Corporate Finance

1. Profit is a story, cash is a fact

Learning goal: Convert an accounting forecast into incremental free cash flow.

Accounting profit mixes cash and non-cash items. A project is valued on incremental cash flow: what the firm gains in cash with the project minus what it would have had without it. Sunk costs already spent and allocated overhead that does not change are excluded.

Start from earnings before interest and tax (EBIT), apply the tax rate to get after-tax operating profit, add back non-cash depreciation, then subtract investment in working capital and capital spending. Interest is left out because financing is handled in the discount rate.

A quick example: EBIT 200, tax 25 percent, depreciation 40, capital spending 60, increase in working capital 10. Free cash flow is 200 x 0.75 + 40 - 60 - 10 = 120. Notice that depreciation is added back only because it was subtracted before tax, and its real benefit is the tax it saves, 0.25 x 40 = 10. Also decide whether side effects belong: if the project takes sales from an existing product, count the lost cash flow as a cost of the project.

Worked example

Revenue rises by 200, cash costs rise by 120, depreciation is 20, tax rate is 25%, and net working capital rises by 10. Find the year-1 incremental FCFF (all in the same currency units, no capex).

1. EBIT = 200 - 120 - 20 = 60.
2. After-tax operating profit = 60 x (1 - 0.25) = 45.
3. Add back depreciation: 45 + 20 = 65.
4. Subtract the working-capital build: 65 - 10 = 55.

Mental model: FCFF = EBIT(1 - t) + depreciation - change in NWC - capex.

Common trap: Adding depreciation before tax, or subtracting interest in an FCFF calculation.

2. The invoice you have not been paid for

Learning goal: Time working capital so cash flow is neither early nor late.

Working capital ties up cash before it comes back. If sales rise by 100 and customers pay in 60 days, part of that revenue is still a receivable, not cash. Inventory builds do the same, and payables work in the other direction.

The standard method forecasts the level of net working capital (NWC) each year and charges the change in that level to cash flow. A growing project keeps absorbing cash. When it winds down, the balance is released as an inflow. Leaving out that release understates value.

Build the schedule year by year. If NWC is 15 percent of sales and sales rise from 400 to 500, NWC rises from 60 to 75, so cash flow falls by 15 in that year. If sales fall back to 400 in the last year, NWC returns to 60 and the 15 comes back as an inflow. Check the days measures too: receivable days, inventory days and payable days give the three components of NWC, and a change in any of them changes the cash flow without any change in sales.

Worked example

NWC levels at the end of years 0 to 3 are 0, 10, 25 and 15. Find the cash effect each year and the total.

1. Year 1: change +10, cash effect -10.
2. Year 2: change +15, cash effect -15.
3. Year 3: change -10, cash effect +10.
4. Total = -10 - 15 + 10 = -15.

Mental model: Cash effect of NWC = minus the change in its level; the last balance is released when the project ends.

Common trap: Charging the level instead of the change.

3. One cash flow, one discount rate

Learning goal: Pair free cash flow to firm or to equity with the matching discount rate.

Cash flows to the firm (FCFF) are before payments to lenders, so they belong to all investors and are discounted at the weighted average cost of capital (WACC). Subtracting debt from the resulting enterprise value gives equity value.

Cash flows to equity (FCFE) are after interest and net borrowing, so they belong to shareholders only and are discounted at the cost of equity. Mixing them, for example FCFF at the cost of equity, gives a value that matches neither claim.

The consistency test is simple. Whatever cash flow you discount, the claim holders must match the discount rate. FCFF discounted at WACC gives enterprise value, then subtract net debt to get equity value. FCFE discounted at the cost of equity gives equity value directly. If the capital structure is stable and assumptions are consistent the two should agree closely, and a large gap is a sign that interest, borrowing or the tax shield was handled differently in the two versions.

Worked example

FCFF is a constant 9 per year forever, WACC is 9%, and debt is 40. Find enterprise and equity value (units of 1 million). Treat net debt as 40.

1. Enterprise value = 9 / 0.09 = 100.
2. Equity value = enterprise value - debt = 100 - 40 = 60.
3. Check logic: the 9 is before interest, so subtracting debt after discounting is correct.

Mental model: Cash flow and discount rate must belong to the same investors.

Common trap: Using one rate for both flows because the formula looks the same.

4. The blended price of money

Learning goal: Build WACC from market-value weights and after-tax debt cost.

WACC = (E/V) x cost of equity + (D/V) x cost of debt x (1 - tax rate). Weights use market values, not book values, because investors price their claims at what they could sell them for today.

Interest is tax deductible in this standard model, so debt costs less after tax. The formula assumes the capital mix and business risk stay stable for the project being valued. A project with different risk needs a different rate.

To compute WACC with market weights: equity 600, debt 400, cost of equity 10 percent, pre-tax cost of debt 5 percent, tax 25 percent. Weights are 0.6 and 0.4. WACC = 0.6 x 10 + 0.4 x 5 x 0.75 = 6 + 1.5 = 7.5 percent. Use the yield on the firm's debt as the cost of debt, not the coupon on old bonds, and revisit the weights if the company plans to change its mix.

Worked example

The firm is 60% equity, 40% debt by market value. Cost of equity 12%, pre-tax cost of debt 5%, tax rate 25%. Find WACC in percent.

1. Equity part = 0.60 x 12% = 7.2%.
2. After-tax cost of debt = 5% x (1 - 0.25) = 3.75%.
3. Debt part = 0.40 x 3.75% = 1.5%.
4. WACC = 7.2% + 1.5% = 8.7%.

Mental model: WACC = E/V x re + D/V x rd x (1 - t), with market weights.

Common trap: Using book weights or the pre-tax cost of debt.

5. Four years and a leftover

Learning goal: Discount a project that pays for years and then has a terminal cash flow.

Net present value (NPV) adds the discounted cash flows. Each flow is divided by (1 + r) raised to the year it arrives. A terminal flow at the end of year 4 is discounted four years, together with the year-4 regular flow.

NPV is positive if the project earns more than the required return, given the assumptions about the cash flows and the discount rate. The rule is only as good as those inputs, so a decision should come with a sensitivity check.

Do a full table. For cash flows of -100, 40, 50, 60 at 10 percent, the discount factors are 0.9091, 0.8264 and 0.7513, so present values are 36.36, 41.32 and 45.08 and NPV = 22.76. Then run a check: at what discount rate does the NPV reach zero? That is the IRR, about 21.6 percent. The distance between 10 and 21.6 percent is a margin of safety against errors in the discount rate.

Worked example

A project costs 100 now, pays 30 at the end of each of years 1 to 4, and 20 more at the end of year 4. The discount rate is 8.7%. Find NPV.

1. Annuity factor for four years at 8.7% = 1/1.087 + 1/1.087^2 + 1/1.087^3 + 1/1.087^4 = 3.2612, so the four 30s are worth 97.84.
2. Terminal 20 at the end of year 4 = 20 / 1.087^4 = 14.33.
3. Using unrounded component PVs: NPV=−100+97.835052...+14.325567...=12.160619..., or 12.16. Independently rounded components need not sum to the rounded total.
4. Accept on NPV > 0 if the cash flow and rate assumptions hold.

Mental model: NPV = sum of flows / (1+r)^t with every flow dated to its own year.

Common trap: Moving a terminal flow to the wrong year.

6. IRR is a rate, NPV is a size

Learning goal: Compare NPV and IRR and know when they disagree.

The internal rate of return (IRR) is the discount rate that sets NPV to zero. It answers "what return does this project earn", while NPV answers "how much value does it add at my required return". For one conventional project (one outflow, then inflows) they agree on accept or reject.

They can disagree when ranking mutually exclusive projects of different size or timing. A small project can have a higher IRR but a lower NPV. With several sign changes in the cash flows there can be more than one IRR. When choosing between projects, rank by NPV, because value is the goal.

A small example: project A costs 100 and returns 130 in one year, so IRR is 30 percent and NPV at 10 percent is 18.2. Project B costs 1,000 and returns 1,200, so IRR is 20 percent and NPV at 10 percent is 90.9. B has the lower IRR and the higher NPV. If you can take only one, B adds more value. A modified IRR that assumes reinvestment at a stated rate can reduce some problems but still depends on that assumption.

Worked example

Cash flows are -100 now, then 60 at the end of years 1 and 2. At 10%, what is NPV, and roughly what is IRR?

1. NPV at 10% = -100 + 60/1.1 + 60/1.21 = -100 + 54.55 + 49.59 = 4.13.
2. At 13% NPV is still slightly positive; at 14% it is negative.
3. Solving gives IRR near 13.07%.
4. Since 13.07% exceeds 10%, accept, matching the positive NPV.

Mental model: NPV is dollars of value; IRR is a percentage and ignores size.

Common trap: Ranking exclusive projects by IRR.

7. Borrowing gives you a coupon, and a bill

Learning goal: Value tax shields separately from operations with APV.

Adjusted present value (APV) values the business as if it were financed only with equity, then adds the value of financing side effects. The two main ones are the interest tax shield (a gain) and the expected costs of financial distress (a loss).

For permanent debt D at rate r, with certain shields discounted at r, the annual shield is t x r x D. Its present value is t x D. That shortcut depends on permanence and the shield being as safe as the debt. It is not a universal identity. If debt is repaid on a schedule, discount each year's shield instead.

Compare the two valuation routes. WACC adjusts the rate for the tax shield and discounts operating cash flows at the lower rate. APV keeps the unlevered rate and adds the shield separately. They agree when the assumptions match. APV is clearer when the debt schedule changes over time, such as in a leveraged buyout where debt is paid down fast, because each year's shield is visible and can carry its own discount rate.

Worked example

Unlevered enterprise value is 80m. Permanent debt is 40m at 5%, tax rate 25%, shields certain and discounted at 5%. Present value of distress cost is 5m. Find APV and equity value.

1. Annual shield = 0.25 x 0.05 x 40 = 0.5m.
2. Present value = 0.5 / 0.05 = 10m.
3. APV = 80 + 10 - 5 = 85m.
4. Equity = 85 - 40 = 45m.

Mental model: APV = unlevered value + PV(tax shield) - PV(distress cost).

Common trap: Applying t x D to debt that is not permanent.

8. How much debt is too much

Learning goal: See the trade-off between tax shield and distress cost, without a magic formula.

A static trade-off view says debt adds a tax shield but also raises expected distress and agency costs. At low debt the shield dominates. At high debt the extra distress cost can outweigh it. The best level is where the marginal gain equals the marginal cost.

The numbers in practice are uncertain. Treat any "optimal" ratio as an estimate with a range, and test it against how stable the firm's cash flows are. A firm with volatile cash flows usually faces distress costs sooner.

Evidence from practice helps interpret the trade-off. Firms with stable, tangible assets tend to carry more debt, while firms with volatile cash flows and intangible assets carry less. The pecking order idea says firms prefer internal funds, then debt, then new equity, because outsiders suspect that managers issue equity when it is overvalued. Neither theory explains all behavior, so compare them as lenses rather than choose a single winner.

Worked example

Tax rate 30%. Debt of 30 has estimated distress cost 4 in present value. Find net gain from debt using PV shield = t x D.

1. PV shield = 0.30 x 30 = 9.
2. Subtract distress cost: 9 - 4 = 5.
3. Net gain 5 is positive, so under these assumptions the debt adds value.
4. Check how the answer changes if distress cost doubles to 8: net 1, still positive but thin.

Mental model: Trade-off theory balances a shield that rises with debt against distress costs that rise faster.

Common trap: Quoting a single optimal ratio as if measured.

9. A dividend is just a withdrawal

Learning goal: Test dividend and buyback decisions in a frictionless benchmark.

In a frictionless benchmark with no taxes, no transaction costs and no information effects, paying a dividend moves value from the firm to shareholders without creating any. The share price falls by the dividend on the ex-dividend date. A buyback at fair price leaves price per share unchanged.

Real firms break the benchmark, which is why payout policy still matters: taxes, signaling, and the agency problem of idle cash in the hands of managers. Use the benchmark to separate what is arithmetic from what is a real-world friction.

Work the arithmetic. A firm worth 1,000 with 100 shares trades at 10. It pays a dividend of 1 per share. After the ex-date value is 900, so each share is worth 9, and a holder has 9 plus 1 cash. A buyback of 100 at 10 retires 10 shares, leaving 900 of value over 90 shares, also 10 per share. A holder who wants cash can sell shares in either case, so absent frictions the policies are equivalent.

Worked example

A firm has equity value 100m and 10m shares (price 10). It pays a 10m dividend. Then, in a second case, it uses 10m to buy back shares at 10. Compare the per-share outcomes in the frictionless benchmark.

1. Dividend case: equity 90m over 10m shares = 9 per share, plus 1 in cash, total 10.
2. Buyback case: 1m shares are retired, leaving 9m shares.
3. Remaining equity 90m over 9m shares = 10 per share.
4. Both leave shareholders no better or worse in the benchmark; differences come from frictions.

Mental model: Without frictions, payout changes form, not value.

Common trap: Believing a dividend creates value in the benchmark.

10. Most of the value is in the part you cannot see

Learning goal: Test terminal value assumptions and report a decision under uncertainty.

A terminal value after the forecast horizon is often a large share of enterprise value. A Gordon-growth terminal value is TV = FCF x (1 + g) / (r - g), valid only when g is below r and is sustainable forever. As r - g shrinks, TV rises sharply.

Good practice is to show a small table of results for a few growth and rate pairs, state which pair you chose and why, and check that the implied terminal assumptions are plausible. The decision rule stays NPV, but the conclusion should read "positive under these assumptions, fragile if g exceeds x".

Test the sensitivity. FCF 100, r 10 percent, g 2 percent gives TV = 100 x 1.02 / 0.08 = 1,275. If g is 3 percent, TV = 103 / 0.07 = 1,471, an increase of 15 percent from a one-point change. A reasonable check is to compare the implied exit multiple, TV divided by the final year's EBITDA, against what comparable firms trade at. If the implied multiple is far outside the range, revisit the growth or rate.

Worked example

FCF in the last forecast year is 10, r is 9% and g is 1%. Find the terminal value at that date.

1. TV = 10 x 1.01 / (0.09 - 0.01).
2. Numerator = 10.1.
3. Denominator = 0.08.
4. TV = 126.25.

Mental model: TV is highly sensitive to r - g; show a grid and state your choice.

Common trap: Using a growth rate close to or above the discount rate.

11. Levering and unlevering beta

Learning goal: Move a beta from one capital structure to another and see its effect on the cost of equity.

Equity beta rises with leverage because shareholders bear the fixed claims of lenders. A common textbook relation, assuming debt beta is zero, is levered beta = unlevered beta x (1 + (1 - tax rate) x D/E). The unlevered, or asset, beta reflects business risk alone.

To value a project in a new industry, find comparable firms, unlever each equity beta using its own debt-to-equity ratio, average the unlevered betas, then relever at the target capital structure. This isolates business risk from financing risk.

With CAPM, the cost of equity is the risk-free rate plus equity beta times the market risk premium. If unlevered beta is 0.8, tax is 25 percent and D/E is 0.5, the levered beta is 0.8 x (1 + 0.75 x 0.5) = 1.10. With a risk-free rate of 4 percent and a premium of 5 percent the cost of equity is 4 + 1.10 x 5 = 9.5 percent.

Treat the output as an estimate. Betas are noisy, the formula assumes a zero debt beta and constant debt, and the premium is an assumption you choose. Show a range.

Worked example

Unlevered beta 0.8, tax 25%, D/E 0.5. Levered beta?

1. 1 + 0.75 x 0.5.
2. = 1.375.
3. 0.8 x 1.375.
4. 1.10.

Mental model: Levered beta = unlevered x (1 + (1 - t) D/E). CAPM then gives the cost of equity.

Common trap: Using a company's own beta for a project in a different industry.

12. Sensitivity and break-even: stress the NPV

Learning goal: Show how NPV changes with the discount rate and the cash flow, and find the break-even.

An NPV profile plots NPV against the discount rate. It falls as the rate rises and crosses zero at the IRR. A flatter profile means the project is less sensitive to the rate.

Break-even analysis asks how much an input can change before the decision flips. If a project has NPV of 22 and each year's cash flow is cut by a fixed percent, the break-even cut is the one that brings NPV to zero. That tells decision-makers how wrong the forecast can be.

Scenario analysis changes several inputs together: a base case, a downside and an upside, each with a plausible story. Sensitivity changes one input at a time. Show both, and avoid presenting a single point estimate as certain.

When you present, give the base NPV, the break-even for the biggest drivers and one downside scenario. State what you would monitor after launch.

Worked example

Outlay 100, inflows 40, 50, 60, rate 10%. NPV to 1 decimal?

1. PVs 36.4, 41.3, 45.1.
2. Sum 122.8.
3. Minus 100.
4. 22.8.

Mental model: Show NPV, break-even and a downside; do not hide behind one number.

Common trap: Presenting a single point estimate as certain.

13. The option to abandon

Learning goal: Value flexibility with a one-step decision tree.

Many projects can be stopped if they go badly. A simple model has one step: invest an outlay now, then next year the cash flow is high with probability p or low with probability 1 - p. If the firm can abandon for a salvage value, in the low state it takes the larger of the low cash flow and the salvage.

Value = (p x High + (1 - p) x max(Low, Salvage)) / (1 + r) - Outlay. The value of the option is the difference from the same project without abandonment. It is never negative, and it is zero when the salvage is below the low cash flow.

The model is crude. Probabilities are guesses, one step hides many paths, and a risk-adjusted rate is used on a payoff that changes risk. Still, the tree shows that flexibility has value, so a rigid NPV can undervalue a project that can be stopped.

Use this to ask better questions. What can we learn early? What is the cost of stopping? Can we scale up if it works?

Worked example

Outlay 100, p = 0.5, High 200, Low 20, salvage 60, r 10%. Value with abandonment?

1. Low state uses max(20, 60) = 60.
2. Expected 0.5 x 200 + 0.5 x 60 = 130.
3. 130 / 1.1 = 118.2.
4. Minus 100 gives 18.2.

Mental model: Flexibility has value: with abandonment the low state is floored at the salvage value.

Common trap: Ignoring flexibility when it exists.