Venture Capital puzzles, solved step by step

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  1. 047A debt-free company has an enterprise value of Rs 1,000 crore and 10 crore shares at Rs 100. It borrows Rs 100 crore and buys back shares at Rs 125. Ignoring taxes, what happens to enterprise value, equity value and the price of the remaining shares, compared with a buyback at Rs 100?Valuation riddlesHardSilver LakeSan Francisco · 2022

    Try it first

    After the buyback at Rs 125, what is each remaining share worth?

    Show the worked solution

    Enterprise value stays at Rs 1,000 crore, equity value falls to Rs 900 crore, and the remaining shares fall to Rs 97.83, against Rs 100 for a buyback at Rs 100. Borrowing Rs 100 crore and paying it out swaps equity for debt without changing what the business is worth. At Rs 100 a share the company retires 1 crore shares and the price holds. At Rs 125 it retires only 0.8 crore, so the Rs 20 crore premium comes out of the holders who stay.

    Why does enterprise value not change?

    A house worth Rs 1 crore is still worth Rs 1 crore after the owner takes out a Rs 10 lakh loan against it and spends the cash; what changes is how much of the house the owner owns outright. Enterprise value is the value of the business, and borrowing to pay shareholders changes who has a claim on it, not what it is worth. The Rs 100 crore of new debt is matched by Rs 100 crore leaving the company, so net debt rises by Rs 100 crore and equity value falls by the same amount, to Rs 900 crore. That holds only while we ignore taxes and the costs of financial distress.

    Borrow Rs 100 crore, buy back shares: what moves and what does notEquity 1,000EV 1,000BeforeEquity 900Debt 100EV 1,000AfterRs crore; ignoring taxesPrice of each remaining shareBuyback priceRs 100Rs 125Shares retired, crore1.00.8Shares left, crore9.09.2Equity Rs 900 cr / sharesRs 100.00Rs 97.83Premium paid to sellers: Rs 20 crore= 9.2 crore shares x Rs 2.17 lost by holders who stay
    Enterprise value stays at Rs 1,000 crore while Rs 100 crore of it moves from equity to debt, and the price of each remaining share holds at Rs 100 for a buyback at Rs 100 but falls to Rs 97.83 for a buyback at Rs 125.

    Why does the buyback price change the price per share?

    Equity is Rs 900 crore after the buyback either way; what differs is how many shares it is split across. Buying at the fair price of Rs 100 retires 1 crore shares and leaves Rs 900 crore over 9 crore shares, Rs 100 a share, so nobody gains or loses. Paying Rs 125 retires only 0.8 crore shares, leaving Rs 900 crore over 9.2 crore, Rs 97.83. The sellers received Rs 100 crore for shares worth Rs 80 crore, a Rs 20 crore premium, and the holders who stay paid it: 9.2 crore shares times Rs 2.17 is Rs 20 crore.

    The relationship
    Pafter=EV−DN−D/Pbb=1,000−10010−100/125=9009.2=97.83P_{\text{after}} = \frac{EV - D}{N - D/P_{bb}} = \frac{1{,}000 - 100}{10 - 100/125} = \frac{900}{9.2} = 97.83
    EVenterprise value, Rs 1,000 crore, unchanged
    Dthe new debt, Rs 100 crore, all paid out
    Nshares before the buyback, 10 crore
    P bbthe buyback price per share
    What it says in wordsThe remaining equity is enterprise value less the new debt, spread over the shares not bought back.

    What changes once you stop ignoring taxes?

    Say the limits, because the interviewer will raise them next. Interest is usually tax-deductible, so the debt creates a tax saving that can lift enterprise value a little above Rs 1,000 crore; against that, more debt raises the risk and cost of distress. Signalling matters too: a buyback can tell the market that management thinks the shares are cheap, which may move the price on its own. None of that changes the core point. A buyback at a premium to fair value moves wealth from holders who stay to holders who sell.

    Where candidates lose it

    The common loss is saying the share price rises because there are fewer shares, or that enterprise value falls because cash left the company. The cash leaving is matched by debt arriving, so EV is unchanged, and fewer shares are matched by less equity.

    The second loss is answering only the Rs 100 case. The question's Rs 125 is there to test whether you see that overpaying for shares hands the Rs 20 crore premium to sellers, and the remaining shares fall to Rs 97.83.

    What the interviewer asks next

    • At what buyback price would the remaining shares rise above Rs 100?
    • With a 25% tax rate and permanent debt, roughly how much does the tax shield add to enterprise value?
    • How would the answer change if the company paid for the buyback from Rs 100 crore of existing cash instead of new debt?

    Asked at Silver Lake, Technology, Media and Telecom, San Francisco, 2022 (Wall Street Oasis): If you raise $100 debt to buy back $100 of shares, how does that affect EV and equity value?

  2. 059A company's assets will be worth either Rs 200 crore or Rs 1,400 crore next year, with equal odds. It owes Rs 1,000 crore of debt, all due then, so the expected asset value of Rs 800 crore is below the debt. Why is the equity still worth something, and how much? Ignore discounting.Valuation riddlesHardSilver LakeSan Francisco · 2022

    Try it first

    What is the equity worth?

    Show the worked solution

    About Rs 200 crore, because equity is a call option on the assets. Shareholders get whatever is left after the debt, but never less than zero. In the bad case the assets of Rs 200 crore all go to lenders and equity gets nothing; in the good case equity keeps Rs 1,400 crore less Rs 1,000 crore, or Rs 400 crore. Half of Rs 400 crore is Rs 200 crore. Lenders, expecting Rs 600 crore on Rs 1,000 crore owed, carry the downside.

    If expected assets are below the debt, where does the equity value come from?

    Think of a lottery ticket bought with borrowed money where the lender can only take the ticket back. If it loses you owe nothing more; if it wins you keep the winnings above the loan. Limited liability makes equity a call optionThe right, not the obligation, to buy an asset at a fixed price. Its payoff is zero below that price and rises one for one above it. on the company's assets with a strike equal to the debt, so equity is worth the average of its payoffs, not the payoff at the average assets. The payoff at Rs 800 crore of assets is zero. The average of the payoffs, 0 and Rs 400 crore, is Rs {ev_eq59:.0f} crore.

    Equity is a call option on the assets, struck at the debt04008001,2001,6000200400600Asset value next year, Rs croreEquity, Rs croreDebt Rs 1,000 crthe strike priceBad: equity 0Good: equity 400Expected equity Rs 200 crpayoff at 800 is 0
    The equity payoff is flat at zero below the Rs 1,000 crore debt and rises above it, so the chord joining the two outcomes passes above the curve: at expected assets of Rs 800 crore the payoff is zero, but expected equity is Rs 200 crore.
    The relationship
    E[max⁡(A−D,0)]=12max⁡(200−1000,0)+12max⁡(1400−1000,0)=0+200=200E[\max(A - D, 0)] = \tfrac{1}{2}\max(200-1000,0) + \tfrac{1}{2}\max(1400-1000,0) = 0 + 200 = 200
    Aasset value next year, Rs 200 or Rs 1,400 crore
    Ddebt due, Rs 1,000 crore
    max(A - D, 0)what shareholders receive; never negative
    What it says in wordsTake what equity gets in each outcome, floored at zero, and average those, rather than averaging the assets first.

    Who pays for the equity's value, and why does it matter in practice?

    Assets are worth Rs 800 crore on average, and equity takes Rs 200 crore of that, so the debt is worth Rs 600 crore against Rs 1,000 crore owed. Equity's option value is paid for by the lenders, which is why shareholders of a distressed company prefer riskier bets: more spread raises the option's value and pushes more of the downside onto the debt. If the outcomes were Rs 0 or Rs 1,600 crore, the same Rs 800 crore average would give equity Rs 300 crore. That is the answer to why a distressed company's stock can trade well above zero, and why lenders write covenants to stop management gambling.

    State the limits. Ignoring discounting, and treating the odds as known, keeps the arithmetic clean; a real valuation would use risk-neutral probabilities and a time value, which is what an option pricing model does. And the debt is assumed to be one bullet due next year; covenants that let lenders take control earlier would cut the equity's option short.

    Where candidates lose it

    The fast wrong answer is zero, or a negative number, from netting Rs 1,000 crore of debt against Rs 800 crore of average assets. It forgets that shareholders cannot owe more than they put in, and that floor is the entire source of value.

    The second trap is getting Rs 200 crore but not naming the mechanism. Say the words call option, struck at the debt, and add who pays for it: the lenders. That is the sentence the interviewer is listening for.

    What the interviewer asks next

    • If the outcomes were Rs 0 or Rs 1,600 crore, what would the equity be worth?
    • Why might management of this company take on a riskier project with a lower expected value?
    • What covenant would you write into the debt to protect against that?

    Asked at Silver Lake, Technology, Media and Telecom, San Francisco, 2022 (Wall Street Oasis): Why would a distressed company have a high equity value?