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. 007A 91-day Treasury bill with a face value of Rs 100 is issued at Rs 98.25. What annualised yield does the buyer earn on a 365-day basis, and why is it not simply 1.75 times four?Bond pricing and yieldWarm upIndian debt capital markets

    Try it first

    Is the true yield above or below 7.00%, which is 1.75 times four?

    Show the worked solution

    About 7.14%. The buyer pays Rs 98.25 and gets Rs 100 back 91 days later, a gain of Rs 1.75 on Rs 98.25 invested, which is 1.781% for the period. Scaling by 365 over 91 gives 7.14% a year. It beats 1.75 times four for two reasons: the return is earned on the price paid, not the face value, and a year holds a little more than four 91 day periods.

    Why divide by 98.25 and not by 100?

    Lend a friend Rs 98 and get Rs 100 back: you made Rs 2 on Rs 98, not on Rs 100. A yield is always the gain divided by the money you actually put in, and on a discounted bill that is the price, not the face value. Dividing by 100 gives the discount rate, a quoting convention that understates what the buyer earns. On a Rs 1.75 gain the difference is small, but interviewers ask this precisely to see whether you know which base is which.

    The yield is earned on what you pay, Rs 98.25, not on the Rs 100 faceDay 0: pay Rs 98.25Day 91: receive Rs 10091 daysgain Rs 1.75 / Rs 98.25= 1.781% for 91 days1.75 x 4, on face value7.00% wrong base, wrong daysx 365/91, on face value7.02% the discount ratex 365/91, on the price paid7.14% the simple yieldcompounded over a year7.34% effective annual ratebars start at 6.50%, so the gaps are stretched to be visible
    The Rs 1.75 gain on Rs 98.25 paid is 1.781% over 91 days; annualised on the price paid it is 7.14%, above both 1.75 times four at 7.00% and the discount rate on face value at 7.02%, and compounding would lift it further to 7.34%.

    Why 365 over 91 instead of 4?

    A quarter is not exactly 91 days: 365 divided by 91 is 4.011. On a 365 day basis a 91 day return is scaled by 365 over 91, so the days in the bill's life, not the word quarter, set the multiplier. That adds only about two basis points here, but on bills of 182 or 364 days, or where day counts differ between markets, getting the multiplier right is exactly what a desk checks.

    The relationship
    y=100−PP×365d=1.7598.25×36591≈7.14%y = \frac{100 - P}{P} \times \frac{365}{d} = \frac{1.75}{98.25} \times \frac{365}{91} \approx 7.14\%
    Pthe price paid, Rs 98.25
    100the face value repaid at maturity
    ddays to maturity, 91
    ythe simple annualised yield
    What it says in wordsGain over price paid, scaled up by how many such periods fit in a 365 day year.

    Is 7.14% what the buyer really earns over a year?

    Only if the gain is not reinvested. If the buyer rolls into a new bill at the same price every 91 days, interest earns interest and the effective annual rate is about 7.34%. Simple and compounded yields answer different questions, so say which one you are quoting. The simple formula here is the convention commonly used to quote Treasury bills; confirm the convention and day count of the market you are pricing in before comparing a bill yield with a bond yield, which may be stated on a different basis.

    Where candidates lose it

    The fast wrong answer is 7%. It uses the face value as the base and four as the multiplier, two small errors that both push the answer down, and it tells the interviewer the candidate has memorised a shortcut without knowing what a yield is.

    The second loss is quoting the compounded figure without saying so, then being unable to compare it with a quoted bill yield. Name the convention with the number.

    What the interviewer asks next

    • What price for the same bill gives a yield of exactly 7.00%? (About Rs 98.28.)
    • Why does a bond yield quoted with semi-annual compounding not compare directly with this bill yield?
    • If yields rise 50 basis points the day after you buy, roughly how much does the bill's price fall?
  2. 049You buy a 6% annual coupon bond at 100. A year later, just after the coupon is paid, it trades at 95. What total return did you earn over the year, and why is the answer not minus 5%?Bond pricing and yieldWarm upFixed income asset management

    Try it first

    Your total return for the year is:

    Show the worked solution

    A total return of +1%. You paid 100. During the year you received a coupon of 6, and the bond is now worth 95, so you hold 101 of value against 100 paid. Total return is income plus price change, 6 minus 5, divided by the price you paid. Quoting minus 5% counts only the price and forgets the coupon, which for a bond is most of the return.

    What counts as return on a bond?

    If you buy a flat for Rs 1 crore, collect Rs 6 lakh of rent over the year and the flat's market value slips to Rs 95 lakh, you are not down 5%. You have Rs 95 lakh of flat and Rs 6 lakh of cash: Rs 1.01 crore. Total return is everything the investment paid you plus the change in what it is worth, divided by what you paid. For a bond, the coupon is the rent.

    The relationship
    TR=C+(P1−P0)P0=6+(95−100)100=+1%TR = \frac{C + (P_1 - P_0)}{P_0} = \frac{6 + (95 - 100)}{100} = +1\%
    Ccoupon received during the year, 6
    P_0price paid, 100
    P_1price a year later, just after the coupon, 95
    What it says in wordsAdd the income received to the change in price, then divide by what you paid.
    Total return = income + price change: 6 - 5 = +1100Price paid+6Coupon received-5Price change101Ending value80bond 95+ cash 6Total return+1.0%Axis starts at 80 so the small moves are visible
    The investor paid 100, received a 6 coupon and saw the price fall 5 to 95, so ends the year with 101 of value, a total return of +1% rather than the -5% the price alone suggests.

    Why would the price have fallen, and what does it mean for next year?

    A fixed coupon bond falls in price when market yields rise. If the bond had four years left after the coupon, a price of 95 means new buyers earn about 7.49% a year to maturity, so the loss this year is partly paid back as extra yield in later years if the bond is held. That is the other reason not to fixate on the minus 5: a holder to maturity still gets 100 at the end, and the price fall is a mark-to-market loss, not money gone for good.

    Two limits worth saying. This assumes the coupon is simply held as cash; reinvesting it would add a little more. And it assumes the issuer is still sound: if the price fell because default risk rose, the loss may not come back. A desk reports both numbers, the price change and the total return, because they answer different questions.

    Where candidates lose it

    The instant wrong answer is minus 5%, because the price is the only number on the screen. For a bond, the coupon is usually the larger part of the return, and ignoring it gets the sign wrong here.

    The other slip is adding the coupon to the new price and dividing by the new price: 101 over 95. Returns are measured on what you paid, so divide by 100.

    What the interviewer asks next

    • What price a year later would have given a total return of zero?
    • If the coupon were reinvested at 7% for half a year before you measure, what changes?
    • Why do index providers publish total return rather than price return for bond indices?