Private Equity puzzles, solved step by step

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

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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
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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?
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?

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

12	cash interest each year, 12% of 100
142	year 5: interest 12, principal 100 and warrants 30
r	the lender's IRR

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

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