rs 10,000 more, you keep what is left after income tax. If you decide not to buy a Rs 10,000 laptop, you keep all Rs 10,000, because buying it was never a deduction in that year. Revenue and cost changes reach free cash flow after tax; a capex change reaches it in full, because capex is subtracted below the tax line. So with a 25% tax rate, Rs 10 crore more revenue or Rs 10 crore less cost of goods sold adds Rs 7.5 crore, and Rs 10 crore less capex adds Rs 10 crore.

One assumption sits under the revenue answer: that the extra Rs 10 crore comes with no extra cost and no extra working capital. In practice more revenue usually needs more receivables and stock, which takes a little more away. Say so in one clause.

What changes when the change repeats every year?

Capex becomes depreciation. Spend Rs 10 crore less every year on assets depreciated over five years, and depreciation falls by Rs 2 crore more each year until it is Rs 10 crore lower, which raises tax by Rs 2.5 crore. In steady state a recurring capex cut adds Rs 10 crore of cash less Rs 2.5 crore of lost tax shield, Rs 7.5 crore a year: exactly the same as the revenue and cost changes. The capex cut's edge is timing, worth something in present value, but not a permanent advantage.

Even a one-off cut keeps only part of its edge. Discounted at 10%, Rs 10 crore saved next year is worth Rs 9.09 crore, but the depreciation shield given up over the following five years is worth Rs 1.72 crore, leaving Rs 7.37 crore against Rs 6.82 crore for a one-year revenue gain. Still ahead, by much less than Rs 2.5 crore.

How does the terminal value change the ranking?

If the terminal value uses an exit multiple of EBITDA, a permanent Rs 10 crore of extra revenue or lower cost raises terminal EBITDA by Rs 10 crore and the terminal value by Rs 80 crore at 8x, while a capex cut does not touch EBITDA and adds nothing through the multiple. With an exit multiple, recurring revenue and cost changes beat capex by a wide margin; with a perpetuity growth terminal, all three are equal in steady state. Name which terminal method you are assuming before you rank them. And add the business limit: capex cut without consequence is rare, because the assets usually drive future revenue.

About 10.4% a year, forever, with the company paying out 48% of its earnings. A forward P/E equals the payout ratio divided by the cost of equity less growth, and with a fixed ROE the payout is set by growth: payout = 1 minus g/ROE. So 30 = (1 minus g/0.20) / (0.12 minus g). Multiply out: 3.6 minus 30g = 1 minus 5g, so 25g = 2.6 and g = 10.4%. Check: a dividend yield of 48% / 30 = 1.6% plus 10.4% growth is the 12% cost of equity.

Where does a multiple hide a growth forecast?

If someone offers to sell you a shop for thirty years of its current profit, they are not quoting a price; they are telling you how fast they expect the profit to grow. The multiple is the forecast. Under the Gordon growth model a share is worth next year's dividend over (r minus g), so dividing by next year's earnings gives P/E = payout / (r minus g), and a given P/E can be solved for g. The one thing people forget is that the payout is not free to choose: growth has to be paid for with retained profit.

With a fixed return on reinvested profit, growth equals ROE times the retention rate, so retention is g/ROE and payout is 1 minus g/ROE. At a 20% ROE, growing at 10% means keeping half the earnings. Put that into the multiple: 30 = (1 minus 5g) / (0.12 minus g). Cross-multiply: 3.6 minus 30g = 1 minus 5g, so 2.6 = 25g and g = 10.4%. The implied payout is 1 minus 0.104/0.20 = 48%, and the implied dividend yield is 48% of a 3.33% earnings yield, 1.6%.

Why does the ROE matter as much as the growth rate?

Because the ROE sets how much growth costs. At a 15% ROE the same 30x needs 11.1% growth, since a bigger share of earnings has to be retained to fund each point of it. At an ROE equal to the cost of equity, 12%, the multiple is 1/r = 8.3x whatever the growth rate, because every retained rupee earns exactly what shareholders could earn elsewhere. Growth adds value only when the reinvested profit earns more than the cost of equity; the multiple prices the spread between ROE and r as much as it prices g. The curve is also steep near the answer: at 10% the multiple is 25x, at 11% it is 45x, so a 30x stock is one point of growth away from either 25x or 45x.

Now say the limitation. 10.4% growth forever, only 1.6 points below the discount rate, is not a forecast anyone would defend; no company outgrows the economy indefinitely. The honest reading is that 30x prices a long period of fast growth that will fade, and a two-stage model, say 10.4% for a decade and a lower rate after, is the next thing to build. The one-line solve is still worth doing, because it converts a multiple into a sentence you can argue with: the market expects this company to compound earnings at about 10% for a very long time while earning 20% on what it retains.