Control Premium vs Minority Discount: One Fact, Two Ends

A control premium and a minority discount describe one movement, measured from opposite ends. For Sankalp Industrial Systems Limited, an invented manufacturer, an offer of Rs 115.00 against an unaffected Rs 90.00 is a premium of 27.78 per cent and implies a discount of 21.74 per cent. Same fact, same rupees, two different numbers. Each divides by a different starting point.

One small observation carries all the arithmetic below: a percentage is meaningless until its denominator is named. Going up from Rs 90.00 to Rs 115.00 is Rs 25.00 measured against Rs 90.00. Coming back down from Rs 115.00 to Rs 90.00 is the same Rs 25.00 measured against Rs 115.00. The rupees never change; the denominator does, and that alone is why one movement carries two percentages. The observation applies twice: once between a premium and a discount, and once between two different bases for the premium itself.

What exactly is a control premium, and what exactly is a minority discount?

Most of the confusion here comes from treating a premium and a discount as a matched pair before either has been pinned down. Take the two separately and finish each one before putting them side by side.

A control premium is the extra money a buyer pays, over the price the shares were changing hands at before anybody knew an offer was coming, to end up holding enough of the company to decide what it does. Three parts are load bearing in that sentence. A control premium is money, so it has a rupee figure before it has a percentage. The premium is measured against an unaffected share priceThe price a share changed hands at while the market still knew nothing about a possible offer for the company., not against yesterday's close and not against a valuation somebody built. And the premium is paid for controlling stakeA holding large enough to decide what a company does, rather than to watch what it does. arithmetic rather than for a slice. Nobody pays a premium to buy a hundred shares on a screen.

A minority discount runs the other way: it is the reduction from a value that carries control down to the value of a holding that carries none of it. A person sitting on a small stake cannot change the plan, cannot move the cash, cannot appoint anybody and cannot decide when the business is sold. A holding without those powers is worth less than the same proportion of a value that assumes them, and the reduction is the minority discount. Notice that the discount is defined by a subtraction from the higher figure. The premium is defined by an addition to the lower one. Premium and discount start from opposite ends by construction.

Here is the everyday version before the rupees arrive. A caterer charges Rs 200.00 a plate and raises it to Rs 250.00. Going up, that is Rs 50.00 on Rs 200.00, a rise of 25 per cent. Coming back down, it is Rs 50.00 on Rs 250.00, a cut of 20 per cent. Nobody finds that mysterious at a wedding. The identical thing happens to a share price, and it stops being obvious only because the two directions have been given two different names and are usually quoted by two different people on two different afternoons.

Why does one fact refuse to be one number?

Because a percentage is a ratio and a ratio has a bottom. Written out as rupees the movement leaves nothing to argue about: Rs 115.00 less Rs 90.00 is Rs 25.00, whichever end it is measured from. The disagreement appears only when the Rs 25.00 is turned into a percentage. The premium puts Rs 90.00 underneath it, the discount puts Rs 115.00 underneath it, and Rs 115.00 is the bigger of the two. Dividing the same numerator by a larger denominator always gives a smaller answer, so the discount is smaller than the premium every single time, at every price, without exception.

The rule that a discount is always smaller than its own premium is worth holding on to. The rule converts a fiddly conversion into a sanity check that can be run in the head. A premium and a discount quoted for the same transaction, with the discount the larger of the two, means one of them has been computed on the wrong base or copied from a different deal. Catching that takes no formula, only the memory of which denominator is bigger.

The size of the gap between the two denominators also marks when the shortcut is nearly harmless and when it is not. At small movements the two denominators are almost the same size, so the two percentages nearly coincide. At Rs 95.00 the premium is 5.56 per cent and the discount is 5.26 per cent, a gap of three tenths of a point that nobody would notice in a meeting. At Rs 130.00 the premium is 44.44 per cent and the discount is 30.77 per cent, a gap of 13.67 points that changes the answer materially. The shortcut survives in practice precisely because it is almost right in the range where people first learn it.

How is a premium turned into a discount?

Divide the premium by one plus the premium. Dividing by one plus the premium is the whole conversion, and writing the division out rather than reaching for it from memory is what stops the shortcut being taken by accident.

Put the case figures through it. The premium is 0.277778. One plus the premium is 1.277778. Dividing gives 0.217391, or 21.74 per cent. Now reverse it. The discount is 0.217391. One less the discount is 0.782609. Dividing gives 0.277778, or 27.78 per cent again. The two directions agree exactly, and that agreement is the only check needed on a conversion that otherwise looks like two arbitrary formulas.

There is a neater way to hold the same fact. Multiplying 1.277778 by 0.782609 gives exactly 1.000000. The factor that takes the price up and the factor that brings it back are true reciprocals. The round trip is exact for that reason. The two percentages themselves are emphatically not reciprocals, and not related in any usable way. Adding 27.78 to 21.74 gives 49.52, and 49.52 is not a fact about this transaction or about anything else. The reciprocity lives in the multipliers, never in the percentages, and mistaking one for the other is where this whole subject goes wrong.

Why does the same offer produce two different premium percentages?

The same observation applies a second time, and the second application catches far more people than the first. Everybody at least knows a premium and a discount are supposed to be different. Almost nobody stops to ask whether two premiums are.

Sankalp Industrial Systems Limited has 20,00,00,000 shares. At the unaffected price of Rs 90.00 that is a market capitalisationThe share price multiplied by the number of shares, being what the whole of the equity is changing hands at on the day. of Rs 18,00,00,00,000. Its traded enterprise value, settled elsewhere and restated here in one line, is Rs 22,40,00,00,000. The offer of Rs 115.00 a share values the equity at Rs 23,00,00,00,000 and the enterprise at Rs 27,40,00,00,000.

Work the premium on each base. On the share price it is Rs 23,00,00,00,000 over Rs 18,00,00,00,000 less one, giving 27.78 per cent. On enterprise value it is Rs 27,40,00,00,000 over Rs 22,40,00,00,000 less one, giving 22.32 per cent. Both figures are correct, both describe the identical offer, and a reader given only one of them and no note of which base it sits on cannot compare it with anything.

Read the fourth column and then the fifth. The step is the same Rs 5,00,00,00,000 in the second and third rows. Only the denominator moved, and the percentage moved with it. The first two rows carry the same percentage because multiplying both a price and a step by the same 20,00,00,000 shares changes nothing about their ratio.

Where does the extra Rs 4,40,00,00,000 of base come from?

The extra base makes the second premium figure feel arbitrary until somebody looks at it, and the extra base is entirely mechanical. Enterprise value is bigger than market capitalisation by the net of four things, and none of those four is bid for by the buyer.

Gross debt of Rs 6,00,00,00,000 comes across at its carrying amountThe figure something already sits at in the books, before anybody revalues it.. The lenders are entitled to their money back and to nothing else; no lender receives a share of what a buyer was prepared to pay for control. The minority interestThe slice of a subsidiary that belongs to shareholders outside the group, which the consolidated accounts show but the group does not hold. of Rs 60,00,00,000 travels the same way. The buyer gets the cash back on completion, so cash of Rs 1,20,00,00,000 comes off. Non-operating assetsThings a company holds that the operating forecast does not earn anything from, such as surplus land or a stake in another business. of Rs 1,00,00,00,000 come off for the same reason.

The four adjustments net to Rs 4,40,00,00,000, and that net figure sits inside the enterprise value base and inside nothing else. The whole gap between 27.78 per cent and 22.32 per cent is that Rs 4,40,00,00,000 diluting the percentage while contributing nothing to the premium, and once that is seen the second figure stops looking like a different opinion and starts looking like the same opinion divided differently.

Does the gap between the two premium bases move as the price moves?

The gap does not move, and the reason is worth deriving rather than believing. The offer enterprise value is always the offer equity plus that fixed Rs 4,40,00,00,000. The traded enterprise value is always the market capitalisation plus the same fixed Rs 4,40,00,00,000. Subtract the second from the first and the Rs 4,40,00,00,000 cancels, leaving exactly the rupee premium on the equity. So both premiums are the same numerator sitting over two denominators that never change.

Check it once on the locked figures. The share price premium is 27.7778 per cent and the enterprise value premium is 22.3214 per cent. Dividing the second by the first gives 0.80357, the ratio of the bases to five places. Two premium percentages for one company move in lockstepTwo quantities move in lockstep when the ratio between them never changes, whatever happens to either one., and the wedge between them is set by the capital structure and by nothing else.

The fixed ratio has a practical consequence. Because the ratio depends on each company's own debt, cash and minority interest, a premium quoted on the share price and one quoted on enterprise value cannot be reconciled with a rule of thumbA rough figure carried in the head because it is usually near enough, and which stops being near enough once the numbers get large.. A business carrying no debt and no cash would have a ratio of one and its two premium figures would be identical. A heavily borrowed business would have a ratio well below 0.80. The correction is per company, always.

Which base is a premium actually quoted on?

On the share price, nearly always. A shareholder is offered that number, and a market can see it for itself. Enterprise value premiums appear in analytical work rather than in announcements, and they appear alongside a multiple far more often than alone.

The rule for a reader is short: before two premium figures are compared, establish what each one was divided by, and where that cannot be established, the two do not belong side by side. A premium quoted without its base is not a weak number or an approximate number; it is not a number that can be set beside another one at all.

What happens to all three figures as the offer price rises?

Two things happen at once, and they are opposite. Moving the price with one control shows both of them at the same time. The two premium figures hold their ratio exactly. The discount drifts steadily away from both.

The second and third columns come first. Any row's third column divided by its second gives 0.8036, at Rs 95.00 and at Rs 130.00 alike. The two premium columns are one column in two units.

Now the fourth column against the second. At Rs 95.00 the discount is 94.7 per cent of the premium. At Rs 115.00 it is 78.3 per cent of it. At Rs 130.00 it is 69.2 per cent of it. The discount is falling away from the premium the whole way up, and it does so along a definite path: the discount is always the premium divided by one plus the premium, so as the premium grows the divisor grows with it. The discount is not a fixed fraction of the premium and never becomes one. No single conversion factor can therefore be memorised and reused.

Why do 22.32 per cent and 21.74 per cent sit so close together, and what does that mean?

Nothing whatever. The two land within six tenths of a point of each other, and a reader who has been half following will take them for two roundings of one calculation. There is an exact reason for the closeness, and the reason turns out to be a coincidence with a precise address.

Put both on a per share footing. The traded enterprise value of Rs 22,40,00,00,000 spread over 20,00,00,000 shares is Rs 112.00 a share. So the enterprise value premium is the Rs 25.00 step divided by Rs 112.00, and the implied minority discount is the same Rs 25.00 step divided by Rs 115.00. One numerator, two denominators, and the denominators happen to be three rupees apart. A three rupee gap between two denominators, and nothing more meaningful than that, is the whole reason 22.32 per cent and 21.74 per cent look like relatives.

The coincidence even has an exact centre. Since one divides by Rs 112.00 and the other by the offer price, the two are equal when the offer price is Rs 112.00 and at no other price at all. Set the offer there and both read 19.64 per cent. The offer actually on the table is Rs 115.00, close enough to Rs 112.00 for the two figures to resemble each other and far enough for them not to match.

Move away from Rs 112.00 and the resemblance goes. At Rs 130.00 the enterprise value premium is 35.71 per cent and the implied discount is 30.77 per cent, five points apart. At Rs 95.00 they cross over, at 4.46 per cent against 5.26 per cent, with the discount now the larger of the two. Two figures that swap places depending on the offer price were never two readings of one thing, and no reader should carry the pair around as though they were.

How fast does the shortcut get expensive?

Faster than most people expect, and the reason is that the cost is not proportional to the premium. The cost is proportional to the square of the step.

Run it along the range. At an offer of Rs 95.00 the shortcut is short by 28 paise a share, an amount no review would ever query. At Rs 110.00 it is short by Rs 4.44. At Rs 120.00 it is short by exactly Rs 10.00. At Rs 130.00 it is short by Rs 17.78, nearly a fifth of the unaffected price. The shortcut is close to harmless where people first meet it and badly wrong exactly where a transaction gets interesting. No error could have a worse shape than that.

The same mistake run backwards, and why it is a different size

Everything so far has gone from a premium to a discount. Run the conversion the other way and the shortcut fails again, and by a different margin. The two unequal margins are what make the shortcut impossible to correct by feel.

Suppose a valuation reports a control premium of 30 per cent and somebody applies a 30 per cent discount to price a holding without control. The true implied discount is 0.30 divided by 1.30, or 23.08 per cent. Taking off 30 gives away 6.92 points too much.

Now suppose the reverse. A valuation reports a minority discount of 30 per cent and somebody applies a 30 per cent premium to get to a control value. The true implied premium is 0.30 divided by 0.70, or 42.86 per cent. Using 30 understates by 12.86 points.

The two errors sit either side of the same shortcut and are not the same size: 6.92 points one way against 12.86 points the other, nearly twice as large. There is no offsetting and no cancelling. A workflow that makes the shortcut in both directions does not come out even; it comes out wrong twice, by unequal amounts, in the direction that suits whoever is holding the pen.

Two figures now sit close together for no reason, so one clarification is worth making while both are on the table. The 23.08 per cent just derived belongs to a general illustration built on a round 30 per cent and has nothing to do with Sankalp Industrial Systems Limited, whose own figures are 27.78, 22.32 and 21.74. The illustration and the case stay apart.

What do these figures refuse to say?

A great deal, and the refusal is part of the answer rather than a disclaimer bolted to the end of it.

A premium of 27.78 per cent does not say that the premium was worth paying. The premium does not say that Rs 115.00 was the right price, a fair price or a defensible price. The premium does not say what control is worth in general, in this segment, or for any other business. No second buyer ever made an offer for Sankalp Industrial Systems Limited, so the premium says nothing about what a different buyer would have paid. A premium is a measurement of one offer against one unaffected price, and a discount is that same measurement read backwards. Neither is a judgement, and neither becomes a general figure by being quoted often enough.

The refusal has teeth in practice. The strongest temptation in this whole area is to collect a handful of reported premiums, take a middle value and carry it around as what control costs. The arithmetic above says nothing against that habit directly, but it does show that such a middle value is unreadable unless every figure in it was quoted on the same base and converted the same way. On the evidence of one invented company, two perfectly correct premium figures for one offer differ by 5.46 points purely on the choice of denominator.