Debt Capital Markets puzzles, solved step by step

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100
Traced to a firm
16
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13
Hard
30
Topic
All topicsLeverage, coverage and cash flow9Mental maths and numeracy8Estimation and market sizing7Logic and brainteasers8Cost of capital and valuation riddles7Bond pricing and yield7Compounding, PIK and fees6Issuance and refinancing arithmetic8Credit spreads and default probability8Duration and convexity8Capital structure and recovery8Probability and expected value10Yield curve and forward rates6
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  1. 005A company can run at 0%, 20%, 40% or 60% debt to capital. Its pre-tax cost of debt would be 7%, 7.5%, 9% and 12% at those levels, and its cost of equity 12%, 12.8%, 14% and 17%. Tax is 25%. Which leverage minimises the weighted average cost of capital?Cost of capital and valuation riddlesHardCorporate banking

    Try it first

    Before you weight anything: where do you expect the minimum?

    Show the worked solution

    40% debt, where WACC bottoms at 11.10%. After tax, debt costs 5.25%, 5.625%, 6.75% and 9% at the four levels. Weighting each with equity gives 12.00%, 11.365%, 11.10% and 12.20%. Cheap, tax-shielded debt pulls WACC down at first; by 60% the rise in both debt and equity costs outweighs the extra cheap debt.

    Why does WACC fall at first?

    Think of funding a house with a mix of a bank loan and money from relatives who expect a share of any gain. The bank is cheaper, and in this example the interest also cuts your tax bill. Replacing expensive equity with cheaper, tax-deductible debt lowers the blended cost, as long as neither cost rises too fast. Going from 0% to 20% debt, the after-tax debt cost is only 5.625% against equity at 12.8%, so WACC drops from 12.00% to 11.365%.

    WACC falls while debt is cheap, then climbs as both costs rise11.0%11.5%12.0%0%20%40%60%Debt as a share of capital12.00%11.37%11.10%, lowest12.20%DebtKd after taxKeWACC0%5.250%12.0%12.00%20%5.625%12.8%11.37%40%6.750%14.0%11.10%60%9.000%17.0%12.20%Kd after tax = pre-tax cost x (1 - 25%)WACC = D/V x Kd after tax + E/V x Ke
    WACC falls from 12.00% with no debt to 11.37% at 20% and 11.10% at 40%, then jumps to 12.20% at 60%, because at that level the after-tax debt cost reaches 9% and equity demands 17%.

    Why does it turn up again?

    Every extra rupee of debt makes the company riskier for everyone. Lenders ask for more, and shareholders, now standing behind a bigger claim, ask for more too. Past some point the rising cost of both sources outweighs the benefit of shifting weight onto the cheaper one. From 40% to 60%, the debt weight rises by 20 points, but debt's after-tax cost jumps from 6.75% to 9% and equity's from 14% to 17%, and the blend climbs 1.10 points. That is the trade-off theoryThe idea that a company balances the tax saving from debt against the rising cost of financial distress as leverage grows. in four data points.

    The relationship
    WACC=DV kd(1−t)+EV ke=0.4×9%×0.75+0.6×14%=2.70%+8.40%=11.10%\text{WACC} = \tfrac{D}{V}\,k_d(1-t) + \tfrac{E}{V}\,k_e = 0.4 \times 9\% \times 0.75 + 0.6 \times 14\% = 2.70\% + 8.40\% = 11.10\%
    D/V, E/Vthe shares of debt and equity in total capital
    k_dpre-tax cost of debt at that leverage
    ttax rate, 25%
    k_ecost of equity at that leverage
    What it says in wordsWACC is each source's cost, debt after tax, weighted by its share of the capital.

    What does a lender add after the number?

    Two caveats show judgement. The minimum is only as good as the cost schedule, and in real life that schedule is an estimate, especially the equity cost at high leverage. Only four points were given, so the true minimum could sit anywhere between 20% and 60%; say that you would want more points before recommending a structure. And a debt desk also cares about what the WACC table cannot show: whether 40% leverage keeps the rating in a band that keeps the market open in a bad year.

    Where candidates lose it

    The first trap is using the pre-tax debt cost. Without the tax shield WACC at 40% is 12.00%, higher than the 11.74% at 20%, so the minimum moves to 20% and the answer changes for a reason the interviewer planted deliberately.

    The second is assuming more debt is always better because debt is cheaper. The question gave rising costs precisely to test whether you weight each level with its own numbers rather than one fixed debt cost.

    What the interviewer asks next

    • What tax rate would make 20% and 40% leverage give the same WACC?
    • If the cost of equity at 60% were 15% instead of 17%, where is the minimum now?
    • Why might a company choose to run below its WACC-minimising leverage?
  2. 050An unlevered firm is worth Rs 2,000 crore. It adds Rs 800 crore of permanent debt at a 25% tax rate. Distress would cost 30% of firm value and the extra debt creates a 10% probability of distress. Is the firm worth more or less after the debt?Cost of capital and valuation riddlesHardCorporate bankingLeveraged finance

    Try it first

    Is the firm worth more or less with the debt?

    Show the worked solution

    Worth more: about Rs 2,134 crore, Rs 134 crore above the unlevered value. Permanent debt of Rs 800 crore at a 25% tax rate creates a tax shield worth 0.25 x 800, Rs 200 crore, lifting value to Rs 2,200 crore. Distress would destroy 30% of that, Rs 660 crore, but with a 10% chance the expected cost is Rs 66 crore. The shield wins unless the distress probability passes about 30%.

    Where does debt add value, and where does it take it away?

    A family that takes a home loan gets a tax deduction on the interest every year, a steady saving. But if the loan is too big and a job is lost, they may have to sell the house in a hurry at a poor price. Debt adds value through the tax saved on interest and subtracts value through the expected cost of financial distress; the net of the two decides whether borrowing helps. That balance is the {term('trade-off theory', 'The idea that a company chooses its debt level by weighing the tax benefit of interest against the expected costs of financial distress.')} of capital structure.

    The relationship
    VL=VU+tD−p⋅c⋅(VU+tD)=2,000+200−0.10×0.30×2,200=2,134V_L = V_U + tD - p \cdot c \cdot (V_U + tD) = 2{,}000 + 200 - 0.10 \times 0.30 \times 2{,}200 = 2{,}134
    V_Uvalue of the firm with no debt, Rs 2,000 crore
    tDpresent value of the tax shield on permanent debt, 0.25 x 800 = 200
    pprobability of distress, 10%
    cshare of firm value lost in distress, 30%
    What it says in wordsLevered value is unlevered value plus the tax shield, less the probability-weighted cost of distress.
    Debt adds value through the tax shield until expected distress costs catch up2,000Unlevered firm+200Tax shield-66Expected distress2,134Levered firm1,800Axis starts at Rs 1,800 crore so the moves are visibleNet gain+134BreakevenPD 30.3%not 10%
    The firm's Rs 2,000 crore unlevered value rises to Rs 2,200 crore with the Rs 200 crore tax shield, and a 10% chance of losing 30% in distress takes off Rs 66 crore, leaving Rs 2,134 crore, a net gain of Rs 134 crore.

    Why is the tax shield worth t times D?

    For permanent debt, the company pays interest of r times D every year forever and saves t times that in tax. Discounting a perpetual saving of t r D at the debt's own rate r gives t D. So Rs 800 crore of permanent debt at a 25% tax rate is worth Rs 200 crore in tax savings, whatever the interest rate, as long as the debt is truly permanent and the company always has profits to deduct against. Both conditions are generous, which is why real tax shields are usually worth less than t D.

    Now find where the answer flips. The shield is 200; the distress cost is the probability times 660. The firm is worse off once the distress probability passes 200 over 660, about 30.3%, so debt at this level is value-adding only while distress stays well below a one-in-three chance. Adding more debt raises both lines, the shield in a straight line and the distress probability faster, which is why an optimal level exists. The limits: the 30% cost and the 10% probability are estimates, and indirect distress costs, lost customers and staff, are the hardest part to measure.

    Where candidates lose it

    One common error is to compare the full Rs 660 crore distress cost with the Rs 200 crore shield and conclude the debt destroys value. The distress cost only happens 10% of the time; it has to be weighted by its probability.

    The opposite error is adding the Rs 200 crore shield and stopping, which is the textbook answer with taxes and no distress. The question gave you the distress numbers so you would net them off.

    What the interviewer asks next

    • At what distress probability does the firm become worth less than Rs 2,000 crore?
    • The debt is only outstanding for five years, not forever. How does that change the value of the tax shield?
    • Why might a stable utility carry more debt than a young technology company with the same value?