Trade-off theory of capital structure

Put the two pieces together and the levered value becomes

V_L = V_U + t·D − p(rating) × k × V_U

p(rating) = the cumulative probability of distress at the rating that D implies
k         = the cost of distress if it happens, as a fraction of unlevered value

The first term is flat, the second rises linearly in debt, and the third rises much faster than linearly, because the probability schedule steepens sharply below investment grade. A linear gain against a convex loss produces an interior maximum, which is the whole shape of the trade-off model.

Worked example: Chandori Forgings

Chandori Forgings Ltd (synthetic) is the anchor. It earns ₹480 crore of EBITDA and ₹360 crore of operating profit, is taxed at 25.17%, and has an unlevered cost of equity of 13.5%, so V_U = 360 × 0.7483 / 0.135 = ₹1,995.466667 crore. The full six-point grid is worked in the first worked example; the shape is what matters here. Value rises from ₹2,028.52 crore at 0.3 turns of net debt to EBITDA, peaks at ₹2,184.20 crore at 1.8 turns, and falls back to ₹2,126.44 crore at 3.0 turns.

Look at the neighbourhood of the peak rather than the peak. The 2.4-turn point is worth ₹2,181.66 crore, which is 0.1163% below the maximum. The 1.2-turn point, where a conservative board would actually stop, is 2.5149% below it. Across the whole grid from 0.3 to 3.0 turns, total value moves 7.1276%. In value per share the plateau between 1.2 and 1.8 turns spans 2.6924%.

What does the flat peak tell an analyst?

Three consequences follow, and they are the reason to build the curve rather than read about it. First, the model over-predicts leverage relative to what firms choose, which is a genuine puzzle and not an error in your arithmetic. Second, precision about the optimum is false precision, because a 15-point change in the distress-cost fraction moves it a full 0.6 turns, from 1.8 to 2.4. Third, and most usefully, the flatness is itself the finding: if a company sits anywhere on the plateau, leverage is not what is wrong with it, and an analyst who recommends a recapitalisation to capture 11.63 basis points of modelled value has mistaken the model for the business.

{
  "skill": "Building a trade-off curve with a solved coverage-to-spread fixed point",
  "use": "Have an assistant write the iteration that solves rating, spread and coverage jointly across a leverage grid, then have it write a SECOND implementation that iterates upward from the worst rating, so the two can be diffed and the step-function multiplicity is exposed rather than hidden.",
  "tools": ["Chat assistants with code execution - Claude, ChatGPT, Gemini", "Python: NumPy or plain loops; pandas for the grid table", "A spreadsheet with Goal Seek, for the by-hand replication of one row"],
  "prompt": "Here is a rating table of (coverage band, spread, cumulative distress probability), a base rate, an EBIT, a tax rate and an unlevered value. For a grid of debt levels, solve the fixed point rd = base + spread(EBIT / (D x rd)) by iterating from the narrowest spread downward, and return debt, rating, spread, rd, coverage, tax shield, expected distress cost and levered value. Then re-solve iterating upward from the widest spread and print every debt level where the two answers differ. Do not round intermediates.",
  "verify": "Recompute two rows by hand, including the coverage that justifies the rating, and confirm the iteration converges rather than oscillating. Trace the spread table and the distress-probability schedule to their primary sources and date both, because spreads are refreshed annually and a stale table silently moves the optimum. Match the tax rate to the marginal statutory rate, not the effective rate. Then re-run the grid at two other distress-cost fractions and confirm the optimum moves; if it does not, the schedule is too flat to be doing any work and the model is a tax-shield calculation wearing a trade-off label.",
  "diy": "You must be able to solve one row of the fixed point with a calculator and read the rating off the table, because that is the gated skill and Worked example 1 is the drill. A machine produces a smooth curve with a confident maximum whatever schedule you feed it. Only you can say whether a 40% distress-cost fraction is defensible for this industry, and that judgement is the analytical content of the exercise.",
  "market": "IN"
}