Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
Topics: 12
Hard: 30

Showing 1–6 of 6 · filtered from 100Clear filters

001You buy a three-year loan at 96. It pays a floating coupon of a 4% base rate plus a 5% margin. Roughly what yield are you earning, and why does the discount matter more on a short loan than a long one?Credit and PIK mathsHardPrivate credit
Try it first
Before you calculate: which is closest to the yield on the loan?
Show the worked solution
About 10.5%, against a 9% coupon. The coupon pays 9 a year. The discount of 4 is collected when the loan repays at 100, which spread over three years adds about 1.33 a year, so the rough yield is 10.33%. Dividing by the average price of 98 refines it to 10.54%, and the exact figure is 10.63%. A shorter life spreads the same discount over fewer years.
Where does the extra yield come from if the coupon is fixed at 9%?
Picture buying a gift voucher worth 100 for 96, redeemable in three years. On top of whatever the voucher pays along the way, you pocket 4 when you redeem it. A loan bought below par pays you twice: the coupon every year, and the gap between the price and par when the borrower repays. The coupon here is the base rate plus the margin, 4% plus 5%, so 9 a year on each 100 of face value. The 4 points of discount are the second source, and the whole question is how to turn that one-off 4 into a per-year rate.
The shortcut is to spread it evenly: 4 divided by 3 years is 1.33 a year, so the rough yield is 9 plus 1.33, which is 10.33%. That figure is measured against 100, but you paid less than 100 for most of the life. Divide the 10.33 of annual income by the average of the purchase price and par, 98, and you get 10.54%. The exact yield, solving for the rate that discounts the coupons and the 100 back to 96, is 10.63%.
The 4-point discount adds 4 points a year to a one-year loan, 1.33 to a three-year loan and only 0.67 to a six-year loan, while the 9% coupon is the same in every case, so the three-year loan bought at 96 yields roughly 10.3% to 10.6%.
Why does the same discount matter more on a short loan?
Because the discount is a fixed sum and the life is the divisor. A discount adds roughly its size divided by the years until repayment, so halving the life doubles what the discount is worth per year. Bought at 96, a one-year loan yields about 13.5%, the three-year loan 10.6% and a six-year loan 9.9%. That is why credit investors care so much about when a loan actually repays. Leveraged loans can usually be prepaid at par, and many desks quote yields to an assumed life shorter than the legal maturity for exactly this reason: a borrower who refinances early hands the discount back sooner, which raises the yield.
The relationship
$y \approx \frac{c + (100 - P)/n}{(100 + P)/2} = \frac{9 + 4/3}{98} \approx 10.5\%$
c the annual coupon per 100 of face value, here 9
P the price paid, here 96
n years until the loan repays at 100, here 3
What it says in wordsAnnual income is the coupon plus the discount spread over the life, and you divide it by the average amount invested.
What assumption sits underneath the number?
The coupon floats, so the 9% is only today's coupon. Every yield on a floating-rate loan assumes the base rate stays where it is, so the honest answer is a spread over the base rate, not a fixed percentage. Here the loan earns the 5% margin plus roughly 1.5 points from the discount, about 6.5 points over the base. Say that framing, then give 10.5% as the figure at a 4% base rate, and the interviewer hears that you know which part of the return the loan locks in and which part moves.
Where candidates lose it
The usual miss is answering 9%, as if the price did not matter, or 13%, adding the whole discount as though it arrived in one year. Both come from treating the discount as a lump rather than as income spread over the life.
The second loss is stopping at the number. The follow-on about short loans is the real test: say that the discount is divided by the life, so early repayment raises the yield, and give the one-year and six-year figures as proof.
What the interviewer asks next
The borrower repays at par after one year. What did you actually earn?
What price would you pay for a 10% yield on the three-year loan?
Why might a lender accept a lower margin in exchange for a bigger discount, called original issue discount?
032A portfolio company has a loan of 300 at 9% with three years left. It can repay early at a 2% call premium and refinance at 7.5%. How much does refinancing save, net of the premium?Credit and PIK mathsCorePrivate creditMid-market buyout fund
Try it first
Ignoring discounting, what is the net saving?
Show the worked solution
About 7.5 before discounting, about 5.7 after. The rate falls 1.5 points on 300, saving 4.5 a year, or 13.5 over three years. The call premium is 2% of 300, which is 6, paid up front. Net, 13.5 less 6 is 7.5. Discount the savings at 7.5% and they are worth 11.7 today, so the net gain is about 5.7.
How should you think about a call premium?
Switching to a cheaper phone plan midway through a contract usually costs an exit fee. You switch only if the monthly saving, times the months left, beats the fee. A call premium is the lender's exit fee: it compensates the lender for losing a loan that pays above today's rate, so the borrower refinances only when the saving over the remaining term beats it. Here the fee is 6 and each year saves 4.5, so the fee is earned back in 1.33 years.
The borrower pays a premium of 6 at the refinancing date and saves 4.5 a year for three years, so the cumulative saving turns positive during year two and ends at 7.5, or about 5.7 discounted at 7.5%.
When would refinancing not be worth it?
Work out the breakeven rate gap. Spread the premium over the years left: 6 over 3 years is 2 a year, which is 0.67% of 300. Refinancing pays only if the rate falls by more than the premium divided by the years remaining, about 67 basis points here. With one year left the breakeven gap would be the full 2 points, and refinancing at 7.5% would lose money.
The relationship
$\text{Net} = L\,(r_{old}-r_{new})\,n - pL = 300 \times 1.5\% \times 3 - 2\% \times 300 = 7.5$
L the loan, 300
r_old - r_new the rate cut, 1.5 points
n years remaining, 3
p the call premium, 2%
What it says in wordsThe saving is the rate cut times the loan times the years left, less the premium paid to get out.
Say what the sums leave out. A new loan carries its own arrangement fee and legal costs, often a point or more, which would eat most of the 7.5. Lenders also build call protection to step down over time, so waiting a year can cut the premium. And if the business improves, a lender may simply reprice the existing loan, which avoids the premium altogether.
Where candidates lose it
The common loss is quoting the gross saving of 13.5 and forgetting the premium, or quoting one year's saving of 4.5. The interviewer wants the full trade: saving over the remaining term against the one-off cost.
The second loss is ignoring the fees on the new loan. Mentioning them, even without a number, shows you know refinancing has a cost on both sides.
What the interviewer asks next
With one year left, does refinancing at 7.5% still make sense?
The new lender charges a 1% arrangement fee. What is the net saving now?
Why do lenders insist on call protection in leveraged loans, and why does it usually step down?
040A company has floating-rate debt of 500 priced at a 5% base rate plus a 4% margin, and EBITDA of 110. The base rate rises by 200 basis points. What happens to interest cover?Credit and PIK mathsWarm upPrivate creditIndian mid-market PE
Try it first
Where does EBITDA interest cover go?
Show the worked solution
Interest cover falls from 2.44x to 2.0x. The all-in rate is 5% plus 4%, so 9% on 500 is 45 of interest, and 110 / 45 is 2.44x. After a 200 basis point rise the rate is 11%, interest is 55, and 110 / 55 is 2.0x. A two-point rise in the base rate lifts the interest bill by 22% and cuts cash left after interest from 65 to 55.
Why does a two-point rate rise do so much damage?
A household with a floating-rate home loan feels every rate rise in the next EMI, while one with a fixed rate does not. On floating-rate debt the borrower carries the rate risk, so a rise in the base rate goes straight into the interest bill. Two points on 500 is 10 more a year, which is 22% more than the 45 being paid today, with no change in how the business is running.
With EBITDA fixed at 110, a 200 basis point rise in the base rate lifts interest on 500 of floating debt from 45 to 55, so cover falls from 2.44x to 2.0x and the cash left after interest falls from 65 to 55.
The relationship
$\text{Cover} = \frac{EBITDA}{D \times (b + m)} = \frac{110}{500 \times 11\%} = 2.0x$
D floating debt, 500
b the base rate, now 7%
m the lender's margin, 4%
What it says in wordsInterest cover is EBITDA over the interest bill, and on floating debt the bill moves with the base rate.
What does a sponsor do about it?
Most leveraged loans are floating, so sponsors usually hedge part of the debt. An interest rate swap fixes the rate on a portion of the loan, and a cap limits how high it can go in return for an upfront premium. Lenders often require a minimum hedged share in the loan agreement. The trade-off is cost and lost upside: a swap set before rates fall locks the borrower into the higher rate.
Say what a credit analyst would look at next. EBITDA interest cover ignores capital spending and tax, so cash cover is tighter than 2.0x. A cover covenant set at, say, 2.0x would now be right at its limit, which turns a market move into a negotiation with lenders.
Where candidates lose it
The common loss is saying cover is unchanged because EBITDA is unchanged. Candidates forget that the coupon on floating debt resets with the base rate.
The second loss is adding the 200 basis points to the margin and calling it a 2% rise in interest. The rate goes from 9% to 11%, which is a 22% rise in the bill. Say the percentage change in the bill, not in the rate.
What the interviewer asks next
How much EBITDA growth would restore 2.44x cover after the rate rise?
If 60% of the debt is swapped to a fixed 7.5%, what is cover after the rise?
Why do lenders often insist on hedging, and who benefits when rates fall?
061A sponsor can fund 4.5x EBITDA with senior debt of 3.0x at 7% plus second lien of 1.5x at 11%, or with a single unitranche loan of 4.5x at 8.5%. Which is cheaper, and why might the sponsor still choose the unitranche?Credit and PIK mathsCorePrivate credit
Try it first
What is the blended rate on the split stack?
Show the worked solution
The split stack is cheaper, at a blended 8.33% against 8.5%. Two thirds of the debt costs 7% and one third costs 11%, so the blend is 7% plus a third of the 4-point gap. On EBITDA of 100, that is 37.5 of interest a year against 38.25, a gap of only 0.75. Sponsors often pay that for one lender, one set of documents and a surer, faster close.
How do you blend two rates quickly?
If you borrow Rs 2 lakh from a bank at 7% and Rs 1 lakh from a relative at 11%, you are not paying 9% on the whole: most of the money is the cheap kind. A blended rate is a weighted average, so start from the cheaper rate and add the gap times the share of debt that is expensive. Here the second lien is a third of the stack, so the blend is 7% plus a third of 4 points, 8.33%.
The split stack of 300 senior at 7% and 150 second lien at 11% costs 37.5 a year, a blended 8.33%, while one unitranche of 450 at 8.5% costs 38.25, so the single lender costs 0.75 a year more.
The relationship
$r = \tfrac{3.0}{4.5}(7\%) + \tfrac{1.5}{4.5}(11\%) = 7\% + \tfrac{1}{3}(4\%) = 8.33\%$
3.0/4.5 senior debt's share of the stack, two thirds
1.5/4.5 second lien's share, one third
4% the gap between the two rates
What it says in wordsThe blended cost is the cheap rate plus the expensive tranche's share of the gap.
Why would a sponsor pay more for one lender?
Because the 0.75 a year buys simplicity that has real value. With two tranches there are two lender groups and an intercreditor agreementThe contract between two groups of lenders to the same company that sets who is paid first, who controls enforcement and what each may do without the other. between them, and every waiver or amendment means two negotiations. A unitranche puts one lender across the table, which usually means a faster close, fewer parties who can block a change, and more certainty in a competitive auction. The limit: compare all-in costs, because upfront fees and call protection can differ between the two and move the answer by more than 0.17 points.
Where candidates lose it
The usual slip is a simple average: 7% and 11% make 9%, so the unitranche looks cheap. Senior is twice the size of the second lien, so the blend sits much nearer 7%.
The other loss is stopping at cheaper means better. The interviewer wants the second half: a gap of 0.17 points is small, and a sponsor will often pay it for speed and a single counterparty.
What the interviewer asks next
At what unitranche rate would the sponsor be indifferent on interest alone?
What changes if the second lien pays 11% as PIK rather than cash?
Why might a unitranche lender split the loan into first-out and last-out pieces behind the scenes?
074A mezzanine lender puts in 100, earns 12% cash interest a year, and also receives warrants for 2% of the company's equity. The loan is repaid at the end of year 5, when the equity is worth 1,500. What is the lender's IRR?Credit and PIK mathsHardPrivate creditSpecial situations
Try it first
Roughly where does the IRR land?
Show the worked solution
About 16.3%. The lender pays 100, receives 12 a year for five years and gets the 100 back at the end. On its own that is exactly a 12% return. The warrants are 2% of equity worth 1,500, which is 30, received with the final payment, taking year 5 to 142. That one extra payment lifts the IRR by about 4.3 points.
What does the lender earn before the warrants?
Think of a fixed deposit at 12% that pays its interest every year and returns the principal at maturity: it earns exactly 12%. A loan bought at par that pays its full coupon in cash and is repaid at par earns exactly its coupon rate, so the first 12 points of the IRR need no calculation at all. The question is only about what the warrants add on top.
The lender pays 100, receives 12 a year for four years, and in year 5 collects 12 of interest, 100 of principal and 30 of warrant value, a stream with an IRR of 16.3% against 12.0% for the coupon alone.
How do you estimate the warrant's effect without a calculator?
The 30 arrives in five years. At around 16%, five years of discounting roughly halves money, so the warrants are worth about 14 to 15 in today's terms. Spread that extra 14 or so of value across five years of a 100 loan and it adds roughly 4 points a year to the coupon, which puts the IRR near 16%. Then confirm by trial: at 16%, the flows of 12 a year and 142 at the end discount to slightly above 100, so the IRR is a touch above 16%, 16.3%.
The relationship
$100 = \sum_{t=1}^{4} \frac{12}{(1+r)^t} + \frac{142}{(1+r)^5} \quad\Rightarrow\quad r \approx 16.3\%$
12 cash interest each year, 12% of 100
142 year 5: interest 12, principal 100 and warrants 30
r the lender's IRR
What it says in wordsThe IRR is the rate at which the coupons, the repayment and the warrant value discount back to the 100 lent.
Say the risk too. The warrant is worth 30 only if the equity really is worth 1,500 at exit. If equity were worth 500, the warrants would bring 10 and the IRR would fall to about 13.5%. That is the point of mezzanine: a contractual coupon plus a slice of the upside, priced for sitting behind the senior lenders.
Where candidates lose it
The common slip is adding the warrants as if they paid every year: 30 on 100 is 30%, plus 12%, gives an absurd 42%. The warrant value arrives once, at the end, and is spread over five years in the IRR.
The other miss is ignoring the coupon's own return and computing only the warrant uplift, or treating the 12% as compounding PIK. Here the interest is paid in cash each year, so it does not accrue onto the principal.
What the interviewer asks next
What if the 12% were PIK, added to the loan each year instead of paid in cash?
Equity is worth 500 at exit instead. What is the IRR?
Why would a sponsor give away warrants rather than pay a higher coupon?
092A private credit fund makes a 3-year loan of 400 at 10%. The borrower has a 5% chance of default each year and lenders would recover 60%. What is the expected annual loss rate, and what does the loan earn after expected losses?Credit and PIK mathsCorePrivate credit
Try it first
What is the expected loss each year, as a share of the loan?
Show the worked solution
Expected loss is about 2% a year, 8 on 400, so the loan earns about 8% after losses. The chance of default is 5% and a default costs the 40% not recovered: 5% times 40% is 2%. That takes a fifth of the 10% coupon. Assuming a 5% base rate, the 5 point spread shrinks to about 3 points once expected losses are paid for.
Why multiply the default chance by the loss, not use it alone?
If you lend a friend Rs 1,000 and think there is a one in twenty chance they cannot repay, but you know you would get Rs 600 back from them anyway, your expected loss is not Rs 50 but one twentieth of Rs 400, Rs 20. Expected loss is the probability of default times the loss given defaultThe share of the loan a lender loses when a borrower defaults, after recoveries. A 60% recovery means a 40% loss given default., so recovery matters as much as the chance of default. Here: 5% times 40%, which is 2% a year.
With a 5% chance of default and 60% recovered, a 400 loan loses 160 if the borrower defaults and nothing otherwise, an expected loss of 8, or 2% a year, a fifth of the 10% coupon, leaving about 8% after losses.
The relationship
$EL = PD \times LGD = 5\% \times 40\% = 2\% \qquad 10\% - 2\% = 8\%$
PD probability of default in a year, 5%
LGD loss given default, one minus the 60% recovery
EL expected loss each year, as a share of the loan
What it says in wordsExpected loss is how often default happens times how much it costs when it does; subtract it from the coupon to get the loss-adjusted yield.
What changes over the full three years?
Default can happen in any year, so the chance of at least one default over three years is 1 minus 0.95 cubed, about 14.3%. Expected loss in rupees falls slightly each year because a loan that has already defaulted cannot default again: 8, then 7.6, then 7.2, about 22.8 over the life. The simple 2% a year is a good approximation. It also ignores the coupon lost in the year of default, which a fuller model would include.
A credit interviewer then asks whether 8% is enough. Compare it with the fund's cost of capital and with what safer loans pay. If senior loans to stronger borrowers yield 9% with expected losses of 0.5%, this loan is the worse risk-adjusted deal despite its higher coupon.
Where candidates lose it
The common slip is using the 5% default probability as the loss, which ignores recovery and overstates the expected loss by two and a half times.
The second is forgetting that expected loss is an average. A single loan either loses 160 or nothing; the 2% only describes a large, diversified book of loans like it.
What the interviewer asks next
What coupon would give the same 8% loss-adjusted yield if recovery fell to 30%?
Why do recoveries tend to fall exactly when defaults rise?
How would a PIK toggle change the expected loss on this loan?

c	the annual coupon per 100 of face value, here 9
P	the price paid, here 96
n	years until the loan repays at 100, here 3

L	the loan, 300
r_old - r_new	the rate cut, 1.5 points
n	years remaining, 3
p	the call premium, 2%

D	floating debt, 500
b	the base rate, now 7%
m	the lender's margin, 4%

3.0/4.5	senior debt's share of the stack, two thirds
1.5/4.5	second lien's share, one third
4%	the gap between the two rates

Try it first

Where does the extra yield come from if the coupon is fixed at 9%?

Why does the same discount matter more on a short loan?

What assumption sits underneath the number?

Where candidates lose it

What the interviewer asks next

Try it first

How should you think about a call premium?

When would refinancing not be worth it?

Where candidates lose it

What the interviewer asks next

Try it first

Why does a two-point rate rise do so much damage?

What does a sponsor do about it?

Where candidates lose it

What the interviewer asks next

Try it first

How do you blend two rates quickly?

Why would a sponsor pay more for one lender?

Where candidates lose it

What the interviewer asks next

Try it first

What does the lender earn before the warrants?

How do you estimate the warrant's effect without a calculator?

Where candidates lose it

What the interviewer asks next

Try it first

Why multiply the default chance by the loss, not use it alone?

What changes over the full three years?

Where candidates lose it

What the interviewer asks next