Investment Banking puzzles, solved step by step

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017Free cash flow of Rs 100 crore arrives every year for five years and the discount rate is 10%. How much does the mid-year convention add to the value, and why?DCF and cost of capitalCoreLazardNew York · 2026
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How much does the mid-year convention add?
Show the worked solution
It adds about Rs 18.5 crore, or 4.9%, taking the value from Rs 379.1 crore to Rs 397.6 crore. The mid-year convention assumes each year's cash arrives evenly through the year, so on average halfway through rather than at the end. Every cash flow is then discounted for half a year less, which multiplies each present value by 1.10 to the power of one half, 1.0488.
Why would cash arrive in the middle of the year?
A shop does not collect its year's takings in one lump on the last day of March; money comes in every day. A DCF that discounts each year's cash flow as if it all arrived on the final day is too harsh. The mid-year convention treats each year's cash flow as arriving, on average, halfway through the year, so it is discounted for t minus one half years instead of t. For a business with steady receipts that is the more accurate assumption, and it is the default in many banks' models.
Moving each Rs 100 crore cash flow from the end of its year to the middle lifts its discount factor by the same ratio, so the five-year value rises from Rs 379.08 crore to Rs 397.58 crore, an uplift of 4.88%.
The relationship
$\sum_{t=1}^{5} \frac{100}{1.10^{\,t-0.5}} = 1.10^{0.5} \times \sum_{t=1}^{5} \frac{100}{1.10^{\,t}} = 1.0488 \times 379.08 = 397.58$
t - 0.5 each year's cash treated as arriving at the middle of the year
1.10^0.5 the gain from discounting every cash flow for half a year less
What it says in wordsShifting every cash flow six months earlier multiplies the whole value by the square root of one plus the discount rate.
Why is the uplift exactly the square root of 1.10 minus one?
Because the shift is the same for every cash flow. Moving a payment six months earlier multiplies its present value by 1.10 to the power of one half, whatever year it sits in. Since every cash flow gets the same uplift, the whole value rises by the same 4.88%, and neither the size nor the pattern of the cash flows changes that. Half the discount rate, 5%, is a fair first approximation to say out loud; at a 20% rate the exact uplift is 9.5%, against the 10% the approximation would give, so the shortcut weakens as rates rise.
Where should you not apply it?
Two cautions. A terminal valueThe value of all cash flows beyond the forecast years, usually from a growing perpetuity or an exit multiple. built on a perpetuity of flows is usually discounted on the same mid-year basis, but one built on an exit multiple is not, because a sale happens at a point in time, the end of the final year. Shifting an exit-multiple terminal value half a year earlier overstates the answer, and an interviewer who asks about the convention is often fishing for exactly that. For seasonal businesses that collect most cash late in the year, the convention itself flatters the value.
Where candidates lose it
The usual slip is to say the convention changes the cash flows, or that it only affects the first year. It changes timing only, and it moves every cash flow by the same six months, so the uplift applies to all of them.
The second slip is applying it to a terminal value built on an exit multiple. A sale happens at a moment, so that value stays discounted for the full five years; candidates who shift it too overstate the value and get caught on the follow-up.
What the interviewer asks next
If the discount rate were 20%, what would the convention add?
Should the mid-year convention apply to a terminal value built on an exit multiple?
For a business that bills most customers in its final quarter, is the convention still appropriate?
Asked at Lazard, Investment Banking, New York, 2026 (Wall Street Oasis): Asked about SBC, LBOs, Advanced accounting questions and mid-year conventions
028A company's tax rate rises from 25% to 30%. Name three ways this moves a DCF and put numbers on each for a business with EBIT of Rs 200 crore, funded 30% by debt at 8% before tax and 70% by equity at 12%.DCF and cost of capitalHardGoldman SachsSan Francisco · 2025
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Which way does the valuation move overall?
Show the worked solution
The value falls, by about 4.8% here, because the cut to cash flow outweighs the cheaper debt. One: unlevered free cash flow falls, as NOPAT drops from Rs 150 crore to Rs 140 crore. Two: the after-tax cost of debt falls from 6.0% to 5.6%, taking the WACC from 10.20% to 10.08%. Three: the terminal value, which carries both effects, falls from about Rs 2,516 crore to Rs 2,395 crore at 4% growth.
Why does a tax rate touch both the top and the bottom of the fraction?
A landlord pays more tax on her rent, so the rent she keeps falls; but the interest on her mortgage is deductible, so the loan costs her a little less after tax. Two opposite effects from one change. A DCF is the same fraction: cash flow on top, discount rate below. A higher tax rate shrinks the cash flows directly and shrinks the WACC indirectly through the interest tax shield, and the first effect is almost always larger because it hits every rupee of operating profit while the second touches only the debt slice. Interviewers ask for three ways so that you show both sides and then say which wins.
The tax rise cuts NOPAT from Rs 150 crore to Rs 140 crore, lowers the WACC from 10.20% to 10.08% through the cheaper after-tax debt, and takes the terminal value from Rs 2,516 crore to Rs 2,395 crore, because the cash flow effect of Rs -168 crore outweighs the WACC relief of Rs 46 crore.
How big is each effect on these numbers?
Cash flow. NOPAT is EBIT times one minus the tax rate: Rs 200 crore x 0.75 = Rs 150 crore before and Rs 200 crore x 0.70 = Rs 140 crore after, a fall of Rs 10 crore, or 6.7%, in every forecast year, treating depreciation, capex and working capital as cancelling out. Discount rate. The after-tax cost of debt is 8% x (1 - t): 6.0% before and 5.6% after. Weighted at 30% debt and 70% equity at 12%, the WACC moves from 10.20% to 10.08%, down 12 basis points. Terminal value. At 4% growth, a Gordon growthA terminal value that treats the final cash flow as growing at a constant rate forever: the next cash flow divided by the discount rate less the growth rate. terminal value is Rs 2,516 crore before and Rs 2,395 crore after. Separating the two, the cash flow cut alone takes Rs 168 crore off the terminal value and the lower WACC alone adds Rs 46 crore back, so the net is Rs 121 crore, 4.8% lower.
Line Tax 25% Tax 30% Change
NOPAT, Rs crore 150 140 -6.7%
After-tax cost of debt 6.0% 5.6% -40 bp
WACC 10.20% 10.08% -12 bp
Terminal value at 4% growth, Rs crore 2,516 2,395 -4.8%
Illustrative inputs from the question: EBIT Rs 200 crore, 30% debt at 8%, 70% equity at 12%, long-run growth 4%.
The relationship
$WACC = 0.30 \times 8\%\,(1-t) + 0.70 \times 12\% \qquad TV = \frac{EBIT\,(1-t)\,(1+g)}{WACC - g}$
(1 - t) the share of each rupee left after tax, 0.75 falling to 0.70
0.30, 0.70 the debt and equity weights
g the long-run growth rate, 4% here
What it says in wordsThe tax rate enters the cash flow once and the discount rate once, and the cash flow effect is the larger of the two.
Is there a subtler channel the interviewer might be after?
Yes: the cost of equity. When you relever a beta, the formula carries a (1 - t) term, so a higher tax rate lowers the levered beta a little and the cost of equity with it. With debt to equity of 3 to 7 and a levered beta of 1.0 at 25%, the unlevered beta is 0.757, and relevering at 30% gives 0.984; with an assumed 6% risk-free rate and 6% equity risk premium, the cost of equity slips from 12% to 11.90% and the WACC to 10.01%. Even with that help the value is still about 3.7% lower, so the ranking of the effects does not change. Say the limitation too: the WACC relief assumes the company actually pays tax and keeps its interest deductible, and interest deductions are capped in many jurisdictions, a rule to confirm for the country in question.
Where candidates lose it
The common loss is naming only the cash flow effect, or naming the WACC effect and then concluding that value rises. Three ways means cash flow, the cost of debt inside the WACC, and the terminal value or cost of equity that carries them, and the net is down.
The second loss is putting no numbers on it. The question hands you EBIT, weights and rates so that you can say Rs 150 to Rs 140 crore and 10.20% to 10.08% out loud; a candidate who stays qualitative has not answered.
What the interviewer asks next
If the company had no debt at all, which of the three effects survive?
How would a large tax-loss carryforward change the answer?
Why might a biotech DCF, with years of losses ahead, be almost indifferent to this change?
Asked at Goldman Sachs, Investment Banking, San Francisco, 2025 (Wall Street Oasis): Three ways in which the tax rate affects a DCF?
043Comparable companies have an unlevered beta of 0.8. Your target will run at a debt-to-equity ratio of 50% with a 25% tax rate. The risk-free rate is 7% and the equity risk premium 6%. What is the target's cost of equity?DCF and cost of capitalCoreMoelis & CompanyLos Angeles · 2026
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What cost of equity do you get?
Show the worked solution
About 13.6%. The peers' 0.8 is an unlevered beta, the risk of the business with no debt. The target carries debt equal to 50% of its equity, so relever it: 0.8 times (1 plus 75% times 0.5) is 1.1. CAPM then gives 7% plus 1.1 times 6%, which is 13.6%. Using the unlevered 0.8 directly would understate the cost at 11.8%.
Why can you not use the peer beta as it is?
Think of two people with the same job and salary, one with no loan and one with a large home loan. A pay cut hurts the borrower more, because the loan payment does not shrink. Debt magnifies the swings that reach shareholders, so the same business carries a higher equity beta the more it borrows. Peers' betas are first stripped of their own debt to give an unlevered betaThe beta a business would have with no debt: the risk of its operations alone., then relevered at the target's own mix of debt and equity.
The peers' unlevered beta of 0.8 relevers to 1.1 at a 50% debt-to-equity ratio and a 25% tax rate, giving a cost of equity of 13.6%, while the three common slips give 11.8%, 13.0% and 14.2%.
How does the relevering work?
Use the standard formula, which assumes the debt itself carries no market risk. Levered beta equals unlevered beta times one plus (one minus the tax rate) times debt over equity: 0.8 times (1 + 0.75 x 0.5), which is 0.8 times 1.375, or 1.1. The tax term is there because interest is tax deductible, so the tax saving softens a quarter of the debt's burden on shareholders. Then CAPM: 7% plus 1.1 times 6% is 13.6%.
The relationship
$\beta_L = \beta_U\left[1 + (1-t)\frac{D}{E}\right] = 0.8 \times 1.375 = 1.1 \qquad k_e = 7\% + 1.1 \times 6\% = 13.6\%$
beta_U the peers' unlevered beta, 0.8
t the tax rate, 25%
D/E the target's debt divided by its equity, 50%
What it says in wordsAdd back the target's own debt to the business risk, then price that risk with CAPM.
Which slips change the answer?
Three, each worth naming. Skipping the relevering gives 11.8%, using debt over total capital instead of debt over equity gives 13.0%, and ignoring tax gives 14.2%. A spread of 2.4 points in the cost of equity moves a DCF a long way, so the interviewer wants to hear which ratio you are plugging in. State the limit too: the rates here are illustrations, and a live model takes the current government bond yield and the premium the bank uses.
Where candidates lose it
The fast loss is plugging 0.8 straight into CAPM. The peers' beta has been cleaned of debt, so using it unadjusted values the target as if it had none, and the cost of equity comes out too low.
The quieter loss is the ratio. The formula takes debt over equity, 50%, not debt over total capital, one third. Say the ratio out loud before you multiply.
What the interviewer asks next
The target's debt rises to 100% of equity. What is the cost of equity now?
What is the WACC if the target's debt costs 10% before tax?
Why do you unlever each peer before averaging, rather than averaging the levered betas?
Asked at Moelis & Company, Investment Banking, Los Angeles, 2026 (Wall Street Oasis): high-level technical questions aimed at accounting (NWC) and valuation topics (EV, DCF, UFCF, Cost of Equity)
085A software company's free cash flow is Rs 130 crore if Rs 30 crore of stock-based compensation is added back as a non-cash charge, and Rs 100 crore if it is treated as a cash cost. With a WACC of 10% and perpetual growth of 3%, how far apart are the two perpetuity values, and which treatment is right?DCF and cost of capitalHardLazardNew York · 2026
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Which treatment gives the right value for today's shareholders?
Show the worked solution
About Rs 441 crore apart: Rs 1,913 crore with the add-back against Rs 1,471 crore without, a 30% gap. Treat the compensation as a real cost. It is paid in shares rather than cash, but existing holders bear it through dilution. Either deduct it from cash flow and use today's diluted share count, or add it back and count every future share it will create. Adding it back and ignoring the dilution counts the cost as zero.
If no cash leaves, why is stock compensation a cost?
Imagine a family business that pays its manager not in salary but by handing over a slice of the shop every year. No cash leaves the till, but each year the family owns less of the shop. Stock compensation is a real expense paid in ownership instead of money, and today's shareholders pay it through dilution. If the company had paid staff Rs 30 crore in cash, everyone would deduct it. Paying in shares changes who bears the cost, not whether there is one.
The two values follow from the perpetuity formula. With the add-back, 130 x 1.03 / (10% - 3%) is about Rs 1,913 crore. Treated as a cost, 100 x 1.03 / 7% is about Rs 1,471 crore. The gap, about Rs 441 crore, is exactly the present value of paying Rs 30 crore a year, growing at 3%, in shares.
Adding back Rs 30 crore of stock compensation lifts the perpetuity value from Rs 1,471 crore to Rs 1,913 crore, and the Rs 441 crore gap is the present value of the shares handed to staff, so it must be deducted or counted as dilution.
Can the add-back version ever be right?
Yes, if you finish the job. Add the Rs 30 crore back, value the firm at Rs 1,913 crore, then recognise that the company will issue Rs 30 crore of new shares every year, growing at 3%, forever. Those future holders own a stream worth Rs 441 crore of the total, leaving Rs 1,471 crore for today's shareholders. Both consistent methods land on the same value for existing holders; the error is adding back the expense and then dividing by today's share count, which counts the cost nowhere. In practice, deducting it is simpler, because projecting every future grant into the share count is hard.
The relationship
$V_{\text{add}} - V_{\text{ded}} = \frac{\text{SBC} \times (1+g)}{r - g} = \frac{30 \times 1.03}{0.07} \approx 441$
SBC stock-based compensation, Rs 30 crore a year
r WACC, 10%
g perpetual growth, 3%
What it says in wordsThe difference between the two values is the present value of paying staff in shares forever.
One honest limitation: existing options and restricted shares already granted are a separate matter, counted in today's diluted share count by the treasury method. The question here is future grants, which the cash flow treatment decides.
Where candidates lose it
The standard slip is saying stock compensation is non-cash, so it is added back like depreciation, and stopping there. Depreciation is the echo of cash already spent; stock compensation is a cost paid now, in a different currency.
The second loss is saying deduct it without being able to explain why the add-back can be made consistent. Name both consistent routes and the one wrong one: that is the answer an associate-level interviewer is looking for.
What the interviewer asks next
How do already-granted options enter the valuation?
If the company buys back shares each year to offset dilution, how does that change the cash flow treatment?
Why do many sell-side models still add stock compensation back?
Asked at Lazard, Investment Banking, New York, 2026 (Wall Street Oasis): SBC treatment in DCF

t - 0.5	each year's cash treated as arriving at the middle of the year
1.10^0.5	the gain from discounting every cash flow for half a year less

Line	Tax 25%	Tax 30%	Change
NOPAT, Rs crore	150	140	-6.7%
After-tax cost of debt	6.0%	5.6%	-40 bp
WACC	10.20%	10.08%	-12 bp
Terminal value at 4% growth, Rs crore	2,516	2,395	-4.8%

(1 - t)	the share of each rupee left after tax, 0.75 falling to 0.70
0.30, 0.70	the debt and equity weights
g	the long-run growth rate, 4% here

beta_U	the peers' unlevered beta, 0.8
t	the tax rate, 25%
D/E	the target's debt divided by its equity, 50%

Try it first

Why would cash arrive in the middle of the year?

Why is the uplift exactly the square root of 1.10 minus one?

Where should you not apply it?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a tax rate touch both the top and the bottom of the fraction?

How big is each effect on these numbers?

Is there a subtler channel the interviewer might be after?

Where candidates lose it

What the interviewer asks next

Try it first

Why can you not use the peer beta as it is?

How does the relevering work?

Which slips change the answer?

Where candidates lose it

What the interviewer asks next

Try it first

If no cash leaves, why is stock compensation a cost?

Can the add-back version ever be right?

Where candidates lose it

What the interviewer asks next