Equity Research puzzles, solved step by step

The terminal value is about 77% of the value, and a one point rise in WACC cuts the total by about 16%. At 10% the five years are worth 379 and the terminal value 1,304 in today's money, total 1,683. At 11% they are worth 370 and 1,039, total 1,408. The explicit years barely move; the terminal value drops by a fifth.

Where does most of the value sit?

Think of valuing a flat you plan to rent out for five years and then keep forever. The five years of rent are real, but the flat itself, the part you keep, is most of what you are paying for. A DCF is the same. The five explicit years are worth 379; everything after year five, the terminal value, is worth 1,304 in today's money, 77.5% of the total. The terminal value at year five is 100 x 1.05 / (0.10 - 0.05) = 2,100, discounted back five years.

Why does one point of WACC move the value so much?

The terminal value divides by the gap between WACC and growth. Moving WACC from 10% to 11% widens that gap from 5% to 6%, cutting the undiscounted terminal value from 2,100 to 1,750, a sixth, before the extra discounting takes more. The explicit years fall only from 379.1 to 369.6. Together the value drops from 1,683 to 1,408, down 16.3%.

This is why a research note shows a sensitivity table of WACC against terminal growth rather than a single number. It is also why you check the implied exit multiple: 2,100 is 21 times year-five cash flow, and if peers trade nowhere near that, the inputs need a second look. The limitation is that the table shows sensitivity; it does not tell you which rate is right.

B trades lower, at 13.3x against A's 16.7x, and they match when B's debt costs 8%. A earns 80 x 0.75 = 60 on equity of 1,000. B pays 30 of interest, earns 37.5 on equity of 500. The P/Es match when the after-tax cost of debt equals A's earnings yield of 6%: r x 0.75 = 6%, so r = 8%. Cheaper debt lowers B's P/E; dearer debt raises it.

Why would the same business carry two different P/Es?

Suppose a flat earns rent of Rs 6 for every Rs 100 of its price. Buy it with half cash and half a loan at 4.5% after tax, and your cash earns more than 6%, because the borrowed half costs less than it yields. P/E divides the equity's value by the equity's earnings, and debt changes both, by different amounts. Swapping equity for debt that costs less than the equity's earnings yield lowers the P/E; swapping for dearer debt raises it.

Why is 8% the crossing point?

Swapping 500 of equity for 500 of debt removes equity that was earning its share of A's 6% earnings yield, 30 of net income, and adds an after-tax interest cost of 500 x r x 0.75. If that cost is exactly 30, net income halves along with equity and the P/E does not move; 30 over 375 is 8% before tax. The general rule: leverage cuts P/E when the after-tax cost of debt is below the earnings yield, E/P. The limitation worth saying: B's lower P/E is not a sign it is cheaper. Its equity is riskier, so investors should demand a lower multiple for the same business.

About 53.64 per share. Equity value is 5,000 plus 500 of net cash, 5,500. If the price is P, the options add 10 shares and their 400 of exercise cash buys back 400 / P shares, so P = 5,500 / (110 - 400/P). Solving gives P = 5,900 / 110 = 53.636. Iterating from the naive 55 gets there in three rounds.

Why can you not just divide by a share count?

Think of splitting a restaurant bill where one late guest pays a fixed Rs 40 whatever the bill, and the rest is shared. How much the others pay depends on the bill, and the bill depends on who is sharing it. Under the treasury stock methodA way to count option dilution: assume in-the-money options are exercised and the exercise cash is used to buy back shares at the current price., how many net new shares the options create depends on the share price, and the share price depends on how many shares there are. The dilution and the answer have to be found together.

How do you solve it in one line instead of looping?

Write P x (110 - 400/P) = 5,500. The P cancels in the second term, leaving 110P - 400 = 5,500, so P = 5,900 / 110 = 53.636. When the options are in the money, the consistent price is simply equity value plus exercise cash, divided by all shares including the options. The check: at 53.636, the 400 of cash buys back 7.458 shares, so 2.542 net new shares take the count to 102.542, and 5,500 over that is 53.636.

Say the limitation. The method ignores the time value of options and assumes exercise today. A model that values the options properly would subtract their value from equity instead, and the answer would come out slightly lower.

About 6.7x, on an implied stub value of Rs 2,000 crore. The 50% stake is worth half of Rs 16,000 crore, Rs 8,000 crore, at the subsidiary's own market price. Take that out of the holding company's Rs 10,000 crore and the market is paying Rs 2,000 crore for everything else. Against Rs 300 crore of earnings that is 6.7 times, assuming the holding company carries no debt or cash of its own.

How do you find the price of a business that has no price of its own?

A thali costs Rs 300 and includes a sweet the same restaurant sells alone for Rs 80. The rest of the meal is costing you Rs 220. When a company's value contains something with its own visible price, subtract that price to see what the market is paying for the rest. Analysts call what is left the stubThe value the market implicitly assigns to a holding company's own business after subtracting the market value of its listed stakes.. Here the visible item is the listed stake, worth Rs 8,000 crore at the subsidiary's share price.

Why might 6.7x not be the whole story?

Holding companies usually trade below the value of what they own. If the market applies a 20% holding-company discount to the stake, it is valuing the stake at Rs 6,400 crore, so the stub rises to Rs 3,600 crore and the implied multiple to 12x. The discount reflects tax on any eventual sale of the stake, dividends that may never reach the holding company's own shareholders, and the cost of running the holding company. Give both numbers and say which assumption produces each.

What would you check before calling the stub cheap?

Three things. Whether the holding company carries debt, which the stub has to absorb; whether the Rs 300 crore of earnings is recurring or flattered by one-off items; and whether the stake is ever likely to be sold or distributed. A discount that never closes is not a mispricing, so a low stub multiple is a question to investigate, not a conclusion.

About 9.7% a year, forever. For a bank, justified price to book is ROE minus growth over cost of equity minus growth. Setting 2.5 equal to (18% minus g) over (13% minus g) gives g of 9.67%. With no growth the bank would deserve 1.38x, so about 45% of today's price is paying for growth, which needs 54% of profits retained every year.

Why does price to book depend on ROE against cost of equity?

Picture a shop that earns Rs 18 a year on every Rs 100 its owner has put in, when the owner could earn Rs 13 elsewhere for the same risk. Nobody would sell that shop for Rs 100. A bank's book value is the owners' capital, so a bank earning more on it than shareholders demand is worth more than its book, and one earning less is worth less. With no growth, every Rs 100 of book produces Rs 18 a year forever, worth 18 over 13, or {pb0_80:.2f} times book.

How do you get the growth out of the price?

Growth needs capital. A bank growing its book at g must retain g over ROE of its profits, so the dividend on each Rs 100 of book is ROE minus g, not ROE. Put that dividend into the Gordon growth modelA valuation of a stream of dividends growing at a constant rate forever: next year dividend divided by cost of equity minus growth. and the price to book falls out. Set the formula equal to 2.5 and solve: 2.5 times (13% minus g) equals 18% minus g, so 1.5g equals 14.5%, and g is 9.67%.

Is 9.7% forever believable?

That is the question the interviewer really wants answered. A reverse valuation is only useful if you then judge the implied number. Growing book at 9.7% while paying out 46% of profits requires the 18% ROE to hold for decades, with asset quality intact. Notice how steep the curve is near the answer: a point more of growth, or a point less of cost of equity, moves the justified multiple a long way, which is why small changes in rate expectations swing bank valuations. The limitation is the model itself: one growth rate forever is a simplification, and a two-stage version is fairer to a bank growing fast today.