Debt Capital Markets puzzles, solved step by step

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All topicsLeverage, coverage and cash flow9Mental maths and numeracy8Estimation and market sizing7Logic and brainteasers8Cost of capital and valuation riddles7Bond pricing and yield7Compounding, PIK and fees6Issuance and refinancing arithmetic8Credit spreads and default probability8Duration and convexity8Capital structure and recovery8Probability and expected value10Yield curve and forward rates6
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  1. 011A 4-year bullet loan pays 10% a year and the borrower pays a 2% upfront fee. Roughly what yield does the lender earn, and how would you estimate it in your head?Compounding, PIK and feesWarm upCorporate bankingLeveraged finance

    Try it first

    Which is closest to the lender's yield?

    Show the worked solution

    About 10.6%, exactly 10.64%. The lender puts out Rs 98, receives Rs 10 a year and gets Rs 100 back in year 4. In your head: spread the 2 point fee over 4 years, about 0.5 a year, giving 10.5, then divide by the average money at work, about 99, which gives 10.61%. An upfront fee is extra yield spread over the life of the loan.

    What does the fee actually change?

    Imagine lending a friend Rs 1,000 for four years at 10%, but handing over only Rs 980 because you keep Rs 20 as a processing charge. Your interest is still on Rs 1,000 and you still get Rs 1,000 back. An upfront fee lowers the money the lender actually puts out, so the same coupons and repayment earn a higher yield on it. Look at the top of the figure: Rs 98 out, then 10, 10, 10 and 110 back.

    A 2 point fee is extra yield spread over the life of the loan-98 today100 lent,2 kept as fee+10Year 1+10Year 2+10Year 3+110Year 4Coupon alone10.00%+ 2 points / 4 years10.50%... on a base of 98 or so10.61%Exact yield10.64%bars start at 9.80%
    The lender pays out Rs 98 and receives 10 a year plus 100 in year 4; the coupon alone is 10.00%, spreading the fee gives about 10.50%, adjusting for the smaller base gives 10.61%, and the exact yield is 10.64%.

    How do you get close to 10.64% without a calculator?

    Use the bond trader's approximation: yearly income over the average money at work. Yearly income is the coupon plus the fee spread evenly over the life, 10 plus 2 over 4, which is 10.5; the average money at work is halfway between 98 and 100, which is 99. 10.5 over 99 is 10.61%, within a few basis points of the exact 10.64%. Say the first step, 10.5, as your quick answer and the second as the refinement; interviewers like hearing both.

    The relationship
    y≈C+F/n(100+P)/2=10+2/4(100+98)/2=10.599≈10.61%y \approx \frac{C + F/n}{(100 + P)/2} = \frac{10 + 2/4}{(100 + 98)/2} = \frac{10.5}{99} \approx 10.61\%
    Cannual coupon, 10
    Fupfront fee, 2 points
    nyears to repayment, 4
    Pmoney actually lent after the fee, 98
    What it says in wordsYield is roughly the yearly income, with the fee spread over the years, divided by the average money the lender has at work.

    What if the loan is repaid early?

    Then the fee is spread over fewer years and is worth more a year. If the borrower repays at par after 2 years, the same 2 points lift the yield to about 11.17%, because the fee is earned over half the time. This is why lenders care about expected life, not stated maturity, and why a loan that is likely to be refinanced early is priced partly on its fee. Say that one sentence and you have shown the interviewer you see the fee as yield, not as a one-off receipt.

    Where candidates lose it

    The two fast wrong answers sit either side. Saying 10% ignores the fee because it is paid once; saying 12% adds the whole fee to one year's coupon. Both miss that the fee belongs to the whole life of the loan.

    The quieter loss is stopping at 10.5 when the interviewer asks for more precision. The base is Rs 98, not Rs 100, and that small adjustment is the difference between 10.50% and 10.64%.

    What the interviewer asks next

    • What upfront fee would lift the yield on this loan to 11%?
    • Why does the same fee matter more on a 2 year loan than on a 7 year loan?
    • How should a bank book this fee in its income: all at once, or over the life of the loan?